Math Reference Book - esc13.netjukebox.esc13.net/geo_eocs/math_reference_book.pdf(B) look for patterns and represent generalizations algebraically. (4) Foundations for functions. The

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Math Reference Book

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Algebra I TEKS........................................................................................4 STAAR Algebra I Eligible TEKS...............................................................10 STAAR Algebra I Blueprint.....................................................................17Geometry TEKS.......................................................................................20STAAR Geometry Eligible TEKS............................................................25STAAR Geometry Blueprint ..................................................................33 Algebra II TEKS......................................................................................35STAAR Algebra II Eligible TEKS .............................................................42STAAR Algebra II Blueprint.....................................................................51IES Practice Guide................................................................................53Texas College and Career Readiness Standards................................158English Language Proficiency Standards...............................................189Reaching for Common Ground...........................................................206

Table of Contents

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Algebra I TEKS

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§111.32. Algebra I (One Credit).

(a) Basic understandings.

(1) Foundation concepts for high school mathematics. As presented in Grades K-8, the basic understandings of number, operation, and quantitative reasoning; patterns, relationships, and algebraic thinking; geometry; measurement; and probability and statistics are essential foundations for all work in high school mathematics. Students will continue to build on this foundation as they expand their understanding through other mathematical experiences.

(2) Algebraic thinking and symbolic reasoning. Symbolic reasoning plays a critical role in algebra; symbols provide powerful ways to represent mathematical situations and to express generalizations. Students use symbols in a variety of ways to study relationships among quantities.

(3) Function concepts. A function is a fundamental mathematical concept; it expresses a special kind of relationship between two quantities. Students use functions to determine one quantity from another, to represent and model problem situations, and to analyze and interpret relationships.

(4) Relationship between equations and functions. Equations and inequalities arise as a way of asking and answering questions involving functional relationships. Students work in many situations to set up equations and inequalities and use a variety of methods to solve them.

(5) Tools for algebraic thinking. Techniques for working with functions and equations are essential in understanding underlying relationships. Students use a variety of representations (concrete, pictorial, numerical, symbolic, graphical, and verbal), tools, and technology (including, but not limited to, calculators with graphing capabilities, data collection devices, and computers) to model mathematical situations to solve meaningful problems.

(6) Underlying mathematical processes. Many processes underlie all content areas in mathematics. As they do mathematics, students continually use problem-solving, language and communication, and reasoning (justification and proof) to make connections within and outside mathematics. Students also use multiple representations, technology, applications and modeling, and numerical fluency in problem-solving contexts.

(b) Knowledge and skills.

(1) Foundations for functions. The student understands that a function represents a dependence of one quantity on another and can be described in a variety of ways. The student is expected to:

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(A) describe independent and dependent quantities in functional relationships;

(B) gather and record data and use data sets to determine functional relationships between quantities;

(C) describe functional relationships for given problem situations and write equations or inequalities to answer questions arising from the situations;

(D) represent relationships among quantities using concrete models, tables, graphs, diagrams, verbal descriptions, equations, and inequalities; and

(E) interpret and make decisions, predictions, and critical judgments from functional relationships.

(2) Foundations for functions. The student uses the properties and attributes of functions. The student is expected to:

(A) identify and sketch the general forms of linear (y = x) and quadratic (y = x2) parent functions;

(B) identify mathematical domains and ranges and determine reasonable domain and range values for given situations, both continuous and discrete;

(C) interpret situations in terms of given graphs or creates situations that fit given graphs; and

(D) collect and organize data, make and interpret scatterplots (including recognizing positive, negative, or no correlation for data approximating linear situations), and model, predict, and make decisions and critical judgments in problem situations.

(3) Foundations for functions. The student understands how algebra can be used to express generalizations and recognizes and uses the power of symbols to represent situations. The student is expected to:

(A) use symbols to represent unknowns and variables; and

(B) look for patterns and represent generalizations algebraically.

(4) Foundations for functions. The student understands the importance of the skills required to manipulate symbols in order to solve problems and uses the necessary algebraic skills required to simplify algebraic expressions and solve equations and inequalities in problem situations. The student is expected to:

(A) find specific function values, simplify polynomial expressions, transform and solve equations, and factor as necessary in problem situations;

6

(B) use the commutative, associative, and distributive properties to simplify algebraic expressions; and

(C) connect equation notation with function notation, such as y = x + 1 and f(x) = x + 1.

(5) Linear functions. The student understands that linear functions can be represented in different ways and translates among their various representations. The student is expected to:

(A) determine whether or not given situations can be represented by linear functions;

(B) determine the domain and range for linear functions in given situations; and

(C) use, translate, and make connections among algebraic, tabular, graphical, or verbal descriptions of linear functions.

(6) Linear functions. The student understands the meaning of the slope and intercepts of the graphs of linear functions and zeros of linear functions and interprets and describes the effects of changes in parameters of linear functions in real-world and mathematical situations. The student is expected to:

(A) develop the concept of slope as rate of change and determine slopes from graphs, tables, and algebraic representations;

(B) interpret the meaning of slope and intercepts in situations using data, symbolic representations, or graphs;

(C) investigate, describe, and predict the effects of changes in m and b on the graph of y = mx + b;

(D) graph and write equations of lines given characteristics such as two points, a point and a slope, or a slope and y-intercept;

(E) determine the intercepts of the graphs of linear functions and zeros of linear functions from graphs, tables, and algebraic representations;

(F) interpret and predict the effects of changing slope and y-intercept in applied situations; and

(G) relate direct variation to linear functions and solve problems involving proportional change.

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(7) Linear functions. The student formulates equations and inequalities based on linear functions, uses a variety of methods to solve them, and analyzes the solutions in terms of the situation. The student is expected to:

(A) analyze situations involving linear functions and formulate linear equations or inequalities to solve problems;

(B) investigate methods for solving linear equations and inequalities using concrete models, graphs, and the properties of equality, select a method, and solve the equations and inequalities; and

(C) interpret and determine the reasonableness of solutions to linear equations and inequalities.

(8) Linear functions. The student formulates systems of linear equations from problem situations, uses a variety of methods to solve them, and analyzes the solutions in terms of the situation. The student is expected to:

(A) analyze situations and formulate systems of linear equations in two unknowns to solve problems;

(B) solve systems of linear equations using concrete models, graphs, tables, and algebraic methods; and

(C) interpret and determine the reasonableness of solutions to systems of linear equations.

(9) Quadratic and other nonlinear functions. The student understands that the graphs of quadratic functions are affected by the parameters of the function and can interpret and describe the effects of changes in the parameters of quadratic functions. The student is expected to:

(A) determine the domain and range for quadratic functions in given situations;

(B) investigate, describe, and predict the effects of changes in a on the graph of y = ax2 + c;

(C) investigate, describe, and predict the effects of changes in c on the graph of y = ax2 + c; and

(D) analyze graphs of quadratic functions and draw conclusions.

(10) Quadratic and other nonlinear functions. The student understands there is more than one way to solve a quadratic equation and solves them using appropriate methods. The student is expected to:

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(A) solve quadratic equations using concrete models, tables, graphs, and algebraic methods; and

(B) make connections among the solutions (roots) of quadratic equations, the zeros of their related functions, and the horizontal intercepts (x-intercepts) of the graph of the function.

(11) Quadratic and other nonlinear functions. The student understands there are situations modeled by functions that are neither linear nor quadratic and models the situations. The student is expected to:

(A) use patterns to generate the laws of exponents and apply them in problem-solving situations;

(B) analyze data and represent situations involving inverse variation using concrete models, tables, graphs, or algebraic methods; and

(C) analyze data and represent situations involving exponential growth and decay using concrete models, tables, graphs, or algebraic methods.

Source: The provisions of this §111.32 adopted to be effective September 1, 1996, 21 TexReg 7371; amended to be effective August 1, 2006, 30 TexReg 1931.

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STAAR Algebra I Eligible TEKS

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Algebra I Assessment

Eligible Texas EssentialKnowledge and Skills

Texas Education AgencyStudent Assessment Division

Fall 2010

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STAAR Algebra I Assessment

Reporting Category 1:Functional Relationships

The student will describe functional relationships in a variety of ways.

(A.1) Foundations for functions. The student understands that a function

represents a dependence of one quantity on another and can be described

in a variety of ways. The student is expected to

(A) describe independent and dependent quantities in functional

relationships; Supporting Standard

(B) gather and record data and use data sets to determine functional

relationships between quantities; Supporting Standard

(C) describe functional relationships for given problem situations and

write equations or inequalities to answer questions arising from the

situations; Supporting Standard

(D) represent relationships among quantities using [concrete] models,

tables, graphs, diagrams, verbal descriptions, equations, and

inequalities; and Readiness Standard

(E) interpret and make decisions, predictions, and critical judgments

from functional relationships. Readiness Standard

STAAR Algebra I Page 2 of 6


Fall 201012

Reporting Category 2:Properties and Attributes of Functions

The student will demonstrate an understanding of the properties and

attributes of functions.

(A.2) Foundations for functions. The student uses the properties and

attributes of functions. The student is expected to

(A) identify and sketch the general forms of linear (y = x) and quadratic

(y = x 2) parent functions; Supporting Standard

(B) identify mathematical domains and ranges and determine reasonable

domain and range values for given situations, both continuous and

discrete; Readiness Standard

(C) interpret situations in terms of given graphs or create situations that

fit given graphs; and Supporting Standard

(D) collect and organize data, make and interpret scatterplots (including

recognizing positive, negative, or no correlation for data

approximating linear situations), and model, predict, and make

decisions and critical judgments in problem situations.

Readiness Standard

(A.3) Foundations for functions. The student understands how algebra can be

used to express generalizations and recognizes and uses the power of

symbols to represent situations. The student is expected to

(A) use symbols to represent unknowns and variables; and

Supporting Standard

(B) look for patterns and represent generalizations algebraically.

Supporting Standard

(A.4) Foundations for functions. The student understands the importance of

the skills required to manipulate symbols in order to solve problems and

uses the necessary algebraic skills required to simplify algebraic

expressions and solve equations and inequalities in problem situations.

The student is expected to

(A) find specific function values, simplify polynomial expressions,

transform and solve equations, and factor as necessary in problem

situations; Readiness Standard

(B) use the commutative, associative, and distributive properties to

simplify algebraic expressions; and Supporting Standard

(C) connect equation notation with function notation, such as y = x + 1

and f (x) = x + 1. Supporting Standard



Fall 201013

Reporting Category 3: Linear Functions

The student will demonstrate an understanding of linear functions.

(A.5) Linear functions. The student understands that linear functions can be

represented in different ways and translates among their various

representations. The student is expected to

(A) determine whether or not given situations can be represented by

linear functions; Supporting Standard

(B) determine the domain and range for linear functions in given

situations; and Supporting Standard

(C) use, translate, and make connections among algebraic, tabular,

graphical, or verbal descriptions of linear functions.

Readiness Standard

(A.6) Linear functions. The student understands the meaning of the slope and

intercepts of the graphs of linear functions and zeros of linear functions

and interprets and describes the effects of changes in parameters of linear

functions in real-world and mathematical situations. The student is

expected to

(A) develop the concept of slope as rate of change and determine slopes

from graphs, tables, and algebraic representations;

Supporting Standard

(B) interpret the meaning of slope and intercepts in situations using

data, symbolic representations, or graphs; Readiness Standard

(C) investigate, describe, and predict the effects of changes in m and b on the graph of y = mx + b; Readiness Standard

(D) graph and write equations of lines given characteristics such as two

points, a point and a slope, or a slope and y-intercept;

Supporting Standard

(E) determine the intercepts of the graphs of linear functions and zeros

of linear functions from graphs, tables, and algebraic

representations; Supporting Standard

(F) interpret and predict the effects of changing slope and y-intercept in

applied situations; and Readiness Standard

(G) relate direct variation to linear functions and solve problems

involving proportional change. Supporting Standard



Fall 201014

Reporting Category 4:Linear Equations and Inequalities

The student will formulate and use linear equations and inequalities.

(A.7) Linear functions. The student formulates equations and inequalities

based on linear functions, uses a variety of methods to solve them, and

analyzes the solutions in terms of the situation. The student is expected to

(A) analyze situations involving linear functions and formulate linear

equations or inequalities to solve problems; Supporting Standard

(B) investigate methods for solving linear equations and inequalities

using [concrete] models, graphs, and the properties of equality,

select a method, and solve the equations and inequalities; and

Readiness Standard

(C) interpret and determine the reasonableness of solutions to linear

equations and inequalities. Supporting Standard

(A.8) Linear functions. The student formulates systems of linear equations

from problem situations, uses a variety of methods to solve them, and


(A) analyze situations and formulate systems of linear equations in two

unknowns to solve problems; Supporting Standard

(B) solve systems of linear equations using [concrete] models, graphs,

tables, and algebraic methods; and Readiness Standard

(C) interpret and determine the reasonableness of solutions to systems

of linear equations. Supporting Standard



Fall 201015

Reporting Category 5:Quadratic and Other Nonlinear Functions

The student will demonstrate an understanding of quadratic and other

nonlinear functions.

(A.9) Quadratic and other nonlinear functions. The student understands

that the graphs of quadratic functions are affected by the parameters of

the function and can interpret and describe the effects of changes in the

parameters of quadratic functions. The student is expected to

(A) determine the domain and range for quadratic functions in given

situations; Supporting Standard

(B) investigate, describe, and predict the effects of changes in a on the

graph of y = ax 2 + c; Supporting Standard

(C) investigate, describe, and predict the effects of changes in c on the

graph of y = ax 2 + c; and Supporting Standard

(D) analyze graphs of quadratic functions and draw conclusions.

Readiness Standard


there is more than one way to solve a quadratic equation and solves them

using appropriate methods. The student is expected to

(A) solve quadratic equations using [concrete] models, tables, graphs,

and algebraic methods; and Readiness Standard

(B) make connections among the solutions (roots) of quadratic

equations, the zeros of their related functions, and the horizontal

intercepts (x-intercepts) of the graph of the function.

Supporting Standard


there are situations modeled by functions that are neither linear nor

quadratic and models the situations. The student is expected to

(A) use patterns to generate the laws of exponents and apply them in

problem-solving situations; Supporting Standard

(B) analyze data and represent situations involving inverse variation

using [concrete] models, tables, graphs, or algebraic methods; and

Supporting Standard

(C) analyze data and represent situations involving exponential growth

and decay using [concrete] models, tables, graphs, or algebraic

methods. Supporting Standard



Fall 201016

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STAAR Algebra I Blueprint

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Texas Education Agency

Student Assessment Division

Fall 2010

STAAR Algebra I Blueprint

Reporting Categories Number of Standards Number of Questions

Reporting Category 1:

Functional Relationships

Readiness Standards 2

8 Supporting Standards 3

Total 5


Properties and Attributes of Functions



Total 9


Linear Functions



Total 10


Linear Equations and Inequalities



Total 6


Quadratic and Other Nonlinear Functions



Total 9

Readiness Standards Total Number of Standards 13 60%–65% 32–35

Supporting Standards Total Number of Standards 26 35%–40% 19–22

Total Number of Questions on Test 49 Multiple Choice

5 Griddable

54 Total

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Geometry TEKS

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§111.34. Geometry (One Credit).


(1) Foundation concepts for high school mathematics. As presented in Grades K-8, the basic understandings of number, operation, and quantitative reasoning; patterns, relationships, and algebraic thinking; geometry; measurement; and probability and statistics are essential foundations for all work in high school mathematics. Students continue to build on this foundation as they expand their understanding through other mathematical experiences.

(2) Geometric thinking and spatial reasoning. Spatial reasoning plays a critical role in geometry; geometric figures provide powerful ways to represent mathematical situations and to express generalizations about space and spatial relationships. Students use geometric thinking to understand mathematical concepts and the relationships among them.

(3) Geometric figures and their properties. Geometry consists of the study of geometric figures of zero, one, two, and three dimensions and the relationships among them. Students study properties and relationships having to do with size, shape, location, direction, and orientation of these figures.

(4) The relationship between geometry, other mathematics, and other disciplines. Geometry can be used to model and represent many mathematical and real-world situations. Students perceive the connection between geometry and the real and mathematical worlds and use geometric ideas, relationships, and properties to solve problems.

(5) Tools for geometric thinking. Techniques for working with spatial figures and their properties are essential in understanding underlying relationships. Students use a variety of representations (concrete, pictorial, numerical, symbolic, graphical, and verbal), tools, and technology (including, but not limited to, calculators with graphing capabilities, data collection devices, and computers) to solve meaningful problems by representing and transforming figures and analyzing relationships.

(6) Underlying mathematical processes. Many processes underlie all content areas in mathematics. As they do mathematics, students continually use problem-solving, language and communication, connections within and outside mathematics, and reasoning (justification and proof). Students also use multiple representations, technology, applications and modeling, and numerical fluency in problem solving contexts.


(1) Geometric structure. The student understands the structure of, and relationships within, an axiomatic system. The student is expected to:

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(A) develop an awareness of the structure of a mathematical system, connecting definitions, postulates, logical reasoning, and theorems;

(B) recognize the historical development of geometric systems and know mathematics is developed for a variety of purposes; and

(C) compare and contrast the structures and implications of Euclidean and non-Euclidean geometries.

(2) Geometric structure. The student analyzes geometric relationships in order to make and verify conjectures. The student is expected to:

(A) use constructions to explore attributes of geometric figures and to make conjectures about geometric relationships; and

(B) make conjectures about angles, lines, polygons, circles, and three-dimensional figures and determine the validity of the conjectures, choosing from a variety of approaches such as coordinate, transformational, or axiomatic.

(3) Geometric structure. The student applies logical reasoning to justify and prove mathematical statements. The student is expected to:

(A) determine the validity of a conditional statement, its converse, inverse, and contrapositive;

(B) construct and justify statements about geometric figures and their properties;

(C) use logical reasoning to prove statements are true and find counter examples to disprove statements that are false;

(D) use inductive reasoning to formulate a conjecture; and

(E) use deductive reasoning to prove a statement.

(4) Geometric structure. The student uses a variety of representations to describe geometric relationships and solve problems. The student is expected to select an appropriate representation (concrete, pictorial, graphical, verbal, or symbolic) in order to solve problems.

(5) Geometric patterns. The student uses a variety of representations to describe geometric relationships and solve problems. The student is expected to:

(A) use numeric and geometric patterns to develop algebraic expressions representing geometric properties;

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(B) use numeric and geometric patterns to make generalizations about geometric properties, including properties of polygons, ratios in similar figures and solids, and angle relationships in polygons and circles;

(C) use properties of transformations and their compositions to make connections between mathematics and the real world, such as tessellations; and

(D) identify and apply patterns from right triangles to solve meaningful problems, including special right triangles (45-45-90 and 30-60-90) and triangles whose sides are Pythagorean triples.

(6) Dimensionality and the geometry of location. The student analyzes the relationship between three-dimensional geometric figures and related two-dimensional representations and uses these representations to solve problems. The student is expected to:

(A) describe and draw the intersection of a given plane with various three-dimensional geometric figures;

(B) use nets to represent and construct three-dimensional geometric figures; and

(C) use orthographic and isometric views of three-dimensional geometric figures to represent and construct three-dimensional geometric figures and solve problems.

(7) Dimensionality and the geometry of location. The student understands that coordinate systems provide convenient and efficient ways of representing geometric figures and uses them accordingly. The student is expected to:

(A) use one- and two-dimensional coordinate systems to represent points, lines, rays, line segments, and figures;

(B) use slopes and equations of lines to investigate geometric relationships, including parallel lines, perpendicular lines, and special segments of triangles and other polygons; and

(C) derive and use formulas involving length, slope, and midpoint.

(8) Congruence and the geometry of size. The student uses tools to determine measurements of geometric figures and extends measurement concepts to find perimeter, area, and volume in problem situations. The student is expected to:

(A) find areas of regular polygons, circles, and composite figures;

(B) find areas of sectors and arc lengths of circles using proportional reasoning;

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(C) derive, extend, and use the Pythagorean Theorem;

(D) find surface areas and volumes of prisms, pyramids, spheres, cones, cylinders, and composites of these figures in problem situations;

(E) use area models to connect geometry to probability and statistics; and

(F) use conversions between measurement systems to solve problems in real-world situations.

(9) Congruence and the geometry of size. The student analyzes properties and describes relationships in geometric figures. The student is expected to:

(A) formulate and test conjectures about the properties of parallel and perpendicular lines based on explorations and concrete models;

(B) formulate and test conjectures about the properties and attributes of polygons and their component parts based on explorations and concrete models;

(C) formulate and test conjectures about the properties and attributes of circles and the lines that intersect them based on explorations and concrete models; and

(D) analyze the characteristics of polyhedra and other three-dimensional figures and their component parts based on explorations and concrete models.

(10) Congruence and the geometry of size. The student applies the concept of congruence to justify properties of figures and solve problems. The student is expected to:

(A) use congruence transformations to make conjectures and justify properties of geometric figures including figures represented on a coordinate plane; and

(B) justify and apply triangle congruence relationships.

(11) Similarity and the geometry of shape. The student applies the concepts of similarity to justify properties of figures and solve problems. The student is expected to:

(A) use and extend similarity properties and transformations to explore and justify conjectures about geometric figures;

(B) use ratios to solve problems involving similar figures;

(C) develop, apply, and justify triangle similarity relationships, such as right triangle ratios, trigonometric ratios, and Pythagorean triples using a variety of methods; and

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(D) describe the effect on perimeter, area, and volume when one or more dimensions of a figure are changed and apply this idea in solving problems.

Source: The provisions of this §111.34 adopted to be effective September 1, 1996, 21 TexReg 7371; amended to be effective August 1, 2006, 30 TexReg 1931; amended to be effective

February 22, 2009, 34 TexReg 1056.

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STAAR Geometry Eligible TEKS

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Geometry Assessment



Fall 2010

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STAAR Geometry Assessment

Reporting Category 1:Geometric Structure

The student will demonstrate an understanding of geometric structure.

(G.1) Geometric structure. The student understands the structure of, and

relationships within, an axiomatic system. The student is expected to

(B) recognize the historical development of geometric systems and know

mathematics is developed for a variety of purposes; and

Supporting Standard

(C) compare and contrast the structures and implications of Euclidean

and non-Euclidean geometries. Supporting Standard

(G.2) Geometric structure. The student analyzes geometric relationships in

order to make and verify conjectures. The student is expected to

(A) use constructions to explore attributes of geometric figures and to

make conjectures about geometric relationships; and

Supporting Standard

(B) make conjectures about angles, lines, polygons, circles, and three-

dimensional figures and determine the validity of the conjectures,

choosing from a variety of approaches such as coordinate,

transformational, or axiomatic. Readiness Standard

(G.3) Geometric structure. The student applies logical reasoning to justify and

prove mathematical statements. The student is expected to

(A) determine the validity of a conditional statement, its converse,

inverse, and contrapositive; Supporting Standard

(B) construct and justify statements about geometric figures and their

properties; Supporting Standard

(C) use logical reasoning to prove statements are true and find counter

examples to disprove statements that are false;

Readiness Standard

(D) use inductive reasoning to formulate a conjecture; and

Supporting Standard

(E) use deductive reasoning to prove a statement.

Supporting Standard

STAAR Geometry Page 2 of 7


Fall 201027

Reporting Category 2:Geometric Patterns and Representations

The student will demonstrate an understanding of geometric patterns andrepresentations.

(G.4) Geometric structure. The student uses a variety of representations to

describe geometric relationships and solve problems. The student is

expected to

(A) select an appropriate representation ([concrete,] pictorial, graphical,

verbal, or symbolic) in order to solve problems.

Supporting Standard

(G.5) Geometric patterns. The student uses a variety of representations to

describe geometric relationships and solve problems. The student is

expected to

(A) use numeric and geometric patterns to develop algebraic expressions

representing geometric properties; Readiness Standard

(B) use numeric and geometric patterns to make generalizations about

geometric properties, including properties of polygons, ratios in

similar figures and solids, and angle relationships in polygons and

circles; Supporting Standard

(C) use properties of transformations and their compositions to make

connections between mathematics and the real world, such as

tessellations; and Supporting Standard

(D) identify and apply patterns from right triangles to solve meaningful

problems, including special right triangles (45-45-90 and 30-60-90)

and triangles whose sides are Pythagorean triples.

Readiness Standard



Fall 201028

Reporting Category 3:Dimensionality and the Geometry of Location

The student will demonstrate an understanding of dimensionality and the geometry of location.

(G.6) Dimensionality and the geometry of location. The student analyzes

the relationship between three-dimensional geometric figures and related

two-dimensional representations and uses these representations to solve

problems. The student is expected to

(A) describe and draw the intersection of a given plane with various

three-dimensional geometric figures; Supporting Standard

(B) use nets to represent and construct three-dimensional geometric

figures; and Supporting Standard

(C) use orthographic and isometric views of three-dimensional geometric

figures to represent and construct three-dimensional geometric

figures and solve problems. Supporting Standard

(G.7) Dimensionality and the geometry of location. The student

understands that coordinate systems provide convenient and efficient

ways of representing geometric figures and uses them accordingly. The

student is expected to

(A) use one- and two-dimensional coordinate systems to represent

points, lines, rays, line segments, and figures;

Supporting Standard

(B) use slopes and equations of lines to investigate geometric

relationships, including parallel lines, perpendicular lines, and special

segments of triangles and other polygons; and Readiness Standard

(C) [derive and] use formulas involving length, slope, and midpoint.

Readiness Standard



Fall 201029

Reporting Category 4:Congruence and the Geometry of Size

The student will demonstrate an understanding of congruence and the geometry of size.

(G.8) Congruence and the geometry of size. The student uses tools to

determine measurements of geometric figures and extends measurement

concepts to find perimeter, area, and volume in problem situations. The


(A) find areas of regular polygons, circles, and composite figures;

Readiness Standard

(B) find areas of sectors and arc lengths of circles using proportional

reasoning; Supporting Standard

(C) [derive,] extend, and use the Pythagorean Theorem;

Readiness Standard

(D) find surface areas and volumes of prisms, pyramids, spheres, cones,

cylinders, and composites of these figures in problem situations;

Readiness Standard

(E) use area models to connect geometry to probability and statistics;

and Supporting Standard

(F) use conversions between measurement systems to solve problems in

real-world situations. Supporting Standard

(G.9) Congruence and the geometry of size. The student analyzes properties

and describes relationships in geometric figures. The student is expected

to

(A) formulate and test conjectures about the properties of parallel and

perpendicular lines based on explorations and [concrete] models;

Supporting Standard

(B) formulate and test conjectures about the properties and attributes of

polygons and their component parts based on explorations and

[concrete] models; Supporting Standard

(C) formulate and test conjectures about the properties and attributes of

circles and the lines that intersect them based on explorations and

[concrete] models; and Supporting Standard

(D) analyze the characteristics of polyhedra and other three-dimensional

figures and their component parts based on explorations and

[concrete] models. Supporting Standard



Fall 201030

(G.10) Congruence and the geometry of size. The student applies the concept

of congruence to justify properties of figures and solve problems. The


(A) use congruence transformations to make conjectures and justify

properties of geometric figures including figures represented on a

coordinate plane; and Supporting Standard

(B) justify and apply triangle congruence relationships.

Readiness Standard



Fall 201031

Reporting Category 5:Similarity and the Geometry of Shape

The student will demonstrate an understanding of similarity and the geometry of shape.

(G.11) Similarity and the geometry of shape. The student applies the

concepts of similarity to justify properties of figures and solve problems.


(A) use and extend similarity properties and transformations to explore

and justify conjectures about geometric figures;

Supporting Standard

(B) use ratios to solve problems involving similar figures;

Supporting Standard

(C) develop, apply, and justify triangle similarity relationships, such as

right triangle ratios, trigonometric ratios, and Pythagorean triples

using a variety of methods; and Readiness Standard

(D) describe the effect on perimeter, area, and volume when one or

more dimensions of a figure are changed and apply this idea in

solving problems. Readiness Standard



Fall 201032

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STAAR GeometryBlueprint

33

Texas Education Agency

Student Assessment Division

Fall 2010

STAAR Geometry Blueprint


Readiness Standards 2 Reporting Category 1:

Geometric Structure Supporting Standards 7 10

Total 9


Geometric Patterns and Representations Supporting Standards 3 8

Total 5


Dimensionality and the Geometry

Location

of



Total 6


Congruence and the Geometry of Size Supporting Standards 8 16

Total 12


Similarity and the Geometry of Shape Supporting Standards 2 8

Total 4




5 Griddable

52 Total

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Algebra II TEKS

35

§111.33. Algebra II (One-Half to One Credit).


(1) Foundation concepts for high school mathematics. As presented in Grades K-8, the basic understandings of number, operation, and quantitative reasoning; patterns, relationships, and algebraic thinking; geometry; measurement; and probability and statistics are essential foundations for all work in high school mathematics. Students continue to build on this foundation as they expand their understanding through other mathematical experiences.

(2) Algebraic thinking and symbolic reasoning. Symbolic reasoning plays a critical role in algebra; symbols provide powerful ways to represent mathematical situations and to express generalizations. Students study algebraic concepts and the relationships among them to better understand the structure of algebra.

(3) Functions, equations, and their relationship. The study of functions, equations, and their relationship is central to all of mathematics. Students perceive functions and equations as means for analyzing and understanding a broad variety of relationships and as a useful tool for expressing generalizations.

(4) Relationship between algebra and geometry. Equations and functions are algebraic tools that can be used to represent geometric curves and figures; similarly, geometric figures can illustrate algebraic relationships. Students perceive the connections between algebra and geometry and use the tools of one to help solve problems in the other.

(5) Tools for algebraic thinking. Techniques for working with functions and equations are essential in understanding underlying relationships. Students use a variety of representations (concrete, pictorial, numerical, symbolic, graphical, and verbal), tools, and technology (including, but not limited to, calculators with graphing capabilities, data collection devices, and computers) to model mathematical situations to solve meaningful problems.

(6) Underlying mathematical processes. Many processes underlie all content areas in mathematics. As they do mathematics, students continually use problem-solving, language and communication, and reasoning (justification and proof) to make connections within and outside mathematics. Students also use multiple representations, technology, applications and modeling, and numerical fluency in problem-solving contexts.


(1) Foundations for functions. The student uses properties and attributes of functions and applies functions to problem situations. The student is expected to:

36

(A) identify the mathematical domains and ranges of functions and determine reasonable domain and range values for continuous and discrete situations; and

(B) collect and organize data, make and interpret scatterplots, fit the graph of a function to the data, interpret the results, and proceed to model, predict, and make decisions and critical judgments.

(2) Foundations for functions. The student understands the importance of the skills required to manipulate symbols in order to solve problems and uses the necessary algebraic skills required to simplify algebraic expressions and solve equations and inequalities in problem situations. The student is expected to:

(A) use tools including factoring and properties of exponents to simplify expressions and to transform and solve equations; and

(B) use complex numbers to describe the solutions of quadratic equations.

(3) Foundations for functions. The student formulates systems of equations and inequalities from problem situations, uses a variety of methods to solve them, and analyzes the solutions in terms of the situations. The student is expected to:

(A) analyze situations and formulate systems of equations in two or more unknowns or inequalities in two unknowns to solve problems;

(B) use algebraic methods, graphs, tables, or matrices, to solve systems of equations or inequalities; and

(C) interpret and determine the reasonableness of solutions to systems of equations or inequalities for given contexts.

(4) Algebra and geometry. The student connects algebraic and geometric representations of functions. The student is expected to:

(A) identify and sketch graphs of parent functions, including linear (f (x) = x), quadratic (f (x) = x2), exponential (f (x) = ax), and logarithmic (f (x) = logax) functions, absolute value of x (f (x) = |x|), square root of x (f (x) = √x), and reciprocal of x (f (x) = 1/x);

(B) extend parent functions with parameters such as a in f (x) = a/x and describe the effects of the parameter changes on the graph of parent functions; and

(C) describe and analyze the relationship between a function and its inverse.

37

(5) Algebra and geometry. The student knows the relationship between the geometric and algebraic descriptions of conic sections. The student is expected to:

(A) describe a conic section as the intersection of a plane and a cone;

(B) sketch graphs of conic sections to relate simple parameter changes in the equation to corresponding changes in the graph;

(C) identify symmetries from graphs of conic sections;

(D) identify the conic section from a given equation; and

(E) use the method of completing the square.

(6) Quadratic and square root functions. The student understands that quadratic functions can be represented in different ways and translates among their various representations. The student is expected to:

(A) determine the reasonable domain and range values of quadratic functions, as well as interpret and determine the reasonableness of solutions to quadratic equations and inequalities;

(B) relate representations of quadratic functions, such as algebraic, tabular, graphical, and verbal descriptions; and

(C) determine a quadratic function from its roots (real and complex) or a graph.

(7) Quadratic and square root functions. The student interprets and describes the effects of changes in the parameters of quadratic functions in applied and mathematical situations. The student is expected to:

(A) use characteristics of the quadratic parent function to sketch the related graphs and connect between the y = ax2 + bx + c and the y = a (x - h)2 + k symbolic representations of quadratic functions; and

(B) use the parent function to investigate, describe, and predict the effects of changes in a, h, and k on the graphs of y = a (x - h)2 + k form of a function in applied and purely mathematical situations.

(8) Quadratic and square root functions. The student formulates equations and inequalities based on quadratic functions, uses a variety of methods to solve them, and analyzes the solutions in terms of the situation. The student is expected to:

38

(A) analyze situations involving quadratic functions and formulate quadratic equations or inequalities to solve problems;

(B) analyze and interpret the solutions of quadratic equations using discriminants and solve quadratic equations using the quadratic formula;

(C) compare and translate between algebraic and graphical solutions of quadratic equations; and

(D) solve quadratic equations and inequalities using graphs, tables, and algebraic methods.

(9) Quadratic and square root functions. The student formulates equations and inequalities based on square root functions, uses a variety of methods to solve them, and analyzes the solutions in terms of the situation. The student is expected to:

(A) use the parent function to investigate, describe, and predict the effects of parameter changes on the graphs of square root functions and describe limitations on the domains and ranges;

(B) relate representations of square root functions, such as algebraic, tabular, graphical, and verbal descriptions;

(C) determine the reasonable domain and range values of square root functions, as well as interpret and determine the reasonableness of solutions to square root equations and inequalities;

(D) determine solutions of square root equations using graphs, tables, and algebraic methods;

(E) determine solutions of square root inequalities using graphs and tables;

(F) analyze situations modeled by square root functions, formulate equations or inequalities, select a method, and solve problems; and

(G) connect inverses of square root functions with quadratic functions.

(10) Rational functions. The student formulates equations and inequalities based on rational functions, uses a variety of methods to solve them, and analyzes the solutions in terms of the situation. The student is expected to:

(A) use quotients of polynomials to describe the graphs of rational functions, predict the effects of parameter changes, describe limitations on the domains and ranges, and examine asymptotic behavior;

39

(B) analyze various representations of rational functions with respect to problem situations;

(C) determine the reasonable domain and range values of rational functions, as well as interpret and determine the reasonableness of solutions to rational equations and inequalities;

(D) determine the solutions of rational equations using graphs, tables, and algebraic methods;

(E) determine solutions of rational inequalities using graphs and tables;

(F) analyze a situation modeled by a rational function, formulate an equation or inequality composed of a linear or quadratic function, and solve the problem; and

(G) use functions to model and make predictions in problem situations involving direct and inverse variation.

(11) Exponential and logarithmic functions. The student formulates equations and inequalities based on exponential and logarithmic functions, uses a variety of methods to solve them, and analyzes the solutions in terms of the situation. The student is expected to:

(A) develop the definition of logarithms by exploring and describing the relationship between exponential functions and their inverses;

(B) use the parent functions to investigate, describe, and predict the effects of parameter changes on the graphs of exponential and logarithmic functions, describe limitations on the domains and ranges, and examine asymptotic behavior;

(C) determine the reasonable domain and range values of exponential and logarithmic functions, as well as interpret and determine the reasonableness of solutions to exponential and logarithmic equations and inequalities;

(D) determine solutions of exponential and logarithmic equations using graphs, tables, and algebraic methods;

(E) determine solutions of exponential and logarithmic inequalities using graphs and tables; and

(F) analyze a situation modeled by an exponential function, formulate an equation or inequality, and solve the problem.

40

Source: The provisions of this §111.33 adopted to be effective September 1, 1996, 21 TexReg 7371; amended to be effective August 1, 2006, 30 TexReg 1931; amended to be effective

February 22, 2009, 34 TexReg 1056.

41

™

™

STAAR Algebra II Eligible TEKS

42

Algebra II Assessment



Fall 2010 43

STAAR Algebra II Assessment

Reporting Category 1:Properties and Attributes of Functions

The student will demonstrate an understanding of the foundational

properties and attributes of functions.

(2A.1) Foundations for functions. The student uses properties and attributes of

functions and applies functions to problem situations. The student is

expected to

(A) identify the mathematical domains and ranges of functions and

determine reasonable domain and range values for continuous and

discrete situations; and Readiness Standard

(B) collect and organize data, make and interpret scatterplots, fit the

graph of a function to the data, interpret the results, and proceed to

model, predict, and make decisions and critical judgments.

Readiness Standard

(2A.4) Algebra and geometry. The student connects algebraic and geometric

representations of functions. The student is expected to

(A) identify and sketch graphs of parent functions, including linear

(f (x) = x), quadratic (f (x) = x 2), exponential (f (x) = ax), and

logarithmic (f (x) = logax) functions, absolute value of x (f (x) = |x|),

square root of x (f (x) = x ), and reciprocal of x (f (x) = 1/x);

Supporting Standard

(B) extend parent functions with parameters such as a in f (x) = a/x and

describe the effects of the parameter changes on the graph of parent

functions; and Readiness Standard

(C) describe and analyze the relationship between a function and its

inverse. Supporting Standard

STAAR Algebra II Page 2 of 8


Fall 201044

Reporting Category 2:Representational Tools to Solve Problems

The student will demonstrate an understanding of the use of

representational tools to solve problems.

(2A.2) Foundations for functions. The student understands the importance of

the skills required to manipulate symbols in order to solve problems and

uses the necessary algebraic skills required to simplify algebraic

expressions and solve equations and inequalities in problem situations.


(A) use tools including factoring and properties of exponents to simplify

expressions and to transform and solve equations; and

Supporting Standard

(B) use complex numbers to describe the solutions of quadratic

equations. Supporting Standard

(2A.3) Foundations for functions. The student formulates systems of equations

and inequalities from problem situations, uses a variety of methods to

solve them, and analyzes the solutions in terms of the situations. The


(A) analyze situations and formulate systems of equations in two or

more unknowns or inequalities in two unknowns to solve problems;

Readiness Standard

(B) use algebraic methods, graphs, tables, or matrices, to solve systems

of equations or inequalities; and Readiness Standard

(C) interpret and determine the reasonableness of solutions to systems

of equations or inequalities for given contexts. Readiness Standard



Fall 201045

Reporting Category 3:Properties of Quadratic Functions

The student will demonstrate an understanding of the properties of

quadratic functions.

(2A.6) Quadratic and square root functions. The student understands that

quadratic functions can be represented in different ways and translates

among their various representations. The student is expected to

(A) determine the reasonable domain and range values of quadratic

functions, as well as interpret and determine the reasonableness of

solutions to quadratic equations and inequalities;

Readiness Standard

(B) relate representations of quadratic functions, such as algebraic,

tabular, graphical, and verbal descriptions; and

Readiness Standard

(C) determine a quadratic function from its roots (real and complex) or a

graph. Supporting Standard

(2A.8) Quadratic and square root functions. The student formulates equations

and inequalities based on quadratic functions, uses a variety of methods to

solve them, and analyzes the solutions in terms of the situation. The


(A) analyze situations involving quadratic functions and formulate

quadratic equations or inequalities to solve problems;

Readiness Standard

(B) analyze and interpret the solutions of quadratic equations using

discriminants and solve quadratic equations using the quadratic

formula; Supporting Standard

(C) compare and translate between algebraic and graphical solutions of

quadratic equations; and Supporting Standard

(D) solve quadratic equations and inequalities using graphs, tables, and

algebraic methods. Readiness Standard



Fall 201046

Reporting Category 4:Multiple Representations of Quadratic Relations

The student will demonstrate an understanding of the connections

between algebraic and geometric representations of quadratic functions

and relations.

(2A.5) Algebra and geometry. The student knows the relationship between the

geometric and algebraic descriptions of conic sections. The student is

expected to

(A) describe a conic section as the intersection of a plane and a cone;

Supporting Standard

(B) sketch graphs of conic sections to relate simple parameter changes

in the equation to corresponding changes in the graph;

Supporting Standard

(C) identify symmetries from graphs of conic sections;

Supporting Standard

(D) identify the conic section from a given equation; and

Supporting Standard

(E) use the method of completing the square. Supporting Standard

(2A.7) Quadratic and square root functions. The student interprets and

describes the effects of changes in the parameters of quadratic functions in

applied and mathematical situations. The student is expected to

(A) use characteristics of the quadratic parent function to sketch the

related graphs and connect between the y = ax 2 + bx + c and the

y = a(x – h)2 + k symbolic representations of quadratic functions;

and Readiness Standard

(B) use the parent function to investigate, describe, and predict the

effects of changes in a, h, and k on the graphs of y = a(x – h)2 + k

form of a function in applied and purely mathematical situations.

Supporting Standard



Fall 201047

Reporting Category 5:Properties of Square Root Functions


square root functions.

(2A.9) Quadratic and square root functions. The student formulates equations

and inequalities based on square root functions, uses a variety of methods

to solve them, and analyzes the solutions in terms of the situation. The


(A) use the parent function to investigate, describe, and predict the

effects of parameter changes on the graphs of square root functions

and describe limitations on the domains and ranges;

Supporting Standard

(B) relate representations of square root functions, such as algebraic,

tabular, graphical, and verbal descriptions; Supporting Standard

(C) determine the reasonable domain and range values of square root


solutions to square root equations and inequalities;

Supporting Standard

(D) determine solutions of square root equations using graphs, tables,

and algebraic methods; Supporting Standard

(E) determine solutions of square root inequalities using graphs and

tables; Supporting Standard

(F) analyze situations modeled by square root functions, formulate

equations or inequalities, select a method, and solve problems; and

Readiness Standard

(G) connect inverses of square root functions with quadratic functions.

Supporting Standard



Fall 201048

Reporting Category 6: Properties of Rational Functions


rational functions.

(2A.10) Rational functions. The student formulates equations and inequalities

based on rational functions, uses a variety of methods to solve them, and


(A) use quotients of polynomials to describe the graphs of rational

functions, predict the effects of parameter changes, describe

limitations on the domains and ranges, and examine asymptotic

behavior; Supporting Standard

(B) analyze various representations of rational functions with respect to

problem situations; Supporting Standard

(C) determine the reasonable domain and range values of rational


solutions to rational equations and inequalities;

Supporting Standard

(D) determine the solutions of rational equations using graphs, tables,

and algebraic methods; Supporting Standard

(E) determine solutions of rational inequalities using graphs and tables;

Supporting Standard

(F) analyze a situation modeled by a rational function, formulate an

equation or inequality composed of a linear or quadratic function,

and solve the problem; and Readiness Standard

(G) use functions to model and make predictions in problem situations

involving direct and inverse variation. Supporting Standard



Fall 201049

Reporting Category 7:Properties of Exponential and Logarithmic Functions


exponential and logarithmic functions.

(2A.11) Exponential and logarithmic functions. The student formulates

equations and inequalities based on exponential and logarithmic functions,

uses a variety of methods to solve them, and analyzes the solutions in

terms of the situation. The student is expected to

(A) develop the definition of logarithms by exploring and describing the

relationship between exponential functions and their inverses;

Readiness Standard

(B) use the parent functions to investigate, describe, and predict the

effects of parameter changes on the graphs of exponential and

logarithmic functions, describe limitations on the domains and

ranges, and examine asymptotic behavior; Supporting Standard

(C) determine the reasonable domain and range values of exponential

and logarithmic functions, as well as interpret and determine the

reasonableness of solutions to exponential and logarithmic equations

and inequalities; Supporting Standard

(D) determine solutions of exponential and logarithmic equations using

graphs, tables, and algebraic methods; Supporting Standard

(E) determine solutions of exponential and logarithmic inequalities using

graphs and tables; and Supporting Standard

(F) analyze a situation modeled by an exponential function, formulate an

equation or inequality, and solve the problem. Readiness Standard



Fall 201050

™

™

STAAR Algebra II Blueprint

51

STAAR Algebra II Blueprint


Reporting Category 1: Properties and Attributes of Functions


8Supporting Standards 2

Total 5

Reporting Category 2: Representational Tools to Solve Problems



Total 5

Reporting Category 3: Properties of Quadratic Functions



Total 7

Reporting Category 4: Representations of Quadratic Relations



Total 7

Reporting Category 5: Properties of Square Root Functions



Total 7

Reporting Category 6: Properties of Rational Functions



Total 7

Reporting Category 7: Properties of Exponential and Logarithmic Functions



Total 6




5 Griddable 50 Total


Fall 201052

™

™

IES Practice Guide

53

IES PRACTICE GUIDE

NCEE 2009-4060U.S. DEPARTMENT OF EDUCATION

WHAT WORKS CLEARINGHOUSE

Assisting Students Struggling with

Mathematics: Response to Intervention

(RtI) for Elementary and Middle Schools

Assisting Students Struggling with

Mathematics: Response to Intervention

(RtI) for Elementary and Middle Schools

54

The Institute of Education Sciences (IES) publishes practice guides in education to bring the best available evidence and expertise to bear on the types of systemic challenges that cannot currently be addressed by single interventions or programs. Authors of practice guides seldom conduct the types of systematic literature searches that are the backbone of a meta-analysis, although they take advantage of such work when it is already published. Instead, authors use their expertise to identify the most important research with respect to their recommendations, augmented by a search of recent publications to ensure that research citations are up-to-date.

Unique to IES-sponsored practice guides is that they are subjected to rigorous exter-nal peer review through the same office that is responsible for independent review of other IES publications. A critical task for peer reviewers of a practice guide is to determine whether the evidence cited in support of particular recommendations is up-to-date and that studies of similar or better quality that point in a different di-rection have not been ignored. Because practice guides depend on the expertise of their authors and their group decisionmaking, the content of a practice guide is not and should not be viewed as a set of recommendations that in every case depends on and flows inevitably from scientific research.

The goal of this practice guide is to formulate specific and coherent evidence-based recommendations for use by educators addressing the challenge of reducing the number of children who struggle with mathematics by using “response to interven-tion” (RtI) as a means of both identifying students who need more help and provid-ing these students with high-quality interventions. The guide provides practical, clear information on critical topics related to RtI and is based on the best available evidence as judged by the panel. Recommendations in this guide should not be construed to imply that no further research is warranted on the effectiveness of particular strategies used in RtI for students struggling with mathematics.

55

Assisting Students Struggling with Mathematics: Response to Intervention (RtI) for Elementary and Middle Schools

April 2009

PanelRussell Gersten (Chair)InstructIonal research Group

Sybilla BeckmannunIversIty of GeorGIa

Benjamin ClarkeInstructIonal research Group

Anne FoegenIowa state unIversIty

Laurel Marshhoward county publIc school system

Jon R. Starharvard unIversIty

Bradley WitzelwInthrop unIversIty

StaffJoseph DiminoMadhavi JayanthiRebecca Newman-Gonchar InstructIonal research Group

Shannon MonahanLibby ScottmathematIca polIcy research

NCEE 2009-4060U.S. DEPARTMENT OF EDUCATION

IES PRACTICE GUIDE

56

This report was prepared for the National Center for Education Evaluation and Re gional Assistance, Institute of Education Sciences under Contract ED-07-CO-0062 by the What Works Clearinghouse, which is operated by Mathematica Policy Research, Inc.

Disclaimer

The opinions and positions expressed in this practice guide are the authors’ and do not necessarily represent the opinions and positions of the Institute of Education Sci-ences or the U.S. Department of Education. This practice guide should be reviewed and applied according to the specific needs of the educators and edu cation agency using it, and with full realization that it represents the judgments of the review panel regarding what constitutes sensible practice, based on the research available at the time of pub lication. This practice guide should be used as a tool to assist in decisionmaking rather than as a “cookbook.” Any references within the document to specific educational products are illustrative and do not imply endorsement of these products to the exclusion of other products that are not referenced.

U.S. Department of Education Arne Duncan Secretary

Institute of Education Sciences Sue Betka Acting Director

National Center for Education Evaluation and Regional Assistance Phoebe Cottingham Commissioner

April 2009

This report is in the public domain. Although permission to reprint this publication is not necessary, the citation should be:

Gersten, R., Beckmann, S., Clarke, B., Foegen, A., Marsh, L., Star, J. R., & Witzel, B. (2009). Assisting students struggling with mathematics: Response to Interven-tion (RtI) for elementary and middle schools (NCEE 2009-4060). Washington, DC: National Center for Education Evaluation and Regional Assistance, Institute of Education Sci ences, U.S. Department of Education. Retrieved from http://ies.ed.gov/ncee/wwc/publications/practiceguides/.

This report is available on the IES website at http://ies.ed.gov/ncee and http://ies.ed.gov/ncee/wwc/publications/practiceguides/.

Alternative formats On request, this publication can be made available in alternative formats, such as Braille, large print, audiotape, or computer diskette. For more information, call the Alternative Format Center at 202–205–8113.

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http://ies.ed.gov/ncee



( iii )


ContentsIntroduction 1

The What Works Clearinghouse standards and their relevance to this guide 3

Overview 4

Summary of the Recommendations 5

Scope of the practice guide 9

Checklist for carrying out the recommendations 11

Recommendation 1. Screen all students to identify those at risk for potential mathematics difficulties and provide interventions to students identified as at risk. 13

Recommendation 2. Instructional materials for students receiving interventions should focus intensely on in-depth treatment of whole numbers in kindergarten through grade 5 and on rational numbers in grades 4 through 8. These materials should be selected by committee. 18

Recommendation 3. Instruction during the intervention should be explicit and systematic. This includes providing models of proficient problem solving, verbalization of thought processes, guided practice, corrective feedback, and frequent cumulative review. 21

Recommendation 4. Interventions should include instruction on solving word problems that is based on common underlying structures. 26

Recommendation 5. Intervention materials should include opportunities for students to work with visual representations of mathematical ideas and interventionists should be proficient in the use of visual representations of mathematical ideas. 30

Recommendation 6. Interventions at all grade levels should devote about 10 minutes in each session to building fluent retrieval of basic arithmetic facts. 37

Recommendation 7. Monitor the progress of students receiving supplemental instruction and other students who are at risk. 41

Recommendation 8. Include motivational strategies in tier 2 and tier 3 interventions. 44

Glossary of terms as used in this report 48

Appendix A. Postscript from the Institute of Education Sciences 52

Appendix B. About the authors 55

Appendix C. Disclosure of potential conflicts of interest 59

Appendix D. Technical information on the studies 61

References 9158

( iv )

ASSISTInG STuDEnTS STRuGGlInG WITh MAThEMATICS: RESPOnSE TO InTERvEnTIOn (RTI) fOR ElEMEnTARy AnD MIDDlE SChOOlS

List of tables

Table 1. Institute of Education Sciences levels of evidence for practice guides 2

Table 2. Recommendations and corresponding levels of evidence 6

Table 3. Sensitivity and specificity 16

Table D1. Studies of interventions that included explicit instruction and met WWC Standards (with and without reservations) 69

Table D2. Studies of interventions that taught students to discriminate problem types that met WWC standards (with or without reservations) 73

Table D3. Studies of interventions that used visual representations that met WWC standards (with and without reservations) 77–78

Table D4. Studies of interventions that included fact fluency practices that met WWC standards (with and without reservations) 83

List of examples

Example 1. Change problems 27

Example 2. Compare problems 28

Example 3. Solving different problems with the same strategy 29

Example 4. Representation of the counting on strategy using a number line 33

Example 5. using visual representations for multidigit addition 34

Example 6. Strip diagrams can help students make sense of fractions 34

Example 7. Manipulatives can help students understand that four multiplied by six means four groups of six, which means 24 total objects 35

Example 8. A set of matched concrete, visual, and abstract representations to teach solving single-variable equations 35

Example 9: Commutative property of multiplication 48

Example 10: Make-a-10 strategy 49

Example 11: Distributive property 50

Example 12: number decomposition 51

59

( 1 )

Introduction

Students struggling with mathematics may benefit from early interventions aimed at improving their mathematics ability and ultimately preventing subsequent failure. This guide provides eight specific recom-mendations intended to help teachers, principals, and school administrators use Response to Intervention (RtI) to identify students who need assistance in mathe-matics and to address the needs of these students through focused interventions. The guide provides suggestions on how to carry out each recommendation and explains how educators can overcome potential roadblocks to implementing the recommendations.

The recommendations were developed by a panel of researchers and practitioners with expertise in various dimensions of this topic. The panel includes a research mathematician active in issues related to K–8 mathematics education, two pro-fessors of mathematics education, sev-eral special educators, and a mathematics coach currently providing professional de-velopment in mathematics in schools. The panel members worked collaboratively to develop recommendations based on the best available research evidence and our expertise in mathematics, special educa-tion, research, and practice.

The body of evidence we considered in de-veloping these recommendations included evaluations of mathematics interventions for low-performing students and students with learning disabilities. The panel con-sidered high-quality experimental and quasi-experimental studies, such as those meeting the criteria of the What Works Clearinghouse (http://www.whatworks.ed.gov), to provide the strongest evidence of effectiveness. We also examined stud-ies of the technical adequacy of batte ries of screening and progress monitoring measures for recommendations relating to assessment.

In some cases, recommendations reflect evidence-based practices that have been demonstrated as effective through rigor-ous research. In other cases, when such evidence is not available, the recommen-dations reflect what this panel believes are best practices. Throughout the guide, we clearly indicate the quality of the evidence that supports each recommendation.

Each recommendation receives a rating based on the strength of the research evi-dence that has shown the effectiveness of a recommendation (table 1). These ratings—strong, moderate, or low—have been de-fined as follows:

Strong refers to consistent and generaliz-able evidence that an intervention pro-gram causes better outcomes.1

Moderate refers either to evidence from studies that allow strong causal conclu-sions but cannot be generalized with as-surance to the population on which a recommendation is focused (perhaps be-cause the findings have not been widely replicated)—or to evidence from stud-ies that are generalizable but have more causal ambiguity than offered by experi-mental designs (such as statistical models of correlational data or group comparison designs for which the equivalence of the groups at pretest is uncertain).

Low refers to expert opinion based on rea-sonable extrapolations from research and theory on other topics and evidence from studies that do not meet the standards for moderate or strong evidence.

1. Following WWC guidelines, we consider a posi-tive, statistically significant effect or large effect size (i.e., greater than 0.25) as an indicator of positive effects.

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InTRODuCTIOn

( 2 )

Table 1. Institute of Education Sciences levels of evidence for practice guides

Strong

In general, characterization of the evidence for a recommendation as strong requires both studies with high internal validity (i.e., studies whose designs can support causal conclusions) and studies with high external validity (i.e., studies that in total include enough of the range of participants and settings on which the recommendation is focused to support the conclu-sion that the results can be generalized to those participants and settings). Strong evidence for this practice guide is operationalized as:

A sys• tematic review of research that generally meets the standards of the What Works Clearinghouse (WWC) (see http://ies.ed.gov/ncee/wwc/) and supports the effectiveness of a program, practice, or approach with no contradictory evidence of similar quality; ORSeveral well-designed, randomized controlled trials or well-designed quasi-experiments •that generally meet the standards of WWC and support the effectiveness of a program, practice, or approach, with no contradictory evidence of similar quality; OROne large, well-designed, randomized controlled, multisite trial that meets WWC standards •and supports the effectiveness of a program, practice, or approach, with no contradictory evidence of similar quality; ORFor assessments, evidence of reliability and validity that meets the Standards for • Educa-tional and Psychological Testing.a

Moderate

In general, characterization of the evidence for a recommendation as moderate requires stud-ies with high internal validity but moderate external validity, or studies with high external validity but moderate internal validity. In other words, moderate evidence is derived from studies that support strong causal conclusions but when generalization is uncertain, or stud-ies that support the generality of a relationship but when the causality is uncertain. Moderate evidence for this practice guide is operationalized as:

Experiments or quasi-experiments generally meeting the standards of WWC and sup-•porting the effectiveness of a program, practice, or approach with small sample sizes and/or other conditions of implementation or analysis that limit generalizability and no contrary evidence; ORComparison group studies that do not demonstrate equivalence of groups at pre-•test and therefore do not meet the standards of WWC but that (a) consistently show enhanced outcomes for participants experiencing a particular program, practice, or approach and (b) have no major flaws related to internal validity other than lack of demonstrated equivalence at pretest (e.g., only one teacher or one class per condition, unequal amounts of instructional time, highly biased outcome measures); ORCorrelational research with strong statistical controls for selection bias and for dis-•cerning influence of endogenous factors and no contrary evidence; ORFor assessments, evidence of reliability that meets the Standards for Educational and •Psychological Testingb but with evidence of validity from samples not adequately rep-resentative of the population on which the recommendation is focused.

Low

In general, characterization of the evidence for a recommendation as low means that the recommendation is based on expert opinion derived from strong findings or theories in related areas and/or expert opinion buttressed by direct evidence that does not rise to the moderate or strong levels. Low evidence is operationalized as evidence not meeting the standards for the moderate or high levels.

a. American Educational Research Association, American Psychological Association, and National Council on Measurement in Education (1999).

b. Ibid.

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The What Works Clearinghouse standards and their relevance to this guide

The panel relied on WWC evidence stan-dards to assess the quality of evidence supporting mathematics intervention pro-grams and practices. The WWC addresses evidence for the causal validity of instruc-tional programs and practices according to WWC standards. Information about these standards is available at http://ies.ed.gov/ncee/wwc/references/standards/. The technical quality of each study is rated and placed into one of three categories:

Meets Evidence Standards• —for random-ized controlled trials and regression discontinuity studies that provide the strongest evidence of causal validity.

Meets Evidence Standards with Reser-•vations—for all quasi-experimental studies with no design flaws and ran-domized controlled trials that have problems with randomization, attri-tion, or disruption.

Does Not Meet Evidence Screens• —for studies that do not provide strong evi-dence of causal validity.

Following the recommendations and sug-gestions for carrying out the recommen-dations, Appendix D presents information on the research evidence to support the recommendations.

The panel would like to thank Kelly Hay-mond for her contributions to the analysis, the WWC reviewers for their contribution to the project, and Jo Ellen Kerr and Jamila Henderson for their support of the intricate logistics of the project. We also would like to thank Scott Cody for his oversight of the overall progress of the practice guide.

Dr. Russell GerstenDr. Sybilla BeckmannDr. Benjamin Clarke

Dr. Anne FoegenMs. Laurel Marsh

Dr. Jon R. StarDr. Bradley Witzel

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Overview

Response to Intervention (RtI) is an early de-tection, prevention, and support system that identifies struggling students and assists them before they fall behind. In the 2004 reauthorization of the Individuals with Dis-abilities Education Act (PL 108-446), states were encouraged to use RtI to accurately identify students with learning disabilities and encouraged to provide additional sup-ports for students with academic difficul-ties regardless of disability classification. Although many states have already begun to implement RtI in the area of reading, RtI ini-tiatives for mathematics are relatively new.

Students’ low achievement in mathemat-ics is a matter of national concern. The re-cent National Mathematics Advisory Panel Report released in 2008 summarized the poor showing of students in the United States on international comparisons of mathematics performance such as the Trends in International Mathematics and Science Study (TIMSS) and the Program for International Student Assessment (PISA).2 A recent survey of algebra teachers as-sociated with the report identified key deficiencies of students entering algebra, including aspects of whole number arith-metic, fractions, ratios, and proportions.3 The National Mathematics Advisory Panel

2. See, for example, National Mathematics Ad-visory Panel (2008) and Schmidt and Houang (2007). For more information on the TIMSS, see http://nces.ed.gov/timss/. For more information on PISA, see http://www.oecd.org.

3. National Mathematics Advisory Panel (2008).

concluded that all students should receive preparation from an early age to ensure their later success in algebra. In particular, the report emphasized the need for math-ematics interventions that mitigate and prevent mathematics difficulties.

This panel believes that schools can use an RtI framework to help struggling students prepare for later success in mathemat-ics. To date, little research has been con-ducted to identify the most effective ways to initiate and implement RtI frameworks for mathematics. However, there is a rich body of research on effective mathematics interventions implemented outside an RtI framework. Our goal in this practice guide is to provide suggestions for assessing students’ mathematics abilities and imple-menting mathematics interventions within an RtI framework, in a way that reflects the best evidence on effective practices in mathematics interventions.

RtI begins with high-quality instruction and universal screening for all students. Whereas high-quality instruction seeks to prevent mathematics difficulties, screen-ing allows for early detection of difficul-ties if they emerge. Intensive interventions are then provided to support students in need of assistance with mathematics learning.4 Student responses to interven-tion are measured to determine whether they have made adequate progress and (1) no longer need intervention, (2) continue to need some intervention, or (3) need more intensive intervention. The levels of intervention are conventionally referred to as “tiers.” RtI is typically thought of as having three tiers.5 Within a three-tiered RtI model, each tier is defined by specific characteristics.

4. Fuchs, Fuchs, Craddock et al. (2008).

5. Fuchs, Fuchs, and Vaughn (2008) make the case for a three-tier RtI model. Note, however, that some states and school districts have imple-mented multitier intervention systems with more than three tiers.

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Tier 1 is the mathematics instruction •that all students in a classroom receive. It entails universal screening of all stu-dents, regardless of mathematics profi-ciency, using valid measures to identify students at risk for future academic failure—so that they can receive early intervention.6 There is no clear consen-sus on the characteristics of instruction other than that it is “high quality.”7

In tier 2 interventions, schools provide •additional assistance to students who demonstrate difficulties on screening measures or who demonstrate weak progress.8 Tier 2 students receive sup-plemental small group mathematics instruction aimed at building targeted mathematics proficiencies.9 These in-terventions are typically provided for 20 to 40 minutes, four to five times each week.10 Student progress is monitored throughout the intervention.11

Tier 3 interventions are provided to •students who are not benefiting from tier 2 and require more intensive as-sistance.12 Tier 3 usually entails one-on-one tutoring along with an appropri-ate mix of instructional interventions. In some cases, special education ser-vices are included in tier 3, and in oth-ers special education is considered an additional tier.13 Ongoing analysis of

6. For reviews see Jiban and Deno (2007); Fuchs, Fuchs, Compton et al. (2007); Gersten, Jordan, and Flojo (2005).

7. National Mathematics Advisory Panel (2008); National Research Council (2001).

8. Fuchs, Fuchs, Craddock et al. (2008); Na-tional Joint Committee on Learning Disabilities (2005).


10. For example, see Jitendra et al. (1998) and Fuchs, Fuchs, Craddock et al. (2008).

11. National Joint Committee on Learning Dis-abilities (2005).


13. Fuchs, Fuchs, Craddock et al. (2008); National Joint Committee on Learning Disabilities (2005).

student performance data is critical in this tier. Typically, specialized person-nel, such as special education teachers and school psychologists, are involved in tier 3 and special education services.14 However, students often receive rele-vant mathematics interventions from a wide array of school personnel, includ-ing their classroom teacher.

Summary of the Recommendations

This practice guide offers eight recom-mendations for identifying and supporting students struggling in mathematics (table 2). The recommendations are intended to be implemented within an RtI framework (typically three-tiered). The panel chose to limit its discussion of tier 1 to universal screening practices (i.e., the guide does not make recommendations for general classroom mathematics instruction). Rec-ommendation 1 provides specific sugges-tions for conducting universal screening effectively. For RtI tiers 2 and 3, recom-mendations 2 though 8 focus on the most effective content and pedagogical prac-tices that can be included in mathematics interventions.

Throughout this guide, we use the term “interventionist” to refer to those teach-ing the intervention. At a given school, the interventionist may be the general class-room teacher, a mathematics coach, a spe-cial education instructor, other certified school personnel, or an instructional as-sistant. The panel recognizes that schools rely on different personnel to fill these roles depending on state policy, school resources, and preferences.

Recommendation 1 addresses the type of screening measures that should be used in tier 1. We note that there is more research on valid screening measures for students in

14. National Joint Committee on Learning Dis-abilities (2005).

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Table 2. Recommendations and corresponding levels of evidence

Recommendation Level of evidence

Tier 1

1. Screen all students to identify those at risk for potential mathematics

difficulties and provide interventions to students identified as at risk.Moderate

Tiers 2 and 3

2. Instructional materials for students receiving interventions should

focus intensely on in-depth treatment of whole numbers in kindergar-

ten through grade 5 and on rational numbers in grades 4 through 8.

These materials should be selected by committee.

Low

3. Instruction during the intervention should be explicit and systematic.

This includes providing models of proficient problem solving, verbal-

ization of thought processes, guided practice, corrective feedback, and

frequent cumulative review.

Strong

4. Interventions should include instruction on solving word problems

that is based on common underlying structures.Strong

5. Intervention materials should include opportunities for students to

work with visual representations of mathematical ideas and interven-

tionists should be proficient in the use of visual representations of

mathematical ideas.

Moderate

6. Interventions at all grade levels should devote about 10 minutes in each

session to building fluent retrieval of basic arithmetic facts.Moderate

7. Monitor the progress of students receiving supplemental instruction

and other students who are at risk.Low

8. Include motivational strategies in tier 2 and tier 3 interventions. Low

Source: Authors’ compilation based on analysis described in text.

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kindergarten through grade 2,15 but there are also reasonable strategies to use for stu-dents in more advanced grades.16 We stress that no one screening measure is perfect and that schools need to monitor the prog-ress of students who score slightly above or slightly below any screening cutoff score.

Recommendations 2 though 6 address the content of tier 2 and tier 3 interventions and the types of instructional strategies that should be used. In recommendation 2, we translate the guidance by the National Mathematics Advisory Panel (2008) and the National Council of Teachers of Math-ematics Curriculum Focal Points (2006) into suggestions for the content of inter-vention curricula. We argue that the math-ematical focus and the in-depth coverage advocated for proficient students are also necessary for students with mathematics difficulties. For most students, the content of interventions will include foundational concepts and skills introduced earlier in the student’s career but not fully under-stood and mastered. Whenever possible, links should be made between founda-tional mathematical concepts in the inter-vention and grade-level material.

At the center of the intervention recom-mendations is that instruction should be systematic and explicit (recommendation 3). This is a recurrent theme in the body of valid scientific research.17 We explore the multiple meanings of explicit instruc-tion and indicate which components of explicit instruction appear to be most re-lated to improved student outcomes. We believe this information is important for districts and state departments to have as they consider selecting materials and

15. Gersten, Jordan, and Flojo (2005); Gersten, Clarke, and Jordan (2007).

16. Jiban and Deno (2007); Foegen, Jiban, and Deno (2007).

17. Darch, Carnine, and Gersten (1984); Fuchs et al. (2003a); Jitendra et al. (1998); Schunk and Cox (1986); Tournaki (2003); Wilson and Sindelar (1991).

providing professional development for interventionists.

Next, we highlight several areas of re-search that have produced promising find-ings in mathematics interventions. These include systematically teaching students about the problem types associated with a given operation and its inverse (such as problem types that indicate addition and subtraction) (recommendation 4).18 We also recommend practices to help students translate abstract symbols and numbers into meaningful visual representations (recommendation 5).19 Another feature that we identify as crucial for long-term success is systematic instruction to build quick retrieval of basic arithmetic facts (recommendation 6). Some evidence exists supporting the allocation of time in the in-tervention to practice fact retrieval using flash cards or computer software.20 There is also evidence that systematic work with properties of operations and counting strategies (for younger students) is likely to promote growth in other areas of math-ematics beyond fact retrieval.21

The final two recommendations address other considerations in implementing tier 2 and tier 3 interventions. Recommenda-tion 7 addresses the importance of moni-toring the progress of students receiving

18. Jitendra et al. (1998); Xin, Jitendra, and Deat-line-Buchman (2005); Darch, Carnine, and Gersten (1984); Fuchs et al. (2003a); Fuchs et al. (2003b); Fuchs, Fuchs, Prentice et al. (2004); Fuchs, Fuchs, and Finelli (2004); Fuchs, Fuchs, Craddock et al. (2008) Fuchs, Seethaler et al. (2008).

19. Artus and Dyrek (1989); Butler et al. (2003); Darch, Carnine, and Gersten (1984); Fuchs et al. (2005); Fuchs, Seethaler et al. (2008); Fuchs, Powell et al. (2008); Fuchs, Fuchs, Craddock et al. (2008); Jitendra et al. (1998); Walker and Po-teet (1989); Wilson and Sindelar (1991); Witzel (2005); Witzel, Mercer, and Miller (2003); Wood-ward (2006).

20. Bernie-Smith (1991); Fuchs, Seethaler et al. (2008); Fuchs et al. (2005); Fuchs, Fuchs, Hamlett et al. (2006); Fuchs, Powell et al. (2008).

21. Tournaki (2003); Woodward (2006).

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interventions. Specific types of formative assessment approaches and measures are described. We argue for two types of ongo-ing assessment. One is the use of curricu-lum-embedded assessments that gauge how well students have learned the material in that day’s or week’s lesson(s). The panel believes this information is critical for in-terventionists to determine whether they need to spend additional time on a topic. It also provides the interventionist and other school personnel with information that can be used to place students in groups within tiers. In addition, we recommend that schools regularly monitor the prog-ress of students receiving interventions

and those with scores slightly above or below the cutoff score on screening mea-sures with broader measures of mathemat-ics proficiency. This information provides the school with a sense of how the overall mathematics program (including tier 1, tier 2, and tier 3) is affecting a given student.

Recommendation 8 addresses the impor-tant issue of motivation. Because many of the students struggling with mathematics have experienced failure and frustration by the time they receive an intervention, we suggest tools that can encourage active engagement of students and acknowledge student accomplishments.

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Scope of the practice guide

Our goal is to provide evidence-based sug-gestions for screening students for mathe-matics difficulties, providing interventions to students who are struggling, and moni-toring student responses to the interven-tions. RtI intentionally cuts across the bor-ders of special and general education and involves school-wide collaboration. There-fore, our target audience for this guide in-cludes teachers, special educators, school psychologists and counselors, as well as administrators. Descriptions of the ma-terials and instructional content in tier 2 and tier 3 interventions may be especially useful to school administrators selecting interventions, while recommendations that relate to content and pedagogy will be most useful to interventionists.22

The focus of this guide is on providing RtI interventions in mathematics for stu-dents in kindergarten through grade 8. This broad grade range is in part a response to the recent report of the National Math-ematics Advisory Panel (2008), which em-phasized a unified progressive approach to promoting mathematics proficiency for elementary and middle schools. Moreover, given the growing number of initiatives aimed at supporting students to succeed in algebra, the panel believes it essential to provide tier 2 and tier 3 interventions to struggling students in grades 4 through 8. Because the bulk of research on mathemat-ics interventions has focused on students in kindergarten through grade 4, some rec-ommendations for students in older grades are extrapolated from this research.

22. Interventionists may be any number of school personnel, including classroom teachers, special educators, school psychologists, paraprofession-als, and mathematics coaches and specialists. The panel does not specify the interventionist.

The scope of this guide does not include recommendations for special education referrals. Although enhancing the valid-ity of special education referrals remains important and an issue of ongoing discus-sion23 and research,24 we do not address it in this practice guide, in part because empirical evidence is lacking.

The discussion of tier 1 in this guide re-volves only around effective screening, be-cause recommendations for general class-room mathematics instruction were beyond the scope of this guide. For this reason, studies of effective general mathematics instruction practices were not included in the evidence base for this guide.25

The studies reviewed for this guide in-cluded two types of comparisons among groups. First, several studies of tier 2 in-terventions compare students receiving multicomponent tier 2 interventions with students receiving only routine classroom instruction.26 This type of study provides evidence of the effectiveness of providing tier 2 interventions but does not permit conclusions about which component is most effective. The reason is that it is not possible to identify whether one particular component or a combination of compo-nents within a multicomponent interven-tion produced an effect. Second, several

23. Kavale and Spaulding (2008); Fuchs, Fuchs, and Vaughn (2008); VanDerHeyden, Witt, and Gilbertson (2007).

24. Fuchs, Fuchs, Compton et al. (2006).

25. There were a few exceptions in which general mathematics instruction studies were included in the evidence base. When the effects of a general mathematics instruction program were specified for low-achieving or disabled students and the intervention itself appeared applicable to teach-ing tier 2 or tier 3 (e.g., teaching a specific opera-tional strategy), we included them in this study. Note that disabled students were predominantly learning disabled.

26. For example, Fuchs, Seethaler et al. (2008) examined the effects of providing supplemen-tal tutoring (i.e., a tier 2 intervention) relative to regular classroom instruction (i.e., tier 1).

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other studies examined the effects of two methods of tier 2 or tier 3 instruction.27 This type of study offers evidence for the effectiveness of one approach to teaching within a tier relative to another approach and assists with identifying the most ben-eficial approaches for this population.

The panel reviewed only studies for prac-tices that sought to improve student math-ematics outcomes. The panel did not con-sider interventions that improved other academic or behavioral outcomes. Instead, the panel focused on practices that ad-dressed the following areas of mathematics proficiency: operations (either computation

27. For example, Tournaki (2003) examined the effects of providing supplemental tutoring in an operations strategy (a tier 2 intervention) relative to supplemental tutoring with a drill and practice approach (also a tier 2 intervention).

or estimation), concepts (knowledge of properties of operations, concepts involv-ing rational numbers, prealgebra con-cepts), problem solving (word problems), and measures of general mathematics achievement. Measures of fact fluency were also included because quick retrieval of basic arithmetic facts is essential for success in mathematics and a persistent problem for students with difficulties in mathematics.28

Technical terms related to mathematics and technical aspects of assessments (psy-chometrics) are defined in a glossary at the end of the recommendations.

28. Geary (2004); Jordan, Hanich, and Kaplan (2003).

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in-depth coverage of rational numbers as well as advanced topics in whole number arithmetic (such as long division).

Districts should appoint committees, including experts in mathematics instruc-tion and mathematicians with knowledge of elementary and middle school math-ematics curricula, to ensure that specific criteria are covered in-depth in the cur-riculum they adopt.

Recommendation 3. Instruction during the intervention should be explicit and systematic. This includes providing models of proficient problem solving, verbalization of thought processes, guided practice, corrective feedback, and frequent cumulative review.

Ensure that instructional materials are systematic and explicit. In particular, they should include numerous clear models of easy and difficult problems, with accom-panying teacher think-alouds.

Provide students with opportunities to solve problems in a group and commu-nicate problem-solving strategies.

Ensure that instructional materials in-clude cumulative review in each session.

Recommendation 4. Interventions should include instruction on solving word problems that is based on common underlying structures.

Teach students about the structure of various problem types, how to categorize problems based on structure, and how to determine appropriate solutions for each problem type.

Teach students to recognize the com-mon underlying structure between famil-iar and unfamiliar problems and to transfer known solution methods from familiar to unfamiliar problems.

Checklist for carrying out the recommendations

Recommendation 1. Screen all students to identify those at risk for potential mathematics difficulties and provide interventions to students identified as at risk.

As a district or school sets up a screen-ing system, have a team evaluate potential screening measures. The team should se-lect measures that are efficient and reason-ably reliable and that demonstrate predic-tive validity. Screening should occur in the beginning and middle of the year.

Select screening measures based on the content they cover, with an emphasis on critical instructional objectives for each grade.

In grades 4 through 8, use screen-ing data in combination with state testing results.

use the same screening tool across a district to enable analyzing results across schools.

Recommendation 2. Instructional materials for students receiving interventions should focus intensely on in-depth treatment of whole numbers in kindergarten through grade 5 and on rational numbers in grades 4 through 8. These materials should be selected by committee.

for students in kindergarten through grade 5, tier 2 and tier 3 interventions should focus almost exclusively on prop-erties of whole numbers and operations. Some older students struggling with whole numbers and operations would also benefit from in-depth coverage of these topics.

for tier 2 and tier 3 students in grades 4 through 8, interventions should focus on

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Recommendation 5. Intervention materials should include opportunities for students to work with visual representations of mathematical ideas and interventionists should be proficient in the use of visual representations of mathematical ideas.

use visual representations such as number lines, arrays, and strip diagrams.

If visuals are not sufficient for develop-ing accurate abstract thought and answers, use concrete manipulatives first. Although this can also be done with students in upper elementary and middle school grades, use of manipulatives with older students should be expeditious because the goal is to move toward understanding of—and facility with—visual representations, and finally, to the abstract.

Recommendation 6. Interventions at all grade levels should devote about 10 minutes in each session to building fluent retrieval of basic arithmetic facts.

Provide about 10 minutes per ses-sion of instruction to build quick retrieval of basic arithmetic facts. Consider using technology, flash cards, and other materi-als for extensive practice to facilitate au-tomatic retrieval.

for students in kindergarten through grade 2, explicitly teach strategies for ef-ficient counting to improve the retrieval of mathematics facts.

Teach students in grades 2 through 8 how to use their knowledge of proper-ties, such as commutative, associative, and distributive law, to derive facts in their heads.

Recommendation 7. Monitor the progress of students receiving supplemental instruction and other students who are at risk.

Monitor the progress of tier 2, tier 3, and borderline tier 1 students at least once a month using grade-appropriate general outcome measures.

use curriculum-embedded assess-ments in interventions to determine whether students are learning from the intervention. These measures can be used as often as every day or as infrequently as once every other week.

use progress monitoring data to re-group students when necessary.

Recommendation 8. Include motivational strategies in tier 2 and tier 3 interventions.

Reinforce or praise students for their effort and for attending to and being en-gaged in the lesson.

Consider rewarding student accom-plishments.

Allow students to chart their progress and to set goals for improvement.

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The panel recommends that schools and districts systematically use universal screening to screen all students to determine which students have mathematics difficulties and require research-based interventions. Schools should evaluate and select screening measures based on their reliability and predictive validity, with particular emphasis on the measures’ specificity and sensitivity. Schools should also consider the efficiency of the measure to enable screening many students in a short time.

Level of evidence: Moderate

The panel judged the level of evidence sup-porting this recommendation to be mod-erate. This recommendation is based on a series of high-quality correlational studies with replicated findings that show the abil-ity of measures to predict performance in mathematics one year after administration (and in some cases two years).29

Brief summary of evidence to support the recommendation

A growing body of evidence suggests that there are several valid and reliable ap-proaches for screening students in the pri-mary grades. All these approaches target

29. For reviews see Jiban and Deno (2007); Fuchs, Fuchs, Compton et al. (2007); Gersten, Jordan, and Flojo (2005).

aspects of what is often referred to as number sense.30 They assess various as-pects of knowledge of whole numbers—properties, basic arithmetic operations, understanding of magnitude, and applying mathematical knowledge to word prob-lems. Some measures contain only one aspect of number sense (such as magni-tude comparison) and others assess four to eight aspects of number sense. The sin-gle-component approaches with the best ability to predict students’ subsequent mathematics performance include screen-ing measures of students’ knowledge of magnitude comparison and/or strategic counting.31 The broader, multicomponent measures seem to predict with slightly greater accuracy than single-component measures.32

Effective approaches to screening vary in efficiency, with some taking as little as 5 minutes to administer and others as long as 20 minutes. Multicomponent measures, which by their nature take longer to ad-minister, tend to be time-consuming for administering to an entire school popu-lation. Timed screening measures33 and untimed screening measures34 have been shown to be valid and reliable.

For the upper elementary grades and mid-dle school, we were able to locate fewer studies. They suggest that brief early screening measures that take about 10 minutes and cover a proportional sam-pling of grade-level objectives are reason-able and provide sufficient evidence of reli-ability.35 At the current time, this research area is underdeveloped.

30. Berch (2005); Dehaene (1999); Okamoto and Case (1996); Gersten and Chard (1999).

31. Gersten, Jordan, and Flojo (2005).


33. For example, Clarke and Shinn (2004).

34. For example, Okamoto and Case (1996).

35. Jiban and Deno (2007); Foegen, Jiban, and Deno (2007).

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How to carry out this recommendation

1. As a district or school sets up a screen-ing system, have a team evaluate potential screening measures. The team should select measures that are efficient and reasonably reliable and that demonstrate predictive va-lidity. Screening should occur in the begin-ning and middle of the year.

The team that selects the measures should include individuals with expertise in mea-surement (such as a school psychologist or a member of the district research and eval-uation division) and those with expertise in mathematics instruction. In the opinion of the panel, districts should evaluate screen-ing measures on three dimensions.

• Predictivevalidity is an index of how well a score on a screening measure earlier in the year predicts a student’s later mathematics achievement. Greater predictive validity means that schools can be more confident that decisions based on screening data are accurate. In general, we recommend that schools and districts employ measures with predictive validity coefficients of at least .60 within a school year.36

• Reliability is an index of the consistency and precision of a measure. We recom-mend measures with reliability coeffi-cients of .80 or higher.37

• Efficiency is how quickly the universal screening measure can be adminis-tered, scored, and analyzed for all the students. As a general rule, we suggest that a screening measure require no

36. A coefficient of .0 indicates that there is no relation between the early and later scores, and a coefficient of 1.0 indicates a perfect positive relation between the scores.

37. A coefficient of .0 indicates that there is no relation between the two scores, and a coeffi-cient of 1.0 indicates a perfect positive relation between the scores.

more than 20 minutes to administer, which enables collecting a substantial amount of information in a reasonable time frame. Note that many screening measures take five minutes or less.38 We recommend that schools select screen-ing measures that have greater effi-ciency if their technical adequacy (pre-dictive validity, reliability, sensitivity, and specificity) is roughly equivalent to less efficient measures. Remember that screening measures are intended for administration to all students in a school, and it may be better to invest more time in diagnostic assessment of students who perform poorly on the universal screening measure.

Keep in mind that screening is just a means of determining which students are likely to need help. If a student scores poorly on a screening measure or screening battery—especially if the score is at or near a cut point, the panel recommends monitoring her or his progress carefully to discern whether extra instruction is necessary.

Developers of screening systems recom-mend that screening occur at least twice a year (e.g., fall, winter, and/or spring).39 This panel recommends that schools alle-viate concern about students just above or below the cut score by screening students twice during the year. The second screen-ing in the middle of the year allows another check on these students and also serves to identify any students who may have been at risk and grown substantially in their mathe-matics achievement—or those who were on-track at the beginning of the year but have not shown sufficient growth. The panel considers these two universal screenings to determine student proficiency as distinct from progress monitoring (Recommenda-tion 7), which occurs on a more frequent

38. Foegen, Jiban, and Deno (2007); Fuchs, Fuchs, Compton et al. (2007); Gersten, Clarke, and Jordan (2007).

39. Kaminski et al. (2008); Shinn (1989).

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basis (e.g., weekly or monthly) with a select group of intervention students in order to monitor response to intervention.

2. Select screening measures based on the content they cover, with an emphasis on crit-ical instructional objectives for each grade.

The panel believes that content covered in a screening measure should reflect the instructional objectives for a student’s grade level, with an emphasis on the most critical content for the grade level. The Na-tional Council of Teachers of Mathematics (2006) released a set of focal points for each grade level designed to focus instruc-tion on critical concepts for students to master within a specific grade. Similarly, the National Mathematics Advisory Panel (2008) detailed a route to preparing all students to be successful in algebra. In the lower elementary grades, the core focus of instruction is on building student under-standing of whole numbers. As students establish an understanding of whole num-bers, rational numbers become the focus of instruction in the upper elementary grades. Accordingly, screening measures used in the lower and upper elementary grades should have items designed to as-sess student’s understanding of whole and rational number concepts—as well as com-putational proficiency.

3. In grades 4 through 8, use screening data in combination with state testing results.

In the panel’s opinion, one viable option that schools and districts can pursue is to use results from the previous year’s state testing as a first stage of screening. Students who score below or only slightly above a benchmark would be considered for sub-sequent screening and/or diagnostic or placement testing. The use of state testing results would allow districts and schools to combine a broader measure that covers more content with a screening measure that is narrower but more focused. Because of the lack of available screening measures at

these grade levels, districts, county offices, or state departments may need to develop additional screening and diagnostic mea-sures or rely on placement tests provided by developers of intervention curricula.

4. use the same screening tool across a district to enable analyzing results across schools.

The panel recommends that all schools within a district use the same screening measure and procedures to ensure ob-jective comparisons across schools and within a district. Districts can use results from screening to inform instructional de-cisions at the district level. For example, one school in a district may consistently have more students identified as at risk, and the district could provide extra re-sources or professional development to that school. The panel recommends that districts use their research and evaluation staff to reevaluate screening measures an-nually or biannually. This entails exam-ining how screening scores predict state testing results and considering resetting cut scores or other data points linked to instructional decisionmaking.

Potential roadblocks and solutions

Roadblock 1.1. Districts and school person-nel may face resistance in allocating time re-sources to the collection of screening data.

Suggested Approach. The issue of time and personnel is likely to be the most sig-nificant obstacle that districts and schools must overcome to collect screening data. Collecting data on all students will require structuring the data collection process to be efficient and streamlined.

The panel notes that a common pitfall is a long, drawn-out data collection process, with teachers collecting data in their class-rooms “when time permits.” If schools are allocating resources (such as providing an intervention to students with the 20 low-est scores in grade 1), they must wait until

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all the data have been collected across classrooms, thus delaying the delivery of needed services to students. Further-more, because many screening measures are sensitive to instruction, a wide gap between when one class is assessed and another is assessed means that many stu-dents in the second class will have higher scores than those in the first because they were assessed later.

One way to avoid these pitfalls is to use data collection teams to screen students in a short period of time. The teams can consist of teachers, special education staff includ-ing such specialists as school psychologists, Title I staff, principals, trained instructional assistants, trained older students, and/or local college students studying child devel-opment or school psychology.

Roadblock 1.2. Implementing universal screening is likely to raise questions such as, “Why are we testing students who are doing fine?”

Suggested Approach. Collecting data on all students is new for many districts and schools (this may not be the case for elementary schools, many of which use screening assessments in reading).40 But screening allows schools to ensure that all students who are on track stay on track and collective screening allows schools to evaluate the impact of their instruction on groups of students (such as all grade 2 students). When schools screen all stu-dents, a distribution of achievement from high to low is created. If students consid-ered not at risk were not screened, the distribution of screened students would consist only of at-risk students. This could create a situation where some students at the “top” of the distribution are in real-ity at risk but not identified as such. For upper-grade students whose scores were

40. U.S. Department of Education, Office of Plan-ning, Evaluation and Policy Development, Policy and Program Studies Service (2006).

high on the previous spring’s state as-sessment, additional screening typically is not required.

Roadblock 1.3. Screening measures may identify students who do not need services and not identify students who do need services.

Suggested Approach. All screening mea-sures will misidentify some students as either needing assistance when they do not (false positive) or not needing assis-tance when they do (false negative). When screening students, educators will want to maximize both the number of students correctly identified as at risk—a measure’s sensitivity—and the number of students correctly identified as not at risk—a mea-sure’s specificity. As illustrated in table 3, screening students to determine risk can result in four possible categories indicated by the letters A, B, C, and D. Using these categories, sensitivity is equal to A/(A + C) and specificity is equal to D/(B + D).

Table 3. Sensitivity and specificity

STUDENTS ACTUALLY AT RISK

Yes No

STUDENTS IDENTIFIED AS BEING AT RISK

Yes A (true positives)

B (false positives)

No C (false negatives)

D (true negatives)

The sensitivity and specificity of a mea-sure depend on the cut score to classify children at risk.41 If a cut score is high (where all students below the cut score are considered at risk), the measure will have a high degree of sensitivity because most students who truly need assistance will be

41. Sensitivity and specificity are also influenced by the discriminant validity of the measure and its individual items. Measures with strong item discrimination are more likely to correctly iden-tify students’ risk status.

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identified as at risk. But the measure will have low specificity since many students who do not need assistance will also be identified as at risk. Similarly, if a cut score is low, the sensitivity will be lower (some students in need of assistance may not be identified as at risk), whereas the specific-ity will be higher (most students who do not need assistance will not be identified as at risk).

Schools need to be aware of this tradeoff between sensitivity and specificity, and the team selecting measures should be aware that decisions on cut scores can be somewhat arbitrary. Schools that set a cut score too high run the risk of spending re-sources on students who do not need help, and schools that set a cut score too low run the risk of not providing interventions to students who are at risk and need extra in-struction. If a school or district consistently finds that students receiving intervention do not need it, the measurement team should consider lowering the cut score.

Roadblock 1.4. Screening data may iden-tify large numbers of students who are at risk and schools may not immediately have the resources to support all at-risk students. This will be a particularly severe problem in low-performing Title I schools.

Suggested Approach. Districts and schools need to consider the amount of resources available and the allocation of

those resources when using screening data to make instructional decisions. Dis-tricts may find that on a nationally normed screening measure, a large percentage of their students (such as 60 percent) will be classified as at risk. Districts will have to determine the resources they have to pro-vide interventions and the number of stu-dents they can serve with their resources. This may mean not providing interven-tions at certain grade levels or providing interventions only to students with the lowest scores, at least in the first year of implementation.

There may also be cases when schools identify large numbers of students at risk in a particular area and decide to pro-vide instruction to all students. One par-ticularly salient example is in the area of fractions. Multiple national assessments show many students lack proficiency in fractions,42 so a school may decide that, rather than deliver interventions at the individual child level, they will provide a school-wide intervention to all students. A school-wide intervention can range from a supplemental fractions program to profes-sional development involving fractions.

42. National Mathematics Advisory Panel (2008); Lee, Grigg, and Dion (2007).

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The panel recommends that individuals knowledgeable in instruction and mathematics look for interventions that focus on whole numbers extensively in kindergarten through grade 5 and on rational numbers extensively in grades 4 through 8. In all cases, the specific content of the interventions will be centered on building the student’s foundational proficiencies. In making this recommendation, the panel is drawing on consensus documents developed by experts from mathematics education and research mathematicians that emphasized the importance of these topics for students in general.43 We conclude that the coverage of fewer topics in more depth, and with coherence, is as important, and probably more important, for students who struggle with mathematics.

43. National Council of Teachers of Mathemat-ics (2006); National Mathematics Advisory Panel (2008).

Level of evidence: Low

The panel judged the level of evidence supporting this recommendation to be low. This recommendation is based on the pro-fessional opinion of the panel and several recent consensus documents that reflect input from mathematics educators and re-search mathematicians involved in issues related to kindergarten through grade 12 mathematics education.44


The documents reviewed demonstrate a growing professional consensus that cov-erage of fewer mathematics topics in more depth and with coherence is important for all students.45 Milgram and Wu (2005) suggested that an intervention curriculum for at-risk students should not be over-simplified and that in-depth coverage of key topics and concepts involving whole numbers and then rational numbers is critical for future success in mathematics. The National Council of Teachers of Math-ematics (NCTM) Curriculum Focal Points (2006) called for the end of brief ventures into many topics in the course of a school year and also suggested heavy emphasis on instruction in whole numbers and rational numbers. This position was reinforced by the 2008 report of the National Mathematics Advisory Panel (NMAP), which provided de-tailed benchmarks and again emphasized in-depth coverage of key topics involving whole numbers and rational numbers as crucial for all students. Although the latter two documents addressed the needs of all students, the panel concludes that the in-depth coverage of key topics is especially

44. National Council of Teachers of Mathemat-ics (2006); National Mathematics Advisory Panel (2008); Milgram and Wu (2005).

45. National Mathematics Advisory Panel (2008); Schmidt and Houang (2007); Milgram and Wu (2005); National Council of Teachers of Math-ematics (2006).

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important for students who struggle with mathematics.


1. for students in kindergarten through grade 5, tier 2 and tier 3 interventions should focus almost exclusively on properties of whole numbers46 and operations. Some older students struggling with whole num-bers and operations would also benefit from in-depth coverage of these topics.

In the panel’s opinion, districts should review the interventions they are con-sidering to ensure that they cover whole numbers in depth. The goal is proficiency and mastery, so in-depth coverage with extensive review is essential and has been articulated in the NCTM Curriculum Focal Points (2006) and the benchmarks determined by the National Mathematics Advisory Panel (2008). Readers are recom-mended to review these documents.47

Specific choices for the content of interven-tions will depend on the grade level and proficiency of the student, but the focus for struggling students should be on whole numbers. For example, in kindergarten through grade 2, intervention materials would typically include significant atten-tion to counting (e.g., counting up), num-ber composition, and number decomposi-tion (to understand place-value multidigit operations). Interventions should cover the meaning of addition and subtraction and

46. Properties of numbers, including the associa-tive, commutative, and distributive properties.

47. More information on the National Mathemat-ics Advisory Panel (2008) report is available at www.ed.gov/about/bdscomm/list/mathpanel/index.html. More information on the National Council of Teachers of Mathematics Curricu-lum Focal Points is available at www.nctm.org/ focalpoints. Documents elaborating the National Council of Teachers of Mathematics Curriculum Focal Points are also available (see Beckmann et al., 2009). For a discussion of why this content is most relevant, see Milgram and Wu (2005).

the reasoning that underlies algorithms for addition and subtraction of whole num-bers, as well as solving problems involv-ing whole numbers. This focus should in-clude understanding of the base-10 system (place value).

Interventions should also include materi-als to build fluent retrieval of basic arith-metic facts (see recommendation 6). Ma-terials should extensively use—and ask students to use—visual representations of whole numbers, including both concrete and visual base-10 representations, as well as number paths and number lines (more information on visual representations is in recommendation 5).

2. for tier 2 and tier 3 students in grades 4 through 8, interventions should focus on in-depth coverage of rational numbers as well as advanced topics in whole number arith-metic (such as long division).

The panel believes that districts should review the interventions they are consid-ering to ensure that they cover concepts involving rational numbers in depth. The focus on rational numbers should include understanding the meaning of fractions, decimals, ratios, and percents, using visual representations (including placing fractions and decimals on number lines,48 see recom-mendation 5), and solving problems with fractions, decimals, ratios, and percents.

In the view of the panel, students in grades 4 through 8 will also require ad-ditional work to build fluent retrieval of basic arithmetic facts (see recommenda-tion 6), and some will require additional work involving basic whole number top-ics, especially for students in tier 3. In the opinion of the panel, accurate and fluent

48. When using number lines to teach rational numbers for students who have difficulties, it is important to emphasize that the focus is on the length of the segments between the whole num-ber marks (rather than counting the marks).

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arithmetic with whole numbers is neces-sary before understanding fractions. The panel acknowledges that there will be periods when both whole numbers and rational numbers should be addressed in interventions. In these cases, the balance of concepts should be determined by the student’s need for support.

3. Districts should appoint committees, in-cluding experts in mathematics instruction and mathematicians with knowledge of el-ementary and middle school mathematics curriculum, to ensure that specific criteria (described below) are covered in depth in the curricula they adopt.

In the panel’s view, intervention materials should be reviewed by individuals with knowledge of mathematics instruction and by mathematicians knowledgeable in el-ementary and middle school mathematics. They can often be experts within the district, such as mathematics coaches, mathematics teachers, or department heads. Some dis-tricts may also be able to draw on the exper-tise of local university mathematicians.

Reviewers should assess how well interven-tion materials meet four criteria. First, the materials integrate computation with solv-ing problems and pictorial representations rather than teaching computation apart from problem-solving. Second, the mate-rials stress the reasoning underlying cal-culation methods and focus student atten-tion on making sense of the mathematics. Third, the materials ensure that students build algorithmic proficiency. Fourth, the materials include frequent review for both consolidating and understanding the links of the mathematical principles. Also in the

panel’s view, the intervention program should include an assessment to assist in placing students appropriately in the in-tervention curriculum.


Roadblock 2.1. Some interventionists may worry if the intervention program is not aligned with the core classroom instruction.

Suggested Approach. The panel believes that alignment with the core curriculum is not as critical as ensuring that instruction builds students’ foundational proficien-cies. Tier 2 and tier 3 instruction focuses on foundational and often prerequisite skills that are determined by the students’ rate of progress. So, in the opinion of the panel, acquiring these skills will be neces-sary for future achievement. Additionally, because tier 2 and tier 3 are supplemental, students will still be receiving core class-room instruction aligned to a school or district curriculum (tier 1).

Roadblock 2.2. Intervention materials may cover topics that are not essential to building basic competencies, such as data analysis, measurement, and time.

Suggested Approach. In the panel’s opin-ion, it is not necessary to cover every topic in the intervention materials. Students will gain exposure to many supplemental top-ics (such as data analysis, measurement, and time) in general classroom instruc-tion (tier 1). Depending on the student’s age and proficiency, it is most important to focus on whole and rational numbers in the interventions.

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The national Mathematics Advisory Panel defines explicit instruction as follows (2008, p. 23):

• “Teachers provide clear models for solving a problem type using an array of examples.”

• “Students receive extensive practice in use of newly learned strategies and skills.”

• “Students are provided with opportunities to think aloud (i.e., talk through the decisions they make and the steps they take).”

• “Students are provided with extensive feedback.”

The nMAP notes that this does not mean that all mathematics instruction should be explicit. But it does recommend that struggling students receive some explicit instruction regularly and that some of the explicit instruction ensure that students possess the foundational skills and conceptual knowledge necessary for understanding their grade-level

mathematics.49 Our panel supports this recommendation and believes that districts and schools should select materials for interventions that reflect this orientation. In addition, professional development for interventionists should contain guidance on these components of explicit instruction.

Level of evidence: Strong

Our panel judged the level of evidence supporting this recommendation to be strong. This recommendation is based on six randomized controlled trials that met WWC standards or met standards with reservations and that examined the ef-fectiveness of explicit and systematic in-struction in mathematics interventions.50 These studies have shown that explicit and systematic instruction can significantly improve proficiency in word problem solv-ing51 and operations52 across grade levels and diverse student populations.


The results of six randomized controlled trials of mathematics interventions show extensive support for various combina-tions of the following components of ex-plicit and systematic instruction: teacher demonstration,53 student verbalization,54



51. Darch, Carnine, and Gersten (1984); Jitendra et al. (1998); Fuchs et al. (2003a); Wilson and Sin-delar (1991).

52. Schunk and Cox (1986); Tournaki (2003).

53. Darch, Carnine, and Gersten (1984); Jitendra et al. (1998); Fuchs et al. (2003a); Schunk and Cox (1986); Tournaki (2003); Wilson and Sindelar (1991).

54. Jitendra et al. (1998); Fuchs et al. (2003a); Schunk and Cox (1986); Tournaki (2003).

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guided practice,55 and corrective feed-back.56 All six studies examined interven-tions that included teacher demonstra-tions early in the lessons.57 For example, three studies included instruction that began with the teacher verbalizing aloud the steps to solve sample mathematics problems.58 The effects of this component of explicit instruction cannot be evaluated from these studies because the demonstra-tion procedure was used in instruction for students in both treatment and compari-son groups.

Scaffolded practice, a transfer of control of problem solving from the teacher to the student, was a component in four of the six studies.59 Although it is not possible to parse the effects of scaffolded instruction from the other components of instruction, the inter-vention groups in each study demonstrated significant positive gains on word problem proficiencies or accuracy measures.

Three of the six studies included opportu-nities for students to verbalize the steps to solve a problem.60 Again, although ef-fects of the interventions were statistically significant and positive on measures of word problems, operations, or accuracy, the effects cannot be attributed to a sin-gle component of these multicomponent interventions.

55. Darch, Carnine, and Gersten (1984); Jiten-dra et al. (1998); Fuchs et al. (2003a); Tournaki (2003).

56. Darch, Carnine, and Gersten (1984); Jitendra et al. (1998); Schunk and Cox (1986); Tournaki (2003).


58. Schunk and Cox (1986); Jitendra et al. (1998); Darch, Carnine, and Gersten (1984).

59. Darch, Carnine, and Gersten (1984); Fuchs et al. (2003a); Jitendra et al. (1998); Tournaki (2003).

60. Schunk and Cox (1986); Jitendra et al. (1998); Tournaki (2003).

Similarly, four of the six studies included immediate corrective feedback,61 and the effects of these interventions were posi-tive and significant on word problems and measures of operations skills, but the ef-fects of the corrective feedback compo-nent cannot be isolated from the effects of other components in three cases.62

With only one study in the pool of six in-cluding cumulative review as part of the intervention,63 the support for this compo-nent of explicit instruction is not as strong as it is for the other components. But this study did have statistically significant pos-itive effects in favor of the instructional group that received explicit instruction in strategies for solving word problems, including cumulative review.


1. Ensure that instructional materials are systematic and explicit. In particular, they should include numerous clear models of easy and difficult problems, with accompa-nying teacher think-alouds.

To be considered systematic, mathematics instruction should gradually build profi-ciency by introducing concepts in a logical order and by providing students with nu-merous applications of each concept. For example, a systematic curriculum builds student understanding of place value in an array of contexts before teaching pro-cedures for adding and subtracting two-digit numbers with regrouping.

Explicit instruction typically begins with a clear unambiguous exposition of con-cepts and step-by-step models of how

61. Darch, Carnine, and Gersten (1984); Jiten-dra et al. (1998); Tournaki (2003); Schunk and Cox (1986).

62. Darch, Carnine, and Gersten (1984); Jitendra et al. (1998); Tournaki (2003).

63. Fuchs et al. (2003a).

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to perform operations and reasons for the procedures.64 Interventionists should think aloud (make their thinking pro-cesses public) as they model each step of the process.65,66 They should not only tell students about the steps and procedures they are performing, but also allude to the reasoning behind them (link to the under-lying mathematics).

The panel suggests that districts select instructional materials that provide inter-ventionists with sample think-alouds or possible scenarios for explaining concepts and working through operations. A crite-rion for selecting intervention curricula materials should be whether or not they provide materials that help intervention-ists model or think through difficult and easy examples.

In the panel’s view, a major flaw in many instructional materials is that teachers are asked to provide only one or two models of how to approach a problem and that most of these models are for easy-to-solve problems. Ideally, the materials will also assist teachers in explaining the reason-ing behind the procedures and problem-solving methods.

2. Provide students with opportunities to solve problems in a group and communicate problem-solving strategies.

For students to become proficient in per-forming mathematical processes, explicit instruction should include scaffolded prac-tice, where the teacher plays an active role and gradually transfers the work to

64. For example, Jitendra et al. (1998); Darch, Carnine, and Gersten (1984); Woodward (2006).

65. See an example in the summary of Tournaki (2003) in appendix D.

66. Darch, Carnine, and Gersten (1984); Jiten-dra et al. (1998); Fuchs et al. (2003a); Schunk and Cox (1986); Tournaki (2003); Wilson and Sin-delar (1991).

the students.67 This phase of explicit in-struction begins with the teacher and the students solving problems together. As this phase of instruction continues, stu-dents should gradually complete more steps of the problem with decreasing guid-ance from the teacher. Students should proceed to independent practice when they can solve the problem with little or no support from the teacher.

During guided practice, the teacher should ask students to communicate the strate-gies they are using to complete each step of the process and provide reasons for their decisions.68 In addition, the panel recommends that teachers ask students to explain their solutions.69 Note that not only interventionists—but fellow students—can and should communicate how they think through solving problems to the inter-ventionist and the rest of the group. This can facilitate the development of a shared language for talking about mathematical problem solving.70

Teachers should give specific feedback that clarifies what students did correctly and what they need to improve.71 They should provide opportunities for students to correct their errors. For example, if a student has difficulty solving a word prob-lem or solving an equation, the teacher should ask simple questions that guide the student to solving the problem correctly. Corrective feedback can also include re-teaching or clarifying instructions when students are not able to respond to ques-tions or their responses are incorrect.

67. Tournaki (2003); Jitendra et al. (1998); Darch, Carnine, and Gersten (1984).

68. For example, Schunk and Cox (1986).

69. Schunk and Cox (1986); Tournaki (2003).

70. For example, Jitendra et al. (1998); Darch, Carnine, and Gersten (1984).

71. Tournaki (2003); Jitendra et al. (1998); Darch, Carnine, and Gersten (1984).

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3. Ensure that instructional materials include cumulative review in each session.

Cumulative reviews provide students with an opportunity to practice topics previ-ously covered in depth. For example, when students are working with fractions, a cumulative review activity could provide them with an opportunity to solve some problems involving multiplication and di-vision of whole numbers. In the panel’s opinion, this review can ensure that the knowledge is maintained over time and helps students see connections between various mathematical ideas.


Roadblock 3.1. Interventionists may be un-familiar with how to implement an interven-tion that uses explicit instruction, and some may underestimate the amount of practice necessary for students in tiers 2 and 3 to master the material being taught.

Suggested Approach. Districts and schools should set up professional devel-opment sessions for interventionists to observe and discuss sample lessons. The panel believes that it is important for pro-fessional development participants to ob-serve the intervention first hand. Watching a DVD or video of the intervention being used with students can give the partici-pants a model of how the program should be implemented.

Interventionists should also have hands-on experience, teaching the lessons to each other and practicing with students. Role-playing can give interventionists practice with modeling and think-alouds, since it is important for them to stop and reflect before formulating an explanation for their thinking processes. The train-ers can observe these activities, provide feedback on what participants did well, and offer explicit suggestions for improv-ing instruction.

As a part of professional development, be sure to convey the benefits that extended practice (not only worksheets) and cumu-lative review can have for student per-formance. If professional development is not an option, teachers can also work with mathematics coaches to learn how to implement the intervention.

Roadblock 3.2. Interventionists may not be expert with the underlying mathemat-ics content.

Suggested Approach. For intervention-ists to explain a mathematical process ac-curately and develop a logical think-aloud, it is important for them to understand the underlying mathematics concept and the mathematical reasoning for the process. Professional development should provide participants with in-depth knowledge of the mathematics content in the interven-tion, including the mathematical reason-ing underlying procedures, formulas, and problem-solving methods.72 The panel be-lieves that when interventionists convey their knowledge of the content, student understanding will increase, misconcep-tions will decrease, and the chances that students solve problems by rote memory will be reduced.

Roadblock 3.3. The intervention materials may not incorporate enough models, think-alouds, practice, and cumulative review.

Suggested Approach. Intervention pro-grams might not incorporate enough mod-els, think-alouds, practice, or cumulative review to improve students’ mathematics performance.73

Consider using a mathematics coach or specialist to develop a template listing the essential parts of an effective lesson,

72. National Mathematics Advisory Panel (2008); Wu (2005) http://math.berkeley.edu/~wu/Northridge2004a2.pdf.

73. Jitendra et al. (1996); Carnine et al. (1997).

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including the number of models, accom-panying think-alouds, and practice and cumulative review items students need to understand, learn, and master the content.

A team of teachers, guided by the math-ematics coach/specialist, can determine the components that should be added to the program.

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Students who have difficulties in mathematics typically experience severe difficulties in solving word problems related to the mathematics concepts and operations they are learning.74 This is a major impediment for future success in any math-related discipline.75

Based on the importance of building proficiency and the convergent findings from a body of high-quality research, the panel recommends that interventions include systematic explicit instruction on solving word problems, using the problems’ underlying structure. Simple word problems give meaning to mathematical operations such as subtraction or multiplication. When students are taught the underlying structure of a word problem, they not only have greater success in problem solving but can also gain insight into the deeper mathematical ideas in word problems.76 The panel also recommends systematic instruction on the structural connections between known, familiar word problems and unfamiliar, new problems. By making explicit the underlying structural connections between familiar and unfamiliar problems, students will know when to apply the solution methods they have learned.77

74. Geary (2003); Hanich et al. (2001).

75. National Mathematics Advisory Panel (2008); McCloskey (2007).

76. Peterson, Fennema, and Carpenter (1989).

77. Fuchs, Fuchs, Finelli et al. (2004).


The panel judged the level of evidence supporting this recommendation to be strong. This recommendation is based on nine randomized controlled trials that met WWC standards or met standards with reservations and that examined the effectiveness of word problem-solving strategies.78 Interventions that teach stu-dents the structure of problem types79—and how to discriminate superficial from substantive information to know when to apply the solution methods they have learned80—positively and marginally or significantly affect proficiency in solving word problems.


Research demonstrates that instruction on solving word problems based on under-lying problem structure leads to statisti-cally significant positive effects on mea-sures of word problem solving.81 Three randomized controlled trials isolated this practice. In these studies, intervention-ists taught students to identify problems of a given type by focusing on the prob-lem structure and then to design and execute appropriate solution strategies for each problem. These techniques typi-cally led to significant and positive effects on word-problem outcomes for students

78. Jitendra et al. (1998); Xin, Jitendra, and Deat-line-Buchman (2005); Darch, Carnine, and Gersten (1984); Fuchs et al. (2003a); Fuchs et al. (2003b); Fuchs, Fuchs, Prentice et al. (2004); Fuchs, Fuchs, Finelli et al. (2004); Fuchs, Fuchs, Craddock et al. (2008); Fuchs, Seethaler et al. (2008).

79. Jitendra et al. (1998); Xin, Jitendra, and Deat-line-Buchman (2005); Darch, Carnine, and Ger-sten (1984).

80. Fuchs et al. (2003a); Fuchs et al. (2003b); Fuchs, Fuchs, Prentice et al. (2004); Fuchs, Fuchs, Finelli et al. (2004); Fuchs, Fuchs, Craddock et al. (2008); Fuchs, Seethaler et al. (2008).


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experiencing difficulties in mathematics across grade levels.82

Six other randomized controlled trials took the instructional intervention on problem structure a step further. They demonstrated that teaching students to distinguish superficial from substantive information in problems also leads to marginally or statistically significant posi-tive effects on measures of word problem solving.83 After students were explicitly taught the pertinent structural features and problem-solution methods for differ-ent problem types, they were taught su-perficial problem features that can change a problem without altering its underlying structure. They were taught to distinguish substantive information from superficial information in order to solve problems that appear new but really fit into one of the categories of problems they already know how to solve. They were also taught that the same underlying problem struc-tures can be applied to problems that are presented in graphic form (for exam-ple, with tables or maps). These are pre-cisely the issues that often confuse and derail students with difficulties in math-ematics. These six studies consistently demonstrated marginally or statistically significant positive effects on an array of word problem-solving proficiencies for students experiencing difficulties in mathematics.84





1. Teach students about the structure of various problem types, how to categorize problems based on structure, and how to determine appropriate solutions for each problem type.

Students should be explicitly taught about the salient underlying structural features of each problem type.85 Problem types are groups of problems with similar math-ematical structures. For example, change problems describe situations in which a quantity (such as children or pencils) is either increased or decreased (example 1). Change problems always include a time element. For these problems, students determine whether to add or subtract by determining whether the change in the quantity is more or less.

85. Xin, Jitendra, and Deatline-Buchman (2005).

Example 1. Change problems

The two problems here are addition and subtraction problems that students may be tempted to solve using an incorrect op-eration. In each case, students can draw a simple diagram like the one shown below, record the known quantities (two of three of A, B, and C) and then use the diagram to decide whether addition or subtraction is the correct operation to use to determine the unknown quantity.

A B

C

Problem 1. Brad has a bottlecap collection. After Madhavi gave Brad 28 more bottle-caps, Brad had 111 bottlecaps. How many bottlecaps did Brad have before Madhavi gave him more?

Problem 2. Brad has a bottlecap collection. After Brad gave 28 of his bottlecaps to Mad-havi, he had 83 bottlecaps left. How many bottlecaps did Brad have before he gave Madhavi some?

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In contrast, compare problems have no time element (example 2). They focus on comparisons between two different types of items in two different sets (pears and apples, boys and girls, hot and cold items). Students add or subtract by determin-ing whether they need to calculate the unknown difference (subtract), unknown compared amount (add), or unknown ref-erent amount (subtract).

2. Teach students to recognize the common underlying structure between familiar and unfamiliar problems and to transfer known solution methods from familiar to unfamil-iar problems.

A known familiar problem often appears as a new and unfamiliar problem to a stu-dent because of such superficial changes as format changes (whether it is written in traditional paragraph form or as an adver-tisement for a brochure), key vocabulary changes (half, one-half, ½), or the inclusion of irrelevant information (additional story elements such as the number of buttons on a child’s shirt or the size of a storage container for a compare problem).88 These superficial changes are irrelevant to un-derstanding the mathematical demands of a problem. But while focusing on these irrelevant superficial changes, students can find it difficult to discern the critical common underlying structure between the new and the old problems and to apply the solution that is part of their repertoire to the new unfamiliar problem.

To facilitate the transfer of the known so-lution from the familiar to the unfamiliar problem, students should first be shown explicitly that not all pieces of information in the problem are relevant to discerning the underlying problem structure.89 Teach-ers should explain these irrelevant superfi-cial features explicitly and systematically, as described in recommendation 3.90 This instruction may be facilitated by the use of a poster displayed in the classroom that lists the ways familiar problems can be-come unfamiliar because of new wording or situations (such as information displayed in




Example 2. Compare problems

There are 21 hamsters and 32 kittens at the pet store. how many more kittens are at the pet store than hamsters?

32

21 ?

Although these problem types seem simple and intuitive to adults and mathematically precocious students, they are not neces-sarily obvious for students requiring math-ematics interventions. To build understand-ing of each problem type, we recommend initially teaching solution rules (or guiding questions that lead to a solution equation) for each problem type through fully and partially worked examples, followed by student practice in pairs.86

Visual representations such as those in ex-ample 2 can be effective for teaching stu-dents how to categorize problems based on their structure and determine a solu-tion method appropriate for the underlying structure (see recommendation 5 for more information on visual representations).87 Teachers can present stories with unknown information and work with students in using diagrams to identify the problem type and transform the information in the diagram into a mathematics equation to solve for the unknown quantity.



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chart versus paragraph form) or the ways relevant to problem type. The students must also be provided with opportunities to explain why a piece of information is relevant or irrelevant.91

We suggest that students practice sets of problems with varied superficial features and cover stories. Students who know how to recognize and solve a “change” problem type with whole numbers should know that they can apply the same strategy to a structurally similar word problem that looks different because of changes in wording and the presence of additional story elements (example 3).92

91. Fuchs, Fuchs, Craddock et al. (2008); Fuchs, Fuchs, Finelli et al. (2004); Fuchs, Fuchs, Prentice et al. (2004).



Roadblock 4.1. In the opinion of the panel, the curricular material may not classify problems into problem types.

Suggested Approach. The interventionist may need the help of a mathematics coach, a mathematics specialist, or a district or state curriculum guide in determining the problem types and an instructional sequence for teaching them to students. The key issue is that students are taught to understand a set of problem structures related to the mathematics they are learn-ing in their intervention.

Roadblock 4.2. As problems get complex, so will the problem types and the task of discriminating among them.

Suggested Approach. As problems get more intricate (such as multistep problems), it becomes more difficult for students to determine the problem type, a critical step that leads to solving the problem correctly. It is important to explicitly and systemati-cally teach students how to differentiate one problem type from another.

Interventionists will need high-quality professional development to ensure that they convey the information clearly and accurately. The professional development program should include opportunities for participants to determine problem types, justify their responses, and practice ex-plaining and modeling problem types to peers and children. Trainers should pro-vide constructive feedback during the practice sessions by telling participants both what they did well and what aspects of their instruction need improvement.

Example 3. Solving different problems with the same strategy

• Mike wants to buy 1 pencil for each of his friends. Each packet of pencils contains 12 pencils. how many pack-ets does Mike have to buy to give 1 pencil to each of his 13 friends?

• Mike wants to buy 1 pencil for each of his friends. Sally wants to buy 10 pen-cils. Each box of pencils contains 12 pencils. how many boxes does Mike have to buy to give 1 pencil to each of his 13 friends?

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A major problem for students who struggle with mathematics is weak understanding of the relationships between the abstract symbols of mathematics and the various visual representations.93 Student understanding of these relationships can be strengthened through the use of visual representations of mathematical concepts such as solving equations, fraction equivalence, and the commutative property of addition and multiplication (see the glossary). Such representations may include number lines, graphs, simple drawings of concrete objects such as blocks or cups, or simplified drawings such as ovals to represent birds.

In the view of the panel, the ability to express mathematical ideas using visual representations and to convert visual representations into symbols is critical for success in mathematics. A major goal of interventions should be to systematically teach students how to develop visual representations and how to transition these representations to standard symbolic representations

93. Hecht, Vagi, and Torgesen (2007).

used in problem solving. Occasional and unsystematic exposure (the norm in many classrooms) is insufficient and does not facilitate understanding of the relationship between the abstract symbols of mathematics and various visual representations.


The panel judged the level of evidence sup-porting this recommendation to be mod-erate. This recommendation is based on 13 randomized controlled trials that met WWC standards or met standards with reservations.94 These studies provide sup-port for the systematic use of visual rep-resentations or manipulatives to improve achievement in general mathematics,95 prealgebra concepts,96 word problems,97 and operations.98 But these representations were part of a complex multicomponent intervention in each of the studies. So, it is difficult to judge the impact of the repre-sentation component alone, and the panel believes that a moderate designation is ap-propriate for the level of evidence for this recommendation.


Research shows that the systematic use of visual representations and manipulatives may lead to statistically significant or sub-stantively important positive gains in math

94. Artus and Dyrek (1989); Butler et al. (2003); Darch, Carnine, and Gertsen (1984); Fuchs et al. (2005); Fuchs, Seethaler et al. (2008); Fuchs, Powell et al. (2008); Fuchs, Fuchs, Craddock et al. (2008); Jitendra et al. (1998); Walker and Poteet (1989); Wilson and Sindelar (1991); Witzel (2005); Witzel, Mercer, and Miller (2003); Woodward (2006).

95. Artus and Dyrek (1989); Fuchs et al. (2005).

96. Witzel, Mercer, and Miller (2003).

97. Darch, Carnine, and Gersten (1984); Fuchs et al. (2005); Fuchs, Seethaler et al. (2008); Fuchs, Fuchs, Craddock et al. (2008); Jitendra et al. (1998); Wilson and Sindelar (1991).

98. Woodward (2006).

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achievement.99 Four studies used visual representations to help pave the way for students to understand the abstract version of the representation.100 For example, one of the studies taught students to use visual representations such as number lines to understand mathematics facts.101 The four studies demonstrated gains in mathematics facts and operations102 and word problem proficiencies,103 and may provide evidence that using visual representations in inter-ventions is an effective technique.

Three of the studies used manipulatives in the early stages of instruction to rein-force understanding of basic concepts and operations.104 One used concrete models such as groups of boxes to teach rules for multiplication problems.105 The three stud-ies largely showed significant and positive effects and provide evidence that using manipulatives may be helpful in the initial stages of an intervention to improve profi-ciency in word problem solving.106

In six of the studies, both concrete and vi-sual representations were used, and over-all these studies show that using some combination of manipulatives and visual representations may promote mathemati-cal understanding.107 In two of the six, instruction did not include fading of the

99. Following WWC guidelines, an effect size greater than 0.25 is considered substantively important.

100. Jitendra et al. (1998); Walker and Poteet (1989); Wilson and Sindelar (1991); Woodward (2006).

101. Woodward (2006).

102. Woodward (2006).

103. Jitendra et al. (1998); Walker and Poteet (1989); Wilson and Sindelar (1991).

104. Darch (1989); Fuchs, Seethaler et al. (2008); Fuchs, Fuchs, Craddock et al. (2008).

105. Darch (1989).

106. Darch (1989); Fuchs, Sethaler et al. (2008); Fuchs, Fuchs, Craddock et al. (2008).

107. Artus and Dyrek (1989); Butler et al. (2003); Fuchs, Powell et al. (2008); Fuchs et al. (2005); Wit-zel (2005); Witzel, Mercer, and Miller (2003).

manipulatives and visual representations to promote understanding of math at a more abstract level.108 One of these inter-ventions positively affected general math achievement,109 but the other had no effect on outcome measures tested.110 In the other four studies, manipulatives and visual rep-resentations were presented to the students sequentially to promote understanding at a more abstract level.111 One interven-tion that used this method for teaching fractions did not show much promise,112 but the other three did result in positive gains.113 One of them taught 1st graders basic math concepts and operations,114 and the other two taught prealgebra concepts to low-achieving students.115


1. use visual representations such as number lines, arrays, and strip diagrams.

In the panel’s view, visual representations such as number lines, number paths, strip diagrams, drawings, and other forms of pic-torial representations help scaffold learn-ing and pave the way for understanding the abstract version of the representation. We recommend that interventionists use such abstract visual representations extensively and consistently. We also recommend that interventionists explicitly link visual rep-resentations with the standard symbolic representations used in mathematics.

108. Artus and Dyrek (1989); Fuchs, Powell et al. (2008).

109. Artus and Dyrek (1989).

110. Fuchs, Powell et al. (2008).

111. Fuchs et al. (2005); Butler et al. (2003); Witzel et al. (2003); Witzel (2005).

112. Butler et al. (2003).

113. Fuchs et al. (2005); Witzel, Mercer, and Miller (2003); Witzel (2005).

114. Fuchs et al. (2005).

115. Witzel, Mercer, and Miller (2003); Witzel (2005).

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In early grades, number lines, number paths, and other pictorial representations are often used to teach students founda-tional concepts and procedural operations of addition and subtraction. Although number lines or number paths may not be a suitable initial representation in some situations (as when working with multipli-cation and division), they can help concep-tually and procedurally with other types of problems. Conceptually, number lines and number paths show magnitude and allow for explicit instruction on magnitude com-parisons. Procedurally, they help teach principles of addition and subtraction op-erations such as “counting down,” “count-ing up,” and “counting down from.”

The figure in example 4 shows how a num-ber line may be used to assist with counting strategies. The top arrows show how a child learns to count on. He adds 2 + 5 = . To start, he places his finger on 2. Then, he jumps five times to the right and lands on 7. The arrows under the number line show how a child subtracts using a counting down strategy. For 10 – 3 = , she starts with her finger on the 10. Then, she jumps three times to the left on the number line, where she finishes on 7.

The goal of using a number line should be for students to create a mental num-ber line and establish rules for movement along the line according to the more or less marking arrows placed along the line. Such rules and procedures should be directly tied to the explicit instruction that guided the students through the use of the visual representation.116

Pictorial representations of objects such as birds and cups are also often used to teach basic addition and subtraction, and simple drawings can help students under-stand place value and multidigit addition

116. Manalo, Bunnell, and Stillman (2000). Note that this study was not eligible for review because it was conducted outside the United States.

and subtraction. Example 5 (p. 34) shows how a student can draw a picture to solve a multidigit addition problem. In the fig-ure, circles represent one unit and lines represent units of 10.

In upper grades, diagrams and pictorial representations used to teach fractions also help students make sense of the basic structure underlying word problems. Strip diagrams (also called model diagrams and bar diagrams) are one type of diagram that can be used. Strip diagrams are drawings of narrow rectangles that show relationships among quantities. Students can use strip diagrams to help them reason about and solve a wide variety of word problems about related quantities. In example 6 (p. 34), the full rectangle (consisting of all three equal parts joined together) represents Shauntay’s money before she bought the book. Since she spent 2⁄3 of her money on the book, two of the three equal parts represent the $26 she spent on the book. Students can then reason that if two parts stand for $26, then each part stands for $13, so three parts stand for $39. So, Shauntay had $39 before she bought the book.

2. If visuals are not sufficient for developing accurate abstract thought and answers, use concrete manipulatives first. Although this can also be done with students in upper el-ementary and middle school grades, use of manipulatives with older students should be expeditious because the goal is to move to-ward understanding of—and facility with—visual representations, and finally, to the abstract.

Manipulatives are usually used in lower grades in the initial stages of learning as teachers introduce basic concepts with whole numbers. This exposure to concrete objects is often fleeting and transitory. The use of manipulatives in upper elementary school grades is virtually nonexistent.117

117. Howard, Perry, and Lindsay (1996); Howard, Perry, and Conroy (1995).

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The panel suggests that the interventionist use concrete objects in two ways.

First, in lower elementary grades, use concrete objects more extensively in the initial stages of learning to reinforce the understanding of basic concepts and operations.118

Concrete models are routinely used to teach basic foundational concepts such as place value.119 They are also useful in teaching other aspects of mathematics such as multiplication facts. When a mul-tiplication fact is memorized by question and answer alone, a student may believe that numbers are to be memorized rather than understood. For example, 4 × 6 equals 24. When shown using manipulatives (as in example 7, p. 35), 4 × 6 means 4 groups of 6, which total as 24 objects.

118. Darch (1989); Fuchs, Seethaler et al. (2008); Fuchs, Fuchs, Craddock et al. (2008).

119. Fuchs et al. (2005); Fuchs, Seethaler et al. (2008); Fuchs, Powell et al. (2008).

Second, in the upper grades, use concrete objects when visual representations do not seem sufficient in helping students understand mathematics at the more ab-stract level.

Use manipulatives expeditiously, and focus on fading them away systematically to reach the abstract level.120 In other words, explicitly teach students the concepts and operations when students are at the con-crete level and consistently repeat the in-structional procedures at the visual and abstract levels. Using consistent language across representational systems (manip-ulatives, visual representations, and ab-stract symbols) has been an important component in several research studies.121 Example 8 (p. 35) shows a set of matched concrete, visual, and abstract representa-tions of a concept involving solving single-variable equations.

120. Fuchs et al. (2005); Witzel (2005); Witzel, Mercer, and Miller (2003).

121. Fuchs et al. (2005); Butler et al. (2003); Witzel (2005); Witzel, Mercer, and Miller (2003).

Example 4. Representation of the counting on strategy using a number line

0 1 2 3 4 5 6 7 8 9 10

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Example 5. Using visual representations for multidigit addition

Example 6. Strip diagrams can help students make sense of fractions

A group of ten can be drawn with a long line to indicate that ten ones are joined to form one ten:

Simple drawingshelp make sense of two-digit addition with regrouping:

36

+27

63

Shauntay spent of the money she had on a book that cost $26. How much money did Shauntay have before she bought the book?

Shauntay’s money at �rst

$26 book

2 parts

1 part

3 parts

$26

$26 ÷ 2 = $13

3 × $13 = $39

Shauntay’s had $39

/2 3

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Example 7. Manipulatives can help students understand that four multiplied by six means four groups of six, which means 24 total objects

Example 8. A set of matched concrete, visual, and abstract representations to teach solving single-variable equations

Solving the Equation withConcrete Manipulatives(Cups and Sticks)

Solving the Equation with Visual Representations of Cups and Sticks

Solving the Equation with Abstract Symbols

Concrete StepsA. 3 sticks plus one group of X equals 7 sticksB. Subtract 3 sticks from each side of the equationC. The equation now reads as one group of X equals 4 sticksD. Divide each side of the equation by one groupE. One group of X is equal to four sticks (i.e., 1X/group = 4 sticks/group; 1X = 4 sticks)

A

B − − − −

X

C X

+ =

=

=

=

=

=

D

XE

X+ =

X

X

3 + 1X = 7

−3 −3

1X

11

4

=X 4

3 + X = 7

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Roadblock 5.1. In the opinion of the panel, many intervention materials pro-vide very few examples of the use of visual representations.

Suggested Approach. Because many cur-ricular materials do not include sufficient examples of visual representations, the interventionist may need the help of the mathematics coach or other teachers in developing the visuals. District staff can also arrange for the development of these materials for use throughout the district.

Roadblock 5.2. Some teachers or interven-tionists believe that instruction in concrete manipulatives requires too much time.

Suggested Approach. Expeditious use of manipulatives cannot be overemphasized. Since tiered interventions often rely on foundational concepts and procedures, the use of instruction at the concrete level allows for reinforcing and making explicit the foundational concepts and operations. Note that overemphasis on manipulatives can be counterproductive, because stu-dents manipulating only concrete objects may not be learning to do math at an ab-stract level.122 The interventionist should


use manipulatives in the initial stages stra-tegically and then scaffold instruction to the abstract level. So, although it takes time to use manipulatives, this is not a major concern since concrete instruction will happen only rarely and expeditiously.

Roadblock 5.3. Some interventionists may not fully understand the mathemati-cal ideas that underlie some of the repre-sentations. This is likely to be particularly true for topics involving negative numbers, proportional reasoning, and interpretations of fractions.

Suggested Approach. If interventionists do not fully understand the mathematical ideas behind the material, they are un-likely to be able to teach it to struggling students.123 It is perfectly reasonable for districts to work with a local university faculty member, high school mathemat-ics instructor, or mathematics special-ist to provide relevant mathematics in-struction to interventionists so that they feel comfortable with the concepts. This can be coupled with professional devel-opment that addresses ways to explain these concepts in terms their students will understand.

123. Hill, Rowan, and Ball (2005); Stigler and Hiebert (1999).

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Quick retrieval of basic arithmetic facts is critical for success in mathematics.124 yet research has found that many students with difficulties in mathematics are not fluent in such facts.125 Weak ability to retrieve arithmetic facts is likely to impede understanding of concepts students encounter with rational numbers since teachers and texts often assume automatic retrieval of facts such as 3 × 9 = and 11 – 7 = as they explain concepts such as equivalence and the commutative property.126 for that reason, we recommend that about 10 minutes be devoted to building this proficiency during each intervention session. Acknowledging that time may be short, we recommend a minimum of 5 minutes a session.


The panel judged the level of evidence supporting this recommendation to be moderate. This recommendation is based on seven randomized controlled trials that met WWC standards or met standards with reservations and that included fact fluency


125. Geary (2004); Jordan, Hanich, and Kaplan (2003); Goldman, Pellegrino, and Mertz (1988).

126. Gersten and Chard (1999); Woodward (2006); Jitendra et al. (1996).

instruction in the intervention.127 These studies reveal a series of small but positive effects on measures of fact fluency128 and procedural knowledge for diverse student populations in the elementary grades.129 In some cases, fact fluency instruction was one of several components in the interven-tion, and it is difficult to judge the impact of the fact fluency component alone.130 However, because numerous research teams independently produced similar findings, we consider this practice worthy of serious consideration. Although the re-search is limited to the elementary school grades, in the panel’s view, building fact fluency is also important for middle school students when used appropriately.


The evidence demonstrates small positive effects on fact fluency and operations for the elementary grades and thus provides support for including fact fluency activi-ties as either stand-alone interventions or components of larger tier 2 interven-tions.131 These positive effects did not, however, consistently reach statistical sig-nificance, and the findings cannot be ex-trapolated to areas of mathematics outside of fact fluency and operations.

127. Bernie-Smith (1991); Fuchs, Seethaler et al. (2008); Fuchs, Fuchs, Hamlett et al. (2006); Fuchs et al. (2005); Fuchs, Powell et al. (2008); Tournaki (2003); Woodward (2006).

128. Fuchs, Seethaler et al. (2008); Fuchs, Fuchs, Hamlet et al. (2006); Fuchs et al. (2005); Fuchs, Powell et al. (2008); Tournaki (2003); Woodward (2006).

129. Bernie-Smith (1991); Fuchs, Seethaler et al. (2008); Fuchs et al. (2005); Tournaki (2003); Woodward (2006).

130. Fuchs, Seethaler et al. (2008); Fuchs et al. (2005).

131. Bernie-Smith (1991); Fuchs et al. (2005); Fuchs, Fuchs, Hamlet et al. (2006); Fuchs, Seetha-ler et al. (2008); Fuchs, Powell et al. (2008); Tour-naki (2003); Woodward (2006).

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Two studies examined the effects of being taught mathematics facts relative to the effects of being taught spelling or word identification using similar methods.132 In both studies, the mathematics facts group demonstrated positive gains in fact flu-ency relative to the comparison group, but the effects were significant in only one of the studies.133

Another two interventions included a facts fluency component in combination with a larger tier 2 intervention.134 For example, in the Fuchs et al. (2005) study, the final 10 minutes of a 40 minute intervention ses-sion were dedicated to practice with addi-tion and subtraction facts. In both stud-ies, tier 2 interventions were compared against typical tier 1 classroom instruc-tion. In each study, the effects on mathe-matics facts were small and not significant, though the effects were generally positive in favor of groups that received the inter-vention. Significant positive effects were detected in both studies in the domain of operations, and the fact fluency compo-nent may have been a factor in improving students’ operational abilities.

Many of the studies in the evidence base included one or more of a variety of com-ponents such as teaching the relationships among facts,135 making use of a variety of materials such as flash cards and com-puter-assisted instruction,136 and teaching math facts for a minimum of 10 minutes

132. Fuchs, Fuchs, Hamlett et al. (2006); Fuchs, Powell et al. (2008).

133. In Fuchs, Fuchs, Hamlett et al. (2006), the effects on addition fluency were statistically sig-nificant and positive while there was no effect on subtraction fluency.


135. Bernie-Smith (1991); Fuchs et al. (2005); Fuchs, Fuchs, Hamlett et al. (2006); Fuchs, Seetha-ler et al. (2008); Woodward (2006).

136. Bernie-Smith (1991); Fuchs et al. (2005); Fuchs, Fuchs, Hamlett et al. (2006); Fuchs, Seetha-ler et al. (2008); Fuchs, Powell et al. (2008).

per session.137 Since these components were typically not independent variables in the studies, it is difficult to attribute any positive effects to the component itself. There is evidence, however, that strategy-based instruction for fact fluency (such as teaching the counting-on procedure) is superior to rote memorization.138


1. Provide about 10 minutes per session of instruction to build quick retrieval of basic arithmetic facts. Consider using technology, flash cards, and other materials for extensive practice to facilitate automatic retrieval.

The panel recommends providing about 10 minutes each session for practice to help students become automatic in retrieving basic arithmetic facts, beginning in grade 2. The goal is quick retrieval of facts using the digits 0 to 9 without any access to pen-cil and paper or manipulatives.

Presenting facts in number families (such as 7 × 8 = 56, 8 × 7 = 56, 56/7 = 8, and 56/8 = 7) shows promise for improving student fluency.139 In the panel’s view, one advantage of this approach is that students simultaneously learn about the nature of inverse operations.

In the opinion of the panel, cumulative re-view is critical if students are to maintain fluency and proficiency with mathematics facts. An efficient way to achieve this is to integrate previously learned facts into practice activities. To reduce frustration and provide enough extended practice so that retrieval becomes automatic (even for

137. Bernie-Smith (1991); Fuchs et al. (2005); Fuchs, Fuchs, Hamlett et al. (2006); Fuchs, Seetha-ler et al. (2008); Fuchs, Powell et al. (2008); Tour-naki (2003); Woodward (2006).

138. Bernie-Smith (1991); Tournaki (2003); Wood-ward (2006).

139. Fuchs et al. (2005); Fuchs, Fuchs, Hamlett et al. (2006); Fuchs, Seethaler et al. (2008).

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those who tend to have limited capacity to remember and retrieve abstract material), interventionists can individualize practice sets so students learn one or two new facts, practice several recently acquired facts, and review previously learned facts.140 If students are proficient in grade-level math-ematics facts, then the panel acknowledges that students might not need to practice each session, although periodic cumulative review is encouraged.

2. for students in kindergarten through grade 2, explicitly teach strategies for effi-cient counting to improve the retrieval of mathematics facts.

It is important to provide students in kin-dergarten through grade 2 with strategies for efficiently solving mathematics facts as a step toward automatic, fluent retrieval. The counting-up strategy has been used to increase students’ fluency in addition facts.141 This is a simple, effective strategy that the majority of students teach them-selves, sometimes as early as age 4.142 But students with difficulties in mathematics tend not to develop this strategy on their own, even by grade 2.143 There is evidence that systematic and explicit instruction in this strategy is effective.144

Students can be explicitly taught to find the smaller number in the mathematics fact, put up the corresponding number of fingers, and count up that number of fin-gers from the larger number. For example, to solve 3 + 5 = , the teacher identifies the smaller number (3) and puts up three fingers. The teacher simultaneously says and points to the larger number before counting three fingers, 6, 7, 8.

140. Hasselbring, Bransford, and Goin (1988). Note that there was not sufficient information to do a WWC review.

141. Bernie-Smith (1991); Tournaki (2003).

142. Siegler and Jenkins (1989).

143. Tournaki (2003).

144. Tournaki (2003).

Note that learning the counting-up strategy not only improves students’ fact fluency145 but also immerses students in the commu-tative property of addition. For example, students learn that when the larger number is presented second (3 + 5 = ), they can rearrange the order and start counting up from 5. In the view of the panel, this linkage is an important part of intervention. After this type of instruction, follow-up practice with flash cards might help students make the new learning automatic.

3. Teach students in grades 2 through 8 how to use their knowledge of properties, such as commutative, associative, and distributive law, to derive facts in their heads.

Some researchers have argued that rather than solely relying on rote memorization and drill and practice, students should use properties of arithmetic to solve com-plex facts involving multiplication and di-vision.146 These researchers believe that by teaching the use of composition and decomposition, and applying the distribu-tive property to situations involving mul-tiplication, students can increasingly learn how to quickly (if not automatically) re-trieve facts. For example, to understand and quickly produce the seemingly difficult multiplication fact 13 × 7 = , students are reminded that 13 = 10 + 3, something they should have been taught consistently during their elementary career. Then, since 13 × 7 = (10 + 3) × 7 = 10 × 7 + 3 × 7, the fact is parsed into easier, known problems 10 × 7 = and 3 × 7 = by applying of the distributive property. Students can then rely on the two simpler multiplication facts (which they had already acquired) to quickly produce an answer mentally.

The panel recommends serious consid-eration of this approach as an option for students who struggle with acquisition of

145. Tournaki (2003).

146. Robinson, Menchetti, and Torgesen (2002); Woodward (2006).

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facts in grades 2 through 8. When choos-ing an intervention curriculum, consider one that teaches this approach to students in this age range. Note, however, that the panel believes students should also spend time after instruction with extensive prac-tice on quick retrieval of facts through the use of materials such as flash cards or technology.

Roadblocks and solutions

Roadblock 6.1. Students may find fluency practice tedious and boring.

Suggested Approach. Games that pro-vide students with the opportunity to practice new facts and review previously learned facts by encouraging them to beat their previous high score can help the

practice be less tedious.147 Players may be motivated when their scores rise and the challenge increases. Further recommenda-tions for motivating students are in recom-mendation 8.

Roadblock 6.2. Curricula may not include enough fact practice or may not have ma-terials that lend themselves to teaching strategies.

Suggested Approach. Some contempo-rary curricula deemphasize fact practice, so this is a real concern. In this case, we recommend using a supplemental program, either flash card or technology based.

147. Fuchs, Seethaler et al. (2008).

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Assess the progress of tier 2 and tier 3 students regularly with general outcome measures and curriculum embedded measures. Also monitor regularly the progress of tier 1 students who perform just above the cutoff score for general outcome measures so they can be moved to tier 2 if they begin to fall behind.

In addition, use progress monitoring data to determine when instructional changes are needed. This includes regrouping students who need continuing instructional support within tier 2 or tier 3, or moving students who have met benchmarks out of intervention groups and back to tier 1.

Information about specific progress monitoring measures is available in Appendix D. A list of online resources is in the text below.


The panel judged the level of evidence supporting this recommendation to be low. No studies that met WWC standards supported this recommendation.148 In-stead, the recommendation is based on the panel’s expert opinion as well as con-

148. The technical adequacy studies of mathe-matics progress monitoring measures were not experimental; the researchers typically used cor-relation techniques to evaluate the reliability and criterion validity of the measures and regression methods to examine sensitivity to growth.

sideration of the standards for measure-ment established by a joint committee of national organizations.149


Although we found no studies that ad-dressed the use of valid measures for strug-gling students within an RtI framework, nonexperimental studies demonstrate the technical adequacy of various progress monitoring measures.150 Measures for the primary grades typically reflect aspects of number sense, including strategic counting, numeral identification, and magnitude com-parisons.151 Studies investigating measures for the elementary grades focus mostly on the characteristics of general outcome mea-sures that represent grade-level mathemat-ics curricula in computation and in mathe-matics concepts and applications.152 Widely used, these measures are recommended by the National Center for Student Progress Monitoring.153 Less evidence is available to support progress monitoring in middle school.154 But research teams have devel-oped measures focusing on math concepts typically taught in middle school,155 basic facts,156 and estimation.157

149. The American Psychological Association, the American Educational Research Associa-tion, and the National Council on Measurement in Education (1999).

150. For example, Clarke et al. (2008); Foegen and Deno (2001); Fuchs et al. (1993); Fuchs, Fuchs, Hamlett, Thompson et al. (1994); Leh et al. (2007); Lembke et al. (2008).

151. For example, Clarke et al. (2008); Lembke et al. (2008); Bryant, Bryant, Gersten, Scammacca, and Chavez (2008).

152. Fuchs and Fuchs (1998); Fuchs et al. (1999).

153. www.studentprogress.org.

154. Foegen (2008).

155. Helwig, Anderson, and Tindal (2002).

156. Espin et al. (1989).

157. Foegen and Deno (2001); Foegen (2000).

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1. Monitor the progress of tier 2, tier 3, and borderline tier 1 students at least once a month using grade-appropriate general out-come measures.

General outcome measures typically take 5 to 10 minutes to administer and should be used at least monthly to monitor tier 2 and tier 3 students. General outcome measures use a sample of items from the array of concepts covered over one year to assess student progress. They provide a broad perspective on student proficiency in mathematics. They target concepts such as magnitude comparison, counting ability, and knowledge of place value for students in kindergarten and grade 1, and increas-ingly complex aspects of place value and proficiency with operations for students in grades 2 through 6. Examining student performance on these measures allows teachers to determine whether students are integrating and generalizing the concepts, skills, and strategies they are learning in the core curriculum and the intervention.158

In addition to monitoring the progress of tier 1 and tier 2 students, the panel recom-mends monitoring the progress of border-line tier 1 students with general outcome measures on a monthly basis. Since these students scored just above the cut score, they were not selected for supplemental instruction. The panel suggests using one standard error of measurement (a statistic available in the technical information for the measures) above the cut score to de-fine the range of scores for borderline stu-dents. Using this approach, teachers can continue to monitor the progress of stu-dents whose scores fell just above the cut score and determine whether they should receive supplemental instruction.

158. Fuchs, Fuchs, and Zumeta (2008).

Choose progress monitoring measures with evidence supporting their reliability, validity, and ability to identify growth. This will require input from individu-als with expertise in these areas, typi-cally school psychologists or members of district research departments. Consider whether the measure produces consistent results (reliability) and provides informa-tion that correlates with other measures of mathematics achievement (criterion validity). Ability to identify growth helps interventionists ensure that students are learning and making progress toward an annual goal through the full array of ser-vices they are receiving.

In some cases, general outcome measures may also be used for screening, as de-scribed in recommendation 1. Resources that teachers can turn to for identifying appropriate measures include the Na-tional Center on Student Progress Moni-toring’s review of available tools (http://www.studentprogress.org/) and the Re-search Institute on Progress Monitoring (http://www.progressmonitoring.org/).

2. use curriculum-embedded assessments in interventions to determine whether stu-dents are learning from the intervention. These measures can be used as often as every day159 or as infrequently as once every other week.160

Many tier 2 and tier 3 intervention pro-grams (commercially developed, re-searcher developed, or district developed) include curriculum-embedded assess-ments (sometimes called unit tests, mas-tery tests, or daily probes). The results of these assessments can be used to deter-mine which concepts need to be reviewed, which need to be re-taught, and which have been mastered. Curriculum-embedded as-sessments are often administered daily

159. Bryant, Bryant, Gersten, Scammacca, and Chavez (2008).

160. Jitendra (2007).

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http://www.studentprogress.org/

http://www.studentprogress.org/

http://www.progressmonitoring.org/

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for students in kindergarten161 and grade 1 and biweekly for students in grades 2 through 6.162 These assessments usually do not possess the same high technical characteristics of the general outcome measures. Curriculum-embedded assess-ments often result in very useful informa-tion for interventionists because they can detect changes in student performance in the concepts and skills being taught at the time. Interventionists need to be cautious about assuming that mastery of individ-ual skills and concepts will translate into improvements in overall proficiency. As a result, the panel recommends using both general outcome measures and curricu-lum-embedded assessments for students receiving interventions.

If the intervention program does not in-clude curriculum-embedded assessments, use efficient, reliable, and valid screen-ing measures, which can also be used as progress monitoring measures (see recom-mendation 1).

3. use progress monitoring data to regroup students when necessary.

Since student skill levels change over time and in varying degrees, the panel

161. For example, one tier 2 intervention pro-gram for 1st and 2nd grade students reported by Bryant, Bryant, Gersten, Scammacca, and Chavez (2008) included daily activity-level progress monitoring that consisted of four oral or written problems drawn from the content focus for that day. Teachers were instructed that a majority of the students in the group had to complete at least three of the four problems correctly to consider the daily lesson successful.

162. A parallel example in grades 3 and beyond can be found in Jitendra’s Solving math word problems instructional materials on teaching word problems (2007). There are many other ex-amples in available commercial programs.

recommends using progress monitoring data to regroup students within tiers so that the small groups used in tier 2 interventions are as homogeneous as possible. If a stu-dent does not fit into any of the intervention groups from his or her class, consider put-ting the child in an intervention group from another class if the schedule permits.

Roadblocks and solutions

Roadblock 7.1. Students within classes are at very different levels. This can make it dif-ficult to group students into appropriate tier 2 and tier 3 intervention groups.

Suggested Approach. If students within a class are at such diverse levels that ap-propriate tier 2 and tier 3 intervention groups cannot be made, consider grouping students across classes. This will facilitate clustering students with similar needs. For example, teachers of upper elementary students may find that students who have not yet mastered basic concepts in a par-ticular area (fractions) are spread across several classrooms. Putting these students in a single tier 2 intervention group would be the most efficient means of meeting their needs, rather than trying to provide one or two students in each class with services duplicated across classrooms. In such a case, a math specialist, parapro-fessional, or other school personnel who have received training can conduct the intervention.

Roadblock 7.2. There is insufficient time for teachers to implement progress monitoring.

Suggested Approach. If teachers are too busy to assess student progress with moni-toring measures, consider training parapro-fessionals or other school staff to do so.

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Adults can sometimes forget how challenging so-called “basic” arithmetic is for students in tier 2 and tier 3 interventions. Many of these students have had experiences of failure and frustration with mathematics by the time they receive an intervention. They may also have a particularly difficult time storing and easily retrieving information in their memories.163 Therefore, it seems particularly important to provide additional motivation for these students.164

Praising students for their effort and for being engaged as they work through mathematics problems is a powerful motivational tool that can be effective in increasing students’ academic achievement.165 Tier 2 and tier 3 interventions should include components that promote student effort (engagement-contingent rewards), persistence (completion-contingent rewards), and achievement (performance-contingent rewards). These components can include praise and rewards. Even a well-designed intervention curriculum may falter without such behavioral supports.

163. Geary (2003).

164. The scope of this practice guide limited the motivational strategies reviewed to strategies used in studies of students struggling with math-ematics. For a wider review of effective motiva-tional strategies used in classrooms, see Epstein et al. (2008) and Halpern et al. (2007).

165. Schunk and Cox (1986); Fuchs et al. (2005). Fuchs, Fuchs, Craddock et al. (2008).


The panel judged the level of evidence supporting this recommendation to be low. This recommendation is based on the professional opinion of the panel, and on nine studies that met WWC standards or met standards with reservations that included motivational strategies in the intervention.166 Although one of these studies demonstrated that praising strug-gling students for their effort significantly improved their ability to solve subtraction problems with regrouping,167 other stud-ies included a motivational component as one of several components of the inter-vention. In the opinion of the panel, these studies did not show that a motivational component is essential but suggest that it may be useful for improving mathematics achievement.168


One study that met WWC standards exam-ined the effects of a motivational component by comparing the performance of students who received praise for their effort dur-ing subtraction instruction with those who did not receive praise.169 This study found significant positive effects on student sub-traction scores in favor of providing effort feedback.170 Although this study provides some evidence of the effectiveness of a mo-tivational strategy, it is the only study that

166. Fuchs et al. (2005); Fuchs, Fuchs, Cradock et al. (2008); Schunk and Cox (1986); Fuchs, Seethaler et al. (2008); Heller and Fantuzzo (1993); Artus and Dyrek (1989); Fuchs, Fuchs et al. (2003b); Fuchs, Fuchs, Hamlett, Phillips et al. (1994); Fuchs, Fuchs, Finelli et al. (2006).

167. Schunk and Cox (1986).

168. There is an extensive literature on motiva-tional strategies outside the scope of this prac-tice guide. For more information on motivational strategies see Epstein et al. (2008) and Halpern et al. (2007).



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explicitly tested the effects of motivational strategies on mathematics outcomes.

In two studies, students received points for engagement and attentiveness,171 and in three studies, students were provided with prizes as tangible reinforcers for ac-curate mathematics problem-solving.172 However, in each of these studies, it was not possible to isolate the effects of rein-forcing attentiveness and accuracy. For example, in two of the studies, students in tier 2 tutoring earned prizes for accura-cy.173 Although in both studies, the tier 2 intervention group demonstrated substan-tively important positive and sometimes significant gains on a variety of mathemat-ics measures relative to the students who remained in tier 1, it is not possible to iso-late the effects of the reinforcers from the provision of tier 2 tutoring. Another study examined the impact of parental involve-ment on students’ mathematics achieve-ment and found statistically significant positive effects on operations and general math achievement.174 However, because the parental involvement component was multifaceted, it is not possible to attribute the positive effects to rewards alone.

Five studies in the evidence base included interventions in which students graphed their progress and in some cases set goals for improvement on future assessments.175 One experimental study examined the effects of student graphing and goal set-ting as an independent variable and found substantively important positive effects on measures of word problems in favor

171. Fuchs et al. (2005); Fuchs, Fuchs, Craddock et al. (2008).

172. Fuchs et al. (2005); Fuchs, Seethaler et al. (2008); Fuchs, Fuchs, Craddock et al. (2008).

173. Fuchs et al. (2005); Fuchs, Seethaler et al. (2008).

174. Heller and Fantuzzo (1993).

175. Artus and Dyrek (1989); Fuchs, Seethaler et al. (2008); Fuchs et al. (2003b); Fuchs, Fuchs, Ham-lett, Phillips et al. (1994); Fuchs, Fuchs, Finelli et al. (2006).

of students who graphed and set goals.176 The other four studies did not isolate the effects of graphing progress.177 Because this recommendation is based primarily on the opinion of the panel, the level of evidence is identified as low.


1. Reinforce or praise students for their effort and for attending to and being engaged in the lesson.

Verbally praise students for their effort178 and for listening carefully and following the lesson in a systematic fashion (engage-ment-contingent rewards).179 The panel believes that praise should be immediate and specific to highlight student effort and engagement. But we also believe that it is ineffective to offer generic and empty praise (“good job!” or “keep up the good work!”) that is not related to actual effort. Instead, praise is most effective when it points to specific progress that students are making and recognizes students’ ac-tual effort.180 Systematically praising stu-dents for their effort and engagement may encourage them to remain focused on the completion of their work.

2. Consider rewarding student accom-plishments.

Consider using rewards to acknowledge completion of math tasks (completion-contingent rewards) and accurate work (performance-contingent rewards). This can be done by applauding or verbally

176. Fuchs et al. (2003b).

177. Artus and Dyrek (1989); Fuchs, Seethaler et al. (2008); Fuchs, Fuchs, Hamlett, Phillips et al. (1994); Fuchs, Fuchs, Finelli et al. (2006).


179. For example, Fuchs et al. (2005); Fuchs, Fuchs, Craddock et al. (2008).

180. See Bangert-Drowns et al. (1991) and Halpern et al. (2007) for a review.

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praising students for actual accomplish-ments, such as finishing assignments, im-proving their score from 70 percent to 80 percent correct, or giving students points or tokens each time they answer a problem correctly, which they can use to “buy” tangi-ble rewards at a later time.181 Again, praise should be specific rather than generic.182 Consider notifying the student’s parents to inform them of their child’s successes in mathematics by phone or email or in a note sent home with the student.183 Remember that parents of these students are likely to receive notification of problems rather than successes, and some evidence suggests that this specific positive attention might sup-port achievement growth.184

3. Allow students to chart their progress and to set goals for improvement.

Several of the interventions in the evi-dence base for this practice guide had stu-dents graph their progress on charts185 and set goals for improving their assessment scores.186 For example, students might graph their scores on a chart showing a se-ries of thermometers, one for each session of the intervention.187 At the beginning of each session, students can examine their charts and set a goal to beat their previ-ous score or to receive the maximum score. This type of goal setting is believed to help students develop self-regulated learning because students take independent respon-sibility for setting and achieving goals.188

181. For example, Fuchs et al. (2005); Fuchs, Seethaler et al. (2008); Fuchs, Fuchs, Craddock et al. (2008).

182. Halpern et al. (2007).

183. For example, Heller and Fantuzzo (1993).


185. Artus and Dyrek (1989); Fuchs, Fuchs, Ham-lett, Phillips et al. (1994); Fuchs, Fuchs, Finelli et al. (2006); Fuchs, Seethaler et al. (2008).

186. Fuchs et al. (2003b).

187. See the procedure in Fuchs et al. (2003b).

188. Fuchs et al. (1997).

Roadblocks and suggested approaches

Roadblock 8.1. Rewards can reduce genu-ine interest in mathematics by directing stu-dent attention to gathering rewards rather than learning math.

Suggested Approach. It is important to inform interventionists that research in other content areas has demonstrated that rewards and praise increase the likelihood of students’ academic success without diminishing their interest in learning.189 Given the frequent history of failure for many of these students, at least in the ele-mentary grades, we suggest using rewards and praise to encourage effort, engage-ment, and achievement. As students learn and succeed more often in mathematics, interventionists can gradually fade the use of rewards because student success will become an intrinsic reward. The WWC ReducingBehaviorProblemsintheElemen-tarySchoolClassroomPracticeGuide190 is a good reference for more information on the use of rewards and praise.

Roadblock 8.2. It is difficult to deter-mine appropriate rewards for individual students.

Suggested Approach. Consider each stu-dent’s interests before choosing an appro-priate reward. Also consider using oppor-tunities to engage in activities students are interested in as rewards to reinforce effort, engagement, and accurate work. Parents may also have ideas for rewards that will help motivate their children. Schools can engage parents in rewarding students and coordinate efforts to reward children at home as well.191

189. Epstein et al. (2008).

190. Epstein et al. (2008).


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Roadblock 8.3.Providingfeedbackandrewarding achievement detracts from class-room instructional time. Itisdifficulttofitit into the classroom schedule.

Suggested Approach. Verbally prais-ing students for their effort individually and their engagement in small group les-sons requires very little time. Awarding points or tokens for correct responses

can be done when the teacher grades the student’s work. To reduce the amount of time it takes for students to “buy” prizes with accumulated points or tokens, ask students to choose which prize they want before they “buy” it. The prizes can then be distributed quickly at the end of the day, so that students are not distracted by the items throughout the school day.

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Glossary of terms as used in this report

The associative property of addition states that (A + B) + C = A + (B + C) for all numbers A, B, C. This property allows for flexibility in calculating sums. For exam-ple, to calculate 85 + 97 + 3, we do not have to add 85 and 97 first but may instead cal-culate the easier sum 85 + (97 + 3), which is 85 + 100, which equals 185. The associa-tive property is also used in deriving basic addition facts from other basic facts and therefore helps with learning these basic facts. For example, to add 8 + 5, a child can think of breaking the 5 into 2 + 3, combin-ing the 2 with the 8 to make a 10, and then adding on the 3 to make 13. In an equation, this can be recorded as: 8 + 5 = 8 + (2 + 3) = (8 + 2) + 3 = 10 + 3 = 13.

The associative property of multiplica-tion states that (A × B) × C = A × (B × C) for all numbers A, B, C. This property allows for flexibility in calculating products. For exam-ple, to calculate 87 × 25 × 4, we do not have to multiply 87 and 25 first but may instead calculate the easier product 87 × (25 × 4), which is 87 × 100, which equals 8,700. The associative property is also used in deriv-ing basic multiplication facts from other basic facts and therefore helps with learn-ing these facts. For example, to calculate 7 × 6, a child who already knows 7 × 3 = 21 can double the 21 to calculate that 7 × 6 = 42. In an equation, this can be recorded as: 7 × 6 = 7 × (3 × 2) = (7 × 3) × 2 = 21 × 2 = 42.

The commutative property of addition states that A + B = B + A for all numbers A, B. This property allows for flexibility in calculating sums and helps lighten the load of learning the basic addition facts. For ex-ample, to add 2 + 9, a child might want to count on 9 from 2 by counting “3, 4, 5, 6, . . . 11,” which is cumbersome. By applying the commutative property, a child can instead add 9 + 2 by counting on 2 from 9, by say-ing “10, 11,” which is much easier (this is the

“minimum addend strategy” or the strategy of “counting on from larger”).

The commutative property of multipli-cation states that A × B = B × A for all num-bers A, B. This property allows for flex-ibility in calculating products and helps lighten the load of learning the basic mul-tiplication facts. As shown in example 9, once a child has learned the multiplication fact 3 × 7 = 21, the child will also know the fact 7 × 3 = 21 if the child understands the commutative property.

3 groups of 7 3 × 7 dots

7 groups of 3 7 × 3 dots

Both use the same array, so the total is the same: 3 × 7 = 7 × 3

Example 9: Commutative property of multiplication

Concurrent validity refers to the correla-tion between the assessment that is being investigated and a similar assessment when the assessments are completed at the same point in time. Correlation coef-ficients range from -1 to 1. A correlation coefficient close to 1 indicates a strong overlap between the assessments.

Counting up/on is a strategy that young children can use to solve addition (and subtraction) problems. To calculate 8 + 3 by counting on, the child starts with 8 and then “counts on” 3 more, saying “9, 10, 11.” The child may use fingers in order to de-termine when to stop counting.

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Counting up/on from a larger addend is a strategy in which children apply count-ing on to solve an addition problem, but first apply the commutative property, if necessary, in order to count on from the larger number. For example, to calculate 3 + 9, a child first changes the problem to 9 + 3 and then counts on 3 from 9.

Counting up/on to solve unknown ad-dend and subtraction problems. Count-ing on can be used to solve unknown ad-dend problems such as 11 + ? = 15. To solve this problem, the child can count on from 11 to 15, counting 12, 13, 14, 15 and raising one finger for each number. Since 4 fingers were raised, 11 + 4 = 15. To solve a sub-traction problem such as 15 minus 11 = ? using counting on, the child must first un-derstand that the problem can be reformu-lated as 11 + ? = 15. Then the child can use counting on as described previously. Note too that solving this subtraction problem using the counting on strategy is much easier than counting down 11. Note too that the reformulation of a subtraction problem as an unknown addend problem is important in its own right because it connects subtraction with addition.

In mathematics assessment, criterion- related validity means that student scores on an assessment should correspond to their scores or performance on other in-dicators of mathematics competence, such as teacher ratings, course grades, or stan-dardized test scores.

Derived fact strategies in addition and subtraction are strategies in which chil-dren use addition facts they already know to find related facts. Especially important among derived fact strategies are the make-a-10 methods because they emphasize base 10 structure. As shown in example 10, to add 8 + 5, a child can think of breaking the 5 into 2 + 3, combining the 2 with the 8 to make a 10, and then adding on the 3 to make 13 (see the associative property of addition). Note that to use this make-a-10

strategy, children must know the “10 part-ner” (number that can be added to make 10) for each number from 1 to 9 and must also know how to break each number into a sum of two (positive whole) numbers in all possible ways. Furthermore, the child must understand all the “teen” numbers (from 11 to 19) as a 10 and some ones (for example, 15 is 10 and 5 ones).

8 + 5

= 10 + 3 = 13

8 + 5

2 3

Example 10: Make-a-10 strategy

Derived fact strategies in multiplica-tion and division are strategies in which children use multiplication facts they al-ready know to find related facts. For exam-ple 5 × 8 is half of 10 × 8, and similarly for all the “5 times” facts (these are examples of applying the associative property). Also, 9 × 8 is 8 less than 10 × 8 , and similarly for all the “9 times” facts (these are examples of applying the distributive property). To calculate 4 × 7, we can double the double of 7, that is, the double of 7 is 14 and the double of 14 is 28, which is 4 times 7. All the “4 times” facts can be derived by dou-bling the double (these are examples of applying the associative property).

The distributive property relates addi-tion and multiplication. It states that A × (B + C) = (A × B) + (A × C) for all numbers A, B, C. This property allows for flexibil-ity in calculating products. For example, to calculate 7 × 13, we can break 13 apart by place value as 10 + 3 and calculate

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7 × 10 = 70 and 7 × 3 = 21 and add these two results to find 7 × 13 = 91. In an equa-tion, this can be recorded as: 7 × 13 = 7 × (10 + 3) = (7 × 10) + (7 × 3) = 70 + 21 = 91. This strategy of breaking numbers apart by place value and applying the distribu-tive property is the basis for the common method of longhand multiplication. The distributive property is also used in deriv-ing basic multiplication facts from other basic facts and therefore helps in learning these facts. As shown in example 11, to calculate 6 × 7, a child who already knows 6 × 5 = 30 and 6 × 2 = 12 can add the 30 and 12 to calculate that 6 × 7 = 30 + 12 = 42. In an equation, this can be recorded as: 6 × 7 = 6 × (5 + 2) = (6 × 5) + (6 × 2) = 30 + 12 = 42.

needing help in mathematics) divided by the sum of this value and the number of false negatives, while specificity is equal to the number of true negatives divided by the sum of this value and the number of false positives (students misidentified during screening).

A general outcome measure refers to a measure of specific proficiencies within a broader academic domain. These profi-ciencies are related to broader outcomes. For example, a measure of oral reading flu-ency serves as a general outcome measure of performance in the area of reading. The measures can be used to monitor student progress over time.

Interventionist refers to the person teaching the intervention. The interven-tionist might be a classroom teacher, in-structional assistant, or other certified school personnel.

The magnitude of a quantity or number is its size, so a magnitude comparison is a comparison of size. The term magnitude is generally used when considering size in an approximate sense. In this case, we often describe the size of a quantity or number very roughly by its order of magnitude, which is the power of ten (namely 1, 10, 100, 1000, . . . . or 0.1, 0.01, 0.001, . . . .) that the quantity or number is closest to.

Number composition and number de-composition are not formal mathematical terms but are used to describe putting num-bers together, as in putting 2 and 3 together to make 5, and breaking numbers apart, as in breaking 5 into 2 and 3. For young children, a visual representation like the one shown on the next page is often used before intro-ducing the traditional mathematical notation 2 + 3 = 5 and 5 = 2 + 3 for number composi-tion and decomposition.

Example 11: Distributive property

We can break6 rows of 7 into6 rows of 5 and6 rows of 2.

6 x 7 = 6 x 5 + 6 x 2 = 30 + 12 = 42

Efficiency is how quickly the universal screening measure can be administered, scored, and analyzed for all the students tested.

False positives and false negatives are technical terms used to describe the mis-identification of students. The numbers of false positives and false negatives are related to sensitivity and specificity. As depicted in table 3 (p. 16) of this guide, sensitivity is equal to the number of true positives (students properly identified as

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A number line is a line on which locations for 0 and 1 have been chosen (1 is to the right of 0, traditionally). Using the distance between 0 and 1 as a unit, a positive real number, N, is located N units to the right of 0 and a negative real number, -N, is located N units to the left of 0. In this way, every real number has a location on the number line and every location on the number line corresponds to a real number.

A number path is an informal precursor to a number line. It is a path of consecu-tively numbered “steps,” such as the paths found on many children’s board games along which game pieces are moved. De-termining locations on number paths only requires counting, whereas determining locations on number lines requires the no-tion of distance.

Reliability refers to the degree to which an assessment yields consistency over time (how likely are scores to be similar if students take the test a week or so later?) and across testers (do scores change when different individuals administer the test?). Alternate form reliability tells us the ex-tent to which an educator can expect sim-ilar results across comparable forms or versions of an assessment.

Predictive validity is the extent to which a test can predict how well students will

do in mathematics a year or even two or three years later.

Response to Intervention (RtI) is an early detection, prevention, and support system in education that identifies strug-gling students and assists them before they fall behind.

Sensitivity indicates how accurately a screening measure predicts which stu-dents are at risk. Sensitivity is calculated by determining the number of students who end up having difficulty in mathemat-ics and then examining the percentage of those students predicted to be at risk on the screening measure. A screening mea-sure with high sensitivity would have a high degree of accuracy. In general, sen-sitivity and specificity are related (as one increases the other usually decreases).

Specificity indicates how accurately a screening measure predicts which stu-dents are not at risk. Specificity is cal-culated by determining the number of students who do not have a deficit in math-ematics and then examining the percent-age of those students predicted to not be at risk on the screening measure. A screening measure with high specificity would have a high degree of accuracy. In general, sen-sitivity and specificity are related (as one increases the other usually decreases).

Strip diagrams (also called model dia-grams and bar diagrams) are drawings of narrow rectangles that show relationships among quantities.

A validity coefficient serves as an index of the relation between two measures and can range from -1.0 to 1.0, with a coeffi-cient of .0 meaning there is no relation between the two scores and increasing positive scores indicating a stronger posi-tive relation.

Example 12: Number decomposition

5

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Appendix A. Postscript from the Institute of Education Sciences

What is a practice guide?

The health care professions have em-braced a mechanism for assembling and communicating evidence-based advice to practitioners about care for specific clini-cal conditions. Variously called practice guidelines, treatment protocols, critical pathways, best practice guides, or simply practice guides, these documents are sys-tematically developed recommendations about the course of care for frequently en-countered problems, ranging from physi-cal conditions, such as foot ulcers, to psy-chosocial conditions, such as adolescent development.192

Practice guides are similar to the prod-ucts of typical expert consensus panels in reflecting the views of those serving on the panel and the social decisions that come into play as the positions of individ-ual panel members are forged into state-ments that all panel members are willing to endorse. Practice guides, however, are generated under three constraints that do not typically apply to consensus panels. The first is that a practice guide consists of a list of discrete recommendations that are actionable. The second is that those recommendations taken together are in-tended to be a coherent approach to a multifaceted problem. The third, which is most important, is that each recommen-dation is explicitly connected to the level of evidence supporting it, with the level represented by a grade (strong, moder-ate, or low).

The levels of evidence, or grades, are usually constructed around the value of

192. Field and Lohr (1990).

particular types of studies for drawing causal conclusions about what works. Thus, one typically finds that a strong level of evidence is drawn from a body of randomized controlled trials, the moder-ate level from well-designed studies that do not involve randomization, and the low level from the opinions of respected authorities (see table 1, p. 2). Levels of evi-dence also can be constructed around the value of particular types of studies for other goals, such as the reliability and va-lidity of assessments.

Practice guides also can be distinguished from systematic reviews or meta-analyses such as What Works Clearinghouse (WWC) intervention reviews or statistical meta-analyses, which employ statistical meth-ods to summarize the results of studies obtained from a rule-based search of the literature. Authors of practice guides sel-dom conduct the types of systematic lit-erature searches that are the backbone of a meta-analysis, although they take ad-vantage of such work when it is already published. Instead, authors use their ex-pertise to identify the most important research with respect to their recommen-dations, augmented by a search of recent publications to ensure that the research citations are up-to-date. Furthermore, the characterization of the quality and direc-tion of the evidence underlying a recom-mendation in a practice guide relies less on a tight set of rules and statistical algo-rithms and more on the judgment of the authors than would be the case in a high- quality meta-analysis. Another distinction is that a practice guide, because it aims for a comprehensive and coherent approach, operates with more numerous and more contextualized statements of what works than does a typical meta-analysis.

Thus, practice guides sit somewhere be-tween consensus reports and meta-analyses in the degree to which systematic processes are used for locating relevant research and characterizing its meaning. Practice guides

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are more like consensus panel reports than meta-analyses in the breadth and complex-ity of the topic that is addressed. Practice guides are different from both consensus reports and meta-analyses in providing advice at the level of specific action steps along a pathway that represents a more-or-less coherent and comprehensive approach to a multifaceted problem.

Practice guides in education at the Institute of Education Sciences

IES publishes practice guides in educa-tion to bring the best available evidence and expertise to bear on the types of sys-temic challenges that cannot currently be addressed by single interventions or pro-grams. Although IES has taken advantage of the history of practice guides in health care to provide models of how to proceed in education, education is different from health care in ways that may require that practice guides in education have some-what different designs. Even within health care, where practice guides now number in the thousands, there is no single tem-plate in use. Rather, one finds descriptions of general design features that permit substantial variation in the realization of practice guides across subspecialties and panels of experts.193 Accordingly, the templates for IES practice guides may vary across practice guides and change over time and with experience.

The steps involved in producing an IES-sponsored practice guide are first to select a topic, which is informed by formal sur-veys of practitioners and requests. Next, a panel chair is recruited who has a national reputation and up-to-date expertise in the topic. Third, the chair, working in collabo-ration with IES, selects a small number of panelists to co-author the practice guide. These are people the chair believes can work well together and have the requisite

193. American Psychological Association (2002).

expertise to be a convincing source of rec-ommendations. IES recommends that at least one of the panelists be a practitioner with experience relevant to the topic being addressed. The chair and the panelists are provided a general template for a practice guide along the lines of the information provided in this appendix. They are also provided with examples of practice guides. The practice guide panel works under a short deadline of six to nine months to pro-duce a draft document. The expert panel members interact with and receive feed-back from staff at IES during the develop-ment of the practice guide, but they under-stand that they are the authors and, thus, responsible for the final product.

One unique feature of IES-sponsored prac-tice guides is that they are subjected to rigorous external peer review through the same office that is responsible for inde-pendent review of other IES publications. A critical task of the peer reviewers of a practice guide is to determine whether the evidence cited in support of particular recommendations is up-to-date and that studies of similar or better quality that point in a different direction have not been ignored. Peer reviewers also are asked to evaluate whether the evidence grade as-signed to particular recommendations by the practice guide authors is appropriate. A practice guide is revised as necessary to meet the concerns of external peer reviews and gain the approval of the standards and review staff at IES. The process of external peer review is carried out independent of the office and staff within IES that insti-gated the practice guide.

Because practice guides depend on the expertise of their authors and their group decisionmaking, the content of a practice guide is not and should not be viewed as a set of recommendations that in every case depends on and flows inevitably from sci-entific research. It is not only possible but also likely that two teams of recognized experts working independently to produce

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a practice guide on the same topic would generate products that differ in important respects. Thus, consumers of practice guides need to understand that they are, in effect, getting the advice of consultants. These consultants should, on average, pro-vide substantially better advice than an

individual school district might obtain on its own because the authors are national authorities who have to reach agreement among themselves, justify their recom-mendations in terms of supporting evi-dence, and undergo rigorous independent peer review of their product.

Institute of Education Sciences

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Appendix B. About the authors

Panel

Russell Gersten, Ph.D., is the director of the Instructional Research Group in Los Alamitos, California, as well as professor emeritus in the College for Education at the University of Oregon. Dr. Gersten re-cently served on the National Mathematics Advisory Panel, where he co-chaired the Work Group on Instructional Practices. He has conducted meta-analyses and research syntheses on instructional approaches for teaching students with difficulties in math-ematics, early screening in mathematics, RtI in mathematics, and research on num-ber sense. Dr. Gersten has conducted nu-merous randomized trials, many of them published in major education journals. He has either directed or co-directed 42 ap-plied research grants addressing a wide array of issues in education and has been a recipient of many federal and non-federal grants (more than $17.5 million). He has more than 150 publications and serves on the editorial boards of 10 prestigious journals in the field. He is the director of the Math Strand for the Center on Instruc-tion (which provides technical assistance to the states in terms of implementation of No Child Left Behind) and the director of research for the Southwest Regional Edu-cational Laboratory.

Sybilla Beckmann, Ph.D., is a profes-sor of mathematics at the University of Georgia. Prior to arriving at the Univer-sity of Georgia, Dr. Beckmann taught at Yale University as a J. W. Gibbs Instructor of Mathematics. Dr. Beckmann has done research in arithmetic geometry, but her current main interests are the mathemati-cal education of teachers and mathematics content for students at all levels, but espe-cially for Pre-K through the middle grades. Dr. Beckmann developed three mathe-matics content courses for prospective

elementary school teachers at the Uni-versity of Georgia and wrote a book for such courses, Mathematics for Elementary Teachers, published by Addison-Wesley, now in a second edition. She is especially interested in helping college faculty learn to teach mathematics content courses for elementary and middle grade teachers, and she works with graduate students and postdoctoral fellows toward that end. As part of this effort, Dr. Beckmann di-rects the Mathematicians Educating Future Teachers component of the University of Georgia Mathematics Department’s VIGRE II grant. Dr. Beckmann was a member of the writing team of the National Council of Teachers of Mathematics Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics, is a member of the Committee on Early Childhood Mathemat-ics of the National Research Council, and has worked on the development of several state mathematics standards. Recently, Dr. Beckmann taught an average grade 6 mathematics class every day at a local public school in order to better understand school mathematics teaching. She has won several teaching awards, including the General Sandy Beaver Teaching Professor-ship, awarded by the College of Arts and Sciences at the University of Georgia.

Benjamin Clarke, Ph.D., is a research associate at the Instructional Research Group and Pacific Institutes for Research. He serves as a co-principal investigator on three federally funded research grants in mathematics instructions and assessment. His current research includes testing the efficacy of a kindergarten mathematics curriculum, evaluating the effectiveness of a grade 1 mathematics intervention program for at-risk students and exam-ining the effects of a computer software program to build student understanding of and fluency with computational proce-dures. He also serves as the deputy direc-tor of the Center on Instruction Mathemat-ics. Dr. Clarke was a 2002 graduate of the University of Oregon School Psychology

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program. He was the recipient of AERA Special Education Dissertation Award for his work in early mathematics assess-ment. He has continued to investigate and publish articles and materials in this area and has presented his work at national conferences.

Anne Foegen, Ph.D., is an associate pro-fessor in the Department of Curriculum and Instruction at Iowa State University. Her research focuses on the mathematics development of students with disabilities, including efforts to develop measures that will allow teachers to monitor the prog-ress of secondary students in mathemat-ics. Dr. Foegen has also been involved in examining the mathematics performance of students with disabilities on large-scale assessments, such as the National Assess-ment of Educational Progress. Her current work in progress monitoring extends re-search in curriculum-based measurement (CBM) in mathematics from kindergarten through grade 12. Her particular focus is on studies at the middle school and high school levels. In a related project, Dr. Foegen developed CBM measures and conducted technical adequacy research on the use of the measures to track sec-ondary students’ progress in algebra and prealgebra.

Laurel Marsh, M.S.E. and M.A.T., is a professional education instructor who serves as a math coach at Swansfield El-ementary School, Howard County School District, in Columbia, Maryland. Also an instructor for the University of Maryland Baltimore County and Johns Hopkins Uni-versity, she has served as an elementary teacher at multiple levels. Ms. Marsh re-ceived a Master of Arts in Teaching from Johns Hopkins University, with a concen-tration in both Early Childhood Education and Elementary Education, as well as a Master of Science in Education from Johns Hopkins University, with a concentration in School Administration and Supervision. As a math support teacher, she provides

professional development to teachers of kindergarten through grade 5 for multiple schools in Howard County. She works with both general educators and special educa-tors through demonstration lessons, co-teaching situations, and school-based pro-fessional development. She also oversees and coordinates interventions for students struggling in mathematics.

Jon R. Star, Ph.D., is an assistant profes-sor of education at Harvard University’s Graduate School of Education. Dr. Star is an educational psychologist who stud-ies children’s learning of mathematics in middle and high school, particularly al-gebra. His current research explores the development of flexibility in mathemati-cal problem solving, with flexibility de-fined as knowledge of multiple strategies for solving mathematics problems and the ability to adaptively choose among known strategies on a particular problem. He also investigates instructional and curricular interventions that may promote the devel-opment of mathematical understanding. Dr. Star’s most recent work is supported by grants from the Institute of Education Sciences at the U.S. Department of Educa-tion and the National Science Foundation. In addition, he is interested in the preser-vice preparation of middle and secondary mathematics teachers. Before his graduate studies, he spent six years teaching middle and high school mathematics.

Bradley Witzel, Ph.D., is an associate pro-fessor and coordinator of special education at Winthrop University in Rock Hill, South Carolina. He has experience in the class-room as an inclusive and self-contained teacher of students with higher incidence disabilities as well as a classroom assistant and classroom teacher of students with low incidence disabilities. Dr. Witzel has taught undergraduate and graduate courses in educational methods for students with disabilities and secondary students with disabilities coupled with the needs of Eng-lish language learning. He has supervised

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interns in elementary, secondary, and spe-cial education certification tracks as well as inclusion practices. He has given numer-ous professional presentations including strategic math, algebra instruction, word problem solving, parent involvement, and motivational classroom management. He has published research and practitioner articles in algebra and math education as well as positive behavior interventions for students with and without learning disabili-ties. He has also written several books and book chapters on mathematics education and interventions. Overall, he is concerned with the development of special education teachers and works to provide researched-validated practices and interventions to professionals and preprofessionals.

Staff

Joseph A. Dimino, Ph.D., is a senior research associate at the Instructional Research Group in Los Alamitos, Cali-fornia, where he is the coordinator of a national research project investigating the impact of teacher study groups as a means to enhance the quality of reading instruction for 1st graders in high-poverty schools and co-principal investigator for a study assessing the impact of collaborative strategic reading on the comprehension and vocabulary skills of English language learners and English-speaking 5th graders. Dr. Dimino has 36 years of experience as a general education teacher, special edu-cation teacher, administrator, behavior specialist, and researcher. He has exten-sive experience working with teachers, parents, administrators, and instructional assistants in instruction, early literacy, reading comprehension strategies, and classroom and behavior management in urban, suburban, and rural communities. He has published in numerous scholarly journals and co-authored books in reading comprehension and early reading inter-vention. Dr. Dimino has delivered papers at various state, national, and interna-tional conferences, including the American

Educational Research Association, Society for the Scientific Study of Reading, Coun-cil for Exceptional Children, International Reading Association, and Association for Supervision and Curriculum Development. He consults nationally in early literacy and reading comprehension instruction.

Madhavi Jayanthi, Ph.D., is a research associate at Instructional Research Group, Los Alamitos, California. Dr. Jayanthi has more than 10 years experience in working on grants by the Office of Special Educa-tion Programs and Institute of Education Sciences. She has published extensively in peer-reviewed journals such as Journal of Learning Disabilities, Remedial and Spe-cial Education, and Exceptional Children. Her research interests include effective instructional techniques for students with disabilities and at-risk learners, both in the areas of reading and mathematics.

Shannon Monahan, Ph.D., is a survey re-searcher at Mathematica Policy Research. She has served as a reviewer for the What Works Clearinghouse for the Reducing Be-haviorProblemsintheElementarySchoolClassroom practice guide and the early childhood interventions topic area, and she coordinated the reviews for this prac-tice guide. Dr. Monahan has worked exten-sively on the development and evaluation of mathematics curricula for low-income children. Currently, she contributes to sev-eral projects that evaluate programs in-tended to influence child development. Her current interests include early childhood program evaluation, emergent numeracy, culturally competent pedagogy, measures development, and research design.

Rebecca A. Newman-Gonchar, Ph.D., is a research associate with the Instruc-tional Research Group. She has experi-ence in project management, study design and implementation, and quantitative and qualitative analysis. Dr. Newman-Gon-char has worked extensively on the de-velopment of observational measures for

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beginning and expository reading instruc-tion for two major IES-funded studies of reading interventions for Title I students. She currently serves as a reviewer for the What Works Clearinghouse for reading and mathematics interventions and Response to Intervention. Her scholarly contribu-tions include conceptual, descriptive, and quantitative publications on a range of top-ics. Her current interests include Response to Intervention, observation measure de-velopment for reading and mathematics instruction, and teacher study groups. She has served as the technical editor for several publications and is a reviewer for Learning Disability Quarterly.

Libby Scott, M.P.P., is a research analyst at Mathematica Policy Research and a for-mer classroom educator. She has experi-ence providing research support and con-ducting data analysis for various projects on topics related to out-of-school time pro-grams, disconnected youth, home school-ing households, and item development for a teacher survey. She also has experience evaluating an out-of-school time program for middle school students. Ms. Scott used her background in classroom teaching and education-related research to support the panel in translating research findings into practitioner friendly text.

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Appendix C. Disclosure of potential conflicts of interest

Practice guide panels are composed of in-dividuals who are nationally recognized experts on the topics about which they are rendering recommendations. The In-stitute of Education Sciences (IES) expects that such experts will be involved profes-sionally in a variety of matters that relate to their work as a panel. Panel members are asked to disclose their professional involvements and to institute deliberative processes that encourage critical exami-nation of the views of panel members as they relate to the content of the practice guide. The potential influence of panel members’ professional engagements is further muted by the requirement that they ground their recommendations in evi-dence documented in the practice guide. In addition, the practice guide undergoes independent external peer review prior to publication, with particular focus on whether the evidence related to the rec-ommendations in the practice guide has been appropriately presented.

The professional engagements reported by each panel member that appear most closely associated with the panel recom-mendations are noted below.

Russell Gersten has written articles on issues related to assessment and screen-ing of young children with potential dif-ficulties in learning mathematics and is currently revising a manuscript on this topic for the scholarly journal, Exceptional Children. However, there is no fiscal re-ward for this or other publications on the topic. He is a royalty author for what may become the Texas or national edition of the forthcoming (2010/11) Houghton Mif-flin reading series, Journeys. At the time of publication, Houghton Mifflin has not determined whether they will ever release

this series. Dr. Gersten provided guidance on the product as it relates to struggling and English language learner students. This topic is not covered in this prac-tice guide. The panel never discussed the Houghton Mifflin series. Russell Gersten has no financial stake in any program or practice mentioned in the practice guide.

Sybilla Beckmann receives royalties on her textbook, Mathematics for Elementary Teachers, published by Addison-Wesley, a division of Pearson Education. This text-book, used in college mathematics courses for prospective elementary teachers, is not within the scope of the review of this practice guide.

Ben Clarke has developed and authored journal articles about early numeracy measures that are referenced in the RtI-Mathematics practice guide. Dr. Clarke does not have a current financial stake in any company or publishing of the measures.

Anne Foegen conducts research and has developed progress monitoring assessments that are referred to in the RtI-Mathematics practice guide. Currently, she has no finan-cial interests in these products, which are not available commercially. The algebra measures are disseminated through Iowa State University on a fee-based schedule to cover the costs of personnel for training and materials (not for profit). Dr. Foegen also has published papers that describe the measures and received a grant that is supporting a portion of the research. She occasionally does private consulting related to research on use of the mathematics prog-ress monitoring measures.

Jon R. Star consults for a company owned by Scholastic, which produces mathemat-ics educational software. Scholastic may produce other curricula related to mathe-matics, but the panel makes no recommen-dations for selecting specific curricula.

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Bradley Witzel wrote the book, Compu-tation of Fractions, and is currently writ-ing Computation of Integers and Solving Equations, through Pearson with Dr. Paul Riccomini. He is also writing the book, RtI Mathematics, through Corwin Press with Dr. Riccomini. Additionally, Dr. Witzel has

delivered workshop presentations on the structure of RtI (not associated with the RtI-Mathematics practice guide). The work on his books is separate from that of the RtI-Mathematics practice guide panel, and he does not share his work from the panel with the books’ co-authors.

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Appendix D. Technical information on the studies



The panel examined reviews of the tech-nical adequacy of screening measures for students identified as at risk when making this recommendation. The panel rated the level of evidence for recommendation 1 as moderate because several reviews were available for evidence on screening mea-sures for younger students. However, there was less evidence available on these measures for older students. The panel relied on the standards of the American Psychological Association, the American Educational Research Association, and the National Council on Measurement in Edu-cation194 for valid screening instruments along with expert judgment to evaluate the quality of the individual studies and to determine the overall level of evidence for this recommendation.

Relevant studies were drawn from recent comprehensive literature reviews and re-ports195 as well as literature searches of databases using key terms (such as “for-mative assessment”). Journal articles sum-marizing research studies on screening in mathematics,196 along with summary information provided by the Research In-

194. American Educational Research Associa-tion, American Psychological Association, and National Council on Measurement in Education (1999).

195. For example, the National Mathematics Advi-sory Panel (2008).

196. Gersten et al. (2005); Fuchs, Fuchs, Compton et al. (2007); Foegen et al. (2007).

stitute on Progress Monitoring197 and the National Center on Progress Monitoring were also used.198

The studies of screening measures all used appropriate correlational designs.199 In many cases, the criterion variable was some type of standardized assessment, often a nationally normed test (such as the Stanford Achievement Test) or a state assessment. In a few cases, however, the criterion measure was also tightly aligned with the screening measure.200 The latter set is considered much weaker evidence of validity.

Studies also addressed inter-tester reliability,201 internal consistency,202 test-retest reliability,203 and alternate form reli-ability.204 Many researchers discussed the content validity of the measure.205 A few even discussed the consequential valid-ity206—the consequences of using screen-ing data as a tool for determining what requires intervention.207 However, these studies all used standardized achievement measures as the screening measure.

In recent years, a number of studies of screening measures have also begun to

197. http://www.progressmonitoring.net/.


199. Correlational studies are not eligible for WWC review.

200. For example, Bryant, Bryant, Gersten, Scam-macca, and Chavez (2008).

201. For example, Fuchs et al. (2003a).

202. For example, Jitendra et al. (2005).

203. For example, VanDerHeyden, Witt, and Gil-bertson (2003).

204. For example, Thurber, Shinn, and Smol-kowski (2002).

205. For example, Clarke and Shinn (2004); Ger-sten and Chard (1999); Foegen, Jiban, and Deno (2007).

206. Messick (1988); Gersten, Keating, and Irvin (1995).

207. For example, Compton, Fuchs, and Fuchs (2007).

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report sensitivity and specificity data.208 Because sensitivity and specificity pro-vide information on the false positive and false negative rates respectively, they are critical in determining the utility of a mea-sure used in screening decisions linked to resource allocation. Note that work on sensitivity and specificity in educational screening is in its infancy and no clear standards have been developed.

The remainder of this section presents evidence in support of the recommenda-tion. We discuss the evidence for measures used in both the early elementary and upper elementary grades and conclude with a more in-depth example of a screen-ing study to illustrate critical variables to consider when evaluating a measure.

Summary of evidence

In the early elementary grades, mea-sures examined included general out-come measures reflecting a sampling of objectives for a grade level that focused on whole numbers and number sense. These included areas of operations and procedures, number combinations or basic facts, concepts, and applications.209 Mea-sures to assess different facets of number sense—including measures of rote and strategic counting, identification of numer-als, writing numerals, recognizing quanti-ties, and magnitude comparisons—were also prevalent.210 Some research teams developed measures focused on a single aspect of number sense (such as strategic counting),211 and others developed batter-ies to create a composite score from single

208. Locuniak and Jordan (2008); VanDerHey-den et al. (2001); Fuchs, Fuchs, Compton et al. (2007).


210. Gersten, Clarke, and Jordan (2007); Fuchs, Fuchs, and Compton et al. (2007).

211. Clarke and Shinn (2004).

proficiency measures.212 Still others de-veloped a broader measure that assessed multiple proficiencies in their screening.213 An example of a single proficiency embed-ded in a broader measure is having stu-dents compare magnitudes of numbers. As an individual measure, magnitude com-parison has predictive validity in the .50 to .60 range,214 but having students make magnitude comparisons is also included in broader measures. For example, the Num-ber Knowledge Test (NKT)215 requires stu-dents to name the greater of two verbally presented numbers and includes problems assessing strategic counting, simple addi-tion and subtraction, and word problems. The broader content in the NKT provided stronger evidence of predictive validity216 than did single proficiency measures.

Further information on the characteristics and technical adequacy of curriculum-based measures (CBM) for use in screen-ing in the elementary grades was summa-rized by Foegen, Jiban, and Deno (2007). They explained that measures primarily assessed the objectives of operations or the concepts and applications standards for a specific grade level. A smaller num-ber of measures assessed fluency in basic facts, problem solving, or word problems. Measures were timed and administration time varied between 2 and 6 minutes for operations probes and 2 to 8 minutes for concepts and applications. Reliability evi-dence included test-retest, alternate form, internal consistency, and inter-scorer, with most reliabilities falling between .80 and .90, meeting acceptable standards for educational decisionmaking. Similar evidence was found for validity with most


213. Okamoto and Case (1996).

214. Lembke et al. (2008); Clarke and Shinn (2004); Bryant, Bryant, Gersten, Scammacca, and Chavez (2008).

215. Okamoto and Case (1996).

216. Chard et al. (2005).

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concurrent validity coefficients in the .50 to .70 range. Lower coefficients were found for basic fact measures ranging from .30 to .60. Researchers have also begun to de-velop measures that validly assess magni-tude comparison, estimation, and prealge-bra proficiencies.217

A study of evaluating a mathematics screening instrument—locuniak and Jordan (2008)

A recent study by Locuniak and Jordan (2008) illustrates factors that districts should consider when evaluating and se-lecting measures for use in screening. The researchers examined early mathematics screening measures from the middle of kindergarten to the end of second grade. The two-year period differs from many of the other screening studies in the area by extending the interval from within a school year (fall to spring) to across sev-eral school years. This is critical because the panel believes the longer the interval between when a screening measure and a criterion measure are administered, the more schools can have confidence that stu-dents identified have a significant deficit in mathematics that requires intervention. The Locuniak and Jordan (2008) study also went beyond examining traditional indices of validity to examine specificity and sen-sitivity. Greater sensitivity and specificity of a measure ensures that schools provide resources to those students truly at risk and not to students misidentified

The various measures studied by Lo-cuniak and Jordan (2008) also reflected mathematics content that researchers consider critical in the development of a child’s mathematical thinking and that many researchers have devised screening

217. Foegen et al. (2007).

measures to assess. Included were number sense measures that assessed knowledge of counting, number combinations, non-verbal calculation, story problems, num-ber knowledge, and short and working memory. The authors used block regres-sion to examine the added value of the math measures in predicting achievement above and beyond measures of cognition, age, and reading ability (block 1), which accounted for 26 percent of the variance on 2nd grade calculation fluency. Adding the number sense measures (block 2) in-creased the variance explained to 42 per-cent. Although the research team found strong evidence for the measures assess-ing working memory (digit span), number knowledge, and number combinations, the array of measures investigated is in-dicative that the field is still attempting to understand which critical variables (math-ematical concepts) best predict future dif-ficulty in mathematics. A similar process has occurred in screening for reading dif-ficulties where a number of variables (such as alphabetic principle) are consistently used to screen students for reading dif-ficulty. Using the kindergarten measures with the strongest correlations to grade 2 mathematics achievement (number knowl-edge and number combinations), the re-searchers found rates of .52 for sensitivity and .84 for specificity.

Another feature that schools will need to consider when evaluating and select-ing measures is whether the measure is timed. The measures studied by Locuniak and Jordan (2008) did not include a timing component. In contrast, general outcome measures include a timing component.218

No studies were found by the panel that examined a timed and untimed version of the same measure.

218. Deno (1985).

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The panel based this recommendation on professional opinion; therefore, the evidence rating is low. The professional opinion included not only the views of the panel members, but also several re-cent consensus documents that reflect input from mathematics educators and research mathematicians involved in is-sues related to K–12 mathematics educa-tion.219 Each of these documents was in-fluenced to some extent by comparisons of curricula standards developed by the 50 states in the United States with nations with high scores on international tests of mathematics performance, such as the Trends in International Mathematics and Science Study (TIMSS) and the Program for International Student Assessment (PISA) (including the Czech Republic, Flemish Belgium, Korea, and Singapore).220 We note, however, that these international compari-sons are merely descriptive and thus do not allow for causal inferences. In other words, we do not know whether their more focused curricula or other factors contrib-ute to higher performance.

We note that some of the other reports we describe here do not directly address the needs of students who receive in-terventions to boost their knowledge of

219. Milgram and Wu (2005); National Council of Teachers of Mathematics (2006); National Math-ematics Advisory Panel (2008).

220. For more information on the TIMSS, see http://nces.ed.gov/timss/. For more information on PISA, see www.oecd.org.

foundational concepts and procedures. However, we concluded that the focus on coverage of fewer topics in more depth, and with coherence, advocated for general education students is as important, and probably more important, for students who struggle with mathematics. We could not locate any experimental research that supported our belief, however. Therefore, we indicate clearly that we are reflecting a growing consensus of professional opin-ion, not a convergent body of scientific evidence—and conclude that the level of evidence is low.

Summary of evidence

Three seminal publications were con-sulted in forming our opinion.221 Milgram and Wu (2005) were among the first to suggest that an intervention curriculum for at-risk students should not be over-simplified and that in-depth coverage of key topics and concepts involving whole numbers and then rational numbers was critical for future success in mathematics. They stressed that mastery of this mate-rial was critical, regardless of how long it takes. Many before had argued about the importance of mastery of units before proceeding forward.222 Milgram and Wu argued that stress on precise definitions and abstract reasoning was “even more critical for at-risk students” (p. 2). They acknowledged this would entail extensive practice with feedback and considerable instructional time.

The National Council of Teachers of Math-ematicsCurriculumFocalPoints (2006) made a powerful statement about reform of mathematics curriculum for all students by calling for the end of brief ventures into many topics in the course of a school year.

221. Milgram and Wu (2005); National Council of Teachers of Mathematics (2006); National Math-ematics Advisory Panel (2008).

222. For example, Bloom (1980); Guskey (1984); Silbert, Carnine, and Stein (1989).

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The topics it suggests emphasize whole numbers (properties, operations, prob-lem solving) and especially fractions and related topics involving rational numbers (proportion, ratio, decimals). The report is equally clear that algorithmic proficiency is critical for understanding properties of operations and related concepts and that algorithmic proficiency, quick re-trieval of mathematics facts, and in-depth knowledge of such concepts as place value and properties of whole numbers are all equally important instructional goals. This position was reinforced by the report of the National Mathematics Advisory Panel (2008) two years later, which provided de-tailed benchmarks and again emphasized in-depth coverage of key topics involving whole numbers and rational numbers as crucial for all students.

In the view of the panel, students in inter-vention programs need to master material on whole numbers and rational numbers, and they must ultimately work with these concepts and principles at an abstract level. We feel that it is less important for 4th graders in an intervention program to cover the entire scope and sequence of topics from the year before. Instead, the aim is to cover the key benchmarks articulated in the National Mathematics Advisory Panel report, involving whole numbers and rational numbers that stu-dents do not fully grasp, and build profi-ciencies they lack.



The panel judged the level of evidence sup-porting the recommendation to be strong.

The panel found six studies223 conducted with low achieving or learning disabled students224 between 2nd and 8th grades that met WWC standards or met standards with reservations and included compo-nents of explicit and systematic instruc-tion.225 Appendix table D1 (p. 69) provides an overview of the components of explicit instruction in each intervention, including the use of teacher demonstration (such as verbalization during demonstration and the use of multiple examples), stu-dent verbalization (either as a procedural requirement or as a response to teacher questions), scaffolded practice, cumula-tive review, and corrective feedback. The relevant treatment and comparison groups compared in each study and the outcomes found for each domain are included in the table, as are grade-level, typical session length, and duration of the intervention.

Because of the number of high-quality randomized and quasi-experimental de-sign studies using explicit and systematic mathematics instruction across grade lev-els and diverse student populations, the frequency of significant positive effects, and the fact that numerous research teams independently produced similar findings, the panel concluded that there is strong evidence to support the recommendation to provide explicit and systematic instruc-tion in tier 2 mathematics interventions.

223. Darch, Carnine, and Gersten (1984); Fuchs et al. (2003a); Jitendra et al. (1998); Tournaki (2003); Schunk and Cox (1986); Wilson and Sin-delar (1991).

224. These students specifically had difficulties with mathematics.

225. For this practice guide, the components of explicit and systematic mathematics instruction are identified as providing models of proficient problem solving, verbalizing teacher and student thought processes, scaffolded practice, cumula-tive review, and corrective feedback.

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Summary of evidence

Teacher demonstrations and think-alouds. The panel suggests that teachers verbal-ize the solution to problems as they model problem solving for students. Tournaki (2003) assessed this approach by compar-ing a group of students whose teachers had demonstrated and verbalized an ad-dition strategy (the Strategy group) against a group of students whose teacher did not verbalize a strategy (the Drill and Practice group). As depicted in appendix table D1, the effects on an assessment of single-digit addition were significant, positive, and substantial in favor of the students whose teacher had verbalized a strategy.226

All six studies examined interventions that included teacher demonstrations early in the mathematics lessons.227 For example, Schunk and Cox (1986), Jitendra et al. (1998), and Darch, Carnine, and Gersten (1984) all conducted studies in which in-struction began with the teacher verbaliz-ing the steps to solve sample mathemat-ics problems. Because this demonstration procedure was used to instruct students in both treatment and comparison groups, the effects of this component of explicit in-struction cannot be evaluated from these studies. However, the widespread use of teacher demonstration in interventions that include other components of explicit instruction supports the panel’s conten-tion that this is a critical component of explicit instructional practice.

For teacher demonstration, the panel spe-cifically recommends that teachers provide

226. Note that during the intervention, students in the Strategy condition were also encouraged to verbalize the problem-solving steps and that this may also be a factor in the success of the inter-vention. The Tournaki (2003) study is described in more detail below.

227. Darch, Carnine, and Gersten (1984); Fuchs et al. (2003a); Jitendra et al. (1998); Tournaki (2003); Schunk and Cox (1986); Wilson and Sin-delar (1991).

numerous models of solving easy and hard problems proficiently. Demonstration with easy and hard problems and the use of numerous examples were not assessed as independent variables in the studies reviewed. However, Wilson and Sindelar (1991) did use numerous examples in in-struction for both groups evaluated. The key difference between the groups was that students in the treatment group were ex-plicitly taught problem-solving strategies through verbal and visual demonstrations while students in the comparison group were not taught these strategies. This study demonstrated substantively important pos-itive effects with marginal significance in favor of the treatment group.228

Scaffolded practice. Scaffolded practice, a transfer of control of problem solving from the teacher to the student, was a component of mathematics interventions in four of the six studies.229 In each study, the intervention groups that included scaf-folded practice demonstrated significant positive effects; however, it is not possible to parse the effects of scaffolded instruc-tion from the other components of ex-plicit instruction in these multicomponent interventions.

Student verbalization. Three of the six studies230 included student verbalization of problem-solution steps in the interven-tions. For example, Schunk and Cox (1986) assessed the effect of having students ver-balize their subtraction problem-solving steps versus solving problems silently. There were significant and substantial positive effects in favor of the group that

228. For this guide, the panel defined margin-ally significant as a p-value in the range of .05 to .10. Following WWC guidelines, an effect size greater than 0.25 is considered substantively important.

229. Darch, Carnine, and Gersten (1984); Fuchs et al. (2003a); Jitendra et al. (1998); Tournaki (2003).

230. Schunk and Cox (1986); Jitendra et al. (1998); Tournaki (2003).

125


( 67 )

verbalized steps. The Tournaki (2003) intervention also included student ver-balization among other components and had significant positive effects. Among other intervention components, Jitendra et al. (1998) included student verbalization through student responses to a teacher’s facilitative questions. Again, the effects were substantively important or statisti-cally significant and positive, but they can-not be attributed to a single component in this multi component intervention.

Corrective feedback. Four of the six stud-ies included immediate corrective feed-back in the mathematics interventions.231 For example, in the Darch, Carnine, and Gersten (1984) study, when a student made an error, teachers in the treatment group would first model the appropriate response, then prompt the students with questions to correct the response, then reinforce the problem-solving strategy steps again. In three of the studies,232 the effects of the corrective feedback compo-nent cannot be isolated from the effects of the other instructional components; however, the effects of the interventions including corrective feedback were posi-tive and significant.

Cumulative review. The panel’s assertion that cumulative review is an important component of explicit instruction is based primarily on expert opinion because only one study in the evidence base included cumulative review as a component of the intervention.233 This study had positive sig-nificant effects in favor of the instructional group that received explicit instruction in strategies for solving word problems.

231. Darch, Carnine, and Gersten (1984); Jiten-dra et al. (1998); Tournaki (2003); Schunk and Cox (1986).

232. Darch, Carnine, and Gersten (1984); Jitendra et al. (1998); Tournaki (2003).

233. Fuchs et al. (2003a).

In summary, the components of explicit and systematic instruction are consistently associated with significant positive effects on mathematics competency, most often when these components are delivered in combination. An example of a study that examines the effects of a combination of these components is described here.

A study of explicit and systematic instruction—Tournaki (2003)

Explicit and systematic instruction is a multicomponent approach, and an in-tervention examined in Tournaki (2003) exemplifies several components in com-bination. This study was conducted with 42 students in grade 2 special education classrooms.234 The students, between 8 and 10 years old, were classified as learn-ing disabled with weaknesses in both read-ing and mathematics. Twenty-nine were boys, and 13 were girls.

Prior to the intervention, the students completed a pretest assessment consist-ing of 20 single-digit addition problems (such as 6 + 3 = ). Internal consistency of the assessment was high (Cronbach’s alpha of .91). Student performance on the assessment was scored for accuracy and latency (the time it took each student to complete the entire assessment). The ac-curacy score is a measure of student abil-ity to perform mathematical operations, and the latency score is an indication of student fluency with single-digit addition facts. After the intervention, students completed a posttest assessment that was identical to the pretest.

Students were randomly assigned to one of three groups (two instruction groups and

234. The sample also included 42 grade 2 stu-dents from general education classrooms, but only the results for the special education students are presented as relevant to this practice guide.

126


( 68 )

a comparison group).235 Students in the two instruction groups met individually with a graduate assistant for a maximum of eight 15-minute supplemental math-ematics sessions on consecutive school days. Instructional materials for both groups consisted of individual worksheets for each lesson with a group of 20 sin-gle-digit addition problems covering the range from 2 + 2 = to 9 + 9 = .

The Strategy instruction group received explicit and systematic instruction to im-prove fact fluency. The instruction began with the teacher modeling the minimum addend strategy for the students and thinking aloud. This strategy is an ef-ficient approach for solving single-digit addition problems (such as 5 + 3 = ). The teacher began by saying, “When I get a problem, what do I do? I read the prob-lem: 5 plus 3 equals how many? Then I find the smaller number.” Pointing to the num-ber, the teacher says, “Three. Now I count fingers. How many fingers am I going to count? Three.” The teacher counts three fingers, points to the larger number and says, “Now, starting from the larger num-ber, I will count the fingers.” The teacher points to the 5, then touches each finger

235. Students in the comparison group received only the pretest and posttest without any supple-mental mathematics instruction outside their classroom. Because the scope of the practice guide is examining the effects of methods of teaching mathematics for low-achieving stu-dents, the comparison group findings are not included here.

as she says, “5, 6, 7, 8. How many did I end up with? Eight. I’ll write 8 to the right of the equal sign.” After writing the num-ber, the teacher finishes modeling by say-ing, “I’ll read the whole problem: 5 plus 3 equals 8.”

The teacher and student solved two prob-lems together through demonstration and structured probing. The student was then asked to solve a third problem indepen-dently while verbalizing the strategy steps aloud. When a student made an error, the teacher gave corrective feedback. The stu-dent was asked to solve the remaining prob-lems without verbalization and to work as fast as possible, but when an error oc-curred, the teacher interrupted the lesson and reviewed the steps in the strategy.

Students in the Drill and Practice group were asked to solve the problems as quickly as possible. At the completion of each lesson, the teacher marked the stu-dent’s errors and asked the student to re-compute. If the error persisted, the teacher told the student the correct answer. Re-sults indicate significant and substantial positive effects in favor of the Strategy group, which received explicit and sys-tematic instruction, relative to Drill and Practice group, which received a more tra-ditional approach. In this study, the com-bination of teacher demonstration, student verbalization, and corrective feedback was successful in teaching students with math-ematics difficulties to accurately complete single-digit addition problems.

127


( 69 )

Table

D1.

Stu

die

s of

inte

rven

tion

s th

at in

cluded

exp

lici

t in

stru

ctio

n a

nd m

et W

WC

Sta

ndar

ds

(w

ith a

nd w

ithou

t re

serv

atio

ns)

Com

pon

en

ts o

f exp

lici

t in

stru

ctio

n i

ncl

ud

ed

in

th

e i

nte

rven

tion

Stu

dy

Com

par

ison

Teac

her

dem

on

-st

rati

on

Stu

den

t verb

aliz

a-ti

on

sG

uid

ed

p

ract

ice

Corr

ect

ive

feed

bac

kC

um

ula

tive

revie

wG

rad

e

level

Du

rati

on

Dom

ain

Ou

tcom

esa

Dar

ch, C

arn

ine,

an

d G

erst

en

(198

4)

Exp

lici

t st

rate

gy

inst

ruct

ion

ver

sus

trad

itio

nal

bas

al

inst

ruct

ion

4

30

min

ute

s/se

ssio

n;

11 s

essi

ons

Wor

d

pro

blem

s1.

79*

Jite

nd

ra e

t al

. (1

998)

Exp

lici

t vi

sual

st

rate

gy

inst

ruct

ion

ver

sus

trad

itio

nal

bas

al

inst

ruct

ion

2–5

40–4

5

min

ute

s/se

ssio

n;

17–2

0

sess

ion

s

Wor

d

pro

blem

s.5

6 (n

.s.)

Tran

sfer

1.01

*

Sch

un

k an

d C

ox

(198

6)C

onti

nu

ous

ve

rbal

izat

ion

s by

stu

den

ts

vers

us

no

stu

den

t ve

rbal

izat

ion

s

6–8

45

min

ute

s/se

ssio

n;

6 se

ssio

ns

Op

erat

ion

s1.

01*

Tou

rnak

i (20

03)

Cou

nti

ng-

on

stra

tegy

inst

ruct

ion

ver

sus

dri

ll a

nd

pra

ctic

e

215

m

inute

s/se

ssio

n;

up

to

8 se

ssio

ns

Op

erat

ion

s2.

21*

Tran

sfer

1.10

*

Wil

son

an

d

Sin

del

ar (1

991)

Stra

tegy

inst

ruct

ion

ver

sus

sequ

enci

ng

of

pra

ctic

e p

robl

ems

2–

430

m

inute

s/se

ssio

n;

14 s

essi

ons

Wor

d

pro

blem

s.8

2~

Fuch

s et

al.

(2

003

a)In

stru

ctio

n o

n

solv

ing

wor

d

pro

blem

s th

at is

b

ased

on

com

mon

u

nd

erly

ing

stru

ctu

res

vers

us

trad

itio

nal

bas

al

inst

ruct

ion

325

–40

m

inute

s/se

ssio

n;

36 s

essi

ons

Wor

d

pro

blem

s2.

09*

a.

Ou

tcom

es a

re r

epor

ted

as

effec

t si

zes.

For

a p

-val

ue

< .0

5, t

he

effec

t si

ze is

sign

ifica

nt

(*);

for

a p

-val

ue

< .1

0, t

he

effec

t si

ze is

mar

gin

ally

sig

nifi

can

t (~

); fo

r a

p-v

alu

e ≥

.10,

th

e eff

ect

size

is

not

sig

nifi

can

t (n

.s.).

Sou

rce:

Auth

ors’

an

alys

is b

ased

on

stu

die

s in

tab

le.

128


( 70 )



The panel rated the level of evidence as strong. We located nine studies that met the standards of the WWC or met the standards with reservations and demonstrated sup-port for the practice of teaching students to solve word problems based on their under-lying structures.236 Appendix table D2 (p. 73) provides an overview of each of the inter-ventions examined in these nine studies.

In all nine interventions, students were taught to recognize the structure of a word problem in order to solve it, and they were taught how to solve each problem type.237 Six of the studies took the instruction on problem structure a step further. Students were taught to distinguish superficial from substantive information in word problems in order to transfer solution methods from familiar problems they already knew how to solve to problems that appeared unfamil-iar.238 Because of the large number of high-quality randomized studies conducted that examined this practice and because most of the interventions examined led to sig-nificant and positive effects on word prob-lem outcomes for children designated as low achieving and/or learning disabled,

236. Jitendra et al. (1998); Xin, Jitendra, and Deat-line-Buchman (2005); Darch, Carnine, and Gersten (1984); Fuchs et al. (2003a); Fuchs et al. (2003b); Fuchs, Fuchs, Prentice et al. (2004); Fuchs, Fuchs, Finelli et al. (2004); Fuchs, Seethaler et al. (2008); Fuchs, Fuchs, Craddock et al. (2008).

237. Jitendra et al. (1998); Xin, Jitendra, and Deat-line-Buchman (2005); Darch, Carnine, and Gersten (1984); Fuchs et al. (2003a); Fuchs et al. (2003b); Fuchs, Fuchs, Prentice et al. (2004); Fuchs, Fuchs, Finelli et al. (2004); Fuchs, Seethaler et al. (2008); Fuchs, Fuchs, Craddock et al. (2008).


we conclude that there is strong evidence to support this recommendation.

Summary of evidence

Teach the structure of problem types. In three of the studies, students were taught to identify problems of a given type and then to design and execute appropriate solution strategies for each type.239 In one of these interventions, students learned to represent the problem using a schematic diagram.240 Once students learned to iden-tify the key problem features and map the information onto the diagram, they learned to solve for unknown quantities in word problems while still representing the problem using a schematic diagram. This intervention had significant and positive effects on a word problem outcome based on a test of problems similar to those taught during the intervention.

In another intervention that also led to a significant and positive effect on a word problem outcome, students were taught to discriminate between multiplication and addition problems, and between multiplica-tion and division problems.241 To discrimi-nate multiplication from addition problems, students were taught that if a problem asks them to use the same number multiple times (sometimes signaled by the words “each” and "every”) to obtain the total number, the problem requires multiplication. If the problem does not ask the student to use the same number multiple times to obtain the total number, the problem requires addition. Next, after students learned the relationship between multiplication and division through the concept of number families, they learned to multiply when the two smaller numbers are given without the big number and to divide when the big number is given.

239. Jitendra et al. (1998); Xin, Jitendra, and Deatline-Buchman (2005); Darch, Carnine, and Gersten (1984).


241. Darch, Carnine, and Gersten (1984).

129


( 71 )

Transfer solution methods from familiar problem types to problems that appear un-familiar. In addition to teaching students to recognize and solve different problem types, six of these studies taught students how to transfer solution methods to prob-lems that appear different but really re-quire the same solution methods as those they already know how to solve.242 In each of these interventions, students were first taught the pertinent structural features and problem-solution methods for differ-ent problem types. Next, they were taught about superficial problem features that can change a problem without altering its struc-ture or solution (for example, different for-mat, different key vocabulary, additional or different question, irrelevant information) and how to solve problems with varied cover stories and superficial features.

In all six studies, word problem outcome measures ranged from those where the only source of novelty was the cover story (immediate transfer), to those that varied one or more superficial features (near or far transfer). In five cases243 the average impact of the intervention on these out-come measures was positive and signifi-cant for the samples designated as low achieving and/or learning disabled, and in one case,244 the impact was marginally significant. These studies show that in-struction on problem structure and trans-ferring known solution methods to unfa-miliar problems is consistently associated with marginally or statistically significant positive effects on word problem solving proficiencies for students experiencing mathematics difficulties.


243. Fuchs et al. (2003a); Fuchs et al. (2003b); Fuchs, Fuchs, Prentice et al. (2004); Fuchs, Fuchs, Finelli et al. (2004); Fuchs, Fuchs, Craddock et al. (2008).

244. Fuchs, Seethaler et al. (2008).

A study of teaching students to transfer solution methods—fuchs, fuchs, finelli, Courey, and hamlett (2004)

Fuchs and colleagues (2004) conducted a study that investigated the effects of teaching students how to transfer known solution methods to problems that are only superficially different from those they already know how to solve.245 The authors randomly assigned 24 teachers to three groups: 1) transfer instruction, 2) ex-panded transfer instruction, and 3) regular basal instruction (comparison group).246 The 351 students in these 24 classes that were present for each of the pretests and posttests were participants in the study.

The intervention included 25- to 40-minute lessons, approximately twice per week for 17 weeks.247 Students in the expanded trans-fer condition learned basic math problem-solving strategies in the first unit of instruc-tion (six sessions over three weeks). They were taught to verify that their answers make sense; line up numbers from text to perform math operations; check operations; and label their work with words, monetary signs, and mathematical symbols.

The remaining units each focused on one of four problem types: 1) shopping list prob-lems (buying multiple quantities of items, each at a different price); 2) buying bag prob-lems (determining how many bags contain-ing a specified number of objects are needed to come up with a desired total number of


246. Since the comparison between the expanded transfer condition and the control condition (reg-ular basal instruction) is most relevant to this practice guide, we do not discuss the transfer instruction condition here.

247. Although this intervention was taught in a whole-class format, the authors reported sepa-rate effects for students classified as low achiev-ing and for students classified as learning dis-abled; therefore, the results are relevant to this practice guide.

130


( 72 )

objects); 3) half problems (determining what half of some total amount is); and 4) picto-graph problems (summing two addends, one derived from a pictograph). There were seven sessions within each unit.

In sessions one through four, with the help of a poster listing the steps, students learned problem-solution rules for solving the type of problem being taught in that particular unit. In the first session, teachers discussed the underlying concepts related to the problem type, presented a worked example, and explained how each step of the solution method was applied in the example. After presenting several worked examples, the teachers presented partially worked examples while the students ap-plied the steps of the solution method. Stu-dents then completed one to four problems in pairs. Sessions two through four were similar, but more time was spent on par-tially worked examples and practice, and at the end of each session, students com-pleted a problem independently.

In sessions five and six, teachers taught students how to transfer the solution meth-ods using problems that varied cover sto-ries, quantities, and one transfer feature per problem. In session five, the teachers began by explaining that transfer means to move and presented examples of how students transfer skills. Then, teachers taught three transfer features that change a problem without changing its type or solution, including formatting, unfamiliar vocabulary, and posing a different ques-tion. These lessons were facilitated by a poster displayed in the classroom about the three ways problems change. Again, teachers presented the information and worked examples, and moved gradually to partially worked examples and prac-tice in pairs. Session six was similar to session five, but the students spent more time working in pairs, and they completed a transfer problem independently.

In the seventh session, teachers instructed students on three additional superficial problem features including irrelevant infor-mation, combining problem types, and mix-ing superficial problem features. Teachers taught this lesson by discussing how prob-lems encountered in “real life” incorporate more information than most problems that the students know how to solve. They used a poster called Real-Life Situations to illus-trate each of these superficial problem fea-tures with a worked example. Next, students worked in pairs to solve problems that var-ied real-life superficial problem features and then completed a problem independently.

The authors used four measures to deter-mine the results of their intervention on word problem-solving proficiencies. The first measure used novel problems struc-tured the same way as problems used in the intervention. The second incorporated novel problems that varied from those used in instruction in terms of the look or the vocabulary or question asked. The third incorporated novel problems that varied by the three additional transfer features taught in session seven. The fourth was a measure designed to approximate real-life problem solving. Although this intervention was taught in a whole-class format, the au-thors separated results for students classi-fied as low performing248 and for students classified as learning disabled. The average impacts on these four outcome measures were positive and significant for both the sample designated as low performing and the sample designated as learning disabled. It is notable that the intervention had a posi-tive and significant impact on the far trans-fer measure (the measure that approximated real-life problem solving). This study dem-onstrates a successful approach for instruct-ing students with mathematics difficulties on solving word problems and transferring solution methods to novel problems.

248. Using pretest scores on the first transfer problem-solving measure, the authors desig-nated each student as low performing, average performing, or high performing.

131


( 73 )

Table

D2. S

tudie

s of

inte

rven

tion

s th

at t

aught

studen

ts t

o d

iscr

imin

ate

pro

ble

m t

ypes

that

met

WW

C s

tandar

ds

(w

ith o

r w

ithou

t re

serv

atio

ns)

Stu

dy

Com

par

ison

Gra

de

level

Du

rati

on

Lear

nin

g d

isab

led

/Low

ach

ievin

gD

om

ain

Ou

tcom

esa

Dar

ch, C

arn

ine,

an

d

Ger

sten

(198

4)In

stru

ctio

n o

n s

olvi

ng

wor

d p

robl

ems

that

is

bas

ed o

n c

omm

on u

nd

erly

ing

stru

ctu

res

vers

us

trad

itio

nal

bas

al in

stru

ctio

n

430

min

ute

s/se

ssio

n; 1

1

sess

ion

s

Low

ach

ievi

ng

Wor

d p

robl

ems

1.79

*

Fuch

s, F

uch

s, P

renti

ce

et a

l. (2

00

4)In

stru

ctio

n o

n s

olvi

ng

wor

d p

robl

ems

that

is

bas

ed o

n c

omm

on u

nd

erly

ing

stru

ctu

res

vers

us

trad

itio

nal

bas

al in

stru

ctio

n

325

–40

min

ute

s/se

ssio

n; 3

2

sess

ion

s

Low

ach

ievi

ng

Wor

d p

robl

ems

4.75

*

Lear

nin

g d

isab

led

bW

ord

pro

blem

s1.

10*

Fuch

s, F

uch

s, F

inel

li

et a

l. (2

00

4)In

stru

ctio

n o

n s

olvi

ng

wor

d p

robl

ems

that

is

bas

ed o

n c

omm

on u

nd

erly

ing

stru

ctu

res

vers

us

trad

itio

nal

bas

al in

stru

ctio

n

325

–40

min

ute

s/se

ssio

n; 3

4

sess

ion

s

Low

ach

ievi

ng

Wor

d p

robl

ems

3.08

*

Lear

nin

g d

isab

led

cW

ord

pro

blem

s1.

23*

Fuch

s, S

eeth

aler

et

al.

(2008

)In

stru

ctio

n o

n s

olvi

ng

wor

d p

robl

ems

that

is

bas

ed o

n c

omm

on u

nd

erly

ing

stru

ctu

res

vers

us

non

rele

vant

inst

ruct

ion

320

–30

min

ute

s/se

ssio

n; 3

6

sess

ion

s

Low

ach

ievi

ng

Wor

d p

robl

ems

.66

~

Con

cep

ts.6

0 (n

.s.)

Fuch

s, F

uch

s, C

rad

doc

k et

al.

(2008

)In

stru

ctio

n o

n s

olvi

ng

wor

d p

robl

ems

that

is

bas

ed o

n c

omm

on u

nd

erly

ing

stru

ctu

res

vers

us

non

rele

vant

inst

ruct

ion

320

–30

min

ute

s/se

ssio

n; 3

8

sess

ion

s

Low

ach

ievi

ng

Wor

d p

robl

ems

.73*

Jite

nd

ra e

t al

. (19

98)

Inst

ruct

ion

on

sol

vin

g w

ord

pro

blem

s th

at

is b

ased

on

com

mon

un

der

lyin

g st

ruct

ure

s ve

rsu

s tr

adit

ion

al b

asal

in

stru

ctio

n

2–5

40–4

5 m

inute

s/se

ssio

n; 1

7-20

se

ssio

ns

Lear

nin

g d

isab

led

d

and

low

ach

ievi

ng

com

bin

ed

Wor

d p

robl

ems

.56

(n.s

.)

Tran

sfer

1.01

*

Xin

, Jit

end

ra, a

nd

D

eatl

ine-

Buch

man

(2

005

)

Inst

ruct

ion

on

sol

vin

g w

ord

pro

blem

s b

ased

on

com

mon

un

der

lyin

g st

ruct

ure

s ve

rsu

s ge

ner

al s

trat

egy

inst

ruct

ion

6–8

60 m

inute

s/se

ssio

n; 1

2

sess

ion

s

Lear

nin

g d

isab

led

eW

ord

pro

blem

s1.

87*

Fuch

s et

al.

(2003

a)In

stru

ctio

n o

n s

olvi

ng

wor

d p

robl

ems

that

is

bas

ed o

n c

omm

on u

nd

erly

ing

stru

ctu

res

vers

us

trad

itio

nal

bas

al in

stru

ctio

n

325

–40

min

ute

s/se

ssio

n; 3

6

sess

ion

s

Low

ach

ievi

ng

Wor

d p

robl

ems

2.09

*

Fuch

s et

al.

(2003

b)In

stru

ctio

n o

n s

olvi

ng

wor

d p

robl

ems

that

is

bas

ed o

n c

omm

on u

nd

erly

ing

stru

ctu

res

vers

us

trad

itio

nal

bas

al in

stru

ctio

n

3N

um

ber

of

min

ute

s n

ot

rep

orte

d; 3

2

sess

ion

s

Low

ach

ievi

ng

Wor

d p

robl

ems

2.05

*

Lear

nin

g d

isab

led

fW

ord

pro

blem

s.8

6*

a.

Ou

tcom

es a

re r

epor

ted

as

effec

t si

zes.

For

a p

-val

ue

< .0

5, t

he

effec

t si

ze is

sign

ifica

nt

(*);

for

a p

-val

ue

< .1

0, t

he

effec

t si

ze is

mar

gin

ally

sig

nifi

can

t (~

); fo

r a

p-v

alu

e ≥

.10,

th

e

effec

t si

ze is

not

sig

nifi

can

t (n

.s.).

b.

Th

irte

en s

tud

ents

in t

his

sam

ple

wer

e cl

assi

fied

as

hav

ing

a le

arn

ing

dis

abil

ity,

on

e as

hav

ing

men

tal r

etar

dat

ion

, eig

ht

as h

avin

g a

spee

ch d

isor

der

, an

d t

wo

as h

avin

g at

tenti

on-

defi

cit/

hyp

erac

tivi

ty d

isor

der

.c.

Fi

ftee

n s

tud

ents

in

th

is s

amp

le w

ere

clas

sifi

ed a

s h

avin

g a

lear

nin

g d

isab

ilit

y an

d fi

ve w

ere

clas

sifi

ed a

s h

avin

g an

“ot

her

” d

isab

ilit

y.d

. Se

ven

teen

stu

den

ts i

n t

his

sam

ple

wer

e cl

assi

fied

as

hav

ing

a le

arn

ing

dis

abil

ity,

five

as

bei

ng

edu

cable

men

tall

y re

tard

ed, a

nd

th

ree

as b

ein

g se

riou

sly

emot

ion

ally

dis

turb

ed.

e.

Eigh

teen

stu

den

ts i

n t

his

sam

ple

wer

e cl

assi

fied

as

hav

ing

a le

arn

ing

dis

abil

ity,

on

e as

bei

ng

seri

ousl

y em

otio

nal

ly d

istu

rbed

, an

d t

hre

e w

ere

not

lab

eled

. f.

Tw

enty

-tw

o st

ud

ents

in

th

is s

amp

le w

ere

clas

sifi

ed a

s h

avin

g a

lear

nin

g d

isab

ilit

y, o

ne

as b

ein

g m

ild

ly m

enta

lly

reta

rded

, on

e as

hav

ing

a b

ehav

ior

dis

ord

er, a

nd

th

ree

as

hav

ing

spee

ch d

elay

.So

urc

e: A

uth

ors’

an

alys

is b

ased

on

stu

die

s in

tab

le.

132


( 74 )



The panel judged the level of evidence for this recommendation to be moderate. We found 13 studies conducted with stu-dents classified as learning disabled or low achieving that met WWC standards or met standards with reservations.249 Four in particular examined the impact of tier 2 in-terventions against regular tier 1 instruc-tion.250 Appendix table D3 (p. 77) provides an overview of these 13 studies. Note that in an attempt to acknowledge meaning-ful effects regardless of sample size, the panel followed WWC guidelines and con-sidered a positive statistically significant effect, or an effect size greater than 0.25, as an indicator of positive effects.251

Summary of evidence

The representations in 11 of the 13 studies were used mainly to teach word problems and concepts (fractions and prealgebra).252

249. Artus and Dyrek (1989); Darch, Carnine, and Gersten (1984); Fuchs, Powell et al. (2008); Fuchs, Fuchs, Craddock et al. (2008); Fuchs, Seethaler et al. (2008); Fuchs et al. (2005); Jitendra et al. (1998); Butler et al. (2003); Walker and Poteet (1989); Wil-son and Sindelar (1991); Witzel, Mercer, and Miller (2003); Witzel (2005); Woodward (2006).

250. Fuchs, Powell et al. (2008); Fuchs, Fuchs, Craddock et al. (2008); Fuchs, Seethaler et al. (2008); Fuchs et al. (2005).

251. For more details on WWC guidelines for substantively important effects, see the What WorksClearinghouseProceduresandStandardsHandbook (WWC, 2008).

252. Jitendra et al. (1998); Butler et al. (2003); Wit-zel (2005); Darch, Carnine, and Gersten (1984); Fuchs, Powell et al. (2008); Fuchs, Fuchs, Crad-dock et al. (2008); Fuchs, Seethaler et al. (2008);

In one study, visual representations were used to teach mathematics facts.253 In all 13 studies, representations were used to understand the information presented in the problem. Specifically, the representa-tions helped answer such questions as what type of problem it is and what oper-ation is required. In all 13 studies, visual representations were part of a complex multicomponent instructional interven-tion. Therefore, it is not possible to ascer-tain the role and impact of the representa-tion component.

Of the 13 studies, 4 used visual represen-tations, such as drawings or other forms of pictorial representations, to scaffold learning and pave the way for the under-standing of the abstract version of the representation.254 Jitendra et al. (1998) examined the differential effects of two instructional strategies, an explicit strat-egy using visual representations and a traditional basal strategy. Students were taught explicitly to identify and differenti-ate among word problems types and map the features of the problem onto the given diagrams specific to each problem type. The intervention demonstrated a nonsig-nificant substantively important positive effect. Wilson and Sindelar (1991) used a diagram to teach students the “big num-ber” rule (e.g., when a big number is given, subtract) (ES = .82~). Woodward (2006) ex-plored the use of visuals such as a number line to help students understand what an abstract fact such as 6 × 7 = meant. The study yielded a substantively impor-tant positive effect on mathematics facts, and a positive and marginally significant average effect on operations.

Fuchs et al. (2005); Walker and Poteet (1989); Wilson and Sindelar (1991); Witzel, Mercer, and Miller (2003).

253. Woodward (2006).

254. Jitendra et al. (1998); Walker and Poteet (1989); Wilson and Sindelar (1991); Woodward (2006).

133


( 75 )

Three studies used manipulatives in the early stages of instruction to reinforce un-derstanding of basic concepts and opera-tions.255 For example, Darch et al. (1984) used concrete models such as groups of boxes to teach rules for multiplication problems. Similarly, Fuchs, Fuchs, Crad-dock et al. (2008) used manipulatives in their tutoring sessions to target and teach the most difficult concepts observed in the classroom. In another study, Fuchs, Seethaler et al. (2008) used concrete ma-terials and role playing to help students understand the underlying mathemati-cal structure of each problem type. In all these studies, manipulatives were one as-pect of a complex instructional package. The studies resulted in mostly significant positive domain average effect sizes in the range of .60 to 1.79.

In six studies, both concrete and visual representations were used to promote mathematical understanding.256 For exam-ple, Artus and Dyrek (1989) used concrete objects (toy cars, empty food wrappers) and visuals (drawings) to help students understand the story content, action, and operation in the word problems (ES = .87~). Likewise, Fuchs, Powell et al. (2008) used manipulatives in the initial stages and later pictorial representations of ones and tens in their software program (ES = .55, n.s.). However, in both studies, concrete objects and visual representations were not part of an instructional format that promoted systematic scaffolded learning. In other words, instruction did not include fading the manipulatives and visual representa-tions to promote understanding of math at the more typical abstract level.

In the remaining four studies, manipu-latives and visual representations were

255. Darch et al. (1984); Fuchs, Seethaler et al. (2008); Fuchs, Fuchs, Craddock et al. (2008).

256. Artus and Dyrek (1989); Butler et al. (2003); Fuchs, Powell et al. (2008); Fuchs et al. (2005); Wit-zel (2005); Witzel, Mercer, and Miller (2003).

presented to the students sequentially to promote scaffolded instruction.257 This model of instruction, with its underpin-ning in Bruner’s (1966) work, is referred to as a concrete to representation to ab-stract (CRA) method of instruction. The CRA method is a process by which stu-dents learn through the manipulation of concrete objects, then through visual rep-resentations of the concrete objects, and then by solving problems using abstract notation.258 Fuchs et al. (2005) taught 1st grade students basic math concepts (e.g., place value) and operations initially using concrete objects, followed by pictorial rep-resentations of blocks, and finally at the abstract level (e.g., 2 + 3 = ) without the use of manipulatives or representations. Butler et al. (2003) examined the differen-tial impact of using two types of scaffolded instruction for teaching fractions, one that initiated scaffolding at the concrete level (concrete-representation-abstract) and the other that started at the repre-sentation level (representation-abstract). Neither variation resulted in significant differences.

Witzel (2005) and Witzel, Mercer, and Miller (2003) investigated the effectiveness of the scaffolded instruction using the CRA method to teach prealgebra (e.g., X – 4 = 6) to low-achieving students and students with disabilities. Using an explicit instruc-tional format, Witzel taught students ini-tially using manipulatives such as cups and sticks. These were replaced with drawings of the same objects and finally faded to typical abstract problems using Arabic symbols (as seen in most textbooks and standardized exams). Both studies resulted in statisti-cally significant or substantively important positive gains (Witzel, Mercer, and Miller, 2003 and ES = .83*; Witzel, 2005 and ES = .54, n.s.). One of these studies is described in more detail here.

257. Butler et al. (2003); Fuchs et al. (2005); Witzel (2005); Witzel, Mercer, and Miller (2003).

258. Witzel (2005).

134


( 76 )

A study of CRA instruction—Witzel, Mercer, and Miller (2003)

In 2003, Witzel, Mercer, and Miller published a study that investigated the effects of using the CRA method to teach prealgebra.259 The participants in the study were teachers and students in 12 grade 6 and 7 classrooms in a southeastern urban county. Each teacher taught one of two math classes using CRA instruction (treatment group) and the other using abstract-only traditional methods (traditional instruction group). Of those participating, 34 students with disabili-ties260 or at risk for algebra difficulty261 in the treatment group were matched with 34 students with similar characteristics across the same teacher’s classes in the traditional instruction group.


260. These students were identified through school services as those who needed additional support, had a 1.5 standard deviation discrepancy between ability and achievement, and had math goals listed in their individualized education plans.

261. These students met three criteria: per-formed below average in the classroom according to the teacher, scored below the 50th percentile in mathematics on the most recent statewide achievement test, and had socioeconomically disadvantaged backgrounds.

The students in both groups were taught to transform equations with single vari-ables using a five-step 19-lesson sequence of algebra equations. In each session, the teacher introduced the lesson, mod-eled the new procedure, guided students through procedures, and began to have students working independently. For the treatment group, these four steps were used for instruction at the concrete, rep-resentational, and abstract stages of each concept. Teachers taught the concrete les-sons using manipulative objects such as cups and sticks, the representational les-sons using drawings of the same objects, and the abstract lessons using Arabic sym-bols. For the traditional instruction group, the teachers covered the same content for the same length of time (50 minutes), but the teachers used repeated abstract lessons rather than concrete objects and pictorial representations.

A 27-item test to measure knowledge on single-variable equations and solving for a single variable in multiple-variable equa-tions was administered to the students one week before treatment (pretest), after the last day of the treatment (posttest), and three weeks after treatment ended (follow-up). The CRA intervention had a positive and significant effect on knowledge of the prealgebra concepts assessed.

135


( 77 )

Stu

dy

Com

par

ison

a

Gra

de

level

Du

rati

on

Dom

ain

Ou

tcom

esb

Art

us

and

Dyr

ek

(198

9)In

stru

ctio

nal

inte

rven

tion

usi

ng

con

cret

e ob

ject

s an

d

rep

rese

nta

tion

al d

raw

ings

ver

sus

trad

itio

nal

lect

ure

fo

rmat

4–6

90

min

ute

s/

sess

ion

; 6

sess

ion

s

Mat

h g

ener

al

ach

ieve

men

t. 8

7~

Butl

er e

t al

. (2

003

)In

stru

ctio

nal

inte

rven

tion

usi

ng

con

cret

e ob

ject

s an

d

rep

rese

nta

tion

al d

raw

ings

ver

sus

rep

rese

nta

tion

al

dra

win

gs o

nly

6–8

45 m

inute

s/

sess

ion

; 10

ses

sion

s

Con

cep

ts-1

4 (n

.s.)

Wor

d p

robl

ems

.07

(n.s

.)

Dar

ch, C

arn

ine,

an

d G

erst

en

(198

4)

Inst

ruct

ion

al inte

rven

tion

usi

ng

con

cret

e ob

ject

s ve

rsu

s tr

adit

ion

al b

asal

in

stru

ctio

n4

30 m

inute

s/

sess

ion

; 11

ses

sion

s

Wor

d p

robl

ems

1.79

*

Fuch

s et

al.

(2

005

)In

stru

ctio

nal

inte

rven

tion

usi

ng

con

cret

e ob

ject

s an

d r

epre

senta

tion

al d

raw

ings

ver

sus

no

inst

ruct

ion

co

nd

itio

n

140

min

ute

s/

sess

ion

;

48 s

essi

ons

Mat

h g

ener

al

ach

ieve

men

t.3

4~

Wor

d p

robl

ems

.56*

Fuch

s, S

eeth

aler

et

al.

(2008

)In

stru

ctio

nal

inte

rven

tion

usi

ng

con

cret

e m

ater

ials

an

d r

ole

pla

yin

g ve

rsu

s n

o in

stru

ctio

n c

ond

itio

n3

20–3

0 m

inute

s/se

ssio

n;

36 s

essi

ons

Wor

d p

robl

ems

.66

~

Con

cep

ts.6

0 (n

.s.)

Fuch

s, P

owel

l et

al.

(2008

) In

stru

ctio

nal

inte

rven

tion

usi

ng

con

cret

e ob

ject

s an

d p

icto

rial

rep

rese

nta

tion

s ve

rsu

s n

o in

stru

ctio

n

con

dit

ion

315

–18

min

ute

s/se

ssio

n;

45 s

essi

ons

Op

erat

ion

s.5

5 (n

.s)

Tran

sfer

—st

ory

pro

blem

s-.0

7 (n

.s.)

Tran

sfer

—m

ath

co

nce

pts

.12

(n.s

.)

Fuch

s, F

uch

s,

Cra

dd

ock

et a

l.

(2008

)

Inst

ruct

ion

al inte

rven

tion

usi

ng

con

cret

e ob

ject

s ve

rsu

s n

o in

stru

ctio

n c

ond

itio

n3

20–3

0 m

inute

s/se

ssio

n;

38 s

essi

ons

Wor

d p

robl

ems

.95*

Tran

sfer

—w

ord

p

robl

ems

.30*

Jite

nd

ra e

t al

. (1

998)

Inst

ruct

ion

al inte

rven

tion

usi

ng

dia

gram

mat

ic

rep

rese

nta

tion

s ve

rsu

s tr

adit

ion

al b

asal

in

stru

ctio

n2–

540

–45

min

ute

s/se

ssio

n;

17–2

0 se

ssio

ns

Wor

d p

robl

ems

.56

(n.s

.)

Tran

sfer

—w

ord

p

robl

ems

1.01

*

Wal

ker

and

Po

teet

(198

9)In

stru

ctio

nal

inte

rven

tion

usi

ng

dia

gram

mat

ic

rep

rese

nta

tion

s ve

rsu

s tr

adit

ion

al in

stru

ctio

n6

–830

min

ute

s/

sess

ion

; 17

ses

sion

s

Wor

d p

robl

ems

.35

(n.s

.)

Table

D3. S

tudie

s of

inte

rven

tion

s th

at u

sed v

isual

rep

rese

nta

tion

s th

at m

et W

WC

sta

ndar

ds

(w

ith a

nd w

ithou

t re

serv

atio

ns)

(con

tinu

ed)

136


( 78 )

Stu

dy

Com

par

ison

a

Gra

de

level

Du

rati

on

Dom

ain

Ou

tcom

esb

Wil

son

an

d

Sin

del

ar (1

991)

Inst

ruct

ion

al inte

rven

tion

usi

ng

dia

gram

mat

ic

rep

rese

nta

tion

s ve

rsu

s in

stru

ctio

n w

ith

out

dia

gram

s2–

430

min

ute

s/

sess

ion

; 14

ses

sion

s

Wor

d p

robl

ems

.82~

Wit

zel (

2005

)In

stru

ctio

nal

inte

rven

tion

usi

ng

con

cret

e ob

ject

s an

d p

icto

rial

rep

rese

nta

tion

s ve

rsu

s tr

adit

ion

al

inst

ruct

ion

6,7

50 m

inute

s/

sess

ion

; 19

ses

sion

s

Con

cep

ts

(pre

alge

bra)

.54

(n.s

.)

Wit

zel,

M

erce

r, a

nd

M

ille

r (2

003

)

Inst

ruct

ion

al inte

rven

tion

usi

ng

con

cret

e ob

ject

s an

d p

icto

rial

rep

rese

nta

tion

s ve

rsu

s tr

adit

ion

al

inst

ruct

ion

6,7

50 m

inute

s/

sess

ion

; 19

ses

sion

s

Con

cep

ts

(pre

alge

bra)

.83*

Woo

dw

ard

(2

006)

Inst

ruct

ion

al inte

rven

tion

usi

ng

pic

tori

al

rep

rese

nta

tion

s ve

rsu

s an

inte

rven

tion

not

usi

ng

rep

rese

nta

tion

s

425

min

ute

s/

sess

ion

; 20

ses

sion

s

Mat

h f

acts

.55

(n.s

.)

Op

erat

ion

s.1

1~

a. I

nst

ruct

ion

al inte

rven

tion

s in

all

th

e st

ud

ies

list

ed w

ere

mu

ltic

omp

onen

t in

nat

ure

, wit

h v

isu

als

bei

ng

one

of t

hos

e co

mp

onen

ts.

b. O

utc

omes

are

rep

orte

d a

s eff

ect

size

s. F

or a

p-v

alu

e <

.05,

th

e eff

ect

size

is s

ign

ifica

nt

(*),

for

a p

-val

ue

< .1

0, t

he

effec

t si

ze is

mar

gin

ally

sig

nifi

cant

(~);

for

a p

-val

ue

≥ .1

0, t

he

effec

t si

ze is

not

sig

nifi

cant

(n.s

.).

Sou

rce:

Auth

ors’

an

alys

is b

ased

on

stu

die

s in

tab

le.

Table

D3. S

tudie

s of

inte

rven

tion

s th

at u

sed v

isual

rep

rese

nta

tion

s th

at m

et W

WC

sta

ndar

ds

(w

ith a

nd w

ithou

t re

serv

atio

ns)

(con

tinue

d)

137


( 79 )



The panel judged the level of evidence supporting the recommendation to be moderate. We found seven studies con-ducted with low-achieving or learning disabled students between grades 1 and 4 that met WWC standards or met stan-dards with reservations and included fact fluency instruction in an intervention.262 Appendix table D4 (p. 83) provides an over-view of the studies and indicates whether fact fluency was the core content of the intervention or a component of a larger intervention. The relevant treatment and comparison groups in each study and the outcomes for each domain are included in the table. Grade level, typical session length, and duration of the intervention are also in the table.

Given the number of high-quality ran-domized and quasi-experimental design studies conducted across grade levels and diverse student populations that in-clude instruction in fact fluency as either an intervention or a component of an in-tervention, the frequency of small but substantively important or significant positive effects on measures of fact flu-ency and mathematical operations (effect sizes ranged from .11 to 2.21), and the fact that numerous research teams inde-pendently produced similar findings, the panel concluded that there is moderate evidence to support the recommendation to provide instruction in fact fluency for both tier 2 and tier 3 mathematics inter-ventions across grade levels. The panel

262. Fuchs, Fuchs, Hamlett et al. (2006); Fuchs, Seethaler et al. (2008); Fuchs et al. (2005). Bernie-Smith (1991); Fuchs, Powell et al. (2008); Tournaki (2003); Woodward (2006).

acknowledges that the broader implication that general mathematics proficiency will improve when fact fluency improves is the opinion of the panel.

Summary of evidence

The panel recognizes the importance of knowledge of basic facts (addition, sub-traction, multiplication, and division) for students in kindergarten through grade 4 and beyond. Two studies examined the ef-fects of teaching mathematics facts relative to the effects of teaching spelling or word identification using similar methods.263 In both studies, the mathematics facts group demonstrated substantively important or statistically significant positive gains in facts fluency relative to the comparison group, although the effects were significant in only one of these two studies.264

Another two interventions included a facts fluency component in combination with a larger tier 2 intervention.265 For example, in the Fuchs et al. (2005) study, the final 10 minutes of a 40-minute intervention ses-sion were dedicated to practice with addi-tion and subtraction facts. In both studies, tier 2 interventions were compared against typical tier 1 classroom instruction. In each study, the effects on mathematics facts were not significant. Significant positive ef-fects were detected in both studies in the domain of operations, and the fact fluency component may have been a factor in im-proving students’ operational abilities.

Relationships among facts. The panel sug-gests emphasizing relationships among basic facts, and five of the studies examined

263. Fuchs, Fuchs, Hamlett et al. (2006); Fuchs, Powell et al. (2008).

264. In Fuchs, Fuchs, Hamlett et al. (2006), the effects on addition fluency were positive while there was no effect on subtraction fluency.


138


( 80 )

exemplify this practice.266 In Woodward (2006), the Integrated Strategies group was specifically taught the connection between single-digit facts (e.g., 4 × 2 = ) and ex-tended facts (40 × 2 = ). In Fuchs et al. (2005, 2006c, 2008e), mathematics facts are presented in number families (e.g., 1 + 2 = 3 and 3 – 2 = 1). Bernie-Smith (1991) examined the effects of a counting up/on procedure that highlighted the relationship between facts versus a rote memorization method that did not highlight this relation-ship. There was a substantively important nonsignificant positive effect in favor of the group that was taught the relationship between facts. Note that fact relationships were not isolated as independent variables in this study.

Materials to teach math facts. The stud-ies used a variety of materials to teach mathematics facts. Woodward (2006) used worksheets, number lines, and arrays of blocks projected on overheads to help stu-dents visualize fact strategies. Tournaki (2003) also used worksheets. Three studies included flash cards.267 For example, the Fuchs, Seethaler et al. (2008) intervention included flash cards with individual ad-dition and subtraction problems on each card. Students had up to two minutes to re-spond to as many cards as they could, and they were provided with corrective feed-back on up to five errors each session.

Three studies included computer assisted instruction to teach mathematics facts.268 In all three interventions, students used a computer program designed to teach addi-tion and subtraction facts. In this program, a mathematics fact was presented briefly on the computer screen. When the fact dis-

266. Bernie-Smith (1991); Fuchs et al. (2005); Fuchs, Fuchs, Hamlett et al. (2006), Fuchs, Seetha-ler et al. (2008); Woodward (2006).

267. Bernie-Smith (1991); Fuchs, Seethaler et al. (2008); Fuchs, Powell et al. (2008).

268. Fuchs et al. (2005); Fuchs, Fuchs, Hamlett et al. (2006); Fuchs, Powell et al. (2008).

appeared, the student typed the fact. If the student made an error, the correct fact was displayed with accompanying audio, and the student had the opportunity to type the fact again. For two of the studies, the duration of the presentation on the screen was tailored to the student’s performance (with less time as the student gained profi-ciency) and the difficulty of facts increased as competency increased.269

Time. The panel advocates dedicating about 10 minutes a session to building fact fluency in addition to the time dedicated to tier 2 and tier 3 interventions. The seven studies supporting this recommendation dedicated a minimum of 10 minutes a ses-sion to fact fluency activities.

Explicit teaching strategies for building fact fluency. Another three studies in the evidence base address methods for teach-ing basic facts to students by comparing instructional approaches.270 Both Bernie-Smith (1991) and Tournaki (2003) investi-gated the effects of being taught a counting up/on strategy relative to a rote memoriza-tion procedure for promoting fact fluency. In the Bernie-Smith (1991) study, perhaps not surprisingly as both interventions were taught to enhance fact fluency, the addition facts competency of students in both groups improved. However, there was a substantively important nonsignifi-cant positive effect in favor of the count-ing-on group when the two groups were compared. In the Tournaki (2003) study, the latency of responses on a fact fluency posttest decreased271 while the accuracy of posttest responses significantly increased

269. Fuchs et al. (2005); Fuchs, Fuchs, Hamlett et al. (2006).

270. Bernie-Smith (1991); Tournaki (2003); Wood-ward (2006).

271. The latency decrease was marginally sig-nificant. A decrease in latency indicates that stu-dents in the counting-on group were answering fact problems more quickly than students in the rote memorization group.

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for the counting-on group relative to the rote memorization group.

Similarly, Woodward (2006) examined an integrated approach that combined instruction in strategies, visual repre-sentations, and timed practice drills (the Integrated Strategies group) versus a tra-ditional timed drill and practice approach for building multiplication fact fluency (the Timed-Practice Only group). In the Inte-grated Strategies group, difficult facts were taught through derived fact strategies or doubling and doubling again strategies. When WWC multiple comparison adjust-ments were applied to outcomes, none of the multiplication fact outcomes were significant, though effects were substan-tively important and positive in favor of the integrated approach.272 The operations domain showed mixed effects with approx-imation scores in favor of the integrated approach and operations scores in favor of the Timed-Practice Only group.

In summary, the evidence demonstrates substantively important or statistically significant positive effects for including fact fluency activities as either stand-alone interventions or components of larger tier 2 interventions. However, because these effects did not consistently reach statisti-cal significance, the panel is cautious and acknowledges that the level of evidence for this recommendation is moderate. There is also evidence that strategy-based instruction for fact fluency (e.g., teaching the counting-on procedure) is a superior approach over rote memorization. Further, many of the studies included here taught the relationships among facts, used a va-riety of materials such as flash cards and computer assisted instruction, and taught math facts for a minimum of 10 minutes a

272. When a study examines many outcomes or findings simultaneously, the statistical signifi-cance of findings may be overstated. The WWC makes a multiple comparison adjustment to pre-vent drawing false conclusions about the number of statistically significant effects (WWC, 2008).

session. Although these components of the interventions were most often not inde-pendent variables in the studies, they are all advocated by the panel. An example of a study investigating the effects of a fact fluency intervention is detailed here.

A study of a fact fluency intervention—fuchs, Powell, hamlett, and fuchs (2008).

This study was conducted with 127 stu-dents in grade 3 classrooms in Tennessee and Texas.273 The students were all iden-tified as having either math difficulties or math and reading difficulties.

Before the intervention, the students com-pleted several pretest assessments. The assessment that related to fact retrieval consisted of one subtest with three sets of 25 addition fact problems and a second subtest with three sets of 25 subtraction fact problems. Students had one minute to write answers for each set within a subtest. Internal consistency for the sets ranged be-tween .88 and .93. Scores on sets of items were combined into a single fact retrieval score. After the intervention, students completed this same fact retrieval assess-ment among a battery of posttests.

Students were randomly assigned to one of four groups (three mathematics instruc-tion groups and a reading instruction com-parison group). For this recommendation, we report only on the comparison between the Fact Retrieval group (n = 32) and the Word Identification comparison group (n = 35).274 Students in both groups met in-

273. This study met standards with reservations because of high attrition. The sample initially included 165 students randomized to the condi-tions and 127 in the postattrition sample. The authors did demonstrate baseline equivalence of the postattrition sample.

274. The third group was Procedural/Estimation Tutoring, which targeted computation of two-digit numbers. The fourth group was a combi-nation of Procedural/Estimation Tutoring and Fact Retrieval.

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dividually with a tutor275 for 15 to 18 min-utes during three sessions each week for 15 weeks.

Sessions for the Fact Retrieval instruction group consisted of three activities. First, the students received computer assisted instruction (CAI). In the computer pro-gram, an addition or subtraction math-ematics fact appeared on the screen for 1.3 seconds. When the fact disappeared, the student typed the fact using short-term memory. A number line illustrated the mathematics fact on the screen with colored boxes as the student typed. If the student typed the fact correctly, applause was heard, and the student was awarded points. Each time the student accumulated five points, animated prizes (e.g., a pic-ture of a puppy) appeared in the student’s animated “treasure chest.” If the student typed the mathematics fact incorrectly, the fact reappeared and the student was prompted to type it again.

The second instructional activity, flash card practice, began after 7.5 minutes of CAI. Flash card practice with corrective feedback included two types of flash cards. The first set of flash cards depicted writ-ten facts without answers. Students were encouraged to answer as many problems

275. There were 22 tutors. Some were masters or doctoral students. Most had teaching or tutor-ing experience.

as possible in two minutes. After three consecutive sessions with a minimum of 35 correct responses, the student was pre-sented with a second set of flash cards that contained a number line similar to the CAI number line. The student was asked to re-spond with the appropriate mathematics facts to accompany the number line for as many cards as possible within the time frame. Corrective feedback was provided for a maximum of five errors per flash card activity. The third activity during Fact Re-trieval instruction focused on cumulative review. Students were allotted two minutes to complete 15 mathematics fact problems using paper and pencil.

Students in the Word Identification com-parison group received computer assisted instruction and participated in repeated reading with corrective feedback during their sessions. The content was tailored to the student’s reading competency level as determined by a pretest.

Results indicated significant positive ef-fects on fact fluency in favor of the group that received fact retrieval instruction relative to the comparison group that re-ceived instruction in word identification. These results suggest that it is possible to teach struggling students mathematics facts in as small an amount of time as 45 minutes of instruction a week when using flash cards and CAI.

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142


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The panel rated the level of evidence for this recommendation as low. The panel relied on the standards of the American Psychological Association, the American Educational Research Association, and the National Council on Measurement Educa-tion276 for valid assessment instruments, along with expert judgment, to evaluate the quality of the individual studies and to determine the overall level of evidence for this recommendation.

Evidence for the recommendation included research studies on mathematics progress monitoring,277 summary reviews of math-ematics progress monitoring research,278 and summary information provided by the Research Institute on Progress Monitor-ing279 and the National Center on Progress Monitoring.280 Very little research evidence specifically addresses the use of math-ematics progress monitoring data within the context of RtI.

Most research on mathematics progress monitoring measures falls into two cat-egories. One group of studies examines the technical adequacy of the measures, including their reliability, validity, and sensitivity to growth. The second investi-gates teacher use of the measures to mod-ify instruction for individual students in order to enhance achievement; the bulk of

276. American Educational Research Association, American Psychological Association, and National Council on Measurement in Education (1999).

277. Clarke et al. (2008); Foegen and Deno (2001); Fuchs et al. (1993); Fuchs, Fuchs, Hamlett, Thompson et al. (1994); Leh et al. (2007); Lembke et al. (2008).

278. Foegen, Jiban, and Deno (2007).

279. www.progressmonitoring.net/.

280. www.studentprogress.org/.

this second body of research has been con-ducted primarily in special education set-tings and therefore is less relevant to the RtI focus of this practice guide. As a result, we focus on the technical adequacy studies in this appendix. Note that because similar and often identical measures are used for screening and progress monitoring, many of the studies reviewed here overlap with those discussed for recommendation 1 on screening. The same measure may be used as both a screening measure and a progress monitoring measure; however, the psychometric properties of these mea-sures are more firmly established when used as screening measures with fewer researchers investigating the function of the measures for modeling growth when used for progress monitoring. This dispar-ity in the research base leads to the panel assigning a moderate level of evidence to Recommendation 1 and a low level of evi-dence to Recommendation 7.

The technical adequacy studies of math-ematics progress monitoring measures were not experimental; the researchers typically used correlational techniques to evaluate the reliability and criterion valid-ity of the measures and regression meth-ods to examine sensitivity to growth. If progress monitoring measures are to be deemed trustworthy, relevant empiri-cal evidence includes data on reliability, concurrent criterion validity, and sensi-tivity to growth. Evidence of reliability generally includes data on inter-scorer agreement,281 internal consistency,282 test-retest reliability,283 and alternate form reliability.284 Evidence of concurrent cri-terion validity is gathered by examining relations between scores on the progress monitoring measures and other indica-

281. For example, Fuchs, Fuchs, Hamlett, Thomp-son et al. (1994).

282. For example, Jitendra, Sczesniak, and Deat-line-Buchman (2005).

283. For example, Clarke and Shinn (2004).

284. VanDerHeyden et al. (2001).

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tors of proficiency in mathematics. Com-mon criterion measures include scores on group and individual standardized tests of mathematics, course grades, and teacher ratings.285

Although issues of reliability and criterion validity are common to both screening and progress monitoring measures, a critical feature specific to progress monitoring is that the measures be sensitive to growth. If teachers are to use progress monitor-ing measures to evaluate the effects of in-struction on student learning, researchers must provide evidence that student scores on the measures change over time, thus providing an indication of their learning. Most research studies have examined sen-sitivity to growth by administering paral-lel forms of a measure across a period of several weeks or months. In some studies, students receive typical instruction, and in others, teachers adapt and refine the instruction in response to the progress monitoring data (often in special educa-tion contexts). In either case, evidence of sensitivity to growth typically involves computing regression equations to deter-mine slopes of improvement and report-ing these as mean weekly growth rates for a group of students. As an example, if a progress monitoring measure has a mean weekly growth rate of .5, teachers could expect that, on average, a student’s score would increase by 1 point every two weeks. Growth rates reported in the litera-ture vary considerably across measures and grade levels; no established stan-dards exist for acceptable rates of student growth under typical instruction.

We discuss the evidence for measures used across the elementary and middle school grades and conclude with a more in-depth example of a technical adequacy study of mathematics progress monitor-ing measures.

285. For example, Foegen and Deno (2001); Fuchs et al. (2003a); Chard et al. (2005).

Summary of evidence

Progressmonitoringintheprimarygrades. Measures for the primary grades typically reflect aspects of number sense, including strategic counting, numeral identification, and magnitude comparison. Among the studies examining sensitivity to growth in the early grades, researchers have re-lied on separate measures for each of the different aspects of numeracy.286 Other researchers have combined individual measures to create composite scores287 or used more comprehensive multiskill measures.288 But so far, the focus of these studies has been on screening rather than on progress monitoring. Reliability coef-ficients for these measures generally ex-ceed .85. Concurrent criterion validity co-efficients with standardized achievement tests are generally in the .5 to .7 range.289 Mean rates of weekly growth reported in the literature vary widely, ranging from .1 to .3290 problems a week to .2 to more than 1.0 problems.291

Progressmonitoringintheelementarygrades. Two types of measures have been investigated for monitoring the mathemat-ics learning of students in the elementary grades. The bulk of the research, con-ducted by a research group led by Dr. Lynn Fuchs, investigates the characteristics of general outcome measures that represent grade-level mathematics curricula in com-putation and in mathematics concepts and applications.292 These measures were de-veloped in the late 1980s and early 1990s,

286. For example, Clarke et al. (2008); Lembke et al. (2008).



289. Chard et al. (2005); Clarke and Shinn (2004); Clarke et al. (2008); Lembke et al. (2008).

290. Lembke et al. (2008).

291. Chard et al. (2005).

292. Fuchs and Fuchs (1998); Fuchs, Hamlett, and Fuchs (1998).

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reflecting the Tennessee state elementary mathematics curriculum of that time. The measures continue to be widely used and are recommended by the National Center for Student Progress Monitoring.293 Teach-ers should carefully examine the content of the measures to ensure that they are representative of the existing mathematics curricula in their states and districts.

The second type of measure is not broadly representative of the instructional curricu-lum as a whole, but instead serves as an indicator of general proficiency in math-ematics. Examples of such measures in-clude basic facts (number combinations)294 and word problem solving.295 Because the general outcome measures are representa-tive of the broader curriculum, they offer teachers more diagnostic information about student performance in multiple aspects of mathematics competence; this advantage is often gained by using mea-sures that are longer and require more ad-ministration time. The indicator measures are more efficient to administer for regular progress monitoring but may be as useful for diagnostic purposes.

Evidence of the reliability of the measures is generally strong, with correlation co-efficients above .8, except for the word problem-solving measures developed by Jitendra’s research team, which are slightly lower.296 Concurrent criterion validity mea-sures have included group and individual achievement tests. Validity correlation coef-ficients range widely across measure types and grade levels. At the lower end, Espin et al. (1989) found correlations between the Wide Range Achievement Test and basic fact measures in the .3 to .5 range for basic


294. Espin et al. (1989); VanDerHeyden, Witt, and Naquin (2003).

295. Jitendra, Sczesniak, and Deatline-Buchman (2005); Leh et al. (2007).

296. Jitendra, Sczesniak, and Deatline-Buchman (2005); Leh et al. (2007).

facts. In contrast, Fuchs, Hamlett, and Fuchs (1998) found correlations between the Stan-ford Achievement Test Math Computation subtest297 and general outcome measures of computation to range from .5 to .9 across grades 2 through 5. In general, concurrent criterion validity coefficients for elementary mathematics progress monitoring measures are in the .5 to .6 range.

Evidence of sensitivity to growth for el-ementary measures exists for the com-putation and concepts/applications mea-sures developed by Fuchs and for the word problem-solving measures developed by Jitendra. Mean growth rates for the Fuchs measures range from .25 to .70. A study by Shapiro and colleagues,298 using the same measures for students with disabilities, re-sulted in mean growth rates of .38 points per week for both types of measures. Mean weekly growth rates for the Jitendra mea-sures were .24 points per week.

Progressmonitoringinmiddleschool. Less evidence is available to support progress monitoring in middle school.299 Research teams have developed measures focus-ing on math concepts typically taught in middle school,300 basic facts301 and esti-mation.302 Criterion validity across the types of measures varies, but the majority of correlations coefficients fall in the .4 to .5 range. Helwig and colleagues303 found higher correlation coefficients with high-stakes state tests in the range of .6 to .8. Reliability estimates including alternate form, inter-rater, and test-retest were all of sufficient quality. Greater rates of growth were found for the non–concept-based measures with rates around .25 units per

297. Gardner, Rudman, Karlsen, and Merwin (1982).

298. Shapiro, Edwards, and Zigmond (2005).

299. Foegen (2008).


301. Espin et al. (1989).

302. Foegen and Deno (2001); Foegen (2000).


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week. A recent study304 compared these measure types with two grade 6 mea-sures305 similar to the measures described above assessing student understanding of operations and concepts for their grade level. In this case, middle school students in grades 6, 7, and 8 were assessed using multiple measures. Evidence was found that even into grades 7 and 8, using grade 6 measures focusing on operations and mathematical concepts still shows reliabil-ity, validity, and sensitivity to growth.

An example of a study of the technical adequacy of mathematics progress monitoring measures—fuchs, fuchs, hamlett, Thompson, Roberts, kupek, and Stecker (1994)

A study conducted by Fuchs, Fuchs, Ham-lett, Thompson, Roberts, Kupek, and Stecker (1994) illustrates the type of tech-nical adequacy evidence that education professionals can use when evaluating and selecting mathematics progress monitor-ing measures. The research team exam-ined the technical features of grade-level general outcome measures of mathematics concepts and applications. The measures were developed by analyzing the Tennes-see mathematics curriculum at grades 2 through 6 to identify critical objectives essential for mathematics proficiency at each grade level. The researchers created 30 alternate forms at each grade level and conducted numerous pilot tests to refine the items and determine appropriate time limits for administration.

A total of 140 students in grades 2 through 4 participated in the study, completing weekly versions of the measures for 20 weeks. All students were enrolled in general education classrooms; about 8 percent of the students had been identified as having learning dis-abilities. The students’ general education

304. Foegen (2008).

305. Fuchs, Hamlett, and Fuchs (1998); Fuchs et al. (1999).

teachers administered the measures using standardized procedures, including admin-istration time limits of 6 to 8 minutes, de-pending on grade level.

The results of the study illustrate the types of data educators should consider to deter-mine if a mathematics progress monitor-ing measure is trustworthy. The authors report evidence of the reliability of the concepts and application measures by de-scribing internal consistency coefficients for students at each grade level (which ranged from .94 to .98). Concurrent crite-rion validity was examined by computing correlations between student scores on the concepts and applications general out-come measures and their scores on three subscales of the Comprehensive Test of Basic Skills (Computation, Concepts and Applications, and Total Math Battery). Results are reported for each subscale at each of the three grade levels, with coef-ficients ranging from .63 to .81. Consider-ing these results in the general context of mathematics progress monitoring mea-sures summarized above, teachers could feel confident that the concepts and ap-plications measures demonstrated strong levels of reliability and criterion validity in this study.

A final consideration is the degree to which the measures are sensitive to student growth. To explore this feature, the re-searchers completed a least-squares re-gression analysis between calendar days and scores on the progress monitoring measures; the scores were then converted to represent weekly rates of improvement. The results ranged from an average in-crease of .40 problems per week in grade 2 to .69 in grade 4. Together with the evi-dence of reliability and criterion validity, the mean growth rate data suggest that teachers can have confidence that students will show improvements in their scores on the measures as their mathematics learn-ing progresses.

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One factor not evident in the technical ad-equacy data in this study, but critical for teachers to consider, is the alignment be-tween the general outcome measure, the instructional curriculum, and expected learning outcomes. This study produced strong technical adequacy data when it was conducted in the early 1990s. Teach-ers considering alternative mathematics progress monitoring measures to rep-resent the instructional curriculum are advised to review the content of these measures in light of current learning ex-pectations for students at each grade level. Given changes in mathematics curricula over the past 10 to 15 years, it is impor-tant to evaluate the degree to which the measures continue to represent important mathematics outcomes.



The panel judged the level of evidence sup-porting this recommendation to be low. The panel found nine studies306 conducted with low-achieving or learning disabled students between grades 1 and 8 that met WWC stan-dards or met standards with reservations and included motivational strategies in an intervention. However, because only two of these studies investigated a motivational strategy in a tier 2 or tier 3 mathematics in-tervention as an independent variable, the panel concluded that there is low evidence to support the recommendation. The panel recommends this practice for students in tier 2 and tier 3 based both on our opinion

306. Fuchs et al. (2005); Fuchs, Fuchs, Craddock et al. (2008); Fuchs, Seethaler et al. (2008); Artus and Dyrek (1989); Fuchs et al. (2003b); Fuchs, Fuchs, Hamlett, Phillips et al. (1994); Fuchs, Fuchs, Finelli et al. (2006); Heller and Fantuzzo (1993); Schunk and Cox (1986).

and on the limited evidence base.307 The evidence base is described below.

Summary of evidence

Reinforce effort. The panel advocates re-inforcing or praising students for their ef-fort. Schunk and Cox (1986) examined the effects of providing effort-attributional feedback (e.g., “You’ve been working hard”) during subtraction instruction versus no effort feedback and found significant posi-tive effects on subtraction posttests in favor of providing effort feedback. This study, described in greater detail below, was one of two studies in the evidence base that examined a motivational strat-egy as an independent variable.

Reinforce engagement. The panel also rec-ommends reinforcing students for attending to and being engaged in lessons. In two of the studies, students received “points” for engagement and attentiveness as well as for accuracy.308 Accumulated points could be applied toward “purchasing” tangible rein-forcers. It is not possible to isolate the effects of reinforcing attentiveness in the studies. In Fuchs et al. (2005), both the treatment and comparison groups received reinforce-ment, and in Fuchs, Fuchs, Craddock et al. (2008), the contrast between reinforcement and no reinforcement was not reported. But the presence of reinforcers for attention and engagement in these two studies echoes the panel’s contention that providing reinforce-ment for attention is particularly important for students who are struggling.

307. The scope of this practice guide limited the evidence base for this recommendation to stud-ies that investigated mathematics interventions for students with mathematics difficulties and included motivational components. There is an extensive literature on motivational strategies outside the scope of this practice guide, and the panel acknowledges that there is considerable debate in that literature on the use of rewards as reinforcers.

308. Fuchs et al. (2005); Fuchs, Fuchs, Craddock et al. (2008).

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Consider rewarding accomplishments. The panel recommends that interventionists consider using rewards to acknowledge accurate work and possibly notifying par-ents when students demonstrate gains. In three of the studies, students were pro-vided prizes as tangible reinforcers for ac-curate mathematics problem solving.309 In both Fuchs et al. (2005) and Fuchs, Seetha-ler et al. (2008), students in tier 2 tutoring earned prizes for accuracy. In both stud-ies, the tier 2 intervention group demon-strated substantively important positive and sometimes significant gains relative to the students who remained in tier 1. But it is not possible to isolate the effects of the reinforcers from the provision of tier 2 tutoring. In Fuchs, Fuchs, Craddock et al. (2008), the authors note that the provision of “dollars” that could be exchanged for prizes was more effective than rewarding students with stickers alone. Because this was not the primary purpose of the study, the reporting of the evidence for that find-ing was not complete; therefore, a WWC re-view was not conducted for that finding.

In a fourth study, Heller and Fantuzzo (1993) examined the impacts of a parental involvement supplement to a mathematics intervention. The parental involvement component included parents providing rewards for student success as well as pa-rental involvement in the classroom. The performance of students who received the parental involvement component in addition to the school-based intervention significantly exceeded the performance of students in only the school-based inter-vention. Because the parental involvement component was multifaceted, it is not pos-sible to attribute the statistically signifi-cant positive effects to rewards alone.

Allow students to chart their progress and to set goals for improvement. Five studies included interventions in which students

309. Fuchs et al. (2005); Fuchs, Seethaler et al. (2008); Fuchs, Fuchs, Craddock et al. (2008).

graphed their progress and in some cases set goals for improvement on future as-sessments.310 One experimental study examined the effects of student graphing and goal setting as an independent vari-able and found substantively important positive nonsignificant effects in favor of students who graphed and set goals.311 In two studies, the interventions included graphing in both groups being compared; therefore, it was not possible to isolate the effects of this practice.312 In another two studies, students in the treatment groups graphed their progress as one component of multicomponent interventions.313 Al-though it is not possible to discern the ef-fect of graphing alone, in Artus and Dyrek (1989), the treatment group made margin-ally significant gains over the comparison group on a general mathematics assess-ment, and in Fuchs, Seethaler et al. (2008), there were substantively important posi-tive non-significant effects on fact retrieval in favor of the treatment group.

In summary, the evidence base for motiva-tional components in studies of students struggling with mathematics is limited. One study that met evidence standards demonstrated benefits for praising strug-gling students for their effort. Other stud-ies included reinforcement for attention, engagement, and accuracy. Because the ef-fects of these practices were not examined as independent variables, no inferences can be drawn about effectiveness based on these studies. Because this recommenda-tion is based primarily on the opinion of the panel, the level of evidence is identi-fied as low.

310. Artus and Dyrek (1989); Fuchs, Seethaler et al. (2008); Fuchs et al. (2003b); Fuchs, Fuchs, Hamlett, Phillips et al. (1994); Fuchs, Fuchs, Fi-nelli et al. (2006).

311. Fuchs et al. (2003b).

312. Fuchs, Fuchs, Hamlett, Phillips et al. (1994); Fuchs, Fuchs, Finelli et al. (2006).

313. Artus and Dyrek (1989); Fuchs, Seethaler et al. (2008).

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An example of a study that investigated a motivational component—Schunk and Cox (1986).

This study was conducted with 90 stu-dents in grades 6 through 8 classrooms in six schools in Texas. The mean age of the students was 13 years and 7 months, and all students were identified as having learning disabilities in mathematics.

Before the intervention, the students com-pleted a pretest assessment that consisted of 25 subtraction problems that required regrouping operations. After the interven-tion, a similar assessment of 25 subtrac-tion problems was completed as a post-test. A separate reliability assessment demonstrated that the two forms of the subtraction assessment were highly cor-related (r = .82).

Students were stratified by gender and school and then randomly assigned to one of nine experimental groups. In all groups, the students received instruction for solv-ing subtraction problems in 45-minute sessions conducted over six consecutive school days. For this recommendation,

we report only on the comparison be-tween three groups. One group (n = 30) received effort feedback in addition to per-formance feedback during the first three sessions. Another group (n = 30) received effort feedback in addition to performance feedback during the last three sessions. A third group (n = 30) did not receive ef-fort feedback (received only performance feedback).314 Effort feedback consisted of the proctor commenting to the student, “You’ve been working hard.” Students in both effort feedback groups received 15 statements of effort feedback across the entire intervention.

Results indicated significant positive ef-fects for effort feedback relative to the comparison group regardless of when the student received the effort feedback. These results suggest that effort feedback is ben-eficial for learning disabled students who may not otherwise recognize the causal link between effort and outcomes.

314. Other group distinctions were related to student verbalization and are described in the discussion of recommendation 3 (explicit instruction).

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™

™

Texas College and Career Readiness Standards

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Texas Higher Education Coordinating BoardDivision for P- Initiatives

College Readiness InitiativesPO Box

Austin, TX [email protected]

Texas Education AgencyDivision of Curriculum

College and Career Readiness Program N. Congress Avenue, Suite -

Austin, TX [email protected]


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MathematicsStandards

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MATHEMATICS STANDARDS

Mathematics as a Way of KnowingMathematics knowledge is essential to becoming a

productive citizen in today’s society. Many factors have increased the level of understanding of mathematics needed by the average adult. Our ever-changing world has become increasingly quantitative in nature. For example, in the physical sciences, social studies, and the business world, a widening array of phenomena is explained with numeric data presented visually in the form of charts and graphs that require interpretation. Mathematical reasoning is key to solving problems, formulating logical arguments, understanding quantitative features of various disciplines, critically analyzing media sources, and searching for patterns. Through mathematics, people become more able to make well-informed decisions by formulating

be viewed solely as a series of stand-alone courses or

source of cross-disciplinary knowledge that is essential for success in numerous areas of study.

Understanding and Using These Standards

The College and Career Readiness Standards (CCRS) were developed as a result of a collaborative effort between secondary and postsecondary faculty.

high school mathematics course titles or to endorse particular sequences. Students may encounter some of the content included in these standards at lower levels and should aim to meet these standards in high school.

These CCRS are designed to help students, parents,

content knowledge and academic skills necessary for college and career readiness. This knowledge enables all stakeholders to determine if the challenge level of any given mathematics course is appropriate to prepare students for college and careers. The CCRS are broad in nature, equipping students for general education college mathematics courses, but are not intended to encompass all skills necessary for students entering

Students who enter college having mastered these standards are likely to be successful in entry-level college mathematics courses and to be prepared for courses in related disciplines that require mathematical

mathematical knowledge and skills will be necessary, although adequate foundation for these future studies would be established.

area topics, while others address global topics. All are viewed as equally important to achieving the

college and career readiness. In addition, students must develop ways of thinking about mathematics. These key cognitive skills elevate mathematics from an exercise in rote memorization to a process of analysis and interpretation that enables the learner to work with a range of complex questions, topics, and issues. The standards contain frequent reference to these key cognitive skills, but always in the context of challenging and appropriate content knowledge. Mathematical thinking never occurs in a vacuum; it is always embedded in appropriate content.

“Students who enter college having mastered these

standards are likely to be successful in entry-level

college mathematics courses and to be prepared for

courses in related disciplines that require mathematical

profi ciency.”

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The use of technology is an instructional decision that facilitates the learning of mathematical concepts and processes. The Vertical Team decided to allow instructors to determine when and how to use technology based on their students’ needs, the instructional resources, and the learning expectation. The growing technological world we live in requires students to embrace technology and the constant changes it brings to daily life.

I. Numeric ReasoningA. Number representation

1. Compare real numbers.

B. Number operations1. Perform computations with real and complex

numbers.

C. Number sense and number concepts1. Use estimation to check for errors and

reasonableness of solutions.

II. Algebraic ReasoningA. Expressions and equations

1. Explain and differentiate between expressions and equations using words such as “solve,” “evaluate,” and “simplify.”

B. Manipulating expressions

concepts, procedures, and algorithms to combine, transform, and evaluate expressions (e.g., polynomials, radicals, rational expressions).

C. Solving equations, inequalities, and systems of equations

concepts, procedures, and algorithms to solve equations, inequalities, and systems of linear equations.

2. Explain the difference between the solution set of an equation and the solution set of an inequality.

D. Representations1. Interpret multiple representations of equations

and relationships.

2. Translate among multiple representations of equations and relationships.

III. Geometric ReasoningA. Figures and their properties

1. Identify and represent the features of plane and

properties.

3. Recognize and apply right triangle relationships including basic trigonometry.

B. Transformations and symmetry

3. Use congruence transformations and dilations to investigate congruence, similarity, and



The use of technology is an instructional decision that facilitates the learning of mathematical concepts and processes. The Vertical Team decided to allow instructors to determine when and how to use technology based on their students’ needs, the instructional resources, and the learning expectation. The growing technological world we live in requires students to embrace technology and the constant changes it brings to daily life.



B. Number operations1. Perform computations with real and complex

numbers.






concepts, procedures, and algorithms to combine, transform, and evaluate expressions (e.g., polynomials, radicals, rational expressions).


concepts, procedures, and algorithms to solve equations, inequalities, and systems of linear equations.


D. Representations1. Interpret multiple representations of equations

and relationships.



1. Identify and represent the features of plane and

properties.


B. Transformations and symmetry

3. Use congruence transformations and dilations to investigate congruence, similarity, and

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C. Connections between geometry and other mathematical content strands1. Make connections between geometry and

algebra.

2. Make connections between geometry, statistics, and probability.

3. Make connections between geometry and measurement.

D. Logic and reasoning in geometry

2. Understand that Euclidean geometry is an axiomatic system.

IV. Measurement ReasoningA. Measurement involving physical and natural

attributes1. Select or use the appropriate type of unit for

the attribute being measured.B. Systems of measurement

1. Convert from one measurement system to another.

2. Convert within a single measurement system.C. Measurement involving geometry and

algebra1. Find the perimeter and area of two-dimensional

2. Determine the surface area and volume of

Pythagorean Theorem, and basic trigonometry.D. Measurement involving statistics and

probability1. Compute and use measures of center and

spread to describe data.2. Apply probabilistic measures to practical

situations to make an informed decision.

V. Probabilistic ReasoningA. Counting principles

1. Determine the nature and the number of

B. Computation and interpretation of probabilities1. Compute and interpret the probability of an

event and its complement.

2. Compute and interpret the probability of conditional and compound events.

VI. Statistical ReasoningA. Data collection

1. Plan a study.

B. Describe data1. Determine types of data.

2. Select and apply appropriate visual representations of data.

3. Compute and describe summary statistics of data.

4. Describe patterns and departure from patterns in a set of data.

C. Read, analyze, interpret, and draw conclusions from data1. Make predictions and draw inferences using

summary statistics.

2. Analyze data sets using graphs and summary statistics.

3. Analyze relationships between paired data using spreadsheets, graphing calculators, or statistical software.

4. Recognize reliability of statistical results.

VII. FunctionsA. Recognition and representation of

functions1. Recognize whether a relation is a function.

2. Recognize and distinguish between different types of functions.

B. Analysis of functions1. Understand and analyze features of a function.

2. Algebraically construct and analyze new functions.

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C. Model real world situations with functions1. Apply known function models.

2. Develop a function to model a situation.

VIII. Problem Solving and Reasoning

A. Mathematical problem solving1. Analyze given information.

2. Formulate a plan or strategy.

3. Determine a solution.

4. Justify the solution.

5. Evaluate the problem-solving process.

B. Logical reasoning1. Develop and evaluate convincing arguments.

2. Use various types of reasoning.

C. Real world problem solving1. Formulate a solution to a real world situation

based on the solution to a mathematical problem.

2. Use a function to model a real world situation.


IX. Communication and Representation

A. Language, terms, and symbols of mathematics1. Use mathematical symbols, terminology, and

notation to represent given and unknown information in a problem.

2. Use mathematical language to represent and communicate the mathematical concepts in a problem.

3. Use mathematics as a language for reasoning, problem solving, making connections, and generalizing.

B. Interpretation of mathematical work1. Model and interpret mathematical ideas and

concepts using multiple representations.

2. Summarize and interpret mathematical information provided orally, visually, or in written form within the given context.

C. Presentation and representation of mathematical work1. Communicate mathematical ideas, reasoning,

and their implications using symbols, diagrams, graphs, and words.

2. Create and use representations to organize, record, and communicate mathematical ideas.

and arguments using precise mathematical language in written or oral communications.

X. ConnectionsA. Connections among the strands of

mathematics1. Connect and use multiple strands of

mathematics in situations and problems.

2. Connect mathematics to the study of other disciplines.

B. Connections of mathematics to nature, real world situations, and everyday life1. Use multiple representations to demonstrate

links between mathematical and real world situations.

2. Understand and use appropriate mathematical models in the natural, physical, and social sciences.

3. Know and understand the use of mathematics in a variety of careers and professions.

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Cross-Disciplinary StandardsCross-Disciplinary Cross-Disciplinary Standards

Cross-DisciplinaryStandards

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CROSS-DISCIPLINARY STANDARDS

Foundations of Learning and Knowing

Although the College and Career Readiness Standards (CCRS) are organized into four distinct disciplinary areas, English/language arts, mathematics, science, and social studies, there are elements that cut across one or more disciplines. In fact, some skill areas

the cross-cutting knowledge and skills that underlie and connect the four disciplinary areas. This important need has been addressed through the addition of a section of cross-disciplinary standards.

Think of cross-disciplinary standards as tools that college instructors in all areas use to challenge, engage,

They include key cognitive skills such as reasoning and problem solving, as well as foundational skills such as reading, writing, data analysis, and conducting research.

Many of these skills are also taught within the

are excellent examples. While the primary responsibility for developing reading and writing skills in secondary school resides within English/language arts courses,

and techniques in all of their courses. For example, they will write a lab report in a biology class or read primary source documents in a history class.

Academic and business leaders emphasize the importance of being able to apply these skills across a

21st century learning and work environments in which the cross-disciplinary skills are prerequisites to solving many of the most important problems students will encounter in college and the workplace. These problems increasingly require applying knowledge

base set of communication and analysis skills that span

content knowledge, but also need to be able to apply key cognitive skills to the academic tasks presented to them, most of which require much more than simple recall of factual knowledge. These cross-disciplinary standards

enable students to engage in deeper levels of thinking

students prepare for the transition from high school’s primary focus on acquiring content knowledge to a postsecondary environment in which complex cognitive skills are necessary to achieve deeper understanding.

Understanding and Using The Cross-Disciplinary Standards

The cross-disciplinary standards are organized

Foundational Skills. The Key Cognitive Skills specify intellectual behaviors that are prevalent in entry-level college courses. The list includes intellectual curiosity, reasoning, problem solving, academic behaviors, work habits, and academic integrity. Foundational Skills

transfer knowledge and apply it across the curriculum. These include reading, writing, conducting research, understanding and using data, and using technology.

The performance indicators found in the appendix illustrate how the cross-disciplinary standards are

least one performance indicator that could be applied

exemplify how the cross-disciplinary standards could be

I. Key Cognitive SkillsA. Intellectual curiosity

1. Engage in scholarly inquiry and dialogue.

2. Accept constructive criticism and revise personal views when valid evidence warrants.

B. Reasoning1. Consider arguments and conclusions of self

and others.

2. Construct well-reasoned arguments to explain

positions.

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CROSS-DISCIPLINARY STANDARDS

or lines of reasoning.4. Support or modify claims based on the results

of an inquiry.

C. Problem solving1. Analyze a situation to identify a problem to be

solved.2. Develop and apply multiple strategies to solve a

problem.3. Collect evidence and data systematically and

directly relate to solving a problem.

D. Academic behaviors1. Self-monitor learning needs and seek assistance

when needed.2. Use study habits necessary to manage academic

pursuits and requirements.3. Strive for accuracy and precision.4. Persevere to complete and master tasks.

E. Work habits1. Work independently.2. Work collaboratively.

F. Academic integrity1. Attribute ideas and information to source

materials and people.2. Evaluate sources for quality of content, validity,

credibility, and relevance.3. Include the ideas of others and the complexities

of the debate, issue, or problem.4. Understand and adhere to ethical codes of

conduct.

II. Foundational SkillsA. Reading across the curriculum

1. Use effective prereading strategies.2. Use a variety of strategies to understand the

meanings of new words.3. Identify the intended purpose and audience of

the text.4. Identify the key information and supporting

details.

5. Analyze textual information critically.6. Annotate, summarize, paraphrase, and outline

texts when appropriate.7. Adapt reading strategies according to structure

of texts.8. Connect reading to historical and current

events and personal interest.

B. Writing across the curriculum1. Write clearly and coherently using standard

writing conventions.2. Write in a variety of forms for various

audiences and purposes.3. Compose and revise drafts.

C. Research across the curriculum1. Understand which topics or questions are to be

investigated. 2. Explore a research topic.

research and devise a timeline for completing work.

4. Evaluate the validity and reliability of sources.5. Synthesize and organize information effectively.6. Design and present an effective product.7. Integrate source material.

D. Use of data1. Identify patterns or departures from patterns

among data.2. Use statistical and probabilistic skills necessary

for planning an investigation and collecting, analyzing, and interpreting data.

3. Present analyzed data and communicate

E. Technology1. Use technology to gather information.2. Use technology to organize, manage, and

analyze information.3. Use technology to communicate and display

4. Use technology appropriately.

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Texas College and Career Readiness Standards a

MathematicsStandardswith performance indicators



a. Classify numbers as natural, whole, integers, rational, irrational, real, imaginary, and/or complex.

b. Use and apply the relative magnitude of real numbers by using inequality symbols to compare them and locate them on a number line.

c. Order real numbers with and without a calculator using relationships involving decimals, rationals, exponents, and radicals.

notation.

numbers.

a. State the standard form used to represent complex numbers and describe their real and imaginary parts.

b. Represent in and square roots of negative numbers as complex numbers.

c. Understand that to solve certain problems and equations, number systems need to be extended from whole numbers to the set of all integers (positive, negative, zero), from integers to rational numbers, from rational numbers to real numbers (rational and irrational numbers), and from real

give examples of each of these types of numbers.

B. Number operations1. Perform computations with real and

complex numbers.

a. Add, subtract, multiply, and divide real numbers accurately, including irrational numbers, numbers with exponents, and absolute value.

b. Transform numerical expressions using

property), order of operations, and properties of exponents.

c. Solve problems involving rational numbers, ratios, percents, and proportions in context of the situation.

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Texas College and Career Readiness Standardsa

MATHEMATICS STANDARDS with performance indicators

d. Calculate the sum, difference, product, and quotient of two complex numbers and express the result in standard form.



a. Identify the most reasonable solution for a given problem from a list of possible

b. Use mental estimates to detect potential errors when using a calculator.

c. Justify the need for an exact answer or an estimate in a given problem (e.g., doing taxes vs. determining amount of paint needed for a room).



represents.

b. Distinguish among and apply different

represent a conditional statement, and represent an identity.


properties, concepts, procedures, and algorithms to combine, transform, and evaluate expressions (e.g., polynomials, radicals, rational expressions).

commutative, associative, distributive) and order of operations to transform expressions to equivalent expressions.

order of operations to evaluate variable expressions when given the value of the variables.

c. Explain why the algorithms and procedures used to transform algebraic expressions are valid.


properties, concepts, procedures, and algorithms to solve equations, inequalities, and systems of linear equations.

a. Solve equations and inequalities in one variable (e.g., numerical solutions, including those involving absolute value, radical, rational, exponential, and logarithmic).

b. Solve for any variable in an equation or inequality that has two or more variables (e.g., literal equations).

properties to solve an equation by constructing a sequence of equivalent equations.

d. Use the elimination, substitution, and/or graphing method to solve a linear system of equations with two variables.

e. Use technology when using matrices to solve linear systems with two or three variables.


a. Represent the solution set of an equation or inequality in various ways (e.g. set notation, interval notation, graphical representation, including shading).

b. Understand that the real solution to an equation can be represented as the x-coordinate of the point of intersection of two graphs.

c. Understand the relationship between a solution of a system of two linear equations with two variables and the graphs of the corresponding lines.

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d. Graph a function and understand the relationship between its real zeros, roots, and the x-intercepts of its graph.

D. Representations1. Interpret multiple representations of

equations and relationships.

a. Interpret graphical representations of equations.

b. Understand how variables can be used to express generalizations and represent situations.

c. Recognize the solution(s) to an equation from a table of values.

d. Describe numerical patterns using algebraic expressions and equations in closed or recursive forms, such as arithmetic sequences.


a. Explain the common information presented in multiple representations of a relationship.

b. Translate one given representation to another representation (e.g., tabular to graphic, graphic to symbolic).

c. Use multiple representations to determine rate of change.

d. Determine if a relationship given in graphical, tabular, or symbolic form is linear or nonlinear.


1. Identify and represent the features of plane

a. Construct and use drawings, models, and coordinate representations of plane and

by hand and using technology.

such as prisms, pyramids, cylinders, and cones.

c. Describe and use cross-sections and nets

d. Describe the conic sections as intersections of a plane with a cone.

e. Recognize and describe orthographic (top, front, side) and isometric views of three-

2. Make, test, and use conjectures about one-,

their properties.

a. Develop and verify attributes of lines and parts of lines in a plane and in space: parallel, intersecting, perpendicular, and skew lines; and angle relationships associated with transversals on parallel lines.

b. Develop and verify angle relationships: vertical, complementary, supplementary, angles on parallel lines, angle-side relations in a triangle, interior/exterior angles on polygons, and angles on circles.

c. Develop, verify, and extend properties of circles, including properties of angles, arcs, chords, tangents, secants, and spheres.

d. Develop and verify properties of triangles and quadrilaterals (e.g., triangle congruence conditions, properties of a parallelogram).

e. Develop and verify properties of parts of prisms, cylinders, pyramids, and cones.

solve problems.


a. Apply the Pythagorean Theorem and its converse to solve real-life situations in two and three dimensions.

b. Apply Pythagorean triples and special right triangle relationships to solve problems.

c. Solve right triangle situations using sine, cosine, and tangent.

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B. Transformations and symmetry1. Identify and apply transformations to

a. Identify whether a transformation is a

b. Find the image or pre-image of a

rotation) or composition of these transformations in coordinate and non-coordinate plane settings.

c. Find the image or pre-image of a

composition of dilations in coordinate and non-coordinate plane settings.

d. Use transformations and compositions of

sum of the three angles inside any triangle is 180 degrees).

a. Identify and distinguish between

b. Identify congruent corresponding parts

symmetry.

3. Use congruence transformations and dilations to investigate congruence,

a. Use congruence transformations to

identify congruent corresponding parts.b. Use dilations and scale factors to

missing image or pre-image dimensions.c. Identify symmetries in design situations

and describe transformations used to create the symmetry and design (e.g., tiling problems).

C. Connections between geometry and other mathematical content strands1. Make connections between geometry and

algebra.

a. Describe lines in the coordinate plane using slope-intercept and point-slope form.

b. Use slopes to describe the steepness and direction of lines in the coordinate plane and to determine if lines are parallel, perpendicular, or neither.

c. Relate geometric and algebraic representations of lines, segments, simple curves, and conic sections [e.g., describe algebraically a circle centered at (h, k) with radius (r)].

of triangles and quadrilaterals using coordinate geometry.

e. Relate the number of solutions to a system of equations of lines to the number of intersections of two or more graphs.

2. Make connections between geometry, statistics, and probability.

a. Compute probabilities using lengths of segments or areas of regions representing desired outcomes.

b. Construct a trend line or a regression line for a scatter plot and use it to make predictions.

3. Make connections between geometry and measurement.

a. Determine perimeter and area of two-

and volume of three-dimensional

formulas.b. Find the measures of the lengths and

surface areas, and volumes of similar solids.

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c. Find arc length and sector area for a given central angle on a circle.

D. Logic and reasoning in geometry1. Make and validate geometric conjectures.

a. Use drawings, manipulatives (e.g., paper folding, transformations) and constructions (e.g., compass/straight-edge, computer graphing utility) to investigate

b. Use counterexamples to verify that a

c. Give a logical argument in a variety of formats to verify that a geometric

d. Use a conditional statement to describe a

investigate the validity of the statement’s converse, inverse, and contrapositive.

e. Make the connection between a biconditional statement and a true conditional statement with a true converse.

2. Understand that Euclidean geometry is an axiomatic system.

a. Distinguish among theorems, properties,

b. Understand that non-Euclidean geometries exist.

IV. Measurement ReasoningA. Measurement involving physical and natural

attributes1. Select or use the appropriate type of unit for

the attribute being measured.

a. Determine appropriate units of

measured in a given situation (e.g., unit analysis, degree, or radian measure of an angle.)

b. Select and accurately use an appropriate tool to make measurements.

determine the accuracy of a measurement in problem situations.

d. Use the appropriate level of precision when providing solutions to measurement problems.

e. Know when to estimate and approximate measurements for given problem situations.

B. Systems of measurement1. Convert from one measurement system to

another.

a. Convert between basic units of measurement from one system to another system (e.g., inches to centimeters, kilometers to miles, pounds to kilograms).

2. Convert within a single measurement system.

a. Convert between basic units of measurement within a system (e.g., inches to feet, square inches to square feet, grams to milligrams).

C. Measurement involving geometry and algebra1. Find the perimeter and area of

a. Describe the difference between perimeter

the units of measurement used in their calculation.

b. Solve problems involving perimeter and area of two-dimensional simple and

dimensions (e.g., triangles, quadrilaterals, circles).

c. Solve problems involving the distance between two points in the coordinate plane and make algebraic and geometric connections.

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2. Determine the surface area and volume of

a. Describe the difference between surface area and volume of three-dimensional

measurement used in their calculation.

b. Solve problems involving surface area and volume of three-dimensional simple and

dimensions, including prisms, pyramids, cylinders, cones, and spheres.

3. Determine indirect measurements of

basic trigonometry.

a. Determine how changes in dimension affect the perimeter, area, and volume of

b. Solve problems using proportional relationships in similar two-dimensional

determine unknown measurements.c. Determine unknown sides and angles

in a right triangle using the Pythagorean Theorem and basic trigonometry.

D. Measurement involving statistics and probability1. Compute and use measures of center and

spread to describe data.

of center (e.g., mean, median, mode)

information.

of variation (e.g., range, IQR, percentiles, variance, standard deviation) based on the

c. Calculate weighted averages, indices, and ratings.

2. Apply probabilistic measures to practical situations to make an informed decision.

a. Justify decisions made from probability measures from a set of data.

b. Interpret given probability measures in a problem.

c. Use and interpret a normal distribution as a mathematical model of measurement for summarizing some sets of data.

V. Probabilistic ReasoningA. Counting principles

1. Determine the nature and the number of

a. Make lists, tables, and tree diagrams to represent all possible outcomes in

b. Determine the number of ways an event may occur using combination and permutation formulas and the Fundamental Counting Principle.

B. Computation and interpretation of probabilities1. Compute and interpret the probability of an

event and its complement.

a. Conduct an experiment or simulation to compute the empirical probability of an event and its complement.

b. Compute and interpret the theoretical probability of a simple event and its complement.

c. Compare the empirical and theoretical probabilities of an event (e.g., experimental probabilities converge to theoretical probability as the number of trials increases).

2. Compute and interpret the probability of conditional and compound events.

a. Distinguish between independent and dependent events.

b. Explain the meaning of conditional probability and know when to use it.

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c. Compute conditional probability.

d. Compute the probability of compound events using tree diagrams, tables, and other methods.

e. Compute the probability for dependent or independent compound events.

VI. Statistical ReasoningA. Data collection

1. Plan a study.

a. Determine question(s) that can be answered with data.

b. Explain the difference between observational and experimental studies.

c. Design and employ a plan of study to collect appropriate data.

d. Use a variety of sampling methods (e.g., census, systematic sampling, random vs. non-random sampling).

e. Identify sampling techniques used in our world (e.g., political polls, medical studies) and determine possible sources of bias.

f. Compare and contrast data variability using different sampling methods.

B. Describe data1. Determine types of data.

a. Recognize and describe the differences between quantitative and qualitative data.

b. Recognize and describe univariate and bivariate data.

2. Select and apply appropriate visual representations of data.

a. Organize and construct graphical displays of data (e.g., line plots, bar graphs, histograms, box plots, scatter plots) to describe the distribution of data.

b. Read and interpret graphical displays of data.

3. Compute and describe summary statistics of data.

a. Calculate, describe, and use the appropriate measure of center (e.g., mean, median, mode) and spread (e.g., range, IQR, percentiles, variance, standard deviation).

b. Describe the effect of outliers on summary statistics.

4. Describe patterns and departure from patterns in a set of data.

a. Describe any natural variability evident in the results within the context of the situation.

induced variability within the context of the situation.

C. Read, analyze, interpret, and draw conclusions from data1. Make predictions and draw inferences using

summary statistics.

a. Make a prediction about long-run behavior (e.g., coin toss).

b. Draw conclusions from analyzing a set of data.

2. Analyze data sets using graphs and summary statistics.

a. Analyze and compare distributions by describing similarities and differences of centers and spreads within and between data sets.

b. Analyze and describe similarities and differences by comparing graphical distributions (e.g., parallel box plots, back-to-back stem-leaf plots) within and between data sets.

3. Analyze relationships between paired data using spreadsheets, graphing calculators, or statistical software.

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a. Describe relationship and trend of paired data observed from scatter plots in the context of the situation.

b. Choose an appropriate linear or non-linear

on graphical analysis.

c. Make a prediction using the appropriate regression model and describe any limitations to the calculated prediction.

4. Recognize reliability of statistical results.

a. Evaluate media reports by analyzing the study design, data source, graphical representation of data, and analyzed data results reported (or not reported).

b. Describe generalizations and limitations of results from observational studies, experiments, and surveys.

c. Identify and explain misleading uses of data.

d. Describe the reliability of statistical results from a set of data.

VII. FunctionsA. Recognition and representation of

functions1. Recognize whether a relation is a function.

a. Determine if a relationship given in tabular, graphic, symbolic, or verbal form

2. Recognize and distinguish between different types of functions.

a. Recognize general forms of linear, quadratic, rational, absolute value, square root, exponential, and logarithmic functions, and other advanced forms such as trigonometric or power functions.

b. Recognize the distinction between a discrete and a continuous function.

c. Recognize a sequence as a function whose domain is a set of whole numbers.

d. Recognize computations (e.g., sums, products, GCF, LCM, mean, surface area) as evaluating a function with two or more inputs and one output.

e. Recognize a plane geometric transformation as evaluating a function with two inputs and two outputs.

B. Analysis of functions1. Understand and analyze features of a

function.

a. Understand functional notation and

its domain.

b. Determine the domain and range of a

graph, symbols, or verbal description.

c. Approximate or determine the x- and y-values of a function given in tabular, graphical, symbolic, or verbal form.

d. Determine and explain if a function,

graphical, or symbolic form, is one-to-one.

2. Algebraically construct and analyze new functions.

a. Determine the domain and range of a combination or composition of two functions.

b. Formulate the composition of two functions.

c. Apply basic transformations to parent functions [e.g., af(x), f(x)+b, f(x+c)] and interpret the results verbally and graphically.

d. Analyze the effects of parameter changes of basic functions, [e.g., f(x)=mx+b, where m and/or b changes].

functions (e.g., step functions).

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f. Determine the inverse function of a given function in tabular, symbolic, or graphical form, if it exists (e.g., the inverse of an exponential function is a logarithmic function).

g. Use properties of inverse functions to solve problems (e.g., inverse trigonometric

C. Model real world situations with functions1. Apply known function models.

a. Apply a linear model for a situation represented by a constant rate of change.

b. Apply given quadratic models to solve

motion).

c. Apply exponential models (e.g., compound interest, growth and decay models) to solve problems.

d. Apply proportional or inverse variation models to solve problems.

e. Recognize and solve problems that can be modeled using a system of two equations in two variables, such as mixture problems.

2. Develop a function to model a situation.

a. Analyze a situation algebraically or graphically and determine if the relationship suggests a linear trend.

b. Use technology to determine a linear regression model for a given situation.

c. Identify real world situations that can be modeled by functions (e.g., situations in science, business, economics).

VIII. Problem Solving and Reasoning

A. Mathematical problem solving1. Analyze given information.

a. Extract needed facts and relationships from given information.

b. Identify what is known, not known, and what one wants to know in a problem.

c. Distinguish relevant from irrelevant information in a given situation.

d. Determine the problem(s) to be solved.e. Identify additional information needed to

reach a solution.

2. Formulate a plan or strategy.

a. Select or develop an appropriate problem-solving strategy (e.g., drawing a picture, looking for a pattern, systematic guessing and checking, acting it out, making a table, working a simpler problem, working backwards).

b. Identify needed algorithms or formulas.c. Determine the nature of a possible

solution and the degree of precision required.

3. Determine a solution.

b. Find an approximate solution with or without technology.

c. Identify and solve sub-problems.d. Use multiple representations (e.g., analytic,

numerical, verbal, and graphical) to support a solution.

4. Justify the solution.

a. Provide a clear explanation of the reasoning used to determine a solution.

b. Evaluate the reasonableness of the solution in the context of the original problem.

c. Verify a general solution in special cases.

d. Review and check strategies and calculations, using an alternative approach when possible.

e. Demonstrate an understanding of the mathematical ideas behind the steps of a

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and use mathematical knowledge to evaluate its effectiveness.

b. Recognize that a mathematical problem can be solved in a variety of ways.

c. Consider extensions and generalizations of the problem, process, or solution.

B. Logical reasoning1. Develop and evaluate convincing

arguments.

b. Use counterexamples to refute

c. Determine the validity of a conditional statement, its converse, its inverse, and its contrapositive.

2. Use various types of reasoning.

a. Use inductive reasoning to formulate a

b. Use deductive reasoning to prove a

c. Use geometric and visual reasoning.

d. Use multiple representations (e.g., analytic, numerical, verbal, graphical) to support an argument.

C. Real world problem solving1. Formulate a solution to a real world

situation based on the solution to a mathematical problem.

a. Make simplifying assumptions about a real world situation to formulate and solve an idealized mathematical problem.

b. Convert given information into an appropriate mathematical model.

c. Interpret results of the mathematical problem in terms of the original real world situation.

2. Use a function to model a real world situation.

a. Choose a function suitable for modeling a real world situation presented using words or data.

b. Determine and interpret the meaning of rates of change, intercepts, zeros, extrema, and trends.

c. Use an appropriate linear or non-linear function (e.g., quadratic, exponential).

d. Use a sequence expressed in recursive or closed form.


a. Evaluate a real world solution for accuracy and effectiveness.

b. Compare and analyze various methods for solving a real world problem.

IX. Communication and Representation

A. Language, terms, and symbols of mathematics1. Use mathematical symbols, terminology,

and notation to represent given and unknown information in a problem.

a. Use variables to represent quantities in contextual situations.

b. Analyze problem situations and represent them using algebraic expressions and equations.

c. Use and understand the many ways an

or formula; to represent an identity; to express a conditional equation; to identify constant and variable terms in expressions, equations, and inequalities).

d. Understand and use interval, set, and function notation.

e. Understand that certain symbols and words can have multiple meanings [e.g., (1, 2) can represent a point or an interval].

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2. Use mathematical language to represent and communicate the mathematical concepts in a problem.

a. Represent information in a problem using algebraic expressions, equations, and inequalities.

b. Recognize contextual problems represented by linear and non-linear models.

3. Use mathematics as a language for reasoning, problem solving, making connections, and generalizing.

a. Use inductive and deductive reasoning to reach valid conclusions.

b. Write the converse, inverse, and contra-positive of any given conditional statement.

B. Interpretation of mathematical work1. Model and interpret mathematical

ideas and concepts using multiple representations.

a. Make tables of inputs and outputs for mathematical relations/functions.

b. Write symbolic representations for a verbal description of a relationship.

c. Construct visual representations (e.g., a graph) of relationships.

d. Describe orally or in written format the behavior of a mathematical idea using graphs, diagrams, tables, and algebraic representations.

e. Represent inequalities using graphs, interval notation, and set notation.

f. Use multiple representations of rate of change.

2. Summarize and interpret mathematical information provided orally, visually, or in written form within the given context.

a. Interpret mathematical information in an article from a media source.

b. Summarize mathematical information given orally and visually in a media report.

C. Presentation and representation of mathematical work

1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, graphs, and words.

a. Communicate ideas mathematically using symbols (e.g., equal signs, parentheses, subscripts, superscripts, order relations, set notation).

b. Develop geometric models to represent concepts and relationships (e.g., scatter plots).

c. Recognize and explain the meaning of information presented using mathematical notation.

2. Create and use representations to organize, record, and communicate mathematical ideas.

a. Use Venn diagrams to represent sets of real numbers, surveys, and other set relationships.

b. Show solutions of equations and inequalities, and solutions of systems of equations and inequalities, using the real number line and rectangular coordinate system.

c. Construct and use graphic organizers (e.g., tables, bubble maps, Venn diagrams, tree diagrams).

3. Explain, display, or justify mathematical ideas and arguments using precise mathematical language in written or oral communications.

a. Explain reasoning in both oral and written forms using notation, terminology, and logic.

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b. Communicate reasons associated with performing steps in algebraic methods (e.g., explaining why a quadratic equation

when solving by factoring).c. Identify units associated with any variables

and constants used in a problem solution.




a. Represent a geometric two-dimensional

using a set of equations or inequalities.b. Connect the concepts of ratios, rates,

proportions, and percents (e.g., show slope as constant rate of change using similar triangles).

c. Compare and contrast different mathematical concepts and procedures that could be used to complete a particular task.

d. Combine appropriate numeric, algebraic, geometric, and statistical/probabilistic methods to solve a given problem.


a. Use mathematical models to solve problems in areas such as science, business, and economics.

b. Use applications of mathematics (e.g., carbon dating, exponential population growth, amortization tables).

c. Use geometric concepts and properties to

architecture.



a. Model a given real world situation using an appropriate combination of sketches, graphs, and algebraic expressions.

b. Describe a given real world situation in algebraic terms, use that description to produce a geometric description, and vice-versa.

c. Connect mathematically created tables,

situations (e.g., download data from the Internet).


a. Identify mathematical sequences, ratios, and patterns in nature (e.g., Fibonacci sequence, golden ratio).

b. Explain the importance of margin of error in results of surveys.

c. Apply known mathematical relations (e.g., Ohm’s Law, Hardy-Weinberg Law, rule for continuously compounded interest) to solve real world problems.


a. Identify mathematics used in several careers and professions.

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b. Communicate reasons associated with performing steps in algebraic methods (e.g., explaining why a quadratic equation

when solving by factoring).c. Identify units associated with any variables

and constants used in a problem solution.




a. Represent a geometric two-dimensional

using a set of equations or inequalities.b. Connect the concepts of ratios, rates,

proportions, and percents (e.g., show slope as constant rate of change using similar triangles).

c. Compare and contrast different mathematical concepts and procedures that could be used to complete a particular task.

d. Combine appropriate numeric, algebraic, geometric, and statistical/probabilistic methods to solve a given problem.


a. Use mathematical models to solve problems in areas such as science, business, and economics.

b. Use applications of mathematics (e.g., carbon dating, exponential population growth, amortization tables).

c. Use geometric concepts and properties to

architecture.



a. Model a given real world situation using an appropriate combination of sketches, graphs, and algebraic expressions.

b. Describe a given real world situation in algebraic terms, use that description to produce a geometric description, and vice-versa.

c. Connect mathematically created tables,

situations (e.g., download data from the Internet).


a. Identify mathematical sequences, ratios, and patterns in nature (e.g., Fibonacci sequence, golden ratio).

b. Explain the importance of margin of error in results of surveys.

c. Apply known mathematical relations (e.g., Ohm’s Law, Hardy-Weinberg Law, rule for continuously compounded interest) to solve real world problems.


a. Identify mathematics used in several careers and professions.

b. Identify several careers or professions that

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Cross-DisciplinaryStandardswith performance indicators

I. Key Cognitive SkillsA. Intellectual curiosity

1. Engage in scholarly inquiry and dialogue.

a. Identify what is known, not known, and what one wants to know in a problem.

b. Conduct investigations and observations.

c. Cite examples or illustrations in which a clear-cut answer cannot be reached.

2. Accept constructive criticism and revise personal views when valid evidence warrants.

a. Articulate a point of view and provide

b. Demonstrate willingness to take intellectual risks by investigating novel, controversial, or unpopular opinions or conclusions.

c. Examine alternative points of view, taking different roles to defend, oppose, and remain neutral on issues.

unexplained phenomena.

B. Reasoning1. Consider arguments and conclusions of self

and others.

a. Know and apply logic to analyze patterns and descriptions and to evaluate conclusions.

b. Cite valid examples or illustrations that support the conclusions.

c. Question whether the claims and conclusions of self and others are supported by evidence.

d. Identify counter examples to disprove a conclusion.

2. Construct well-reasoned arguments to explain phenomena, validate conjectures, or support positions.

a. Participate in a debate that is based on facts and has a logical structure.

b. Construct a visual presentation, including hypothesis, data, results, and conclusion.

c. Write a paper that addresses counter-arguments to advocated positions.

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CROSS-DISCIPLINARY STANDARDS with performance indicators


d. Recognize and apply techniques of

reliability of information.e. Organize an argument separating fact

from opinion.

3. Gather evidence to support arguments,

a. Use different kinds of data (e.g., case studies, statistics, surveys, documents) to support an argument.

b. Evaluate evidence in terms of quality and quantity.

c. Describe limitations of data collection methods.

4. Support or modify claims based on the results of an inquiry.

response to inquiry.b. Review and check strategies and

calculations, using alternative approaches when possible.

C. Problem solving1. Analyze a situation to identify a problem to

be solved.

a. Represent and/or restate the problem in one or more ways (e.g., graph, table, equation), showing recognition

parameters.b. Break complex problems into component

parts that can be analyzed and solved separately.

c. Apply previously learned knowledge to new situations.

d. Analyze a media report, identify any misuse of statistics, and suggest ways to more accurately depict this information.

2. Develop and apply multiple strategies to solve a problem.

a. Use a range of standard methods, devices, techniques, and strategies to gather and analyze information.

areas to solve a given problem.

3. Collect evidence and data systematically and directly relate to solving a problem.

a. Use general and specialized reference works and databases to locate sources.

b. Collect evidence and data directly related to solving the problem and eliminate irrelevant information.

c. Produce charts, graphs, and diagrams accurately, including scale, labeling, units, and organization.

d. Present the collected data visually, describe the data collection procedure, and defend choosing that procedure over other possibilities.

D. Academic behaviors1. Self-monitor learning needs and seek

assistance when needed.

a. Ask questions to check for understanding or to clarify information.

b. Use a systematic method for recording, storing, and organizing materials and resources; avoid haphazard or messy accumulation of information.

2. Use study habits necessary to manage academic pursuits and requirements.

a. Manage time effectively to complete tasks on time.

b. Demonstrate accurate note-taking.c. Use the appropriate level of detail

necessary to complete an assigned task.d. Balance academic and non-academic

activities to successfully participate in both.

3. Strive for accuracy and precision.

a. Collect and report experimental data carefully and correctly.

b. Produce charts, graphs, and diagrams accurately, including scale, labeling, units, and organization.

c. Eliminate irrelevant information from an assignment.

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4. Persevere to complete and master tasks.

a. Persevere until a task is completed by working even when faced with uncertainty or open-ended assignments.

b. Seek assistance when needed to complete the assignment.

c. Recognize when a task is completed.

E. Work habits1. Work independently.

and complete it with minimal supervision, seeking assistance accordingly.

b. Follow directions or procedures independently.

c. Complete assignments outside the classroom setting in a timely manner.

2. Work collaboratively.

a. Work collaboratively with students from various cultural and ethnic backgrounds.

b. Distinguish between situations where collaborative work is appropriate and where it is not.

c. Work in small groups to investigate a problem or conduct an experiment.

F. Academic integrity1. Attribute ideas and information to source

materials and people.

a. Document the work of others, giving credit where credit is due and never claim credit for work that is not one’s own.

b. Use standard bibliographic and reference citation formats, choosing the style

audience.

consequences of academic dishonesty.

2. Evaluate sources for quality of content, validity, credibility, and relevance.

a. Verify validity of a source within a submitted work.

b. Compare and contrast coverage of a single topic from multiple media sources.

3. Include the ideas of others and the complexities of the debate, issue, or problem.

a. Present multiple perspectives of an issue.b. Represent accurately the data, conclusions,

or opinions of others.

4. Understand and adhere to ethical codes of conduct.

a. Follow copyright laws and restrictions.b. Use technology responsibly (e.g., avoiding

malice, misrepresentation, or misleading use of information).

II. Foundational SkillsA. Reading across the curriculum

1. Use effective prereading strategies.

a. Use the title, knowledge of the author, and place of publication to make predictions about a text.

b. Use a table of contents to preview a text and understand its design.

c. Scan headline sections or other division markers, graphics, or sidebars to form an overview of a text.

2. Use a variety of strategies to understand the meanings of new words.

examples, comparison, contrast, cause and effect, and details provided in surrounding text.

b. Consult references (e.g., dictionary, thesaurus) effectively.

symbols).

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3. Identify the intended purpose and audience of the text.

a. Predict purpose and audience of a text based on the title, preface, and other features of a text.

b. Explain how the language of an effective text targets an intended audience.

c. Explain the importance of a technical

4. Identify the key information and supporting details.

a. Outline a chapter of an informational text.

use graphic organizers (e.g., concept maps, diagrams) to organize ideas and concepts in a visual manner.

minor ideas.

5. Analyze textual information critically.

a. Identify faulty premises in an argument.

b. Identify stated and implied assumptions.

c. Identify conclusions unsupported by

d. Use inductive and deductive reasoning.

e. Draw conclusions based on evidence, support, or data through logical reasoning.

f. Compare a primary source and an interpretation in a textbook.

6. Annotate, summarize, paraphrase, and outline texts when appropriate.

a. Outline an informational or literary text.

b. Annotate text for comprehension and analysis.

c. Summarize an article to demonstrate comprehension.

7. Adapt reading strategies according to structure of texts.

a. Identify a variety of textual forms and genres (e.g., long and short texts) and adapt reading strategies accordingly.

b. List strategies to use during reading, including:

information the text is likely to contain.

questioning.

paraphrasing, information in glossaries) to re-examine the text if comprehension fails.

information gained.

c. Explain how form or genre communicates meaning.

8. Connect reading to historical and current events and personal interest.

a. Locate an article or source that relates to a class topic and explain the relevance.

B. Writing across the curriculum1. Write clearly and coherently using standard

writing conventions.

audience to achieve that purpose.

b. Craft a thesis statement that articulates a position and list relevant evidence and examples in logical groupings.

c. Use symbols, diagrams, graphs, and words to communicate ideas.

d. Use appropriate terminology and data expression to communicate information in a concise manner.

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e. Use a variety of reference guides for citation conventions, grammar, mechanics, and punctuation.

2. Write in a variety of forms for various audiences and purposes.

a. Present an argument supported by relevant evidence, examples, and counterarguments.

article or report, extracting in brief form the pertinent information.

c. Evaluate articles by analyzing the study design, data source, graphical representation of data, and analyzed data results reported (or not reported).

selected to conduct research or solve a problem.

e. Write accurate and understandable lab reports and technical documents.

3. Compose and revise drafts.

a. Submit a writing assignment to be proofread by a teacher, parent, or other student. Revise the paper, incorporating constructive criticism when appropriate.

b. Edit text for correct spelling, capitalization, and punctuation.

c. Edit for appropriate tense and voice.d. Edit for correct word use.e. Use a variety of reference guides for

citation conventions, grammar, mechanics, and punctuation.

has few or no grammatical or spelling errors.

C. Research across the curriculum1. Understand which topics or questions are to

be investigated.

a. Formulate research questions.b. Use strategies like those in the writing

process to generate questions and areas to pursue.

c. Consult previous studies or conduct interviews with experts to identify questions central to a research topic.

d. Propose explicit, testable hypotheses, using the “if ..., then ...” format.

2. Explore a research topic.

a. Produce an annotated list of sources consulted, differentiating among primary, secondary, and other sources and explain their relevance to the research topic.

or questions on a research topic.c. Plan an investigative study.d. Explain reasons for valid competing points

of view on a given topic.

research and devise a timeline for completing work.

a. Gather information from a variety of relevant sources.

b. Use general and specialized reference works and databases to locate sources.

c. Locate electronic sources, when appropriate, using advanced search strategies.

d. Select an appropriate range of source materials.

e. Analyze a wide range of sources, including technical texts, primary and secondary

interdisciplinary research when appropriate. f. Design and carry out hands-on

experimental investigations, choosing appropriate apparatuses, identifying controls and variables, tentatively predicting the outcome of the procedures, and evaluating whether actual results agree with predicted results.

g. Use numerical and mathematical tools such as software, including databases, spreadsheets, and other tools, in investigations and explanations.

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4. Evaluate the validity and reliability of sources.

a. State explicitly characteristics or identifying features that indicate accuracy or reliability of sources, to determine whether sources are biased, incomplete, or otherwise unreliable.

b. Follow a set of criteria to determine the validity and reliability of sources.

c. Identify claims found in one or more of the sources that require support or

validity.

d. Evaluate the data presented in graphics, tables, charts, and maps when appropriate to the topic.

5. Synthesize and organize information effectively.

a. Select quotations and evidence that support the thesis.

b. Determine what evidence best supports conclusions.

c. Use well-organized strategies to collect and organize information gathered.

d. Determine the best order for presenting evidence that supports conclusions.

6. Design and present an effective product.

a. Determine the best order for presenting

b. Design a report using features such as headings and graphics appropriate to the writing task.

appropriate to the assignment.

7. Integrate source material.

a. Integrate source material into text by a combination of accurately summarizing, paraphrasing, and quoting.

b. Balance use of source material with relevant explanations.

c. Use source material ethically.

d. Understand and avoid all types of plagiarism.

a. Use appropriate media for presentation of research results.

b. Document sources using a standard

D. Use of data1. Identify patterns or departures from

patterns among data.

a. Identify patterns from multiple representations of data such as graphical and tabular forms.

b. Review current news events and evaluate possible connections (e.g., linking economic data with political events).

2. Use statistical and probabilistic skills necessary for planning an investigation and collecting, analyzing, and interpreting data.

a. Create representations of data (e.g., data tables, correctly labeled and scaled graphs, narrative descriptions).

b. Evaluate a given published report for missing information and misuse of data.

3. Present analyzed data and communicate

a. Compose a written document detailing a

b. Use appropriate visuals and statistical

audience.

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E. Technology1. Use technology to gather information.

a. Use the Internet or other appropriate technologies to post survey questions on an assigned topic.

b. Use devices to measure physical properties.

c. Use online databases to access scholarly work on an assigned research topic.

2. Use technology to organize, manage, and analyze information.

a. Use data analysis software to analyze survey results.

b. Use spreadsheets to manage and organize statistical data.

3. Use technology to communicate and

manner.

a. Create spreadsheets and graphs to

that includes graphics, visuals, or other supporting images.

b. Utilize technology to present information and/or data in a variety of ways.

4. Use technology appropriately.

a. Explain how technology is a useful and

b. Identify when technology may not be necessary or appropriate to communicate

c. Formulate strategies to communicate

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Development of this document facilitated by the Educational Policy Improvement Center

188

™

™

English Language Proficiency Standards

189

Standards for Ensuring Student Success From Kindergarten to College and Career ©2009 University of Texas System/Texas Education Agency

190

This document is a slightly reformatted version of the English Language Proficiency Standards (ELPS) rule text, which is available on the Texas Education Agency Web site, http://ritter.tea.state.tx.us/rules/tac/chapter074/ch074a.html#74.4.

This version of the ELPS has been designed for the Standards for Ensuring Student Success From Kindergarten to College and Career—English and Spanish Language Arts and Reading professional development series. The proficiency level descriptors are formatted like the charts found in the Texas English Language Proficiency Assessment System (TELPAS), rather than the lists found in section (d) of the ELPS rule text.

The ELPS consist of the following sections:

• (a) Introduction • (b) School district responsibilities• (c) Cross-curricular second language acquisition essential knowledge and skills• (d) Proficiency level descriptors*

*Note: For ease of use, the proficiency-level descriptor charts included in the following pages are from TELPAS, an assessment used to determine a student’s level of English language proficiency in four domains: listening, speaking, reading and writing. Information about TELPAS can be found on the TEA Web site, http://www.tea.state.tx.us/index3.aspx?id=3300&menu_id3=793.

About the ELPS

Standards for Ensuring Student Success From Kindergarten to College and Career ©2009 University of Texas System/Texas Education Agency191

English Language Proficiency Standards | 1

§74.4. English Language Proficiency Standards.

(a) Introduction.

(1) The English language proficiency standards in this section outline English language proficiency level descriptors and student expectations for English language learners (ELLs). School districts shall implement this section as an integral part of each subject in the required curriculum. The English language proficiency standards are to be published along with the Texas Essential Knowledge and Skills (TEKS) for each subject in the required curriculum.

(2) In order for ELLs to be successful, they must acquire both social and academic language proficiency in English. Social language proficiency in English consists of the English needed for daily social interactions. Academic language proficiency consists of the English needed to think critically, understand and learn new concepts, process complex academic material, and interact and communicate in English academic settings.

(3) Classroom instruction that effectively integrates second language acquisition with quality content area instruction ensures that ELLs acquire social and academic language proficiency in English, learn the knowledge and skills in the TEKS, and reach their full academic potential.

(4) Effective instruction in second language acquisition involves giving ELLs opportunities to listen, speak, read, and write at their current levels of English development while gradually increasing the linguistic complexity of the English they read and hear, and are expected to speak and write.

(5) The cross-curricular second language acquisition skills in subsection (c) of this section apply to ELLs in Kindergarten-Grade 12.

(6) The English language proficiency levels of beginning, intermediate, advanced, and advanced high are not grade-specific. ELLs may exhibit different proficiency levels within the language domains of listening, speaking, reading, and writing. The proficiency level descriptors outlined in subsection (d) of this section show the progression of second language acquisition from one proficiency level to the next and serve as a road map to help content area teachers instruct ELLs commensurate with students’ linguistic needs.

ELPS INTRODUCTION



(b) School district responsibilities. In fulfilling the requirements of this section, school districts shall:

(1) identify the student’s English language proficiency levels in the domains of listening, speaking, reading, and writing in accordance with the proficiency level descriptors for the beginning, intermediate, advanced, and advanced high levels delineated in subsection (d) of this section;

(2) provide instruction in the knowledge and skills of the foundation and enrichment curriculum in a manner that is linguistically accommodated (communicated, sequenced, and scaffolded) commensurate with the student’s levels of English language proficiency to ensure that the student learns the knowledge and skills in the required curriculum;

(3) provide content-based instruction including the cross-curricular second language acquisition essential knowledge and skills in subsection (c) of this section in a manner that is linguistically accommodated to help the student acquire English language proficiency; and

(4) provide intensive and ongoing foundational second language acquisition instruction to ELLs in Grade 3 or higher who are at the beginning or intermediate level of English language proficiency in listening, speaking, reading, and/or writing as determined by the state’s English language proficiency assessment system. These ELLs require focused, targeted, and systematic second language acquisition instruction to provide them with the foundation of English language vocabulary, grammar, syntax, and English mechanics necessary to support content-based instruction and accelerated learning of English.

ELPS INTRODUCTION



(c) Cross-curricular second language acquisition essential knowledge and skills.

(1) Cross-curricular second language acquisition/learning strategies. The ELL uses language learning strategies to develop an awareness of his or her own learning processes in all content areas. In order for the ELL to meet grade-level learning expectations across the foundation and enrichment curriculum, all instruction delivered in English must be linguistically accommodated (communicated, sequenced, and scaffolded) commensurate with the student’s level of English language proficiency. The student is expected to:

(A) use prior knowledge and experiences to understand meanings in English;

(B) monitor oral and written language production and employ self-corrective techniques or other resources;

(C) use strategic learning techniques such as concept mapping, drawing, memorizing, comparing, contrasting, and reviewing to acquire basic and grade-level vocabulary;

(D) speak using learning strategies such as requesting assistance, employing non-verbal cues, and using synonyms and circumlocution (conveying ideas by defining or describing when exact English words are not known);

(E) internalize new basic and academic language by using and reusing it in meaningful ways in speaking and writing activities that build concept and language attainment;

(F) use accessible language and learn new and essential language in the process;

(G) demonstrate an increasing ability to distinguish between formal and informal English and an increasing knowledge of when to use each one commensurate with grade-level learning expectations; and

(H) develop and expand repertoire of learning strategies such as reasoning inductively or deductively, looking for patterns in language, and analyzing sayings and expressions commensurate with grade-level learning expectations.

ELPS LEARNING STRATEGIES



(2) Cross-curricular second language acquisition/listening. The ELL listens to a variety of speakers including teachers, peers, and electronic media to gain an increasing level of comprehension of newly acquired language in all content areas. ELLs may be at the beginning, intermediate, advanced, or advanced high stage of English language acquisition in listening. In order for the ELL to meet grade-level learning expectations across the foundation and enrichment curriculum, all instruction delivered in English must be linguistically accommodated (communicated, sequenced, and scaffolded) commensurate with the student’s level of English language proficiency. The student is expected to:

(A) distinguish sounds and intonation patterns of English with increasing ease;

(B) recognize elements of the English sound system in newly acquired vocabulary such as long and short vowels, silent letters, and consonant clusters;

(C) learn new language structures, expressions, and basic and academic vocabulary heard during classroom instruction and interactions;

(D) monitor understanding of spoken language during classroom instruction and interactions and seek clarification as needed;

(E) use visual, contextual, and linguistic support to enhance and confirm understanding of increasingly complex and elaborated spoken language;

(F) listen to and derive meaning from a variety of media such as audio tape, video, DVD, and CD ROM to build and reinforce concept and language attainment;

(G) understand the general meaning, main points, and important details of spoken language ranging from situations in which topics, language, and contexts are familiar to unfamiliar;

(H) understand implicit ideas and information in increasingly complex spoken language commensurate with grade-level learning expectations; and

(I) demonstrate listening comprehension of increasingly complex spoken English by following directions, retelling or summarizing spoken messages, responding to questions and requests, collaborating with peers, and taking notes commensurate with content and grade-level needs.

ELPS LISTENING



(3) Cross-curricular second language acquisition/speaking. The ELL speaks in a variety of modes for a variety of purposes with an awareness of different language registers (formal/informal) using vocabulary with increasing fluency and accuracy in language arts and all content areas. ELLs may be at the beginning, intermediate, advanced, or advanced high stage of English language acquisition in speaking. In order for the ELL to meet grade-level learning expectations across the foundation and enrichment curriculum, all instruction delivered in English must be linguistically accommodated (communicated, sequenced, and scaffolded) commensurate with the student’s level of English language proficiency. The student is expected to:

(A) practice producing sounds of newly acquired vocabulary such as long and short vowels, silent letters, and consonant clusters to pronounce English words in a manner that is increasingly comprehensible;

(B) expand and internalize initial English vocabulary by learning and using high-frequency English words necessary for identifying and describing people, places, and objects, by retelling simple stories and basic information represented or supported by pictures, and by learning and using routine language needed for classroom communication;

(C) speak using a variety of grammatical structures, sentence lengths, sentence types, and connecting words with increasing accuracy and ease as more English is acquired;

(D) speak using grade-level content area vocabulary in context to internalize new English words and build academic language proficiency;

(E) share information in cooperative learning interactions;

(F) ask and give information ranging from using a very limited bank of high-frequency, high-need, concrete vocabulary, including key words and expressions needed for basic communication in academic and social contexts, to using abstract and content-based vocabulary during extended speaking assignments;

(G) express opinions, ideas, and feelings ranging from communicating single words and short phrases to participating in extended discussions on a variety of social and grade-appropriate academic topics;

(H) narrate, describe, and explain with increasing specificity and detail as more English is acquired;

(I) adapt spoken language appropriately for formal and informal purposes; and

(J) respond orally to information presented in a wide variety of print, electronic, audio, and visual media to build and reinforce concept and language attainment.

ELPS SPEAKING

Standards for Ensuring Student Success From Kindergarten to College and Career ©2009 University of Texas System/Texas Education Agency 196


(4) Cross-curricular second language acquisition/reading. The ELL reads a variety of texts for a variety of purposes with an increasing level of comprehension in all content areas. ELLs may be at the beginning, intermediate, advanced, or advanced high stage of English language acquisition in reading. In order for the ELL to meet grade-level learning expectations across the foundation and enrichment curriculum, all instruction delivered in English must be linguistically accommodated (communicated, sequenced, and scaffolded) commensurate with the student’s level of English language proficiency. For Kindergarten and Grade 1, certain of these student expectations apply to text read aloud for students not yet at the stage of decoding written text. The student is expected to:

(A) learn relationships between sounds and letters of the English language and decode (sound out) words using a combination of skills such as recognizing sound-letter relationships and identifying cognates, affixes, roots, and base words;

(B) recognize directionality of English reading such as left to right and top to bottom;

(C) develop basic sight vocabulary, derive meaning of environmental print, and comprehend English vocabulary and language structures used routinely in written classroom materials;

(D) use prereading supports such as graphic organizers, illustrations, and pretaught topic-related vocabulary and other prereading activities to enhance comprehension of written text;

(E) read linguistically accommodated content area material with a decreasing need for linguistic accommodations as more English is learned;

(F) use visual and contextual support and support from peers and teachers to read grade-appropriate content area text, enhance and confirm understanding, and develop vocabulary, grasp of language structures, and background knowledge needed to comprehend increasingly challenging language;

(G) demonstrate comprehension of increasingly complex English by participating in shared reading, retelling or summarizing material, responding to questions, and taking notes commensurate with content area and grade level needs;

(H) read silently with increasing ease and comprehension for longer periods;

ELPS READING



(I) demonstrate English comprehension and expand reading skills by employing basic reading skills such as demonstrating understanding of supporting ideas and details in text and graphic sources, summarizing text, and distinguishing main ideas from details commensurate with content area needs;

(J) demonstrate English comprehension and expand reading skills by employing inferential skills such as predicting, making connections between ideas, drawing inferences and conclusions from text and graphic sources, and finding supporting text evidence commensurate with content area needs; and

(K) demonstrate English comprehension and expand reading skills by employing analytical skills such as evaluating written information and performing critical analyses commensurate with content area and grade-level needs.

ELPS READING



(5) Cross-curricular second language acquisition/writing. The ELL writes in a variety of forms with increasing accuracy to effectively address a specific purpose and audience in all content areas. ELLs may be at the beginning, intermediate, advanced, or advanced high stage of English language acquisition in writing. In order for the ELL to meet grade-level learning expectations across foundation and enrichment curriculum, all instruction delivered in English must be linguistically accommodated (communicated, sequenced, and scaffolded) commensurate with the student’s level of English language proficiency. For Kindergarten and Grade 1, certain of these student expectations do not apply until the student has reached the stage of generating original written text using a standard writing system. The student is expected to:

(A) learn relationships between sounds and letters of the English language to represent sounds when writing in English;

(B) write using newly acquired basic vocabulary and content-based grade-level vocabulary;

(C) spell familiar English words with increasing accuracy, and employ English spelling patterns and rules with increasing accuracy as more English is acquired;

(D) edit writing for standard grammar and usage, including subject-verb agreement, pronoun agreement, and appropriate verb tenses commensurate with grade-level expectations as more English is acquired;

(E) employ increasingly complex grammatical structures in content area writing commensurate with grade-level expectations, such as:

(i) using correct verbs, tenses, and pronouns/antecedents;

(ii) using possessive case (apostrophe s) correctly; and

(iii) using negatives and contractions correctly;

(F) write using a variety of grade-appropriate sentence lengths, patterns, and connecting words to combine phrases, clauses, and sentences in increasingly accurate ways as more English is acquired; and

(G) narrate, describe, and explain with increasing specificity and detail to fulfill content area writing needs as more English is acquired.

ELPS WRITING



©2008 TEXAS EDUCATION AGENCY ALL RIGHTS RESERVED.

TE

LPA

S P

rofi

cien

cy L

evel

Des

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Eng

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©2008 TEX

AS ED

UC

ATION

AG

ENC

Y A

LL RIG

HTS R

ESERVED

.

TELPAS Proficiency Level DescriptorsGrades K–12 Listening

Beginning Intermediate Advanced Advanced High

Beginning English languagelearners (ELLs) have little or noability to understand spokenEnglish used in academic andsocial settings.

Intermediate ELLs have the ability to understand simple,high-frequency spoken Englishused in routine academic andsocial settings.

Advanced ELLs have the ability to understand, with secondlanguage acquisition support,grade-appropriate spoken English used in academic andsocial settings.

Advanced high ELLs have theability to understand, with minimal second languageacquisition support, grade-appropriate spoken English usedin academic and social settings.

These students: These students: These students: These students:• struggle to understand simple • usually understand simple or • usually understand longer, more • understand longer, elaborated

conversations and simple routine directions, as well as short, elaborated directions, directions, conversations, anddiscussions even when the topics simple conversations and short, conversations, and discussions on discussions on familiar andare familiar and the speaker uses simple discussions on familiar familiar and some unfamiliar unfamiliar topics with onlylinguistic supports (e.g., visuals, topics; when topics are unfamiliar, topics, but sometimes need occasional need for processingslower speech and other verbal require extensive linguistic processing time and sometimes time and with little dependence oncues, gestures) supports and adaptations (e.g., depend on visuals, verbal cues, visuals, verbal cues, and gestures;

• struggle to identify and distinguishindividual words and phrasesduring social and instructionalinteractions that have not been

visuals, slower speech and otherverbal cues, simplified language,gestures, preteaching to previewor build topic-related vocabulary)

and gestures to supportunderstanding

• understand most main points,most important details, and some

some exceptions when complexacademic or highly specializedlanguage is used

• understand main points, importantintentionally modified for ELLs • often identify and distinguish key implicit information during social details, and implicit information at

• may not seek clarification inEnglish when failing tocomprehend the English they hear;frequently remain silent, watchingothers for cues

words and phrases necessary tounderstand the general meaning(gist) during social and basicinstructional interactions that havenot been intentionally modified forELLs

and basic instructional interactionsthat have not been intentionallymodified for ELLs

• occasionally require/request thespeaker to repeat, slow down, orrephrase to clarify the meaning of

a level nearly comparable to nativeEnglish-speaking peers duringsocial and instructionalinteractions

• rarely require/request the speakerto repeat, slow down, or rephrase

• have the ability to seek clarification in English when failingto comprehend the English theyhear by requiring/requesting thespeaker to repeat, slow down, orrephrase speech

the English they hear to clarify the meaning of theEnglish they hear

TELPAS LISTENING




TE

LPA

S P

rofi

cien

cy L

evel

Des

crip

tors

Gra

des

K–1

2 L

iste

nin

g

Beg

inni

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term

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teA

dvan

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Adv

ance

d H

igh

Beg

inni

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nglis

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lear

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(E

LLs)

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nE

nglis

h us

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iate

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s ha

ve t

he

abili

ty t

o un

ders

tand

sim

ple,

high

-fre

quen

cy s

poke

n E

nglis

hus

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tine

aca

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d E

LLs

have

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abi

lity

to u

nder

stan

d,w

ith

seco

ndla

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ge

acqu

isit

ion

supp

ort,

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e-ap

prop

riat

e sp

oken

E

nglis

h us

ed in

aca

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d hi

gh E

LLs

have

the

abili

ty t

o un

ders

tand

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h m

inim

al s

econ

d la

ngua

ge

acqu

isit

ion

supp

ort,

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e-ap

prop

riat

e sp

oken

Eng

lish

used

in a

cade

mic

and

soc

ial s

etti

ngs.

Thes

e st

uden

ts:

Thes

e st

uden

ts:

Thes

e st

uden

ts:

Thes

e st

uden

ts:

•st

rugg

le t

o un

ders

tand

sim

ple

•us

ually

und

erst

and

sim

ple

or•

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lly u

nder

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d lo

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, m

ore

•un

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tand

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are

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hen

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re u

nfam

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met

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nee

doc

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nee

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verb

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divi

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wor

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soci

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stru

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tha

t ha

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rve

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mos

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port

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som

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som

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whe

n co

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nglis

h w

hen

faili

ng t

oco

mpr

ehen

d th

e E

nglis

h th

ey h

ear;

freq

uent

ly r

emai

n si

lent

, w

atch

ing

othe

rs fo

r cu

es

wor

ds a

nd p

hras

es n

eces

sary

to

unde

rsta

nd t

he g

ener

al m

eani

ng(g

ist)

dur

ing

soci

al a

nd b

asic

inst

ruct

iona

l int

erac

tions

tha

t ha

veno

t be

en in

tent

iona

lly m

odifi

ed fo

rE

LLs

and

basi

c in

stru

ctio

nal i

nter

actio

nsth

at h

ave

not

been

inte

ntio

nally

mod

ified

for

ELL

s

•oc

casi

onal

ly r

equi

re/r

eque

st t

hesp

eake

r to

rep

eat,

slow

dow

n, o

rre

phra

se t

o cl

arify

the

mea

ning

of

a le

vel n

early

com

para

ble

to n

ativ

eE

nglis

h-sp

eaki

ng p

eers

dur

ing

soci

al a

nd in

stru

ctio

nal

inte

ract

ions

•ra

rely

req

uire

/req

uest

the

spe

aker

to r

epea

t, sl

ow d

own,

or

reph

rase

•ha

ve t

he a

bilit

y to

see

k cl

arifi

catio

n in

Eng

lish

whe

n fa

iling

to c

ompr

ehen

d th

e E

nglis

h th

eyhe

ar b

y re

quiri

ng/r

eque

stin

g th

esp

eake

r to

rep

eat,

slow

dow

n, o

rre

phra

se s

peec

h

the

Eng

lish

they

hea

rto

cla

rify

the

mea

ning

of

the

Eng

lish

they

hea

r

©2008 TEX

AS ED

UC

ATION

AG

ENC

Y A

LL RIG

HTS R

ESERVED

.

TELPAS Proficiency Level DescriptorsGrades K–12 Speaking


Beginning English languagelearners (ELLs) have little or noability to speak English inacademic and social settings.

Intermediate ELLs have the ability to speak in a simplemanner using English commonlyheard in routine academic andsocial settings.

Advanced ELLs have the ability to speak using grade-appropriateEnglish, with second languageacquisition support, in academicand social settings.

Advanced high ELLs have theability to speak using grade-appropriate English, with minimalsecond language acquisitionsupport, in academic and socialsettings.

These students: These students: These students: These students:• mainly speak using single words • are able to express simple, • are able to participate comfortably • are able to participate in extended

and short phrases consisting of original messages, speak using in most conversations and discussions on a variety of socialrecently practiced, memorized, or sentences, and participate in short academic discussions on familiar and grade-appropriate academichighly familiar material to get conversations and classroom topics, with some pauses to topics with only occasionalimmediate needs met; may be interactions; may hesitate restate, repeat, or search for disruptions, hesitations, or pauseshesitant to speak and often giveup in their attempts tocommunicate

frequently and for long periods tothink about how to communicatedesired meaning

words and phrases to clarifymeaning

• discuss familiar academic topics

• communicate effectively usingabstract and content-basedvocabulary during classroom

• speak using a very limited bank of • speak simply using basic using content-based terms and instructional tasks, with somehigh-frequency, high-need, vocabulary needed in everyday common abstract vocabulary; can exceptions when low-frequency orconcrete vocabulary, including key social interactions and routine usually speak in some detail on academically demandingwords and expressions needed for academic contexts; rarely have familiar topics vocabulary is needed; use many basic communication in academicand social contexts

• lack the knowledge of English

vocabulary to speak in detail

• exhibit an emerging awareness ofEnglish grammar and speak using

• have a grasp of basic grammarfeatures, including a basic abilityto narrate and describe in present,

of the same idioms andcolloquialisms as their nativeEnglish-speaking peers

grammar necessary to connect mostly simple sentence structures past, and future tenses; have an • can use English grammarideas and speak in sentences; can and simple tenses; are most emerging ability to use complex structures and complex sentencessometimes produce sentences comfortable speaking in present sentences and complex grammar to narrate and describe at a levelusing recently practiced, tense features nearly comparable to nativememorized, or highly familiarmaterial

• exhibit second languageacquisition errors that may hinder

• make errors that interferesomewhat with communication

English-speaking peers

• make few second language• exhibit second language overall communication when trying when using complex grammar acquisition errors that interfere

acquisition errors that may hinder to use complex or less familiar structures, long sentences, and with overall communicationoverall communication, particularlywhen trying to convey informationbeyond memorized, practiced, orhighly familiar material

English

• use pronunciation that can usuallybe understood by peopleaccustomed to interacting with

less familiar words andexpressions

• may mispronounce words, but usepronunciation that can usually be

• may mispronounce words, butrarely use pronunciation thatinterferes with overallcommunication

• typically use pronunciation that ELLs understood by people notsignificantly inhibits accustomed to interacting withcommunication ELLs

TELPAS SPEAKING




TE

LPA

S P

rofi

cien

cy L

evel

Des

crip

tors

Gra

des

K–1

2 L

iste

nin

g

Beg

inni

ngIn

term

edia

teA

dvan

ced

Adv

ance

d H

igh

Beg

inni

ng E

nglis

h la

ngua

ge

lear

ners

(E

LLs)

hav

e lit

tle

or n

oab

ility

to

unde

rsta

nd s

poke

nE

nglis

h us

ed in

aca

dem

ic a

ndso

cial

set

ting

s.

Inte

rmed

iate

ELL

s ha

ve t

he

abili

ty t

o un

ders

tand

sim

ple,

high

-fre

quen

cy s

poke

n E

nglis

hus

ed in

rou

tine

aca

dem

ic a

ndso

cial

set

ting

s.

Adv

ance

d E

LLs

have

the

abi

lity

to u

nder

stan

d,w

ith

seco

ndla

ngua

ge

acqu

isit

ion

supp

ort,

grad

e-ap

prop

riat

e sp

oken

E

nglis

h us

ed in

aca

dem

ic a

ndso

cial

set

ting

s.

Adv

ance

d hi

gh E

LLs

have

the

abili

ty t

o un

ders

tand

,wit

h m

inim

al s

econ

d la

ngua

ge

acqu

isit

ion

supp

ort,

grad

e-ap

prop

riat

e sp

oken

Eng

lish

used

in a

cade

mic

and

soc

ial s

etti

ngs.

Thes

e st

uden

ts:

Thes

e st

uden

ts:

Thes

e st

uden

ts:

Thes

e st

uden

ts:

•st

rugg

le t

o un

ders

tand

sim

ple

•us

ually

und

erst

and

sim

ple

or•

usua

lly u

nder

stan

d lo

nger

, m

ore

•un

ders

tand

long

er,

elab

orat

edco

nver

satio

ns a

nd s

impl

ero

utin

e di

rect

ions

, as

wel

l as

shor

t,el

abor

ated

dire

ctio

ns,

dire

ctio

ns,

conv

ersa

tions

, an

ddi

scus

sion

s ev

en w

hen

the

topi

cssi

mpl

e co

nver

satio

ns a

nd s

hort

,co

nver

satio

ns,

and

disc

ussi

ons

ondi

scus

sion

s on

fam

iliar

and

are

fam

iliar

and

the

spe

aker

use

ssi

mpl

e di

scus

sion

s on

fam

iliar

fam

iliar

and

som

e un

fam

iliar

unfa

mili

ar t

opic

s w

ith o

nly

lingu

istic

sup

port

s (e

.g.,

visu

als,

topi

cs;w

hen

topi

cs a

re u

nfam

iliar

,to

pics

, bu

t so

met

imes

nee

doc

casi

onal

nee

d fo

r pr

oces

sing

slow

er s

peec

h an

d ot

her

verb

alre

quire

ext

ensi

ve li

ngui

stic

proc

essi

ng t

ime

and

som

etim

estim

e an

d w

ith li

ttle

depe

nden

ce o

ncu

es,

gest

ures

)su

ppor

ts a

nd a

dapt

atio

ns (

e.g.

,de

pend

on

visu

als,

ver

bal c

ues,

visu

als,

ver

bal c

ues,

and

ges

ture

s;

•st

rugg

le t

o id

entif

y an

d di

stin

guis

hin

divi

dual

wor

ds a

nd p

hras

esdu

ring

soci

al a

nd in

stru

ctio

nal

inte

ract

ions

tha

t ha

ve n

ot b

een

visu

als,

slo

wer

spe

ech

and

othe

rve

rbal

cue

s, s

impl

ified

lang

uage

,ge

stur

es,

pret

each

ing

to p

revi

ewor

bui

ld t

opic

-rel

ated

voc

abul

ary)

and

gest

ures

to

supp

ort

unde

rsta

ndin

g

•un

ders

tand

mos

t m

ain

poin

ts,

mos

t im

port

ant

deta

ils,

and

som

e

som

e ex

cept

ions

whe

n co

mpl

exac

adem

ic o

r hi

ghly

spe

cial

ized

lang

uage

is u

sed

•un

ders

tand

mai

n po

ints

, im

port

ant

inte

ntio

nally

mod

ified

for

ELL

s•

ofte

n id

entif

y an

d di

stin

guis

h ke

yim

plic

it in

form

atio

n du

ring

soci

alde

tails

, an

d im

plic

it in

form

atio

n at

•m

ay n

ot s

eek

clar

ifica

tion

inE

nglis

h w

hen

faili

ng t

oco

mpr

ehen

d th

e E

nglis

h th

ey h

ear;

freq

uent

ly r

emai

n si

lent

, w

atch

ing

othe

rs fo

r cu

es

wor

ds a

nd p

hras

es n

eces

sary

to

unde

rsta

nd t

he g

ener

al m

eani

ng(g

ist)

dur

ing

soci

al a

nd b

asic

inst

ruct

iona

l int

erac

tions

tha

t ha

veno

t be

en in

tent

iona

lly m

odifi

ed fo

rE

LLs

and

basi

c in

stru

ctio

nal i

nter

actio

nsth

at h

ave

not

been

inte

ntio

nally

mod

ified

for

ELL

s

•oc

casi

onal

ly r

equi

re/r

eque

st t

hesp

eake

r to

rep

eat,

slow

dow

n, o

rre

phra

se t

o cl

arify

the

mea

ning

of

a le

vel n

early

com

para

ble

to n

ativ

eE

nglis

h-sp

eaki

ng p

eers

dur

ing

soci

al a

nd in

stru

ctio

nal

inte

ract

ions

•ra

rely

req

uire

/req

uest

the

spe

aker

to r

epea

t, sl

ow d

own,

or

reph

rase

•ha

ve t

he a

bilit

y to

see

k cl

arifi

catio

n in

Eng

lish

whe

n fa

iling

to c

ompr

ehen

d th

e E

nglis

h th

eyhe

ar b

y re

quiri

ng/r

eque

stin

g th

esp

eake

r to

rep

eat,

slow

dow

n, o

rre

phra

se s

peec

h

the

Eng

lish

they

hea

rto

cla

rify

the

mea

ning

of

the

Eng

lish

they

hea

r

©2008 TEX

AS ED

UC

ATION

AG

ENC

Y A

LL RIG

HTS R

ESERVED

.

TELPAS Proficiency Level DescriptorsGrades K–1 Reading


Beginning English languagelearners (ELLs) have little or noability to use the English languageto build foundational readingskills.

Intermediate ELLs have a limitedability to use the English languageto build foundational readingskills.

Advanced ELLs have the ability to use the English language, withsecond language acquisitionsupport, to build foundationalreading skills.

Advanced high ELLs have theability to use the Englishlanguage, with minimal secondlanguage acquisition support, tobuild foundational reading skills.

These students: These students: These students: These students:• derive little or no meaning from • demonstrate limited • demonstrate comprehension of • demonstrate, with minimal second

grade-appropriate stories read comprehension (key words and most main points and most language acquisition support andaloud in English, unless the stories general meaning) of grade- supporting ideas in grade- at a level nearly comparable toare appropriate stories read aloud in appropriate stories read aloud in native English-speaking peers,

– read in short “chunks”– controlled to include the little

English, unless the stories include

– predictable story lines

English, although they may stilldepend on visual and linguisticsupports to gain or confirm

comprehension of main points andsupporting ideas (explicit andimplicit) in grade-appropriate

English they know such as – highly familiar topics meaning stories read aloud in Englishlanguage that is high-frequency,concrete, and recently practiced

– accompanied by ample visualsupports such as illustrations,gestures, pantomime, andobjects and by linguistic

– primarily high-frequency,concrete vocabulary

– short, simple sentences– visual and linguistic supports

• regularly recognize and

• recognize some basic Englishvocabulary and high-frequencywords in isolated print

• with second language acquisitionsupport, are able to decode most

• with some exceptions, recognizesight vocabulary and high-frequency words to a degreenearly comparable to that of nativeEnglish-speaking peers

supports such as careful understand common grade-appropriate English text • with minimal second languageenunciation and slower speech environmental print in English because they * acquisition support, have an ability

• begin to recognize and understandenvironmental print in English(e.g., signs, labeled items, namesof peers, logos)

• have difficulty decoding most

(e.g., signs, labeled items, namesof peers, logos)

• have difficulty decoding grade-appropriate English text becausethey *

– understand the meaning ofmost grade-appropriate Englishwords

– have little difficulty with Englishsounds and sound-symbol

to decode and understand grade-appropriate English text at a levelnearly comparable to nativeEnglish-speaking peers *

grade-appropriate English text – understand the meaning of only relationships that result frombecause they * those English words they hear differences between their

– understand the meaning of veryfew words in English

– struggle significantly withsounds in spoken English wordsand with sound-symbolrelationships due to differences

frequently– struggle with some sounds in

English words and some sound-symbol relationships due todifferences between theirprimary language and English

primary language and English

between their primary languageand English

* The last descriptor applies only to students who are at the developmental stage of decoding written text (i.e., they have “cracked the code”necessary for learning to read).

TELPAS READING




TE

LPA

S P

rofi

cien

cy L

evel

Des

crip

tors

Gra

des

K–1

2 L

iste

nin

g

Beg

inni

ngIn

term

edia

teA

dvan

ced

Adv

ance

d H

igh

Beg

inni

ng E

nglis

h la

ngua

ge

lear

ners

(E

LLs)

hav

e lit

tle

or n

oab

ility

to

unde

rsta

nd s

poke

nE

nglis

h us

ed in

aca

dem

ic a

ndso

cial

set

ting

s.

Inte

rmed

iate

ELL

s ha

ve t

he

abili

ty t

o un

ders

tand

sim

ple,

high

-fre

quen

cy s

poke

n E

nglis

hus

ed in

rou

tine

aca

dem

ic a

ndso

cial

set

ting

s.

Adv

ance

d E

LLs

have

the

abi

lity

to u

nder

stan

d,w

ith

seco

ndla

ngua

ge

acqu

isit

ion

supp

ort,

grad

e-ap

prop

riat

e sp

oken

E

nglis

h us

ed in

aca

dem

ic a

ndso

cial

set

ting

s.

Adv

ance

d hi

gh E

LLs

have

the

abili

ty t

o un

ders

tand

,wit

h m

inim

al s

econ

d la

ngua

ge

acqu

isit

ion

supp

ort,

grad

e-ap

prop

riat

e sp

oken

Eng

lish

used

in a

cade

mic

and

soc

ial s

etti

ngs.

Thes

e st

uden

ts:

Thes

e st

uden

ts:

Thes

e st

uden

ts:

Thes

e st

uden

ts:

•st

rugg

le t

o un

ders

tand

sim

ple

•us

ually

und

erst

and

sim

ple

or•

usua

lly u

nder

stan

d lo

nger

, m

ore

•un

ders

tand

long

er,

elab

orat

edco

nver

satio

ns a

nd s

impl

ero

utin

e di

rect

ions

, as

wel

l as

shor

t,el

abor

ated

dire

ctio

ns,

dire

ctio

ns,

conv

ersa

tions

, an

ddi

scus

sion

s ev

en w

hen

the

topi

cssi

mpl

e co

nver

satio

ns a

nd s

hort

,co

nver

satio

ns,

and

disc

ussi

ons

ondi

scus

sion

s on

fam

iliar

and

are

fam

iliar

and

the

spe

aker

use

ssi

mpl

e di

scus

sion

s on

fam

iliar

fam

iliar

and

som

e un

fam

iliar

unfa

mili

ar t

opic

s w

ith o

nly

lingu

istic

sup

port

s (e

.g.,

visu

als,

topi

cs;w

hen

topi

cs a

re u

nfam

iliar

,to

pics

, bu

t so

met

imes

nee

doc

casi

onal

nee

d fo

r pr

oces

sing

slow

er s

peec

h an

d ot

her

verb

alre

quire

ext

ensi

ve li

ngui

stic

proc

essi

ng t

ime

and

som

etim

estim

e an

d w

ith li

ttle

depe

nden

ce o

ncu

es,

gest

ures

)su

ppor

ts a

nd a

dapt

atio

ns (

e.g.

,de

pend

on

visu

als,

ver

bal c

ues,

visu

als,

ver

bal c

ues,

and

ges

ture

s;

•st

rugg

le t

o id

entif

y an

d di

stin

guis

hin

divi

dual

wor

ds a

nd p

hras

esdu

ring

soci

al a

nd in

stru

ctio

nal

inte

ract

ions

tha

t ha

ve n

ot b

een

visu

als,

slo

wer

spe

ech

and

othe

rve

rbal

cue

s, s

impl

ified

lang

uage

,ge

stur

es,

pret

each

ing

to p

revi

ewor

bui

ld t

opic

-rel

ated

voc

abul

ary)

and

gest

ures

to

supp

ort

unde

rsta

ndin

g

•un

ders

tand

mos

t m

ain

poin

ts,

mos

t im

port

ant

deta

ils,

and

som

e

som

e ex

cept

ions

whe

n co

mpl

exac

adem

ic o

r hi

ghly

spe

cial

ized

lang

uage

is u

sed

•un

ders

tand

mai

n po

ints

, im

port

ant

inte

ntio

nally

mod

ified

for

ELL

s•

ofte

n id

entif

y an

d di

stin

guis

h ke

yim

plic

it in

form

atio

n du

ring

soci

alde

tails

, an

d im

plic

it in

form

atio

n at

•m

ay n

ot s

eek

clar

ifica

tion

inE

nglis

h w

hen

faili

ng t

oco

mpr

ehen

d th

e E

nglis

h th

ey h

ear;

freq

uent

ly r

emai

n si

lent

, w

atch

ing

othe

rs fo

r cu

es

wor

ds a

nd p

hras

es n

eces

sary

to

unde

rsta

nd t

he g

ener

al m

eani

ng(g

ist)

dur

ing

soci

al a

nd b

asic

inst

ruct

iona

l int

erac

tions

tha

t ha

veno

t be

en in

tent

iona

lly m

odifi

ed fo

rE

LLs

and

basi

c in

stru

ctio

nal i

nter

actio

nsth

at h

ave

not

been

inte

ntio

nally

mod

ified

for

ELL

s

•oc

casi

onal

ly r

equi

re/r

eque

st t

hesp

eake

r to

rep

eat,

slow

dow

n, o

rre

phra

se t

o cl

arify

the

mea

ning

of

a le

vel n

early

com

para

ble

to n

ativ

eE

nglis

h-sp

eaki

ng p

eers

dur

ing

soci

al a

nd in

stru

ctio

nal

inte

ract

ions

•ra

rely

req

uire

/req

uest

the

spe

aker

to r

epea

t, sl

ow d

own,

or

reph

rase

•ha

ve t

he a

bilit

y to

see

k cl

arifi

catio

n in

Eng

lish

whe

n fa

iling

to c

ompr

ehen

d th

e E

nglis

h th

eyhe

ar b

y re

quiri

ng/r

eque

stin

g th

esp

eake

r to

rep

eat,

slow

dow

n, o

rre

phra

se s

peec

h

the

Eng

lish

they

hea

rto

cla

rify

the

mea

ning

of

the

Eng

lish

they

hea

r

TELPAS R

eading Information B

ookletC

hapter 2: Test Design

7

TELPAS Grades 2–12 Reading Proficiency Level Descriptors


Beginning English language learners (ELLs) have little or no ability to read and understand English used in academic and social contexts.

Intermediate ELLs have the ability to read and understand simple, high-frequency English used in routine academic and social contexts.

Advanced ELLs have the ability to read and understand, with second language acquisition support, grade-appropriate English used in academic and social contexts.

Advanced high ELLs have the ability to read and understand, with minimal second language acquisition support, grade-appropriate English used in academic and social contexts.

These students:• read and understand the very limited

recently practiced, memorized, or highly familiar English they have learned; vocabulary predominantly includes

- environmental print

- some very high-frequency words

- concrete words that can be represented by pictures

• read slowly, word by word

• have a very limited sense of English language structures

• comprehend predominantly isolated familiar words and phrases; comprehend some sentences in highly routine contexts or recently practiced, highly familiar text

• are highly dependent on visuals and prior knowledge to derive meaning from text in English

• are able to apply reading comprehension skills in English only when reading texts written for this level

These students:• read and understand English vocabulary

on a somewhat wider range of topics and with increased depth; vocabulary predominantly includes

- everyday oral language

- literal meanings of common words

- routine academic language and terms

- commonly used abstract language such as terms used to describe basic feelings

• often read slowly and in short phrases; may re-read to clarify meaning

• have a growing understanding of basic, routinely used English language structures

• understand simple sentences in short, connected texts, but are dependent on visual cues, topic familiarity, prior knowledge, pretaught topic-related vocabulary, story predictability, and teacher/peer assistance to sustain comprehension

• struggle to independently read and understand grade-level texts

• are able to apply basic and some higher-order comprehension skills when reading texts that are linguistically accommodated and/or simplified for this level

These students:• read and understand, with second language

acquisition support, a variety of grade-appropriate English vocabulary used in social and academic contexts:

- with second language acquisition support, read and understand grade-appropriate concrete and abstract vocabulary, but have difficulty with less commonly encountered words

- demonstrate an emerging ability to understand words and phrases beyond their literal meaning

- understand multiple meanings of commonly used words

• read longer phrases and simple sentences from familiar text with appropriate rate and speed

• are developing skill in using their growing familiarity with English language structures to construct meaning of grade-appropriate text

• are able to apply basic and higher-order comprehension skills when reading grade-appropriate text, but are still occasionally dependent on visuals, teacher/peer assistance, and other linguistically accommodated text features to determine or clarify meaning, particularly with unfamiliar topics

These students:• read and understand vocabulary at a

level nearly comparable to that of their native English-speaking peers, with some exceptions when low-frequency or specialized vocabulary is used

• generally read grade-appropriate, familiar text with appropriate rate, speed, intonation, and expression

• are able to, at a level nearly comparable to native English-speaking peers, use their familiarity with English language structures to construct meaning of grade-appropriate text

• are able to apply, with minimal second language acquisition support and at a level nearly comparable to native English-speaking peers, basic and higher-order comprehension skills when reading grade-appropriate text

TELPAS READING




TE

LPA

S P

rofi

cien

cy L

evel

Des

crip

tors

Gra

des

K–1

2 L

iste

nin

g

Beg

inni

ngIn

term

edia

teA

dvan

ced

Adv

ance

d H

igh

Beg

inni

ng E

nglis

h la

ngua

ge

lear

ners

(E

LLs)

hav

e lit

tle

or n

oab

ility

to

unde

rsta

nd s

poke

nE

nglis

h us

ed in

aca

dem

ic a

ndso

cial

set

ting

s.

Inte

rmed

iate

ELL

s ha

ve t

he

abili

ty t

o un

ders

tand

sim

ple,

high

-fre

quen

cy s

poke

n E

nglis

hus

ed in

rou

tine

aca

dem

ic a

ndso

cial

set

ting

s.

Adv

ance

d E

LLs

have

the

abi

lity

to u

nder

stan

d,w

ith

seco

ndla

ngua

ge

acqu

isit

ion

supp

ort,

grad

e-ap

prop

riat

e sp

oken

E

nglis

h us

ed in

aca

dem

ic a

ndso

cial

set

ting

s.

Adv

ance

d hi

gh E

LLs

have

the

abili

ty t

o un

ders

tand

,wit

h m

inim

al s

econ

d la

ngua

ge

acqu

isit

ion

supp

ort,

grad

e-ap

prop

riat

e sp

oken

Eng

lish

used

in a

cade

mic

and

soc

ial s

etti

ngs.

Thes

e st

uden

ts:

Thes

e st

uden

ts:

Thes

e st

uden

ts:

Thes

e st

uden

ts:

•st

rugg

le t

o un

ders

tand

sim

ple

•us

ually

und

erst

and

sim

ple

or•

usua

lly u

nder

stan

d lo

nger

, m

ore

•un

ders

tand

long

er,

elab

orat

edco

nver

satio

ns a

nd s

impl

ero

utin

e di

rect

ions

, as

wel

l as

shor

t,el

abor

ated

dire

ctio

ns,

dire

ctio

ns,

conv

ersa

tions

, an

ddi

scus

sion

s ev

en w

hen

the

topi

cssi

mpl

e co

nver

satio

ns a

nd s

hort

,co

nver

satio

ns,

and

disc

ussi

ons

ondi

scus

sion

s on

fam

iliar

and

are

fam

iliar

and

the

spe

aker

use

ssi

mpl

e di

scus

sion

s on

fam

iliar

fam

iliar

and

som

e un

fam

iliar

unfa

mili

ar t

opic

s w

ith o

nly

lingu

istic

sup

port

s (e

.g.,

visu

als,

topi

cs;w

hen

topi

cs a

re u

nfam

iliar

,to

pics

, bu

t so

met

imes

nee

doc

casi

onal

nee

d fo

r pr

oces

sing

slow

er s

peec

h an

d ot

her

verb

alre

quire

ext

ensi

ve li

ngui

stic

proc

essi

ng t

ime

and

som

etim

estim

e an

d w

ith li

ttle

depe

nden

ce o

ncu

es,

gest

ures

)su

ppor

ts a

nd a

dapt

atio

ns (

e.g.

,de

pend

on

visu

als,

ver

bal c

ues,

visu

als,

ver

bal c

ues,

and

ges

ture

s;

•st

rugg

le t

o id

entif

y an

d di

stin

guis

hin

divi

dual

wor

ds a

nd p

hras

esdu

ring

soci

al a

nd in

stru

ctio

nal

inte

ract

ions

tha

t ha

ve n

ot b

een

visu

als,

slo

wer

spe

ech

and

othe

rve

rbal

cue

s, s

impl

ified

lang

uage

,ge

stur

es,

pret

each

ing

to p

revi

ewor

bui

ld t

opic

-rel

ated

voc

abul

ary)

and

gest

ures

to

supp

ort

unde

rsta

ndin

g

•un

ders

tand

mos

t m

ain

poin

ts,

mos

t im

port

ant

deta

ils,

and

som

e

som

e ex

cept

ions

whe

n co

mpl

exac

adem

ic o

r hi

ghly

spe

cial

ized

lang

uage

is u

sed

•un

ders

tand

mai

n po

ints

, im

port

ant

inte

ntio

nally

mod

ified

for

ELL

s•

ofte

n id

entif

y an

d di

stin

guis

h ke

yim

plic

it in

form

atio

n du

ring

soci

alde

tails

, an

d im

plic

it in

form

atio

n at

•m

ay n

ot s

eek

clar

ifica

tion

inE

nglis

h w

hen

faili

ng t

oco

mpr

ehen

d th

e E

nglis

h th

ey h

ear;

freq

uent

ly r

emai

n si

lent

, w

atch

ing

othe

rs fo

r cu

es

wor

ds a

nd p

hras

es n

eces

sary

to

unde

rsta

nd t

he g

ener

al m

eani

ng(g

ist)

dur

ing

soci

al a

nd b

asic

inst

ruct

iona

l int

erac

tions

tha

t ha

veno

t be

en in

tent

iona

lly m

odifi

ed fo

rE

LLs

and

basi

c in

stru

ctio

nal i

nter

actio

nsth

at h

ave

not

been

inte

ntio

nally

mod

ified

for

ELL

s

•oc

casi

onal

ly r

equi

re/r

eque

st t

hesp

eake

r to

rep

eat,

slow

dow

n, o

rre

phra

se t

o cl

arify

the

mea

ning

of

a le

vel n

early

com

para

ble

to n

ativ

eE

nglis

h-sp

eaki

ng p

eers

dur

ing

soci

al a

nd in

stru

ctio

nal

inte

ract

ions

•ra

rely

req

uire

/req

uest

the

spe

aker

to r

epea

t, sl

ow d

own,

or

reph

rase

•ha

ve t

he a

bilit

y to

see

k cl

arifi

catio

n in

Eng

lish

whe

n fa

iling

to c

ompr

ehen

d th

e E

nglis

h th

eyhe

ar b

y re

quiri

ng/r

eque

stin

g th

esp

eake

r to

rep

eat,

slow

dow

n, o

rre

phra

se s

peec

h

the

Eng

lish

they

hea

rto

cla

rify

the

mea

ning

of

the

Eng

lish

they

hea

r

©2008 TEX

AS ED

UC

ATION

AG

ENC

Y A

LL RIG

HTS R

ESERVED

.

TELPAS Proficiency Level DescriptorsGrades K–1 Writing


Beginning English languagelearners (ELLs) have little or noability to use the English language to build foundationalwriting skills.

Intermediate ELLs have a limitedability to use the English language to build foundationalwriting skills.

Advanced ELLs have the ability to use the English language tobuild, with second languageacquisition support, foundationalwriting skills.

Advanced high ELLs have theability to use the English language to build, with minimalsecond language acquisitionsupport, foundational writingskills.

These students: These students: These students: These students:• are unable to use English to • know enough English to explain • use predominantly grade- • use English at a level of

explain self-generated writing briefly and simply self-generated appropriate English to explain, in complexity and detail nearly(e.g., stories they have created writing, including emergent forms some detail, most self-generated comparable to that of nativeor other personal expressions), of writing, as long as the topic is writing, including emergent forms English-speaking peers whenincluding emergent forms of highly familiar and concrete and of writing explaining self-generated writing,writing (pictures, letter-like forms,mock words, scribbling, etc.)

requires very high-frequencyEnglish

• can participate meaningfully, withsecond language acquisition

including emergent forms ofwriting

• know too little English to • can participate meaningfully in support, in most grade-appropriate • can participate meaningfully inparticipate meaningfully in grade- grade-appropriate shared writing shared writing activities using the most grade-appropriate sharedappropriate shared writing activities using the English English language writing activities using the Englishactivities using the Englishlanguage

• cannot express themselvesmeaningfully in self-generated,

language only when the writingtopic is highly familiar andconcrete and requires very high-frequency English

• although second languageacquisition support is needed,have an emerging ability toexpress themselves in self-

language

• although minimal second language acquisition support maybe needed, express themselves in

connected written text in English • express themselves meaningfully generated, connected written text self-generated, connected writtenbeyond the level of high-frequency, in self-generated, connected in English in a grade-appropriate text in English in a manner nearlyconcrete words, phrases, or short written text in English when their manner * comparable to their native English-sentences that have been recentlypracticed/memorized *

• may demonstrate little or noawareness of English print

writing is limited to shortsentences featuring simple,concrete English used frequentlyin class *

• occasionally exhibit secondlanguage acquisition errors whenwriting in English *

speaking peers *

conventions • frequently exhibit features of theirprimary language when writing inEnglish (e.g., primary languagewords, spelling patterns, wordorder, literal translating) *

* These descriptors apply only to students who are at the developmental stage of generating original written text using a standardwriting system.

TELPAS WRITING




TE

LPA

S P

rofi

cien

cy L

evel

Des

crip

tors

Gra

des

K–1

2 L

iste

nin

g

Beg

inni

ngIn

term

edia

teA

dvan

ced

Adv

ance

d H

igh

Beg

inni

ng E

nglis

h la

ngua

ge

lear

ners

(E

LLs)

hav

e lit

tle

or n

oab

ility

to

unde

rsta

nd s

poke

nE

nglis

h us

ed in

aca

dem

ic a

ndso

cial

set

ting

s.

Inte

rmed

iate

ELL

s ha

ve t

he

abili

ty t

o un

ders

tand

sim

ple,

high

-fre

quen

cy s

poke

n E

nglis

hus

ed in

rou

tine

aca

dem

ic a

ndso

cial

set

ting

s.

Adv

ance

d E

LLs

have

the

abi

lity

to u

nder

stan

d,w

ith

seco

ndla

ngua

ge

acqu

isit

ion

supp

ort,

grad

e-ap

prop

riat

e sp

oken

E

nglis

h us

ed in

aca

dem

ic a

ndso

cial

set

ting

s.

Adv

ance

d hi

gh E

LLs

have

the

abili

ty t

o un

ders

tand

,wit

h m

inim

al s

econ

d la

ngua

ge

acqu

isit

ion

supp

ort,

grad

e-ap

prop

riat

e sp

oken

Eng

lish

used

in a

cade

mic

and

soc

ial s

etti

ngs.

Thes

e st

uden

ts:

Thes

e st

uden

ts:

Thes

e st

uden

ts:

Thes

e st

uden

ts:

•st

rugg

le t

o un

ders

tand

sim

ple

•us

ually

und

erst

and

sim

ple

or•

usua

lly u

nder

stan

d lo

nger

, m

ore

•un

ders

tand

long

er,

elab

orat

edco

nver

satio

ns a

nd s

impl

ero

utin

e di

rect

ions

, as

wel

l as

shor

t,el

abor

ated

dire

ctio

ns,

dire

ctio

ns,

conv

ersa

tions

, an

ddi

scus

sion

s ev

en w

hen

the

topi

cssi

mpl

e co

nver

satio

ns a

nd s

hort

,co

nver

satio

ns,

and

disc

ussi

ons

ondi

scus

sion

s on

fam

iliar

and

are

fam

iliar

and

the

spe

aker

use

ssi

mpl

e di

scus

sion

s on

fam

iliar

fam

iliar

and

som

e un

fam

iliar

unfa

mili

ar t

opic

s w

ith o

nly

lingu

istic

sup

port

s (e

.g.,

visu

als,

topi

cs;w

hen

topi

cs a

re u

nfam

iliar

,to

pics

, bu

t so

met

imes

nee

doc

casi

onal

nee

d fo

r pr

oces

sing

slow

er s

peec

h an

d ot

her

verb

alre

quire

ext

ensi

ve li

ngui

stic

proc

essi

ng t

ime

and

som

etim

estim

e an

d w

ith li

ttle

depe

nden

ce o

ncu

es,

gest

ures

)su

ppor

ts a

nd a

dapt

atio

ns (

e.g.

,de

pend

on

visu

als,

ver

bal c

ues,

visu

als,

ver

bal c

ues,

and

ges

ture

s;

•st

rugg

le t

o id

entif

y an

d di

stin

guis

hin

divi

dual

wor

ds a

nd p

hras

esdu

ring

soci

al a

nd in

stru

ctio

nal

inte

ract

ions

tha

t ha

ve n

ot b

een

visu

als,

slo

wer

spe

ech

and

othe

rve

rbal

cue

s, s

impl

ified

lang

uage

,ge

stur

es,

pret

each

ing

to p

revi

ewor

bui

ld t

opic

-rel

ated

voc

abul

ary)

and

gest

ures

to

supp

ort

unde

rsta

ndin

g

•un

ders

tand

mos

t m

ain

poin

ts,

mos

t im

port

ant

deta

ils,

and

som

e

som

e ex

cept

ions

whe

n co

mpl

exac

adem

ic o

r hi

ghly

spe

cial

ized

lang

uage

is u

sed

•un

ders

tand

mai

n po

ints

, im

port

ant

inte

ntio

nally

mod

ified

for

ELL

s•

ofte

n id

entif

y an

d di

stin

guis

h ke

yim

plic

it in

form

atio

n du

ring

soci

alde

tails

, an

d im

plic

it in

form

atio

n at

•m

ay n

ot s

eek

clar

ifica

tion

inE

nglis

h w

hen

faili

ng t

oco

mpr

ehen

d th

e E

nglis

h th

ey h

ear;

freq

uent

ly r

emai

n si

lent

, w

atch

ing

othe

rs fo

r cu

es

wor

ds a

nd p

hras

es n

eces

sary

to

unde

rsta

nd t

he g

ener

al m

eani

ng(g

ist)

dur

ing

soci

al a

nd b

asic

inst

ruct

iona

l int

erac

tions

tha

t ha

veno

t be

en in

tent

iona

lly m

odifi

ed fo

rE

LLs

and

basi

c in

stru

ctio

nal i

nter

actio

nsth

at h

ave

not

been

inte

ntio

nally

mod

ified

for

ELL

s

•oc

casi

onal

ly r

equi

re/r

eque

st t

hesp

eake

r to

rep

eat,

slow

dow

n, o

rre

phra

se t

o cl

arify

the

mea

ning

of

a le

vel n

early

com

para

ble

to n

ativ

eE

nglis

h-sp

eaki

ng p

eers

dur

ing

soci

al a

nd in

stru

ctio

nal

inte

ract

ions

•ra

rely

req

uire

/req

uest

the

spe

aker

to r

epea

t, sl

ow d

own,

or

reph

rase

•ha

ve t

he a

bilit

y to

see

k cl

arifi

catio

n in

Eng

lish

whe

n fa

iling

to c

ompr

ehen

d th

e E

nglis

h th

eyhe

ar b

y re

quiri

ng/r

eque

stin

g th

esp

eake

r to

rep

eat,

slow

dow

n, o

rre

phra

se s

peec

h

the

Eng

lish

they

hea

rto

cla

rify

the

mea

ning

of

the

Eng

lish

they

hea

r

©2008 TEX

AS ED

UC

ATION

AG

ENC

Y A

LL RIG

HTS R

ESERVED

.

TELPAS Proficiency Level DescriptorsGrades 2–12 Writing


Beginning English language learners(ELLs) lack the English vocabulary andgrasp of English language structuresnecessary to address grade-appropriatewriting tasks meaningfully.

Intermediate ELLs have enough Englishvocabulary and enough grasp of Englishlanguage structures to address grade-appropriate writing tasks in a limited way.

Advanced ELLs have enough Englishvocabulary and command of Englishlanguage structures to address grade-appropriate writing tasks, althoughsecond language acquisition support isneeded.

Advanced high ELLs have acquired theEnglish vocabulary and command ofEnglish language structures necessary toaddress grade-appropriate writing taskswith minimal second languageacquisition support.

These students: These students: These students: These students:• have little or no ability to use the English • have a limited ability to use the English • are able to use the English language, with • are able to use the English language, with

language to express ideas in writing and language to express ideas in writing and second language acquisition support, to minimal second language acquisitionengage meaningfully in grade-appropriate engage meaningfully in grade-appropriate express ideas in writing and engage support, to express ideas in writing andwriting assignments in content area writing assignments in content area meaningfully in grade-appropriate writing engage meaningfully in grade-appropriateinstruction instruction assignments in content area instruction writing assignments in content area

• lack the English necessary to develop or • are limited in their ability to develop or • know enough English to be able to instructiondemonstrate elements of grade- demonstrate elements of grade-appropriate develop or demonstrate elements of • know enough English to be able toappropriate writing (e.g., focus and writing in English; communicate best when grade-appropriate writing in English, develop or demonstrate, with minimalcoherence, conventions, organization, topics are highly familiar and concrete, and although second language acquisition second language acquisition support,voice, and development of ideas) in require simple, high-frequency English support is particularly needed when topics elements of grade-appropriate writing English

Typical writing features at this level:• ability to label, list, and copy

Typical writing features at this level:• simple, original messages consisting of

short, simple sentences; frequent

are abstract, academically challenging, orunfamiliar

Typical writing features at this level:

in English

Typical writing features at this level:• nearly comparable to writing of native

• high-frequency words/phrases and short, inaccuracies occur when creating or taking • grasp of basic verbs, tenses, grammar English-speaking peers in clarity andsimple sentences (or even short risks beyond familiar English features, and sentence patterns; partial precision with regard to Englishparagraphs) based primarily on recently • high-frequency vocabulary; academic grasp of more complex verbs, tenses, vocabulary and language structures, withpracticed, memorized, or highly familiar writing often has an oral tone grammar features, and sentence patterns occasional exceptions when writing aboutmaterial; this type of writing may be quiteaccurate

• present tense used primarily

• loosely connected text with limited use ofcohesive devices or repetitive use, whichmay cause gaps in meaning

• emerging grade-appropriate vocabulary;academic writing has a more academictone

academically complex ideas, abstractideas, or topics requiring low-frequencyvocabulary

• frequent primary language features(spelling patterns, word order, literal

• repetition of ideas due to lack of vocabularyand language structures

• use of a variety of common cohesivedevices, although some redundancy may

• occasional difficulty with naturalness ofphrasing and expression

translations, and words from the student’sprimary language) and other errorsassociated with second languageacquisition may significantly hinder or

• present tense used most accurately; simplefuture and past tenses, if attempted, areused inconsistently or with frequentinaccuracies

occur• narrations, explanations, and descriptions

developed in some detail with emergingclarity; quality or quantity declines when

• errors associated with second languageacquisition are minor and usually limitedto low-frequency words and structures;errors rarely interfere with communication

prevent understanding, even forindividuals accustomed to the writing ofELLs

• descriptions, explanations, and narrationslacking detail; difficulty expressing abstractideas

abstract ideas are expressed, academicdemands are high, or low-frequencyvocabulary is required

• primary language features and errorsassociated with second languageacquisition may be frequent

• some writing may be understood only byindividuals accustomed to the writing ofELLs; parts of the writing may be hard tounderstand even for individuals accustomedto the writing of ELLs

• occasional second language acquisitionerrors

• communications are usually understoodby individuals not accustomed to thewriting of ELLs

TELPAS WRITING


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Reaching for Common Ground

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OCTOBER 2005 NOTICES OF THE AMS 1055

Reaching for CommonGround in K–12

Mathematics EducationDeborah Loewenberg Ball, Joan Ferrini-Mundy,

Jeremy Kilpatrick, R. James Milgram, Wilfried Schmid, and Richard Schaar

Over the past decade, much debate has arisen be-tween mathematicians and mathematics educa-tors. These debates have significantly distracted theattention of key players at all levels and have im-peded efforts to improve mathematics learning inthis country. This document represents an attemptto identify a preliminary list of positions on whichmany may be able to agree.

Our effort arose out of discussions betweenRichard Schaar and major players in both com-munities. He suspected that some of these dis-agreements might be more matters of languageand lack of communication than representative offundamental differences of view. To test this idea,he convened a small group of mathematicians andmathematics educators.1

We tried to bring clarity to key perspectives onK–12 mathematics education. We began by ex-ploring typical “flashpoint” topics and probed ourown positions on each of these to determinewhether and where we agreed or disagreed. For thefirst meeting, held in December 2004, we began withsummary statements drawn from prior exchangesamong the members of our group. We affirmedsome agreements in this meeting and “discovered”others. We listened closely to one another, fre-quently asking for clarification or for examples. Wetested our understanding of others’ points of viewby proposing statements that we then examined col-lectively. We drafted this document as a group,composing actual text as we worked. One of us

typed, and our emerging draft was projected ontoa screen in the meeting room. The process enabledus to take issue with particular words and termsand then reshape them until all of us were satis-fied. We were forced to look closely at our own lan-guage and to seek common ground, not only in theterms we used but even in their nuanced meaning.

This document was completed at our secondmeeting, in June 2005. All of us are encouraged bythe extent of our agreement. The document treatsonly a subset of the controversial issues, many ofwhich arise in K–8 mathematics. We expect to con-tinue the process by examining a wider range ofmajor issues in mathematics education. We havenecessarily limited ourselves to questions

1We are grateful to the National Science Foundation andTexas Instruments, Inc. for funding this portion of our work.

Deborah Lowewnberg Ball is the William H. Payne Colle-giate Professor of Mathematics Education and Teacher Ed-ucation and director of teacher education at the Univer-sity of Michigan. Her email address is [email protected].

Joan Ferrini-Mundy is associate dean for science and math-ematics education and outreach at Michigan State Uni-versity. Her email address is [email protected].

Jeremy Kilpatrick is Regents Professor of mathematics education at the University of Georgia. His email addressis [email protected].

R. James Milgram is professor of mathematics at StanfordUniversity. His email address is [email protected].

Wilfried Schmid is the Dwight Parker Robinson Professorof Mathematics at Harvard University. His email addressis [email protected].

Richard Schaar is executive advisor at Texas Instruments,Inc. His email address is [email protected].

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depending primarily on disciplinary judgment, asopposed to those requiring empirical evidence.

We begin with three fundamental assertions andcontinue with a list of areas in which we found com-mon ground. For each, we have written a shortparagraph that captures the fundamental points ofour agreement. Our next step is to explore how oth-ers respond to the document and to use their re-sponses to decide how best to make progress onthe aims of this project. Our goal is to forge newalliances, across communities, necessary to de-velop effective solutions to the serious problemsthat plague mathematics education in this country.

Fundamental PremisesAll students must have a solid grounding in math-ematics to function effectively in today’s world. Theneed to improve the learning of traditionally un-derserved groups of students is widely recognized;efforts to do so must continue. Students in the topquartile are underserved in different ways; atten-tion to improving the quality of their learning op-portunities is equally important. Expectations forall groups of students must be raised. By the timethey leave high school, a majority of studentsshould have studied calculus.1. Basic skills with numbers continue to be vitally

important for a variety of everyday uses. Theyalso provide a crucial foundation for the higher-level mathematics essential for success in theworkplace which must now also be part of a basiceducation. Although there may have been a timewhen being able to perform extensive paper-and-pencil computations mechanically was suf-ficient to function in the workplace, this is nolonger true. Consequently, today’s students needproficiency with computational procedures. Pro-ficiency, as we use the term, includes both com-putational fluency and understanding of the un-derlying mathematical ideas and principles.2

2. Mathematics requires careful reasoning aboutprecisely defined objects and concepts. Mathe-matics is communicated by means of a power-ful language whose vocabulary must be learned.The ability to reason about and justify mathe-matical statements is fundamental, as is theability to use terms and notation with appro-priate degrees of precision. By precision, wemean the use of terms and symbols, consistentwith mathematical definitions, in ways appro-priate for students at particular grade levels. Wedo not mean formality for formality’s sake.

3. Students must be able to formulate and solveproblems. Mathematical problem solving includesbeing able to (a) develop a clear understanding

of the problem that is being posed, (b) translatethe problem from everyday language into a pre-cise mathematical question, (c) choose and useappropriate methods to answer the question, (d)interpret and evaluate the solution in terms ofthe original problem, and (e) understand that notall questions admit mathematical solutions andrecognize problems that cannot be solved math-ematically.

Areas of AgreementDiscussions of the following items are often rid-dled with difficulties in communication, making itsometimes confusing to determine whether andhow much disagreement exists. Issues also arisefrom a confounding of a mathematical idea withits implementation in the classroom. For example,the fact that algorithms have often been taughtbadly does not imply that algorithms themselvesare bad. We worked to clarify issues and termsand arrived at statements with which we agreed.A. Automatic recall of basic facts: Certain proce-

dures and algorithms in mathematics are sobasic and have such wide application that theyshould be practiced to the point of automatic-ity. Computational fluency in whole numberarithmetic is vital. Crucial ingredients of com-putational fluency are efficiency and accuracy.Ultimately, fluency requires automatic recall ofbasic number facts: by basic number facts wemean addition and multiplication combinationsof integers 0 through 10. This goal can be ac-complished using a variety of instructional meth-ods.

B. Calculators: Calculators can have a useful roleeven in the lower grades, but they must be usedcarefully so as not to impede the acquisition offluency with basic facts and computational pro-cedures. Inappropriate use of calculators mayalso interfere with students’ understanding ofthe meaning of fractions and their ability tocompute with fractions. Along the same lines,graphing calculators can enhance students’ un-derstanding of functions, but students must de-velop a sound idea of what graphs are and howto use them independently of the use of a graph-ing calculator.

C. Learning algorithms: Students should be able touse the basic algorithms of whole number arith-metic fluently, and they should understand howand why the algorithms work. Fluent use and un-derstanding ought to be developed concurrently.These basic algorithms were a major intellectualaccomplishment. Because they embody the struc-ture of the base-ten number system, studyingthem can reinforce students’ understanding ofthe place value system.

More generally, an algorithm is a systematicprocedure involving mathematical operations

2J. Kilpatrick, J. Swafford, and B. Findell (eds.), Adding itup: Helping children learn mathematics, Washington, DC,National Academy Press, 2001.

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that uses a finite number of steps to produce adefinite answer. An algorithm can be imple-mented in different ways; different recordingmethods for the same algorithm do not consti-tute different algorithms. The idea of an algo-rithm is fundamental in mathematics. Studyingalgorithms beyond those of whole number arith-metic provides opportunities for students toappreciate the diversity and importance of al-gorithms. Examples include constructing the bi-sector of an angle, solving two linear equationsin two unknowns, and calculating the square rootof a number by a succession of dividing and av-eraging.

D. Fractions: Understanding the number meaningof fractions is critical. Ratios, proportions, andpercentages cannot be properly understood

without fractions. The arithmetic of fractions isimportant as a foundation for algebra.

E. Teaching mathematics in “real-world” contexts:It can be helpful to motivate and introduce math-ematical ideas through applied problems. How-ever, this approach should not be elevated to ageneral principle. If all school mathematics istaught using real-world problems, then someimportant topics may not receive adequate at-tention. Teachers must choose contexts withcare. They need to manage the use of real-worldproblems or mathematical applications in waysthat focus students’ attention on the mathe-matical ideas that the problems are intended todevelop.

F. Instructional methods: Some have suggested theexclusive use of small groups or discovery

The document “Reaching for common ground in K–12 mathematics education” had its origin in conversationsbetween myself and James Milgram, professor of mathematics at Stanford University. While I am a mathematician,in the late 1970s I left academic life for the business world. After serving as president of the business unit at TexasInstruments, Inc. that developed graphing calculators for mathematical education use and as a senior vice presi-dent of the corporation, I was asked to focus on issues of high-tech workforce development which has at its coreK–12 mathematics education. Because the “math wars” are part of the K–12 landscape, I decided to meet with Jim,who, besides being a researcher in algebraic and geometric topology, has been deeply involved in these issues. Inthe 1990s, he worked with others to first write the current California Mathematics Standards and then to write theCalifornia Mathematics Framework. In addition to contributing to other state standards, he has worked with thesecretary’s office at the U.S. Department of Education on K–12 education issues.

An outcome of these discussions was the hypothesis that there might be common ground in K–12 mathematicseducation that would begin the process of changing the focus toward student outcomes instead of controversy. Toprove this hypothesis, I decided to assemble a small, yet very diverse, blue ribbon panel to develop common ele-ments of understanding. After face-to-face conversations with a number of people, in addition to James Milgram,the following people agreed to join the group.

Deborah Loewenberg Ball is William H. Payne Collegiate Professor of Mathematics Education and Teacher Edu-cation and director of teacher education at the University of Michigan. Ball’s work focuses on studies of instruc-tion and of the processes of learning to teach. She directs the Mathematics Teaching and Learning to Teach Pro-ject, which studies classroom teaching and analyzes the mathematical demands of the work, and several researchprojects that investigate efforts to improve teaching through policy, reform initiatives, and teacher education.

Joan Ferrini-Mundy is a University Distinguished Professor and also associate dean for science and mathemat-ics education in the College of Natural Science at Michigan State University. Her faculty appointments are in math-ematics and teacher education. She chaired the Writing Group for the National Council of Teachers of Mathemat-ics’ (NCTM) 2000 Principles and Standards for School Mathematics. She directs PROM/SE, an NSF comprehensivemathematics and science partnership, studying the improvement of mathematics and science teaching and learn-ing in sixty-two school districts.

Jeremy Kilpatrick is Regents Professor of mathematics education at the University of Georgia. He chaired the Na-tional Research Council’s Committee on Mathematics Learning, which produced the 2001 report Adding it up. Healso served on the RAND Mathematics Study Panel, which produced Mathematical proficiency for all students in2002.

Wilfried Schmid is the Dwight Parker Robinson Professor of Mathematics at Harvard University. After becominginvolved in mathematics education because of his daughter, a second grader at the time, Wilfried was the lead math-ematician in the development of the Massachusetts Mathematics Framework, which was adopted in 2000. Since thattime, he has served on the National Assessment of Educational Progress Steering Committee and was the repre-sentative of the International Union of Mathematicians on the program committee of the 10th International Con-gress on Mathematical Education.

As the document indicates, after two sessions, the group was able to coalesce around some very specific prin-ciples which should serve as common ground.

—Richard Schaar

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learning at the expense of direct instruction inteaching mathematics. Students can learn ef-fectively via a mixture of direct instruction,structured investigation, and open exploration.Decisions about what is better taught throughdirect instruction and what might be bettertaught by structuring explorations for studentsshould be made on the basis of the particularmathematics, the goals for learning, and thestudents’ present skills and knowledge. For ex-ample, mathematical conventions and defini-tions should not be taught by pure discovery.Correct mathematical understanding and con-clusions are the responsibility of the teacher.Making good decisions about the appropriatepedagogy to use depends on teachers havingsolid knowledge of the subject.

G. Teacher knowledge: Teaching mathematics ef-fectively depends on a solid understanding ofthe material. Teachers must be able to do themathematics they are teaching, but that is notsufficient knowledge for teaching. Effectiveteaching requires an understanding of the un-derlying meaning and justifications for the ideasand procedures to be taught and the ability tomake connections among topics. Fluency, ac-curacy, and precision in the use of mathemati-cal terms and symbolic notation are also crucial.Teaching demands knowing appropriate repre-sentations for a particular mathematical idea, de-ploying these with precision, and bridging be-tween teachers’ and students’ understanding. Itrequires judgment about how to reduce math-ematical complexity and manage precision inways that make the mathematics accessible tostudents while preserving its integrity.

Well-designed instructional materials, such astextbooks, teachers’ manuals, and software, mayprovide significant mathematical support, butthey cannot substitute for highly qualified,knowledgeable teachers. Teachers’ mathemati-cal knowledge must be developed through solidinitial teacher preparation and ongoing, sys-tematic professional learning opportunities.

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