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School Leaders, Pre-K - 5

Understanding and Supporting Teachers’ Enactment of the Effective

Teaching PracticesSession 1

Diane J. Briars

Immediate Past President

National Council of Teachers of Mathematics

dbriars@nctm.org

• Increase your understanding of effective teaching practices to promote high quality mathematics learning by all students and how to support teachers successful implementation of these practices.

• Prepare you to use the principles and practices in Principles to Actions: Ensuring Mathematical Success for All as a framework for improving mathematics teaching and learning in your school or district.

Our Work

Sessions 1, 2, & 3: Teaching and Learning: Understanding the Eight Effective Mathematics Teaching Practices

Session 4: Professionalism and Assessment

Session 5: Action planning

Introductions

• Name

• District

• Role

• 1 challenge/need you’re trying to address

Effective Instructional Leaders:

• Make high achievement by allstudents a priority.

• Recognize and value high quality content, instruction and assessment.

• Focus: Focus strongly where the

standards focus.

• Coherence: Think across grades, and link

to major topics

• Rigor: In major topics, pursue conceptual

understanding, procedural skill and

fluency, and application

Key Features of CCSS-M

• Focus: Focus strongly where the

standards focus.

• Coherence: Think across grades, and link

to major topics

understanding, procedural skill and

fluency, and application

Curriculum Standards, Not Assessment Standards

Understand and apply properties of operations and the relationship between addition and subtraction. (1.OA)

3. Apply properties of operations as strategies to add and subtract. Examples: If 8 + 3 = 11 is known, then 3 + 8 = 11 is also known. (Commutative property of addition.) To add 2 + 6 + 4, the second two numbers can be added to make a ten, so 2 + 6 + 4 = 2 + 10 = 12. (Associative property of addition.)

4. Understand subtraction as an unknown-addend problem. For example, subtract 10 – 8 by finding the number that makes 10 when added to 8.

Curriculum Standards, Not Assessment Standards

Develop understanding of fractions as numbers.(3.NF)

1. Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b.

2. Understand a fraction as a number on the number line; represent fractions on a number line diagram.

3. Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size.

a. Understand two fractions as equivalent (equal) if they are the same size, or the same point on a number line.

Skill-algorithm understandingfrom the rote application of an algorithm through the selection and comparison of algorithms to

the invention of new algorithms (calculators and computers included)

Property-proof understandingfrom the rote justification of a property through the derivation of properties to the proofs of new

properties

Use-application understandingfrom the rote application of mathematics in the real world through the use of mathematical

models to the invention of new models

Representation-metaphor understandingfrom the rote representations of mathematical ideas through the analysis of such representations

to the invention of new representations

Vocabulary

Problem Solving

Zalman Usiskin, 2012

Dimensions of Mathematical Understanding in CCSS

Why Focus on Understanding?

• Understanding facilitates initial learning and retention.

• Understanding supports appropriate application and transfer.

Phil Daro, 2010

Other “Butterflies”?

• FOIL

• Cross multiplication

• Division of fractions: Keep-change-flip, KFC, etc.

• a ─ b = a + -b: Keep-change-change

• Key words

• Division algorithm: Does McDonalds Sell Cheese Burgers?

Key Instructional Shift

From emphasis on:

How to get answers

To emphasis on:

Understanding mathematics

Guiding Principles for School Mathematics

1. Teaching and Learning

Effective teaching is the non-negotiable core that ensures that all students learn

mathematics at high levels.

We Must Focus on Instruction

Student learning of mathematics “depends fundamentally on what happens inside the classroom as teachers and learners interact over the curriculum.”

(Ball & Forzani, 2011, p. 17)

“Teaching has 6 to 10 times as much impact on achievement as all other factors combined ... Just three years of effective teaching accounts on average for an improvement of 35 to 50 percentile points.”

Schmoker (2006, p.9)

Common Core State Standards for Mathematics

Two type of standards:

• Standards for Mathematical Practice

• Standards for Mathematical Content

Common Core State Standards for Mathematics

Two type of standards:

• Standards for Mathematical Content

Standards for Mathematical Practice

1. Make sense of problems and persevere in solving them.

2. Reason abstractly and quantitatively.

3. Construct viable arguments and critique the reasoning of others.

4. Model with mathematics.

5. Use appropriate tools strategically.

6. Attend to precision.

7. Look for and make use of structure.

8. Look for and express regularity in repeated reasoning.

How familiar are you with the Standards for Mathematical Practice?

Rate your familiarity on a scale of 1 to 6.

Standards for Mathematical Practice K-5 Elaborations

• Read your assigned practice

• What are the key student expectations/outcomes of your practice?

• What would you “look for” to see this in the classroom—teacher actions/students actions.

To what extent are each and every student in your school/district achieving proficiency in your

practice?

• Describe the thinking processes, habits of mind and dispositions that students need to develop a deep, flexible, and enduring understanding of mathematics; in this sense they are also a means to an end.

SP1. Make sense of problems

“….they [students] analyze givens, constraints, relationships and goals. ….they monitor and evaluate their progress and change course if necessary. …. and they continually ask themselves “Does this make sense?”

AND….

• Describe mathematical content students need to learn. SP1. Make sense of problems

“……. students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends.”

Developing Proficiency in the Standards for Mathematical Practice Requires

Explicit attention to development of students’ proficiency in the Standards for Mathematical

Practice.

Diane J. Briars, January 2014

• Focus: Focus strongly where the standards

focus.

• Coherence: Think across grades, and link to

major topics

understanding, procedural skill and fluency,

and application

• Focus: Focus strongly where the standards

focus.

• Coherence: Think across grades, and link to

major topics

understanding, procedural skill and fluency,

and application

Discuss at Your Table

What does it mean to be fluent with computational procedures?

What is Procedural Fluency?

Which students demonstrate procedural fluency? Evidence for your answer?

• Alan

• Ana

• Marissa

1000 - 98 100 - 18

https://mathreasoninginventory.com/Home/VideoLibrary

Source: The Marilyn Burns Math Reasoning Inventory

1000 - 98 99 + 17

https://mathreasoninginventory.com/Home/VideoLibrary

What is Procedural Fluency?

Which students demonstrate procedural fluency? Evidence for your answer?

• Alan

• Ana

Marissa

295 students, 25 on each bus

Procedural Fluency

• Efficiency—can carry out easily, keep track of subproblems, and make use of intermediate results to solve the problem.

• Accuracy—reliably produces the correct answer.

• Flexibility—knows more than one approach, chooses appropriate strategy, and can use one method to solve and another method to double-check.

• Appropriately—knows when to apply a particular procedure.

Read & Discuss:Developing Computational Fluency with Whole Numbers

• As you read, identify key ideas

• Discuss: First word/last word protocol

– First person shares one key idea without comments;

– Each of the other people comment on this idea;

– First person comments on the discussion

– Repeat for each person at the table.

What Research Tells Us

• When procedures are connected with the underlying concepts, students have better retention of the procedures and are more able to apply them in new situations

• Informal methods general methods formal algorithms is more effective than rote instruction.

• Engaging students in solving challenging problems is essential to build conceptual understanding.

Developing Procedural Fluency

1. Develop conceptual understanding building on students’ informal knowledge

2. Develop informal strategies to solve problems

3. Refine informal strategies to develop fluency with standard methods and procedures (algorithms)

Developing Proficiency in the Standards for Mathematical Practice Requires

Instructional practices that promote students’ development of conceptual

understanding, procedural fluency, and proficiency in the Standards for

Mathematical Practice.

Teaching and LearningAn excellent mathematics program requires effective teaching that engages students in meaningful learning through individual and collaborative experiences that promote their

ability to make sense of mathematical ideas and reason mathematically.

EffectiveMathematics Teaching Practices

1. Establish mathematics goals to focus learning.

2. Implement tasks that promote reasoning and problem solving.

3. Use and connect mathematical representations.

4. Facilitate meaningful mathematical discourse.

5. Pose purposeful questions.

6. Build procedural fluency from conceptual understanding.

7. Support productive struggle in learning mathematics.

8. Elicit and use evidence of student thinking.

The Band Concert

The third-grade class is responsible for setting up the chairs for their spring band concert. In preparation, they need to determine the total number of chairs that will be needed and ask the school’s engineer to retrieve that many chairs from the central storage area.

The class needs to set up 7 rows of chairs with 20 chairs in each row, leaving space for a center aisle.

How many chairs does the school’s engineer need to retrieve from the central storage area?

The Case of Mr. Harris and the Band Concert Task

• Read the Case of Mr. Harris and study the strategies used by his students.

• Make note of what Mr. Harris did before or during instruction to support his students’ developing understanding of multiplication.

• Talk with the people at your table about the “Teaching Practices” Mr. Harris is using and how they support students’ progress in their learning.

• Be prepared to share the Teaching Practices that you discussed.

Relating the Case to the Effective Mathematics Teaching Practices

Establish Mathematics Goals To Focus Learning

Learning Goals should:

• Clearly state what it is students are to learn and understand about mathematics as the result of instruction;

• Be situated within learning progressions; and

• Frame the decisions that teachers make during a lesson.

Formulating clear, explicit learning goals sets the stage for everything else.

(Hiebert, Morris, Berk, & Janssen, 2007, p.57)

Mr. Harris’ Math Goals

Students will recognize the structure of multiplication as equal groups within and among different representations, focusing on identifying the number of equal groups and the size of each group within collections or arrays.

Student-friendly version ...

We are learning to represent and solve word problems and explain how different representations match the story situation and the math operations.

Standard 3.OA.3. Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.

Standard 3.NBT. 3. Multiply one-digit whole numbers by multiples of 10 in the range 10–90 (e.g., 9 x 80, 5 x 60) using strategies based on place value and properties of operations.

Alignment to the Common Core State Standards

• In what ways did Mr. Harris’ math goal focus his

interactions with students throughout the lesson?

Consider Case Lines 4-9, 21-24, 27-29.

Mr. Harris’ Goal: Students will recognize the structure of multiplication as equal groups within and among different representations, focusing on identifying the number of equal groups and the size of each group within collections or arrays.

Establish Mathematics Goals To Focus Learning

Implement Tasks that Promote Reasoning and Problem Solving

Mathematical tasks should:

• Provide opportunities for students to engage in exploration or encourage students to use procedures in ways that are connected to concepts and understanding;

• Build on students’ current understanding; and

• Have multiple entry points.

Why Tasks Matter

• Tasks form the basis for students’ opportunities to learn what mathematics is and how one does it;

• Tasks influence learners by directing their attention to particular aspects of content and by specifying ways to process information;

• The level and kind of thinking required by mathematical instructional tasks influences what students learn; and

• Differences in the level and kind of thinking of tasks used by different teachers, schools, and districts, is a major source of inequity in students’ opportunities to learn mathematics.

Why Tasks Matter

Compare Band Concert to:

Find the product

5 x 20 =

6 x 80 =

4 x 70 =

3 x 50 =

9 x 20 =

2 x 60 =

8 x 30 =

• How is the Band Concert task similar to or different from the find the product problems?

• Which one is more likely to promote problem solving?

Principles to Actions, p. 24

Core Instructional Issue

Do all students have the opportunity to engage in mathematical tasks that promote students’ attainment of the mathematical practices on a regular

basis?

Disclaimer

The National Council of Teachers of Mathematics is a public voice of mathematics education, providing vision, leadership, and professional development to support teachers in ensuring equitable mathematics learning of the highest quality for all students. NCTM’s Institutes, an official professional development offering of the National Council of Teachers of Mathematics, supports the improvement of pre-K-6 mathematics education by serving as a resource for teachers so as to provide more and better mathematics for all students. It is a forum for the exchange of mathematics ideas, activities, and pedagogical strategies, and for sharing and interpreting research. The Institutes presented by the Council present a variety of viewpoints. The views expressed or implied in the Institutes, unless otherwise noted, should not be interpreted as official positions of the Council.

Diane J. Briars

dbriars@nctm.org

Principle on Teaching and Learning

An excellent mathematics program requires

effective teaching that engages students in

meaningful learning through individual and

collaborative experiences that promote their

ability to make sense of mathematical ideas

and reason mathematically.

Use and Connect Mathematical Representations

Different Representations should:

• Be introduced, discussed, and connected;

• Focus students’ attention on the structure or essential features of mathematical ideas; and

• Support students’ ability to justify and explain their reasoning.

Strengthening the ability to move between and among these representations improves the growth of children’s concepts.

Lesh, Post, Behr, 1987

Contextual

Physical

Visual

Symbolic

Verbal

Important Mathematical Connections between and withindifferent types of representations

Principles to Actions (NCTM, 2014, p. 25)

(Adapted from Lesh, Post, & Behr, 1987)

What mathematical representations were students working with in the lesson?

How did Mr. Harris support students in making connections between andwithin different types of representations?

How did students benefit from making these connections?

Contextual

Physical

Visual

Symbolic

Verbal

Consider Lines 43-48. In what ways did comparing representations strengthen the understanding of these students?

Jasmine Kenneth

Consider Lines 48-49. How did comparing representations benefit Molly?

Mathematical Discourse should:

• Build on and honor students’ thinking.

• Let students share ideas, clarify understandings, and develop convincing arguments.

• Engage students in analyzing and comparing student approaches.

• Advance the math learning of the whole class.

Facilitate Meaningful Mathematical Discourse

Facilitate MeaningfulMathematical Discourse

Discussions that focus on cognitively challenging mathematical tasks, namely

those that promote thinking, reasoning, and problem solving, are a primary mechanism for promoting conceptual understanding of

mathematics (Hatano & Inagaki, 1991; Michaels, O’Connor, & Resnick, 2008).

Smith, Hughes, Engle & Stein, 2009, p. 549

Meaningful Discourse

What did Mr. Harris do (before or during the discussion) that may have

positioned him to engage students in a productive discussion?

Structuring Mathematical Discourse...

During the whole class discussion

of the task, Mr. Harris was strategic in:

• Selecting specific student representations and strategies for discussion and analysis.

• Sequencing the various student approaches for analysis and comparison.

• Connecting student approaches to key math ideas and relationships.

Consider Lines 52-57. Why did Mr. Harris select and sequence the work of these three

students and how did that support student learning?

JasmineKenneth

Teresa

1. Anticipating

2. Monitoring

3. Selecting

4. Sequencing

5. Connecting

5 Practices for OrchestratingProductive MathematicsDiscussions

(Smith & Stein, 2011)

Planning with the Student in Mind

• Anticipate solutions, thoughts, and responses that students might develop as they struggle with the problem.

• Generate questions that could be asked to promote student thinking during the lesson, and consider the kinds of guidance that could be given to students who showed one or another types of misconception in their thinking.

• Determine how to end the lesson so as to advance students’ understanding

Stigler & Hiebert, 1997

Pose Purposeful Questions

Effective Questions should:

• Reveal students’ current understandings;

• Encourage students to explain, elaborate, or clarify their thinking; and

• Make the mathematics more visible and accessible for student examination and discussion.

Teachers’ questions are crucial in helping students make connections and learn

important mathematics and science concepts. Teachers need to know how students typically

think about particular concepts, how to determine what a particular student or group of students thinks about those ideas, and how to help students deepen their understanding.

Weiss & Pasley, 2004

In what ways did Mr. Harris’ questioning supports students’

learning about important mathematical ideas and relationships?

Lines 33-36

“How does your drawing show 7 rows?”

“How does your drawing show that there are 20 chairs in each row?

“How many twenties are you adding, and why?”

“Why are you adding all those twenties?

Purposeful Questions

Math Learning Goal

Students will recognize the structure of multiplication as equal

groups within and among different representations—identify the

number of equal groups and the size of each group within

collections or arrays.

• How did you get that?

• How do you know that?

• Can you explain your idea?

• Why?

• Can you convince us?

• Did anyone get something else?

• Can someone tell me or share with me another way?

• Do you think that means the same things?

• Is there another opinion about this?

• Why did you say that, Justin?

Boaler, J., & Brodie, K. (2004)

Build Procedural Fluency from Conceptual Understanding

Procedural Fluency should:

• Build on a foundation of conceptual understanding;

• Result in generalized methods for solving problems; and

• Enable students to flexibly choose among methods to solve contextual and mathematical problems.

Students must be able to do much more than carry out mathematical procedures.

They must know which procedure is appropriate and most productive in a given situation, what a procedure accomplishes,

and what kind of results to expect. Mechanical execution of procedures

without understanding their mathematical basis often leads to

bizarre results.

Martin, 2009, p. 165

In what ways did this lesson develop a foundation of conceptual

understanding for building toward procedural fluency in multiplying with

multiples of ten?

What foundational understandings were students developing at each of these points in the lesson that are critical for moving toward procedural fluency?

Lines 59-69: Discussion of skip counting.

Lines 70-76: Wrote the multiplication equation.

Lines 78-81: Asked students to compare Tyrell and Ananda’s work.

Tyrell Ananda

Discuss ways to use this student work to develop informal ideas of the distributive property—how

numbers can be decomposed, combined meaningfully in parts, and then recomposed to find the total.

Tyrell Ananda

Discuss ways to use this student work to develop the understanding that 14 tens = 140 and to

meaningfully to build toward fluency in working with multiples of ten.

“Fluency builds from initial exploration and discussion of number concepts to using informal reasoning strategies based on meanings and properties of the operations to the eventual use of general methods as tools in solving problems.”

Principles to Actions (NCTM, 2014, p. 42)

Support Productive Struggle in Learning Mathematics

Productive Struggle should:

• Be considered essential to learning mathematics with understanding;

• Develop students’ capacity to persevere in the face of challenge; and

• Help students realize that they are capable of doing well in mathematics with effort.

The struggle we have in mind comes from solving problems that are within reach and grappling with key mathematical ideas that are comprehendible but not yet well formed

Hiebert et al., 1996

By struggling with important mathematics we mean the opposite of simply being presented information to be memorized or being asked only to practice what has been demonstrated.

Hiebert & Grouws, 2007, pp. 387-388

How did Mr. Harris support productive struggle among his students, individually and collectively, as they grappled with important mathematical ideas and relationships?

At which points in the lesson might Mr. Harris have consciously restrained himself from “taking over” the thinking of his students?

Elicit and Use Evidence of Student Thinking

Evidence should:

• Provide a window into students’ thinking;

• Help the teacher determine the extent to which students are reaching the math learning goals; and

• Be used to make instructional decisions during the lesson and to prepare for subsequent lessons.

Harold Asturias, 1996

Formative assessment is an essentially interactive process, in which the teacher

can find out whether what has been taught has been learned, and if not, to do something about it. Day-to-day formative assessment is one of the most powerful

ways of improving learning in the mathematics classroom.

Wiliam, 2007, pp. 1054; 1091

Identify specific places during the lesson (cite line numbers) in which Mr. Harris elicited evidence of student learning.

Discuss how he used or might use that evidence to adjust his instruction to support and extend student learning.

Throughout the lesson, Mr. Harris was eliciting and using evidence of student thinking.

Lines 33-36: Purposeful questioning as students worked individually.

Lines 43-51: Observations of student pairs discussing and comparing their representations.

Lines 59-74: Whole class discussion.

Lines 78-80: Asked students to respond in writing.

Examples of Eliciting and Using Evidence

Promoting Proficiency in the Standards for Mathematical Practice

“Not all tasks are created equal, and different

tasks will provoke different levels and kinds

of student thinking.”

Stein, Smith, Henningsen, & Silver, 2000

“The level and kind of thinking in which students engage determines what they

will learn.”Hiebert, Carpenter, Fennema, Fuson, Wearne, Murray, Oliver, & Human, 1997

What Research Tells Us

Learner’s should:

• Acquire conceptual knowledge as well as skills to enable them to organize their knowledge, transfer knowledge to new situations, and acquire new knowledge.

• Engage with challenging tasks that involve active meaning-making to build conceptual knowledge.

Heibert and Grouws, 2007

What Are Mathematical Tasks?

Mathematical tasks are a set of problems or a single complex problem the purpose of which is to focus students’ attention on a particular mathematical idea.

Why Focus on Mathematical Tasks?

Implement Tasks that Promote Reasoning and Problem Solving

Mathematical tasks should:

• Provide opportunities for students to engage in exploration or encourage students to use procedures in ways that are connected to concepts and understanding;

• Build on students’ current understanding; and

• Have multiple entry points.

The Band Concert

The third-grade class is responsible for setting up the chairs for their spring band concert. In preparation, they need to determine the total number of chairs that will be needed and ask the school’s engineer to retrieve that many chairs from the central storage area.

The class needs to set up 7 rows of chairs with 20 chairs in each row, leaving space for a center aisle.

How many chairs does the school’s engineer need to retrieve from the central storage area?

The QUASAR Project

• Assisted schools in economically disadvantaged communities to develop instructional programs that emphasize thinking, reasoning and problem solving in mathematics.

• Worked with lowest achieving middle schools in six urban sites.

• Studied the impact of high quality curricula and professional development upon student achievement.

Edward Silver, Margaret S. Smith, Mary Kay Stein

Comparing Two Mathematical Tasks

Martha was re-carpeting her bedroom which was 15 feet long and 10 feet wide. How many

square feet of carpeting will she need to purchase?

Smith, Stein, Arbaugh, Brown, and Mossgrove, 2004

Comparing Two Mathematical Tasks

Ms. Brown’s class will raise rabbits for their spring science fair. They have 24 feet of fencing with which to build a rectangular rabbit pen in which to keep the rabbits.

1. If Ms. Brown's students want their rabbits to have asmuch room as possible, how long would each of the sidesof the pen be?

2. How long would each of the sides of the pen be if theyhad only 16 feet of fencing?

3. How would you go about determining the pen with the most room for any amount of fencing? Organize your work so that someone else who reads it will understand it.

Smith, Stein, Arbaugh, Brown, and Mossgrove, 2004

Compare the Two Tasks

• Work each task.

• Share solution strategies.

• Discuss: How are Martha’s Carpeting Task and the Fencing Task the same and how are they different?

Solution Strategies: Martha’s Carpeting Task

Martha’s Carpeting TaskUsing the Area Formula

A = l x w

A = 15 x 10

A = 150 square feet

Martha’s Carpeting TaskDrawing a Picture

Solution Strategies:The Fencing Task

The Fencing TaskDiagrams on Grid Paper

The Fencing TaskUsing a Table

Length Width Perimeter Area

1 11 24 11

2 10 24 20

3 9 24 27

4 8 24 32

5 7 24 35

6 6 24 36

7 5 24 35

The Fencing TaskGraph of Length and Area

0 1 2 3 4 5 6 7 8 9 10 11 12 13

Length

The Fencing TaskUsing Calculus

A = lw

A = w(12-w)

A = 12w – w2

A’ = 12 – 2w

0 = 12 – 2w

2w = 12

Cognitive Level of Tasks

• Lower-Level Tasks

(e.g., Martha’s Carpeting Task)

• Higher-Level Tasks

(e.g., The Fencing Task)

The Quasar Project

With a Partner:

• Categorize Tasks A – L into two

categories: high level cognitive demand

and low level cognitive demand.

• Develop a list of criteria that describe

the tasks in each category.

Characterizing Tasks

Lower-Level Tasks

• Memorization

– What are the decimal equivalents for the fractions ½ and ¼?

• Procedures without connections

– Convert the fraction 3/8 to a decimal.

Higher-Level Tasks

• Procedures with connections

– Using a 10 x 10 grid, identify the decimal and percent equivalents of 3/5.

• Doing mathematics

– Shade 6 small squares in a 4 x 10 rectangle. Using the rectangle, explain how to determine:

a) The decimal part of area that is shaded;

b) The fractional part of area that is shaded.

Principles to Actions, pp 18-19

Task Analysis Guide

Lower-level Demands(Memorization)

• Reproducing previously learned facts, rules, formulas, definitions or committing them to memory

• Cannot be solved with a procedure• Have no connection to concepts or meaning

that underlie the facts rules, formulas, or definitions

Higher-level Demands(Procedures with Connections)

• Use procedure for deeper understanding of concepts

• Broad procedures connected to ideas instead narrow algorithms

• Usually represented in different ways• Require some degree of cognitive effort;

procedures may be used but not mindlessly

Lower-level Demands (Procedures without Connections)

• Are algorithmic• Require limited cognitive demand• Have no connection to the concepts or

meaning that underlie the procedure• Focus on producing correct answers instead

of understanding• Require no explanations

Higher-level Demands(Doing Mathematics)

• Require complex non-algorithmic thinking• Require students to explore and understand

the mathematics• Demand self-monitoring of one’s cognitive

process• Eequire considerable cognitive effort and may

involve some level of anxiety b/c solution path isn’t clear

NCTM (2014) Principles to Action., pp 18-19

Level of Cognitive Demand

Is different than:

• Difficulty

• Importance

Implementation Issue

Do all students have the opportunity to engage in mathematical tasks that

promote reasoning and problem solving on a regular basis?

Opportunities for all students to engage in challenging tasks?

• Examine tasks in your instructional materials:

– Higher cognitive demand?

– Lower cognitive demand?

• Where are the challenging tasks?

• Do all students have the opportunity to grapple with challenging tasks?

• Examine the tasks in your assessments:

– Higher cognitive demand?

– Lower cognitive demand?

Disclaimer

Diane J. Briars

dbriars@nctm.org

Task Analysis Guide

Lower-level Demands(Memorization)

• Reproducing previously learned facts, rules, formulas, definitions or committing them to memory

• Cannot be solved with a procedure• Have no connection to concepts or meaning

that underlie the facts rules, formulas, or definitions

Higher-level Demands(Procedures with Connections)

• Use procedure for deeper understanding of concepts

• Broad procedures connected to ideas instead narrow algorithms

• Usually represented in different ways• Require some degree of cognitive effort;

procedures may be used but not mindlessly

Lower-level Demands (Procedures without Connections)

• Are algorithmic• Require limited cognitive demand• Have no connection to the concepts or

meaning that underlie the procedure• Focus on producing correct answers instead

of understanding• Require no explanations

Higher-level Demands(Doing Mathematics)

• Require complex non-algorithmic thinking• Require students to explore and understand

the mathematics• Demand self-monitoring of one’s cognitive

process• Eequire considerable cognitive effort and may

involve some level of anxiety b/c solution path isn’t clear

NCTM (2014) Principles to Action., pp 18-19

Increasing Cognitive Demand

How could the cognitive demand of this task be increased?

Increasing Cognitive Demand“Algebrafying”

“I want to buy a t-shirt that costs $14. I have $8 already saved. How much more money do I need to buy the

shirt?”

“Suppose the T-shirt costs $15, $16, $17 or $26. For each of these cases, write a number sentence that describes how much more money I need to by the shirt. If P stands for the price of any T-shirt, write a number sentence using P that describes how much

more money I need to buy the T-shirt.”

Blanton & Kaput, TCM, 2003

Increasing Cognitive Demand“Algebrafying”

“Jack wants to save up to buy a tablet that costs $45. He makes $5 per hour babysitting. How

many hours will he need to work in order to buy the tablet?”

Blanton & Kaput, TCM, 2003

Graphing Dice Throws

2 3 4 5 6 7 8 9 10 11 12

“My children are “low level”,

so I do the activity using 1 die.”

1 2 3 4 5 6

The Implementation of the Task

• Read the Case of Mr. Stevenson. Compare it to the Case of Mr. Harris.

Consider:

• How are the implementations in the two classrooms the same and how are they different?

• Which of the effective teaching practices did each of the two teachers engage in?

• In what ways did the use of the effective teaching practices support student learning?

Be sure to use evidence from the case to support your claims.

Disclaimer

Diane J. Briars

dbriars@nctm.org

Classroom “Look Fors”

• What is the teacher doing?

• What are students doing?

• What happens to the demand during the lesson?

• To what extent are all students engaged in the mathematics learning?

The Mathematical Tasks Framework

Stein, Grover & Henningsen (1996)Smith & Stein (1998)Stein, Smith, Henningsen & Silver (2000)

Tasks as

set up by

teachers

Tasks as they appear in curricular

materials

Tasks as enacted by

teachers and

students Studentlearning

LSC Evaluation Study

While teachers were using the materials more extensively in their classrooms, there was a wide variation in how well they were implementing these materials. Teachers were often content to omit rich activities, skip over steps and jump to higher level concepts, or leave little time for students to ‘make sense’ of the lessons.

Weiss, et al, 2006

LSC Evaluation Study

In fact, classroom observations indicated that the lessons taught as the developers intended were more likely to provide students with learning opportunities than those that were “adapted.”

Weiss, et al, 2006

Stein & Lane, 1996

Maintaining Demand Matters!

High High

Low Low

High Low

Moderate

Task Set-Up Task Implementation Student Learning

Classroom “Look Fors”

• Who is doing the mathematical thinking?

• What is the instructional goal—understanding mathematics or getting answers?

• What is the cognitive demand of the tasks?

• What happens to the demand during the lesson?

• To what extent are all students engaged in the mathematics learning?

A LOOK INTO A CLASSROOM

Sharing Brownies

Four friends want to share 7 brownies equally.

How many brownies, or what fractional part of a brownie, will each person get?

Sharing Brownies

1. Individually do the task.

2. Then compare your work with others.

3. Consider each of the following questions and be prepared to share your thinking with the group:

a) What mathematics content does the task involve?

b) Which mathematical practices are needed to complete the task?

c) What is the level of demand of the task?

About the Lesson

• Grade 4 classroom.

• The teacher, Ms. Katherine Casey, has been working several years on standards-based mathematics instruction.

• Ms. Lucy West, a mathematics coach, is working with Ms. Casey during this lesson.

Norms for Watching Video

• Video clips are examples, not exemplars. – To spur discussion not criticism

• Video clips are for investigation of teaching and learning, not evaluation of the teacher. – To spur inquiry not judgment

• Video clips are snapshots of teaching, not an entire lesson. – To focus attention on a particular moment not what came before

or after

• Video clips are for examination of a particular interaction.– Cite specific examples (evidence) from the video clip, transcript

and/or lesson graph.

A Look into a Classroom…

Look for:

• What happened to the cognitive demand when the task was implemented?

• What specific teacher actions affected the cognitive demand and supported/hindered students’ engagement reasoning and problem solving?

• Debriefing Activity

– In pairs…

• One person is the teacher

• The other person is the coach/administrator

• The coach engages the teacher in reflection and moves the person toward one or two short-term goals tied to implementing tasks that promote reasoning and problem solving.

Sharing Brownies

..\..\..\..\Brownie lesson\M_E_2407_Brownie_Problem_NCTM.mp4

• Debriefing Activity

– In pairs…

• One person is the teacher

• The other person is the coach/administrator

• The coach engages the teacher in reflection and moves the person toward one or two short-term goals tied to implementing tasks that promote reasoning and problem solving.

Teacher Actions that Affect Cognitive Demand

• Task set-up

• Supporting students’ exploration of the task

• Orchestrating debriefing discussion

Factors Affecting Implementation of High-Level Tasks

Factors associated with maintaining demand:• Scaffolding (i.e., task is simplified so student can solve it; complexity is

maintained, but greater resources are made available). Could occur during whole class discussion, presentations, or during group or pair work.

• Students are provided with the means of monitoring their own progress (e.g., rubrics are discussed and used to judge performance; means for testing conjectures are made explicit and used).

• The teacher or capable students model high- level performance.

• Sustained press for justifications, explanations, meaning through teacher questioning, comments, feedback.

• Tasks are selected that build on students’ prior knowledge.

• Teacher draws frequent conceptual connections.

• Sufficient time to explore (not too little, not too much).

Huinker & Bill, NCTM, 2017 (forthcoming)

Factors Affecting Task Implementation

Factors associated with decreasing demand:• Problematic aspects of the task become routinized (e.g., teacher “takes over”

difficult pieces of the task and performs them for the students or tells them how to do it).

• Teacher shifts emphasis from meaning, concepts, or understanding to correctness or completeness of the answer.

• Not enough time is provided for students to wrestle with the demanding aspects of the task or too much time is provided and students flounder or drift off task.

• Classroom management problems prevent sustained engagement.

• Task is inappropriate for the group of students.

• Students not held accountable for high-level products or processes (e.g., although asked to explain their thinking, unclear or incorrect student explanations are accepted; students were given the impression that their work would not “count” (i.e., be used to determine grades).

Huinker & Bill, NCTM, 2017 (forthcoming)

Mathematical Discourse should:

• Build on and honor students’ thinking.

• Let students share ideas, clarify understandings, and develop convincing arguments.

• Engage students in analyzing and comparing student approaches.

• Advance the math learning of the whole class.

Facilitate Meaningful Mathematical Discourse

Leaves and Caterpillars

A fourth-grade class needs five leaves each day to feed its 2 caterpillars.

How many leaves would they need each day for 12 caterpillars?

David Crane and the Caterpillar Task

Read David Crane and the Caterpillar Task

• What were the strengths of this lesson?

• How might this lesson be revised to increase student learning?

Five Practices for Orchestrating Effective Discussions

• Anticipating likely student responses

• Monitoring students’ actual responses

• Selecting particular students to present their mathematical work during the whole class discussion

• Sequencing the student responses

• Connecting different students’ responses—to each other and to key mathematical ideas.

Smith & Stein, 2011

Leaves and Caterpillars

Assume the mathematical goal is for students to use ratio and rate reasoning to solve problems, and introduce the strategies of unit rates, scale factors, and scaling up.

• Which students would you ask to present their solutions?

• In what order would you have them present? Why?

• What questions would you want to be sure to ask?

Thinking Through a Lesson Protocol (TTLP) Planning Template

Adapted from Smith, Bill, and Hughes, 2008

Solution Questions Students Order

Scaling Up—picture

Scaling Up—table

Unit rate – picture

Unit rate --

calculation

Effective Questions should:

• Reveal students’ current understandings;

• Encourage students to explain, elaborate, or clarify their thinking; and

• Make the mathematics more visible and accessible for student examination and discussion.

The Hungry Caterpillar Task

• Work the task and write at least two different solutions you think first graders would create.

• Share with people at your table.

On Monday, the hungry caterpillar ate through one apple, but he was still hungry. On Tuesday he ate through two pears, but he was still hungry. On Wednesday he ate through three plums. On Thursday he ate through four strawberries. On Friday he ate through five oranges. How many pieces of fruit did the hungry caterpillar eat during the week?

The Case of Ms. Bouchard

Read The Case of Ms. Bouchard and the Hungry Caterpillar Task. Pay close attention to the teacher’s questions.

• What do you notice about the questions the teacher asked?

• What purpose did each question appear to serve?

• Which questions reveal insights into students’ understanding and strategies for addition?

• Which questions orient students to each other’s reasoning?

Question Types

Funneling vs. Focusing

Funneling pattern of questioning:

Teacher uses a set of questions to lead students to a desired procedure or conclusion, while giving limited attention to student responses that veer from the desired path. The teacher has decided on a particular path for the discussion to follow and leads the students along that path, not allowing students to make their own connections or build their own understanding of

the targeted mathematical concepts.

Principles to Actions (2014), p. 37

Typical Teacher Questioning-IRE

• Teacher initiates a question.

• Student responds (usually in one or two words).

• Teacher evaluates the response as right or wrong.

Authority for deciding whether answer is right or wrong lies with the teacher—instead of with discipline-based reasoning

Funneling vs. Focusing

Focusing pattern of questioning:

Teacher attends to what the students are thinking, pressing them to communicate their thoughts clearly, and expecting them to reflect on their thoughts and those of their classmates. The teacher is open to a task being investigated in multiple ways. On the basis of content knowledge related to the topic and knowledge of student learning, the teacher plans questions and outlines key points that should become salient in the lesson.Principles to Actions (2014), p. 37

Purposeful Questions

• How did you get that?

• How do you know that?

• Can you explain your idea?

• Why?

• Can you convince us?

• Did anyone get

something else?

• Can someone tell me or share with me another way?

• Do you think that means the same things?

• Is there another opinion about this?

• Why did you say that, Justin?

Boaler, J., & Brodie, K. (2004).

Productive Struggle should:

• Be considered essential to learning mathematics with understanding;

• Develop students’ capacity to persevere in the face of challenge; and

• Help students realize that they are capable of doing well in mathematics with effort.

Promoting Productive Struggle

https://www.youtube.com/watch?v=TTXrV0_3UjY

Students’ Beliefs about Their Intelligence Affect Their Academic Achievement

• Fixed mindset:

– Avoid learning situations if they might make mistakes

– Try to hide, rather than fix, mistakes or deficiencies

– Decrease effort when confronted with challenge

• Growth mindset:

– Work to correct mistakes and deficiencies

– View effort as positive; increase effort when challenged

Dweck, 2007

Students Can Develop Growth Mindsets

• Teacher praise influences mindsets

– Fixed: Praise refers to intelligence

– Growth: Praise refers to effort, engagement, perseverance

• Explicit instruction about the brain, its function, and that intellectual development is the result of effort and learning has increased students’ achievement in middle school mathematics.

• Reading stories of struggle by successful individuals can promote a growth mindset

Disclaimer

Guiding Principles to Support Effective Teaching and Learning:

Professionialism & AssessmentAction Planning

Diane J. Briars

dbriars@nctm.org

Beliefs about Teaching and Learning?

Complete the survey (p. 1):

• Rate the extent to which you agree with each statement.

• Compare with others sitting near you.

• How might these beliefs affect your work?

• Who else’s beliefs about teaching and learning affects your work?

• Which beliefs might be the most challenging?

Unproductive vs ProductiveBeliefs about Teaching and Learning

• Beliefs should not be viewed as good or bad.

• Beliefs are unproductive when they hinder implementation of effective instructional practice or limit student access to important mathematics content and practices.

Your Feelings Looking Ahead?

1. Teaching and

Learning

2. Access and Equity

3. Curriculum

4. Tools and Technology

5. Assessment

6. Professionalism

Essential Elementsof Effective MathPrograms

ProfessionalismIn an excellent mathematics program, educators hold themselves and their

colleagues accountable for the mathematical success of every student

and for their personal and collective professional growth toward effective

teaching and learning of mathematics.

ProfessionalismIn an excellent mathematics program, educators hold themselves and their

colleagues accountable for the mathematical success of every student

and for their personal and collective professional growth toward effective

teaching and learning of mathematics.

Professionalism Obstacle

In too many schools, professional isolation severely undermines attempts to significantly

increase professional collaboration … some teachers actually embrace the norms of

isolation and autonomy. A danger in isolation is that it can lead to teachers developing

inconsistencies in their practice that in turn can create inequities in student learning.

Principles to Actions, p. 100

Incremental Change

• The social organization for improvement is a profession learning community organized around a specific instructional system.

A. S. Bryk (2009)

• The unit of change is the teacher team.

Principles to Actions, pp. 103-104

Collaboration Team Work

• An examination and prioritization of the mathematics content and mathematics practices students are to learn.

• The development and use of common assessments to determine if students have learned the agreed-on content and related mathematical practices.

• The use of data to drive continuous reflection and instructional decisions.

• The setting of both long-term and short-term instructional goals.

• Development of action plans to implement when students demonstrate they have or have not attained the standards.

• Discussion, selection, and implementation of common research-informed instructional strategies and plans.

PLC Collaborative Team Lesson Planning

• Read the collaborative team illustration on pp. 106-107.

• Discuss with people at your table:

– How does this team’s work support/undermine the enactment of the effective teaching practices?

– How is this team’s work similar to the work of teachers in your school?

– How is this team’s work different from the work of teachers in your school?

– Implications for your work?

Key Features of the Team’s Work

• Collaborative professional learning

• Collaborative lesson planning—implementing effective teaching practices.

• Implement and refine lesson (mini-lesson study)

• Teaching practice is public

• Repository of collaborative lessons

Thinking Through a Lesson Protocol (TTLP) Planning Template

Solution Questions Students Order

Scaling Up—picture

Scaling Up—table

Unit rate – picture

Unit rate --

calculation

Rate Your Teams

Critical Issues for Team Collaboration

Collaborative Team Work

Principal’s Tasks

1. Create a schedule that includes regular grade-level and cross-level collaborative planning

2. Set explicit expectations about how to use this planning time, as well as what products should result

3. Monitor the collaborative work

Effective Instructional Leaders:

• Make high achievement by all students a priority.

• Recognize and value high quality content, instruction and assessment.

• Expect all faculty to implement effective mathematical teaching practices.

• Create a learning community that supports teachers and administrators as they work to improve instructional practices, including teachers working in collaborative teams.

AssessmentAn excellent mathematics program ensures that

assessment is an integral part of instruction, provides evidence of proficiency with important mathematics content and practices, includes a

variety of strategies and data sources, and informs feedback to students, instructional

decisions and program improvement.

What is Assessment?

Assessment is the process of

gathering evidence about student’s

knowledge of, ability to use, and

disposition towards mathematics and

of making inferences based on that

evidence for a variety of purposes.

NCTM Assessment Standards for School Mathematics, 1995.

How Good Are Your Assessments?

Evaluate a Grade 3 End-of-Unit Test

• Rate the Grade 3 end-of-unit assessment using the rubric?

• What are the strengths of the assessment?

• What are the weaknesses?

• What would you do to improve the assessment?

How Good Are Your Assessments?

Assessment Tasks that Support Valid Inferences

Learning target:

Understanding the definition of a triangle.

Performance task:

Draw a triangle.

Grade 2

2.G.3. Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words halves, thirds, half of, a third of, etc., and describe the whole as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shape.

Grade 2

Understanding a Concept

• Explain it to someone else

• Represent it in multiple ways

• Apply it to solve simple and complex problems

• Reverse givens and unknowns

• Compare and contrast it to other concepts

• Use it as the foundation for learning other concepts

PLC Collaborative Team Assessments

• Read the collaborative team end-of-unit assessment illustration on pp. 96-98.

• Discuss with people at your table:

– How does this team’s assessment and analysis of results support/hinder students’ learning?

– How is this team’s work similar to/different from the work of teachers in your school?

– Implications for your work?

What about “Test-Prep”?

Too often, teachers are putting regular instruction “on hold” to spend class time

practicing test questions. While on the surface this may appear to make sense, research

indicates just the opposite—

test scores are lower in schools where teachers

spend large amounts of time on test prep.

(Allensworth, Correa, & Ponisciak, 2008)

Intensive Test Prep Produces the Same or Lower Scores as Little or No Test Prep

All Test-Prep

Intensive Test Prep Produces the Same or Lower Scores as Little or No Test Prep

ACT Test-Prep Materials

Effective Assessment PracticeOngoing Review and Practice

Providing students with periodic opportunities to practice using concepts and skills, along with

feedback about their performance, helps students solidify their knowledge and promotes retention, reflection, generalization, and transfer

of knowledge and skill.

IES Practice Guide, 2007

Distributed Practice

• Openers

• Homework

• Incorporate intoinstructional and/orassessment tasks

Good Instruction is the Best Test-Prep

• Students acquire conceptual knowledge as well as skills to enable them to organize their knowledge, transfer knowledge to new situations, and acquire new knowledge.

• Students engage with challenging tasks that involve active meaning-making.

• Students know what is expected

Hiebert & Grouws, 2007

Effective Assessment Practices

1. Create and use common high-quality assessments

2. Use tasks that assess conceptual understanding and mathematical practices

3. Teach students to take responsibility for their learning

4. Use assessment results formatively; i.e., use errors and misconceptions as instructional opportunities

5. Provide opportunities for on-going review and practice instead of stopping instruction for “test prep.”

• Teaching and Learning

• Access and Equity

• Curriculum

• Tools and Technology

• Assessment

• Professionalism

http://www.nctm.org/PtA/

Principles to Actions Resources

• Principles to Actions Executive Summary (in English and Spanish)

• Principles to Actions overview presentation

• Principles to Actions professional development guide (Reflection Guide)

• Mathematics Teaching Practices presentations

– Elementary case, multiplication (Mr. Harris)

– Middle school case, proportional reasoning (Mr. Donnelly) (in English and Spanish)

– High school case, exponential functions (Ms. Culver)

• Principles to Actions Spanish translation

http://www.nctm.org/PtAToolkit/

Principles to Actions ElaborationsComing in April, 2017

• Teaching and Learning

– Taking Action: Implementing Effective Mathematics Teaching Practices in Grades K-5

– Taking Action: Implementing Effective Mathematics Teaching Practices in Grades 6-8

– Taking Action: Implementing Effective Mathematics Teaching Practices in High School

Principles to Actions ElaborationsComing Soon

• Access and Equity

• Curriculum

• Tools and Technology

• Assessment

• Professionalism

Collaborative Team Tools

Available at nctm.org

NCTM-Hunt Institute Video Series:Teaching and Learning Mathematics with

the Common Core

• Enhance public understanding of what students need to know for college and career

• Why conceptual understanding requires a different approach

• Teachers, educators, leaders, and parents with classroom video

• Primarily for the public; useful to educator outreach

NCTM-Hunt Institute Video Series:Teaching and Learning Mathematics with

the Common Core

• Mathematics in the Early Grades

• Developing Mathematical Skills in Upper Elementary Grades

• Building Conceptual Understanding in Mathematics

• Mathematical Foundations for Success in Algebra

• Preparation for Higher Level Mathematics

• Parents: Supporting Mathematics Learning

http://www.nctm.org/Standards-and-Positions/Common-Core-State-Standards/Teaching-and-Learning-

Mathematics-with-the-Common-Core/

e-Book Series

Clicking on the title brings up the article (if you subscribe to TCM)

NCTM Position Statementshttp://www.nctm.org/Standards-and-Positions/NCTM-Position-Statements/

NCTM Research BriefsBriefshttp://www.nctm.org/Research-and-Advocacy/research-briefs-and-clips/

Inside Mathematicshttp://www.insidemathematics.org/

Mathematics Assessment Projecthttp://map.mathshell.org/

The Title Is Principles to Actions

What actions will you take to improve mathematics teaching and

learning in your setting?

Action Planning Worksheet

Thank You!

Diane Briarsdbriars@nctm.org

nctm.org/briars

Disclaimer

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