Assessing Math DifficultiesEDSP 651, Session 6

Agenda for Tonight

• Homework Collection• Guest Presenter: Mark Semmler

– Assessing Math

• Oral Presentation: Erik Gose (CoAlt) • IEP Evaluation Report Review & Practice• Homework for Session #6

Homework

1.  Based on the data provided in the article, does Danny have a disability?  Is there sufficient data to warrant a referral to special education for evaluation?  Why or why not?

2.  What role does a multidimensional assessment in math play in determining a disability and potential specialized programming?

Guest Presenter

Mark Semmler• SpEd Coordinator (K-12 Mathematics), Cherry Creek Schools

• Online Instructor, Colorado Dept. Ed

• Colorado Metro Math Intervention Team member

Outline• Number Knowledge – What do we mean by number

knowledge?• Quick Colorado Academic Standards overview –

What’s the big deal?• Diagnostic assessment – What are a student’s

specific learning needs?• Establishing learning goals – Toward what should we

focus our instruction?• Progress monitoring – How will we know if students

are progressing toward our learning goals?

NUMBER KNOWLEDGEWhat do we mean by number knowledge?

Compare and ContrastHow are these similar? How are they different?

h 3

What are the phonemes represented by the grapheme“h”?

• House• Weight• With• Wither

What are the quantities the character “3” can represent?

• 3• 30• 300,000• 13

Three Aspects of Number

• Verbal• Symbol• Quantity

Quantity

• “Sometimes the subject of arithmetic is imagined as memorizing the basic facts and mastering the algorithms for the four operations.

• “By contrast, imagine a subject that develops strong integrated mathematical knowledge. We want students to know a lot about numbers.”

Developing Number Knowledge, p. 11

What do you know about this number?

23

2/3

Quantity Assessment

Subitizing

Number Talks

1. Students are given a problem to solve. Almost any mental math problem can be used in a number talk.

2. Students spend about a minute thinking of all the different ways they could solve the problem. Typically students will work through problems mentally (w/o paper). Some problems might permit the use of paper.

3. Students discuss solutions and strategies with a partner

4. Class discusses the variety of ways to solve the problem.

Solve:

8+3=

Solve:

19+25

Solve:

19 x 4 =

What is Number Sense?• An understanding of

– basic counting techniques – size/quantity associated with numbers – relationships between numbers – number patterns – place value (NCTM, 2000)

• It is a stronger predictor of math achievement than – age – gender – socio-economic status – reading ability – oral vocabulary – memory – spatial reasoning

(Locuniak & Jordan, 2008; Jordan, Glutting, & Ramineni, 2010; Jordan, Kaplan, Locuniak, & Ramineni, 2007)

QUICK COLORADO ACADEMIC STANDARDS OVERVIEW

What’s the big deal?

Standards Quiz

• Has Colorado adopted the Common Core standards?

• Were the Common Core standards developed at the federal level?

• Are Colorado’s standards the same as other states who have adopted the Common Core?

• How much are the Common Core Math standards reflected in the Colorado Academic Standards?

Colorado Academic Standards

5. Integration of the Common Core State Standards. These revised standards reflect the inclusion of the Common Core State Standards in Mathematics.

Overview of ChangesMathematics Standards

Common Core Math Emphases• Focus

– Doing fewer things at any given time – Building mastery in order to build a

foundation – Focusing on arithmetic in the early

grades• Coherence and Progressions

– Following the progressions of ideas and skills within a strand across the grades

– Seeing how math fits together• Standards of Math Practice

– The thinking skills of math – Includes

• Ability to construct arguments • Modeling (applying math to a real-

world problem) • Use of technology

Applying the Common Core State Standards in Math

Math Standards Tour

Flow of the Common Core

Operations and Algebraic Thinking (OA)

Expressions and

Equations (EE)

Algebra

Number and Operations in Base Ten (NBT)

Number System

(NSS)

Number and

Operations—

Fractions (NF)

K 1 2 3 4 5 6 7 8 High School

Three Strands…

Operations and Algebraic Thinking (OA)

K–5

Concrete uses and meanings of the basic operations (word problems)

Mathematical meaning and formal properties of the basic operations

Prepare for later work with expressions and equations in middle school

Number and Operations in Base Ten (NBT)

K–5

Place value understanding Develop base-ten algorithms using place value and properties of

operations Computation competencies (fluency, estimation)

Number and Operations—Fractions (NF)

3–5

Enlarge concept of number beyond whole numbers, to include fractions

Use understanding of the four operations to extend arithmetic to fractions

Solve word problems related to the equation ax = b (a and b fractions)

DIAGNOSTIC ASSESSMENTWhat are a student’s specific learning needs?

Diagnostic Assessment

• What is the student’s present level of math achievement?1. What is the student’s standing relative to same-

age peers? (Or stated more accurately, What is the student’s achievement level relative to grade level expectations on important elements of mathematics?)

2. Where is the student developmentally? i.e., On what should I focus my instruction, and where should I start?

IES Guide Recommendation 2

• For students in kindergarten through grade 5, tier 2 and tier 3 interventions should focus almost exclusively on properties 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 in-depth coverage of rational numbers as well as advanced topics in whole number arithmetic (such as long division).

Samantha

“Numbers scare me.”

Colorado Academic Standards

• Use place value understanding and properties of operations to add and subtract. (CCSS: 2.NBT) – Fluently add and subtract within 100 using strategies

based on place value, properties of operations, and/or the relationship between addition and subtraction. (CCSS: 2.NBT.5)

– Add and subtract within 1000, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method.2 (CCSS: 2.NBT.7)

Colorado Academic Standards

• Represent and use the digits of a two-digit number. (CCSS: 1.NBT.2) – Represent the digits of a two-digit number as tens

and ones.1 (CCSS: 1.NBT.2)

Colorado Academic Standards

• Use place value and properties of operations to add and subtract. (CCSS: 1.NBT) – Add within 100, including adding a two-digit

number and a one-digit number and adding a two-digit number and a multiple of ten, using concrete models or drawings, and/or the relationship between addition and subtraction. (CCSS: 1.NBT.4)

Multidimensional Assessment

• Individually administered interview-type assessments

• Not normed, as they are not intended to determine a ranking, but to determine developmental level/instructional needs

• Several options– Add+Vantage Math Assessments (AVMR)– Math Reasoning Inventory– NZ Maths– Kathy Richardson assessments

Reflection on the Multidimensional Assessment

• In what ways does this assessment address our initial questions of a diagnostic assessment, and in what ways does it not?1. What is the student’s achievement level relative to grade level

expectations on important elements of mathematics?2. On what should I focus my instruction, and where should I

start?• In what ways does this assessment and associated resources

correlate to the guiding principles of the common core standards: focus, coherence, and math practices, and in what ways does it not?

• Compare and contrast traditional standardized diagnostic assessments (WJ-III, WIAT, KeyMath, etc.) to Multidimensional Assessment as a diagnostic assessment tool

Math Reasoning Inventory(mathreasoninginventory.com)

Whole Number InterviewWhole Number Written ComputationWhole Number Reasoning Strategies

Decimals Interview Decimals Written Computation Decimals Reasoning Strategies

Fractions InterviewFractions Written ComputationFractions Reasoning Strategies

NZ Maths• GloSS assesses the student’s strategy stage in three strategy domains:

– Addition & Subtraction– Multiplication & Division– Proportions & Ratios

• IKAN assesses student’s knowledge stage across five domains– Number Identification, e.g. 5 means “five”, or 7/8 means “seven-eighths”– Number Order and Sequence, e.g. the number after 16 is 17, or the multiple of

10 after 490 is 500– Grouping and Place Value, e.g. 10 can be grouped as 7 and 3, or 23.4 is 234

tenths– Number Facts, e.g. 3 + 3 = 6, or the square root of 100 is 10– Written Recording, e.g. use of empty number lines or written working forms

• Diagnostic Interview– In-depth diagnostic assessment intended to evaluate all three strategy domains

and all five knowledge domains

LEARNING GOALSToward what should we be working?

Process to Establish Learning Goals

• Identify the Math big ideas to focus on for intervention

• Locate current instructional level in the Colorado Academic Standards

• Establish an ambitious yet realistic target from the standards

• Determine the conditions required to reach the target

• Select a tool to monitor progress, establish a baseline and set a target on the tool

Goal Format

• Given condition, Samantha will improve conceptual understanding and procedural fluency in specific need from baseline score to target score as measured by progress monitoring measure.

• Given condition, Samantha will improve conceptual understanding and strategic thinking in specific need from baseline score to target score as measured by progress monitoring measure.

Identify the Math big ideas to focus on for intervention (from

Diagnostic Assessment(s)

–Structuring number–Addition/subtraction–Multiplication/division–Place value

Locate current instructional level in the Colorado Academic Standards

• Counting– Count within 1000. (CCSS: 2.NBT.2)– Skip-count by 5s, 10s, and 100s. (CCSS: 2.NBT.2)

• Addition/Subtraction– Solve addition and subtraction word problems, and add and

subtract within 10.7 (CCSS: K.OA.2)• Place Value

– Represent and use the digits of a two-digit number. (CCSS: 1.NBT.2)• Multiplication and Division

– Interpret products of whole numbers (CCSS: 3.OA.1) e.g., interpret 5 × 7 as the total number of objects in 5 groups of 7 objects each. (CCSS: 3.OA.1)

Establish an ambitious yet realistic target from the standards

• Target: Achieve a third grade level in counting, addition/subtraction, place value, and multiplication/division.

• Standard: Use place value and properties of operations to perform multi-digit arithmetic. (CCSS: 3.NBT) – Use place value to round whole numbers to the nearest 10 or 100. (CCSS: 3.NBT.1)– Fluently add and subtract within 1000 using strategies and algorithms based on

place value, properties of operations, and/or the relationship between addition and subtraction. (CCSS: 3.NBT.2)

– Multiply one-digit whole numbers by multiples of 10 in the range 10–90 using strategies based on place value and properties of operations.1 (CCSS: 3.NBT.3)

• Specific Need: round whole numbers to nearest 10 or 100, and using strategies based on place value and properties of operations, add and subtract within 1000 and multiply one-digit whole numbers by multiples of 10 in the range 10-90

Determine the conditions required to reach the target

• What level of instruction is required for Samantha to achieve the ambitious goal?

• Given core math instruction and focused supplemental instruction in whole number operations

Our Goal So Far…

Given core math instruction and focused supplemental instruction in whole number operations, Samantha will improve conceptual understanding and procedural fluency in rounding whole numbers to nearest 10 or 100, and – using strategies based on place value and properties of operations – adding and subtracting within 1000 and multiplying one-digit whole numbers by multiples of 10 in the range 10-90 from __ to __ as measured by __.

PROGRESS MONITORINGHow will we know if students are progressing toward our learning goals?

IES Guide Progress Monitoring Recommendation

• 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 assessments 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 regroup students when necessary.

General Outcome Measures

• Researched reliability and validity• Provides representative picture of standing

relative to grade level expectations• Typically uses a standards-sampling format• Common examples

Select a tool to monitor progress, establish a baseline and set a target

Using easyCBM• Baseline: 7 points on level 2• Target: 13 points on level 3• Tool: easyCBM, Number, Operations and

Algebra

Our Goal…

• Given core math instruction and focused supplemental instruction in whole number operations, Samantha will improve conceptual understanding and procedural fluency in rounding whole numbers to nearest 10 or 100, and – using strategies based on place value and properties of operations – adding and subtracting within 1000 and multiplying one-digit whole numbers by multiples of 10 in the range 10-90 from 7 points on level 2 to 13 points on level 3 as measured by easyCBM, Number, Operations and Algebra.

Curriculum-Embedded Assessments

• Included with many intervention programs• Alternative: Scoring Tool highlighting targeted

standards scored against a standard rubric.– Can and should guide lesson planning– Adds a necessary urgency to daily work

Reflection• Number Knowledge – What do we mean by number

knowledge?• Quick Colorado Academic Standards overview –

What’s the big deal?• Diagnostic assessment – What are a student’s

specific learning needs?• Establishing learning goals – Toward what should we

focus our instruction?• Progress monitoring – How will we know if students

are progressing toward our learning goals?