IES Practice Guide – Response to Intervention Mathematics

Ben Clarke, Ph.D.University of Oregon

May 22, 2014

• National Research Council: Adding it Up

• National Council Teachers of Mathematics: Focal Points

• National Mathematics Advisory Report

High Level of Interest in Mathematics

• Achievement on the NAEP trending upward for 4th/8th grade and steady for 12th grade– Large numbers of students still lacking proficient

skills– Persistent income and ethnicity gaps– Drop in achievement at the time algebra instruction

begins

State of Mathematics (1)

• TIMS data indicate significant lower levels of achievement between US and other nations–Gap increase over time

• Jobs requiring intensive mathematics and science knowledge will outpace job growth 3:1 (STEM) and everyday work will require greater mathematical understanding

State of Mathematics (2)

• “For people to participate fully in society, they must know basic mathematics. Citizens who cannot reason mathematically are cut off from whole realms of human endeavor. Innumeracy deprives them not only of opportunity but also of competence in everyday tasks.” (Kilpatrick, Swafford, & Findell, Adding It Up, 2001)

Need for mathematical knowledge

• Reauthorization of IDEA (2004) allowed for RtI to be included as a component in special education evaluations

• Premised on the use of research-based interventions and student response to intervention– Students who respond are not identified as learning

disabled– Students who do not respond are referred for a

complete evaluation and potential identification as learning disabled

Response to Intervention

Tier 3: Tertiary/Intensive• Specialized, individualized interventions

for students with significant needs

Tier 2: Secondary/Targeted• Specialized interventions for students at-risk for failure

Tier 1: Primary/Universal• School-wide system of

support• Designed to support the

needs of all students

RtI: A Framework

Figure courtesy of Nelson-Walker (2009)

• Requires high-quality interventions and research-based instruction

• Identifies students based on risk rather than deficits– Prevention model vs. Remediation model

• Links identification assessments and progress monitoring tools with instructional planning (i.e., data-based decision making)

Foundations of RtI (1)

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

Middle Schools

Copies available at the IES website:http://ies.ed.gov/nceehttp://ies.ed.gov/ncee/wwc/publications/practiceguides/

• Russell Gersten (Chair) • Sybilla Beckman• Ben Clarke• Anne Foegen• Laurel Marsh• Jon R. Star• Bradley Witzel

Panelists

Panel works to develop 5 to 10 assertions that are: • Forceful and useful• And COHERENT• Do not encompass all things for all people• Do not read like a book chapter or article

Challenges for the panel:• State of math research• Distinguishing between tiers of support

Jump start the process by using individuals with topical expertise and complementary views

Search for Coherence

• Recommendations

• Levels of evidence

• How to carry out the recommendations

• Potential roadblocks & suggestions

Structure of the Practice Guide

• The panel considered:– High-quality experimental and quasi-experimental

studies– Also examined studies of screening and progress

monitoring measures for recommendations relating to assessment

The Research Evidence

• Each recommendation receives a rating based on the strength of the research evidence

– Strong

– Moderate

– Low

Evidence Rating

Recommendation Level of Scientific Evidence

1. Universal screening (Tier I) Moderate

2. Focus instruction on whole number for grades K-5 and rational number for grades 6-8

Low

3. Systematic instruction Strong

4. Solving word problems Strong

5. Visual representations Moderate

6. Building fluency with basic arithmetic facts

Moderate

7. Progress monitoring Low

8. Use of motivational strategies Low

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.

– Level of Evidence: Low

Evidence

• Consensus across mathematicians, professional organizations, and research panels– Adding it Up (NRC, 2001) covering fewer topics with

greater depth– National Council Teachers of Mathematics (NCTM)

and National Mathematics Advisory Panel (NMAP)– International comparisons– We made the leap to nature of intervention

curricula………

Suggestions

• For tier 2 and 3 students in grades k-5, interventions should focus on the properties of whole number and operations. Some older students would also benefit from this approach.

• For tier 2 and 3 students in grades 4-8, interventions should focus on in depth coverage of rational number and advanced topics in whole number (e.g. long division).

Example: FUSION Curriculum

• Whole number content aligned with Grade 1 Common Core (CCSS-M, 2011)

• 60 small-group lessons • 30-minutes, 3 times per week• 4-5 students per group

• Scope and Sequence: Skill development over time building toward key objectives

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Scope and Sequence: Content Strands of FUSION

• Three Strands:• Base 10 and Place Value: numbers to 100• Number Combinations• Multi-digit addition and subtraction

• Integrated across strands:• Problem Solving• Number Properties• Math related Vocabulary and Discourse

Alignment with key standards: CCSS

CCSS cont.

Focused Content:

• Increasingly found in researcher developed mathematics interventions– Number Rockets (Fuchs and colleagues)– Number Sense (Jordan and colleagues)– Work by Bryant and colleagues

• Efforts are expanding for the upper elementary grades

Suggestions• Districts should appoint committees with

experts in mathematics instruction to ensure specific criteria are covered in-depth in adopted curriculums.

– Integrate conceptual understanding with procedural fluency (e.g .”Basic Facts”)

• Stress reasoning underlying calculation methods• Build algorithmic proficiency

– Contain frequent review of mathematical principles

– Contain assessments to appropriately place students in the program

Roadblocks

• Concern about alignment between the intervention and the core classroom instruction.

• Suggested Approach: The alignment between the two is not critical and intervention programs should focus on building foundational knowledge that will enhance future achievement.

Roadblocks

• Intervention materials may cover topics that are not essential to building basic competencies such as data/probability, measurement, and time.

• Suggested Approach: Students will gain exposure to supplemental topics through tier 1 instruction.

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• Interventions should include instruction on solving word problems that is based on common underlying structures.– Level of evidence: Strong

Recommendation 4

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• Word problems are defined as mathematical problems that entail a short story, contain underlying mathematical structures, and require a mathematical method and solution.

• Simple word problems give meaning to mathematical operations, such as subtraction and multiplication (Gersten et al., 2009; p. 26).

The Utility of Word Problems

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• Word problems rank as the most difficult type of mathematics problems for students with or at-risk for mathematics difficulties (Bryant & Bryant, 2008)

• Contributing factors to these difficulties:– Ambiguous instruction– Reading difficulties– Language comprehension deficits (Powell, Fuchs, & Fuchs, 2013)

• A solution framework:– Provide systematic, explicit instruction that focuses on

• Underlying structures of problems• Structural connections between familiar and unfamiliar problems

Word Problem Solving: A Common Area of Difficulty

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• Group Problems– Put Together– Take Apart

• Change Problems– Add To– Take From

• Compare Problems

Common Problem Types Involving Addition and Subtraction

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Grades 1-2

Grade 5 with rational numbers

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• Group Problems– Two smaller groups/ make up a larger group or

whole/total• Change Problems

– Beginning quantity followed by an action that increases (adds to) or decreases (takes from)

• Compare Problems– Two quantities are compared with the same unit– One quantity is more, one is less, difference

Underlying Structures of Addition and Subtraction Problems

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(Jitendra, 2008; Jungjohann, 2010)

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• Multiplication and Division– Equal Groups– Arrays/Area– Multiplicative Compare

Common Problem Types Involving Multiplication and Division

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Grades 3-4

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Examples of Multiplication & DivisionWord Problem Types

(Learning Progressions, 2011; NRC, 2009)

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IES Practice Guide on Math Problem Solving

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– Curricular material may not classify problems into problem types

– As problems get more complex, so will the problem types and the task of discriminating among them

Roadblocks to Recommendation 4

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• Interventions at all grades should devote about 10 minutes in each session to building fluent retrieval of basic arithmetic facts– Level of Evidence: Moderate

Recommendation 6

• Provide 10 minutes per session of instruction to build quick retrieval of basic facts. Consider the use of technology, flash cards, and other materials to support extensive practice to facilitate automatic retrieval

• For students in grades K-2, explicitly teach strategies for efficient counting to improve the retrieval of math facts

• Teach students in grades 2-8 how to use their knowledge of math properties to derive facts in their heads

Suggestions

Activity

• How well do your current intervention materials

– Focus on critical content– Incorporate a focus on word problems– Build procedural fluency with basic number

combinations

Recommendation 3 Instruction during the intervention should be

explicit and systematic. This includes providing models of proficient problem-solving, verbalization of though processes, guided practice, corrective feedback, and frequent cumulative review.

– Level of Evidence: Strong

Suggestions

• Ensure that intervention materials are systematic and explicit and include numerous models of easy and difficult problems with accompanying teacher think-alouds.

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

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

– Explicit instruction is a structured, systematic instructional methodology for teaching foundational concepts, principles, and skills in the most effective and efficient manner possible (Archer & Hughes, 2010; Carnine, Silbert, Kame’enui, & Tarver, 2004)

– The National Mathematics Advisory Panel stated that “Explicit systematic instruction typically entails teachers explaining and demonstrating specific strategies and allowing students many opportunities to ask and answer questions and to think about the decisions they make while solving problems” (p.48).

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Explicit Instruction

1. Prioritize the most critical content (“big ideas”)2. Specify learning objectives and interaction expectations 3. Address students’ background knowledge and pre-requisite

skills (scope and sequence) 4. Provide vivid, step-by-step demonstrations (models) with

think alouds (I Do)5. Facilitate frequent instructional interactions (We Do)6. Provide and monitor independent practice opportunities

(You Do) including math verbalizations7. Deliver timely, academic feedback8. Engage students in daily, weekly, and monthly review

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Using an Explicit Instructional Approach

Archer & Hughes, 2010; Baker, Fien, Baker, 2010; Doabler et al., 2012b

Scaffolding Instruction for Learning Success

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Time & Mathematical Proficiency

I do

Student Independence

You do

We do

Teacher Support

Adapted from Chard & Jungjohann, 2006

– Use think alouds so that students can hear your thought process.

• Justifications, solution methods, and math reasoning

– Foster opportunities for students to use think alouds on their own (mathematical verbalizations).

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Think Alouds

– Math verbalizations permit students to interact with the teacher and peers around critical mathematics content.

– Specifically, verbalizing can be viewed as a way to process and practice math content and, in this manner, becomes a critical component for supporting early development of mathematical proficiency .

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Mathematical Verbalizations

– Guided practice is defined as a practice opportunity in which the teacher provides active guidance throughout the response cycle, for example responding along with the students.

– Independent practice is defined as a practice opportunity without teacher guidance

– Cumulative review should be interspersed with practice on new content to ensure retention and mastery

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Practice and Review

Roadblocks

• Interventions may not have training in implementing interventions with explicit instruction and may underestimate the amount of practice and review needed by tier 2 and 3 students.

• Suggested Approach: Districts and schools should provide professional development that allows interventionists to observe, discuss, and practice lesson delivery. PD should emphasize the importance of practice and cumulative review.

Roadblocks

• Interventionists may not be expert with the underlying mathematics content.

• Suggested Approach: Professional development shoujld provide interventionists with in-depth knowledge of the mathematics content of interventions including the mathematical reasoning underlying procedures, formulas and problem-solving models.

Roadblocks

• Intervention materials may not incorporate enough models, think-alouds, practice, and cumulative review.

• Suggested Approach: Use a math specialist or coach to develop a template listing the essential parts of an effective lesson.

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• Interventions 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.– Level of evidence: Moderate

Recommendation 5

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• Effective and strategic use of mathematics models can help students understand the relationship between abstract symbols and visual representations.

• “They help clarify ideas in ways that support reasoning and building understanding.” (NRC, 2011)

The Potential Yield of Math Models

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tens ones

1 3

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• Students may require CRA stages of learning

• CRA is an interconnected sequence of instruction that helps students to build conceptual understanding

Concrete Representational Abstract

Incorporating RepresentationsInto Instruction

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• Concrete – Place value models– Counting blocks

• Representational– Number lines– Simple drawings– Graphs

• Abstract– Equations– Verbal description

Examples of Math Models

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CRA Example: Multi-digit Addition

A. The equation reads 35 + 12 equalsB. Represent 35 with place value models; represent 12 with place value

modelsC. Solve problem using abstract symbolsD. 35 + 12 equals 47 OR Three ten-sticks and five cubes plus one-stick and

two cubes equals four ten sticks and 7 cubes

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A B C D

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(Gersten et al, 2009, p.33)

Conceptual/Procedural Visuals

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– Many intervention materials provide few accurate visual representations

– Teachers or interventionists may be resistant to use of math representations, given the demands of school schedules and available resources

– Some interventionists may not fully understand the mathematics content that underlies the representations

Roadblocks to Recommendation 5

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• No math model teaches itself

• The use of visual representations, though necessary, is not sufficient by itself to facilitate conceptual understanding

• Compliment such visual representations with sound, high-quality instruction

Using Visual Representations Wisely

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- How well do your current intervention materials incorporate critical instructional design elements including:

- Explicit and systematic teaching sequences (with teacher models and multiple opportunities for math verabalizations

- Robust math models

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Activity

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For interventions consider:- The focus of the content (and underlying scope and sequence)- The instructional design of the materials- You can overcome some issues with materials but not all

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Concluding thoughts

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