Russell Gersten, Ph.D. Director, Instructional Research Group Professor Emeritus, University of Oregon MiBLSi State Implementer's Conference Lansing, Michigan March 12, 2013

1. The case for early intervention in mathematics 2. The case for intervention in intermediate grades 3. The IES Practice Guide and Evidence Standards  Screening and Progress monitoring  Nature of explicit instruction in mathematics intervention: Controversies

 New research on best screening measures  Key constructs to assess  Most recent knowledge of quality of current progress monitoring measures  What is Smart RtI?

1. Morgan, Farkas & Wu: Examined growth from K through 5 th grade Low in preK and no growth thru K augurs badly for future success in mathematics  Better predictor than early reading  Attentiveness also solid predictor (in K) 2. Duncan et al. (2012) – two large datasets from U.S. and U.K. Early mathematics better predictor than reading as well as mathematics Fuchs/Compton finding number facts intervention accelerating reading growth

1. Fractions knowledge (understanding and procedural but especially understanding of the ideas) is critical for success in algebra (National Mathematics Panel, 2009) mathematically 2. Fractions predictive work of Siegler/Duncan et al (in press) using large data sets supported this empirically

 Many American students unable to solve fractions problems in middle or even high school Example: NAEP Grade 8 in 2007: Pass rate = 49% In which of the following are the three fractions arranged from least to greatest? A. D. B. E. C.  Most think that the reason for poor performance on these items is that students never understood the mathematical ideas relating to fractions

 Geary linkage of early mathematics to career related measures  Algebra as a gateway has been much discussed (Moses etc.)  Algebra 2 as a gateway to career – e.g. Achieve  Focus on college and career readiness – e.g. Conley

 Overview of Practice Guide: interactive (think pair share)  Introduction to evidence base  Linkage to Common Core standards and Learning progressions as resources for RtI  Key principles and concepts that underlie each

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

Mandate:  Create a framework for establishing/refining instruction that is clear and practical  Consist of action-based recommendations that can be implemented in practice  Take risks: don’t equivocate!  Create a coherent document: common themes should underlie the various specific suggestions

 Recommendations  How to carry out the recommendations  Levels of evidence  Potential roadblocks & suggestions

Practice Guides

 Curricula may not include enough fact practice or may not have materials that lend themselves to teaching strategies.  Suggested Approach: Some contemporary 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. Or strategy based (Woodward, 2006).

Each recommendation receives a rating based on the strength of the research evidence.  Strong  Moderate  Minimal simply means no rigorous evidence, not contradictory evidence or negative…..

Recommendation Level of Scientific Evidence 1.Universal screening (Tier I)Moderate 2.Focus instruction on whole number for grades k-5 and rational number and whole number for grades 4-8 Minimal 3.Systematic, focused instructionStrong 4.Solving word problemsStrong 5.Visual representationsModerate 6.Building fluency with basic arithmetic facts Moderate 7.Progress monitoring of all students receiving intervention or at risk Minimal 8.Use of motivational strategiesMinimal

Which level of evidence is the biggest surprise for you? Why?

1. Mathematics progress monitoring measures tend to be weaker than reading (Foegen, Jiban & Deno, 2007) 2. Increasingly, research suggests at least 12 data points are needed to determine slope (Chris, 2013; Compton & Fuchs, 2012)

 Limited lens issue (computation, facts, oral reading fluency) leads to limited instruction  Lack of general outcome measures in math (Foegen et al., 2007)

 Since these assessments lead to instruction and often minimally successful instruction (e.g. heavy emphasis on silent and oral reading fluency without vocabulary and comprehension components), we need to be careful and cautious about how progress monitoring measures guide instruction.

 Need to rethink efficient screening, given current technology Level (i.e. Adaptive) testing on computer vs. quick measures? How to assess understanding (formative assessments) As compared to brief timed worksheets

 Problems in assessing understanding Analog from Reading Maze items for Comprehension  Possible limited relevance to Common Core

 Siegler’s measures whole number, NLE predicts fractions knowledge 1. Measures using Number line seem to be strongest predictors 2. They are better than general mathematics achievement measures 3. Both Siegler and Geary note that predictive validity up through 8 th grade and/or high school career readiness measures.

 Administered in Kindergarten and 1 st Grade  Requires students to name the larger of two visually presented numbers from 0 to 20 in Kindergarten and 0 to 99 in 1 st grade Example Items - Grade 1:

 Are Number Line estimation items appropriate GOMs? Related to instructional goals?  Is there support for the idea of development of a “mental number line” as Robbie Case theorized?

Geary (2013): Three key components from 2013 article (Beginnings of Math Proficiency) 1. Attention (+) Persistence (+) Impulsivity (-) 2. Ability to understand and manipulate sets of numbers ….. More than adding or subtracting or naming digits. 3. Mental Number Line Ability to quickly and accurately compare magnitudes Estimation This is core of number sense 4. Working memory critical starting perhaps end of first grade

Gersten, R., Chard,. D., Jayanthi, M., Baker, S., Morphy, P., & Flojo, J. (2009). Teaching mathematics to students with learning disabilities: A meta-analysis of the intervention research. Review of Educational Research, 79,

 Instruction includes: procedures AND concepts AND word problems  Research by Siegler/ Bethany Rittle- Johnson  Reciprocal relationship

Instruction during the intervention should be systematic and include models of proficient problem-solving, verbalization of thought processes, guided practice, corrective feedback, and frequent cumulative review. Level of Evidence: Strong

 Six randomized controlled trials met standards  Key themes Extensive practice with feedback Let students provide rationale for their decisions Instructors and fellow students model approaches to problem solving

Intervention materials should include opportunities for the student 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

 Use visual representations such as number lines, arrays, and strip diagrams.  If necessary consider expeditious use of concrete manipulatives before visual representations. The goal should be to move toward abstract understanding.

 Assignment: Use the lowest common denominator when appropriate ½ + ⅓ =  Student Response ½ + ⅓ = ⅖

ConcreteVisual Representations

 Temporarily decrease cognitive load for students  Example from Fuchs (2007)

THINGS TO LOOK FOR: Linkage of visual representations to mathematical procedures and ideas Think alouds (as opposed to procedural modeling  Note: this is a simulation.

 Difficulties encountered by some students Extraneous information Different wording  Even though the problems have a common underlying structure  Creates problems for any student who needs intervention Source: Fuchs et al. (2007 )

 How to support students’ talking about mathematics  How to capitalize on helping students learn from hearing peers explain their mathematical reasoning

 Instructional materials for students receiving interventions should focus in depth on: Whole numbers in kindergarten through grade 6 Rational numbers in grades 4 through 8 Applications to geometry and measurement Level of Evidence: minimal

WHAT TO TEACH WHAT TO TEACH Thus, intervention content closely aligned to Common Core BUT Likely to need to backfill with related material from earlier grades

TOOLS TOOLS Progressions for the Common Core State Standards in Mathematics (draft) by The Common Core Standards Writing Group DRAFT 21 April Supported by Brookhill Foundation

Consensus across mathematicians, professional organizations, and research panels –  National Council Teachers of Mathematics (NCTM) and National Mathematics Advisory Panel (NMAP)  International comparisons  We made the leap to nature of intervention curricula………

How Common Core can guide intervention and assessments (interim, progress monitoring etc.)

 Covers fractions more than 1 and less than 1 concurrently  Word problems integrated with symbols/operations from the start  Ideas (concepts) and procedures linked  Limited array of mathematical models

What are Learning Progressions? and What do they have to do with RtI?

 “A picture of the path students typically follow as they learn... (Masters and Forster, 1996)  The organization of learning experiences designed to help students go ahead in different subjects as rapidly as they can (Masters & Forster, 1996)  Descriptions of the successively more sophisticated ways of thinking about an idea that follow one another as students learn (Wilson & Bertenthal, 2005) (All from Gong -CCSSO -Learning progressions -6/19/07)

EXAMPLE OF A LEARNING PROGRESSION EXAMPLE OF A LEARNING PROGRESSION Standard: 3.NF.3abc 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. b. Recognize and generate simple equivalent fractions, e.g., 1/2 2/4, 4/6 2/3. Explain why the fractions are equivalent, e.g., by using a visual fraction model.

DEMONSTRATES APPROPRIATE MATHEMATICAL MODELS DEMONSTRATES APPROPRIATE MATHEMATICAL MODELS Using the number line and fraction strips to see fraction equivalence

LEARNING PROGRESSIONS LEARNING PROGRESSIONS Explicitly link this material to grade 4: Standard 4.NF.3: Understand a fraction a/b with a >1 as a sum of fractions 1/b. b. Decompose a fraction into a sum of fractions with the same denominator in more than one way, recording each decomposition by an equation. Justify decompositions, e.g., by using a visual fraction model.

USE A VISUAL FRACTION MODEL

  Drafts of Learning Progressions on fractions and several other topics  Supported by Brookhill Foundation  Based on: Mathematical analysis Developmental and cognitive psych (still in draft form)

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

 Provide 10 minutes per session of instruction to build quick retrieval of basic facts.  For students in K-2 grade 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.

Include motivational strategies in tier 2 and tier 3 interventions. Level of Evidence: Minimal

 Rewards can reduce genuine interest in mathematics by directing student attention to gathering rewards rather than learning math.  Suggested Approach: Rewards have not shown to reduce intrinsic interest. As students become more successful, rewards can be faded so student success becomes an intrinsic reward.

 No firm research but a life and death issue  Especially for grades 5 and up INTRINSIC MOTIVATION

Thank You