Developing Diagnostic Interviews in Mathematics Karen Karp Amy

Developing Diagnostic Interviews in Mathematics Karen Karp Amy Lingo Project ABRI University of Louisville

Topics Overview of Rt. I Model, a multi-tiered intervention approach Diagnostic interview: a component of formative assessment Designing and implementing diagnostic interviews

Rt. I: 3 -Tiered Model ~5% ~15% Tertiary Prevention: specialized & individualized strategies for students with continued failure Secondary Prevention: supplementary strategies for students who do not respond to primary Prevention: school-wide or class-wide systems for all students and staff ~80% of Students

Components of A Strong Rt. I Model Incorporates a regular screening process Includes evidenced based practices Integrates progress monitoring Instructs with preventative methodology Uses diagnostic assessment to align intervention Newman-Gonchar, R. , Clarke, B. , & Gersten, R. (2009). A summary of nine key studies: Multi-tier intervention and response to interventions for students struggling in mathematics. Portsmouth, NH: RMC Research Corporation, Center on Instruction.

Tier 1 - Universal Mathematics Instruction Implementation of core mathematics instruction Instruction with methodology addressing both conceptual and procedural understanding. Implementation of instruction with fidelity

Identification of Students Through Screening Universal Screener Building level team to facilitate the implementation of the screening and progress monitoring Use benchmarks or growth rates to identify students at low, moderate, or high risk for developing mathematics difficulties.

Student Body is given a Universal Screening X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X

Universal Screenings (What we hear schools are using) DIBELS AIMS Web Think Link GRADE MAP Universal Screening: Determines students with possible mathematics difficulties. This is not an endorsement of the products, but a listing of those described

Students Identified by the Universal Screening are given more in-depth mathematics assessment. XXXXXXXXXXXXXXXX XXXXXXXX In-Depth Assessment determines: Tier of Support Specific Areas of Need Plan of Intervention

In-Depth Assessment CCSS- Math Domains Counting and Cardinality Operations and Algebraic Thinking Number and Operations in Base 10 Number and Operations- Fractions Geometry Data and Measurement

Standards of Mathematical Practice 1. 2. 3. 4. 5. 6. 7. 8. Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision Look for and make use of structure. Look for and express regularity in repeated reasoning.

C R U A E

What Does It Mean to Understand Mathematics? n n n Understanding is the measure of quality and quantity of connections between new ideas and existing ideas Knowing ¹ Understanding (students may know something about fractions, for example, but not understand them) Richard Skemp named the ends of the continuum of understanding Instrumental Relational understanding What to do and why Just doing it

Implications for Teaching n n n The need to replace the question “Does the student know it? ” with the question “How does the student understand it? ” Early number concepts Computation Copyright © Allyn and Bacon 2010

I thought they had it! Petit, M. , Laird, R. , & Marsden, E. (2010). A focus on fractions: Bringing Research to the Classroom. New York: Routledge. .

Diagnostic Interview • • • Gathers in-depth information about an individual student’s knowledge and mental strategies. Provides evidence of prior knowledge, naïve understandings and students’ ways of thinking about concepts. Focuses on a task or problem where students are asked to either verbalize their thinking or demonstrate ideas through models or drawings Emphasizes the collection of evidence Is not a teaching opportunity Uses errors to identify barriers to understanding, to inform instructional decisions

Diagnostic Interview - Example Philip, R. & Cabral, C. (2005). IMAP: Integrating mathematics and pedagogy to illustrate children’s reasoning. San Diego State University Foundation: Pearson.

Student Teacher Interaction Philip, R. & Cabral, C. (2005). IMAP: Integrating mathematics and pedagogy to illustrate children’s reasoning. San Diego State University Foundation: Pearson.

3 3/8 Philip, R. & Cabral, C. (2005). IMAP: Integrating mathematics and pedagogy to illustrate children’s reasoning. San Diego State University Foundation: Pearson.

Procedural Error Philip, R. & Cabral, C. (2005). IMAP: Integrating mathematics and pedagogy to illustrate children’s reasoning. San Diego State University Foundation: Pearson.

Talk Aloud-Verbalization Philip, R. & Cabral, C. (2005). IMAP: Integrating mathematics and pedagogy to illustrate children’s reasoning. San Diego State University Foundation: Pearson.

Drawing-Visual Representation Philip, R. & Cabral, C. (2005). IMAP: Integrating mathematics and pedagogy to illustrate children’s reasoning. San Diego State University Foundation: Pearson.

Difference in Solutions Philip, R. & Cabral, C. (2005). IMAP: Integrating mathematics and pedagogy to illustrate children’s reasoning. San Diego State University Foundation: Pearson.

Student Decision-making Philip, R. & Cabral, C. (2005). IMAP: Integrating mathematics and pedagogy to illustrate children’s reasoning. San Diego State University Foundation: Pearson.

How to design a diagnostic assessment Look at what students need to know common core Give a task Record and/or make notes Have materials available for students to use The difference between a regular task and a diagnostic interview is that you are sitting and listening to the student talk about their thinking and how they are solving the task. Questions are used to probe and materials and other representations are used to see the depth of understanding.

Interventions Following Diagnostics Use of screening information Use of diagnostic interview Create a plan for intervention

Recommendations for identifying and supporting students struggling in mathematics Recommendations are based on strong, moderate and low levels of evidence resulting from comprehensive reviews of current research literature. Gersten, R. , Beckmann, S. , Clarke, B. , Foegen, A. , Marsh, L. , Star, J. R. , & Witzel, B. (2009). Assisting students struggling with mathematics: Response to Intervention (Rt. I) for elementary and middle schools (NCEE 2009 -4060). Washington, DC: National Center for Education Evaluation and Regional Assistance, Institute of Education Sciences, U. S. Department of Education. Retrieved from http: //ies. ed. gov/ncee/wwc/publications/practiceguides/.

General Screening and Intervention Recommendations Screen all students Choose appropriate instructional materials Intervention with explicit instruction Modeling Talk aloud (verbalization) Guided practice Feedback (correction of errors) Frequent review of progress Strong Moderate Low Problem solving instruction based on common underlying structures Visual representations

Screening and Intervention Recommendations 10 minutes per session devoted to fluency building of basic mathematics facts Progress monitoring Strong Integration of motivational strategies Moderate Low

Recommendations for students identified as lowachieving On a regular basis; and For the purpose of building computation and problem solving proficiency; Explicit instruction including opportunities for asking and answering questions Think aloud opportunities regarding decisions during problem solving Dedicated time to foundational skills necessary for grade level mathematics learning National Mathematics Advisory Panel. Foundations for Success: The Final Report of the National Mathematics Advisory Panel, U. S. Department of Education: Washington, DC, 2008.

Recommendations from research involving small group interventions Explicit instruction Concrete--Semi-concrete--Abstract approach Modeling Underlying mathematical structures Examples (consideration of range and sequence) Independent work with immediate corrective feedback Visuals (drawings & diagrams) Note: The interventions may be effective for other student groupings. This listing specifically targets small groups. Newman-Gonchar, R. , Clarke, B. , & Gersten, R. (2009). A summary of nine key studies: Multi-tier intervention and response to interventions for students struggling in mathematics. Portsmouth, NH: RMC Research Corporation, Center on Instruction.

Hedden’s Continuum - CSA

Effective Practices for Teachers Explicit Instruction A range of instructional examples, a sequence from concrete-representational-abstract Verbalization by the students and the teacher Use of visual representation Multiple heuristic strategies Formative assessment information provided to teachers Peer-assisted learning (1: 1 tutoring) Cross age (more effective) Within classroom same grade, role exchange Performance based Jayanthi, M. , Gersten, R. , Baker, S. (2008). Mathematics instruction for students with learning disabilities or difficulty learning mathematics: A guide for teachers. Portsmouth, NH: RMC Research Corporation, Center on Instruction.

Thank you Contact information. Karen Karp Amy Lingo karen@louisville. edu amy. lingo@louisville. edu