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Inclusive Education, Intensive, Personalized Education, and Mathematics Instruction 2012 OSEP Project Directors Conference Washington, DC July 24, 2012 Russell Gersten, Ph.D. Director, Instructional Research Group Professor Emeritus, University of Oregon Pick one: 1. I like mathematics 2. I love mathematics 3. I can live with it or without it 4. I try to avoid mathematics What do special educators need to know in the context of Common Core State Standards & Inclusive Education? 1. 2. 3. 4. 5. Goals of the Common Core in Mathematics FRACTIONS (teacher understanding and proficiency must precede improved instruction) Other aspects of algebra readiness Research base on effective instruction (and the gaping holes) RtI in mathematics: strengths and vulnerabilities Focus on fewer Mathematical Ideas and Procedures in depth • Why? To ensure understanding as well as proficiency & fluency • Why? Understanding of arithmetic is key to success in algebra & more advanced mathematics • Why? In part mathematics proficiency involves higher levels of abstraction & abstracting out • Why else more depth & time on each topic? Mastery & proficiency critical…endless spiraling not good Capitalize on insight from cognitive research (Rittle-Johnson & Siegler, 2001) and working knowledge of mathematicians • Conceptual & procedural knowledge develop in a reciprocal fashion • Problem solving must be integrated into the mix as it enriches both (provides meaning to the computational & procedural work, provides a logic for why things are done) Coherence • Includes demonstrating connections between the various ideas in geometry & arithmetic/algebra & between arithmetic & algebra • Involves allowing students to solve problems, at times, in a variety of ways Precision of mathematical language Special education students must work on all these fronts. Older belief that practice on computation is the only means to teach does not address Common Core standards and will be increasingly frowned on. Many American teachers do not know much about fractions (Ball, 1990; Ma, 1999) Many teachers struggle with both solving fraction problems & explaining, for example, what division of a fraction is Many teachers do not have automatic access to various common interpretations of fractions • Part of whole • Parts of a whole (e.g. 7/3) • Parts of a set • A specific point on a number line (measurement implications) • Equivalent to division Students learn fractions as part of a whole in grade 3. Example: • Half of the class went to museum the first day. There are 18 students in the class. How many went? • Put 9/4 on a number line To teach fractions well, teachers must understand the mathematics Special education students often are unlikely to intuit all the interpretations & nuances At a minimum, special educators should know the interpretations of fractions: • part(s) of units such as circles, pizzas, buildings • parts of a set • equivalent to division • a point on a number line determined by numerator & denominator Achievable objectives through PD & work with mathematics educators or texts Properties of Arithmetic & Multiplication • Commutative • Associative Property • Distributive Property Fractions • It is beginning of road to abstraction. Example: one number but looks like two numbers (e.g. 4/7) Word Problems (path to abstraction) Fact Fluency (to be able to understand mathematics) Gateway course…true for both algebra 1&2 Both require a full year of successful navigation of abstractions Universal graduation requirement & high failure rates Gersten, R., Beckman, S., Clarke, B., Foegen, A., Marsh, L., Star, J. R., Witzel, B., Dimino, J., Jayanthi, M., Newman-Gonchar, R., Monahan, S., & Scott, L. (2009). Assisting students struggling with Mathematics: Response to intervention (RtI) 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. Gersten, R., Chard, D. J., Jayanthi, M., Baker, S. K., Morphy, P., Flojo, J. (2009). Mathematics Instruction for Students with Learning Disabilities: A Meta-Analysis of Instructional Components. Review of Educational Research, 79, 1202-1242. Also see Center on Instruction website at http://www.centeroninstruction.org/ Or google Center on Instruction, look for mathematics (Note: may be removed Sept. 30, 2012) Must include foundational skills relevant to grade level content e.g. division for fractions, fractions for simple linear equations Common Core Progressions are excellent source http://ime.math.arizona.edu/progressions/# Interventions must include activities that cover mathematical ideas but use some of the empirical base on effective instruction for atrisk learners Screening measures strongest in K & 1 In my view, grades 3 & up need rethinking for both screening & progress monitoring Do they measure what Common Core sets as proficiency (understanding, problem solving, representation & modeling of mathematical ideas)? What do grades 4 to 8 screening measures add above & beyond prior years’ state assessment? Key question to ask Will interventionists teach to the test? i.e. if progress monitoring measures focus heavily on easy to score items (computations, easy problems) will this determine content of intervention? Given technology & advances with item response theory, can progress monitoring provide diagnostic information & placement information?