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David H. Allsopp, Ph.D. University of South Florida dallsopp@tempest.coeduf

Effective Mathematics Teaching Practices: Explicit CRA Practices – Making Mathematics Transparent. David H. Allsopp, Ph.D. University of South Florida dallsopp@tempest.coedu.usf.edu. “Explicit” Means To Provide Students…. ACCESS. ACCESS. ACCESS. ACCESS. ACCESS.

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David H. Allsopp, Ph.D. University of South Florida dallsopp@tempest.coeduf

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  1. Effective Mathematics Teaching Practices:Explicit CRA Practices – Making Mathematics Transparent • David H. Allsopp, Ph.D. • University of South Florida • dallsopp@tempest.coedu.usf.edu

  2. “Explicit” Means To Provide Students… ACCESS ACCESS ACCESS ACCESS ACCESS to the target mathematics concept.

  3. Providing Access Student Mathematics Concept • “Students can’t hit what they can’t see...”

  4. Make mathematics concepts accessible to your students by... • Student Mathematics Concept • Authentic Contexts • Visuals • Language Experiences • Teach Problem Solving Strategies • Multiple Opportunities to Apply Understandings • Data-based Decision-making Concrete-to-Representational-to-Abstract Experiences

  5. What is CRA? 17 • It is not a “natural” process for some students • Systematically teaching mathematics through a Concrete-to-Representational-to-Abstract Sequence • Concrete Level - materials that students can manipulate to represent mathematical concepts and to problem solve. • Representational Level - teaching drawing strategies to represent mathematical concepts and to problem solve. • Abstract Level - representing mathematical concepts and problem solving using numbers and mathematical symbols without the use of concrete materials and drawings.

  6. 17 Systematic CRA Teaching Process Identify learner objectives CRA CRA Monitor Progress Make instructional decisions Provide advance organizer Specific Corrective Feedback Specific Positive Reinforce-ment CRA CRA Provide models Evaluate learning Provide guided practice & independent practice CRA CRA

  7. 17 What is CRA? Numbers and other mathematical symbols should be used at all three levels and should be explicitly associated with the concrete materials and drawings that represent them.

  8. Example of Explicit CRA Sequence • Let’s examine the CRA sequence for the following algebraic expression: 4x = 8

  9. Concrete Level:Start with Concrete Experiences • Use simple (discrete) materials to represent the abstract numbers and symbols. • Model how to manipulate the materials in ways to problem solve.

  10. Model how to draw a representation of the concept/problem solving situation. Model how to manipulate the drawings in order to do the mathematics involved. Representational Level:Teach Strategies for Drawing Representations of the Same Concept or Problem Solving Situation 4x = 8 X=2

  11. Abstract Level:Gradually Fade Use of Drawings So Students Do Mathematics Using Numbers & Symbols Only 4x = 8 x = 2 Concrete 4x = 8 x = 2 Representational 4x = 8 x = 2 Abstract

  12. CRA Levels Other Examples in Resource Packet CRA Examples pp. 65-68

  13. 18-19 Part 2: Effectively Implementing Explicit CRA Instruction for Struggling Learners • Important Considerations • Effective Explicit CRA Instructional Practices

  14. 18 Important Considerations for Struggling Learners • Use a variety of appropriate concrete materials. • Use appropriate drawings. • Use appropriate strategies for helping students transition from concrete to abstract levels of understanding. • Allow students to reach mastery at each level of understanding before moving to the next level.

  15. Appropriate Use of Concrete Materials • Accurate representation of concept • Discrete objects • Attributes more “accessible” • Can be “manipulated” more easily • Use a variety over time to stimulate generalization

  16. Manipulative Examples: (More Abstract) • Potential benefits? • Potential barriers?

  17. More About Concrete Materials • Discrete vs. Continuous • Proportional vs. Non-Proportional • Linked vs. Non-linked

  18. Drawing Examples (see MathVIDS website)

  19. Other Drawing Examples (see MathVIDS website)

  20. Individually or with a group… • Draw solutions for one or more of the following operations: • ½ x ¼ = • 1 ¼ + ¼ = • -8 + 7 = • 3x = 24 Activity • Individually or with a group… • Draw solutions for one or more of the following operations: • 14 + 5 = • 3 x 6 = • 15 ÷ 3 = • 7 – 4 =

  21. Periodically “Move Down” Levels to Reinforce Conceptual Understandings Gained At Prior Levels • Maintenance Activities • Periodic opportunities for students to revisit previously mastered abstract level knowledge and skills through 5-10 minute activities. • Teacher prepares a response prompt that engages students in thinking about/describe one or more key features of the previously mastered concept or skill. • Students use concrete materials and/or drawings to emphasize important features of the previously mastered concept or skill. • Purpose is to help students maintain what they have previously mastered AND to further enhance conceptual understanding. • The link below will take you to the MathVIDS Teaching Plans site where you can read about this practice activity: • http://coe.jmu.edu/mathvids2/plans/cflud/C_plan4.html

  22. Reflection Activity • Individually or with a group… • Write and/or share your observations from reviewing the information on Effective CRA Instructional Practices

  23. How Does Systematic CRA Instruction Address Needs of Struggling Learners? ?

  24. How Does Systematic CRA Instruction Address Needs of Struggling Learners? By Providing StudentsMultisensory ACCESSTo The Distinctive Features of Mathematical Concepts & Processes C - R - A Student Mathematics Concept

  25. From the MathVIDS Professional Development Website: CRA Assessment and Teaching In Action - Multimedia Models • http://fcit.usf.edu/mathvids/index.html CRA Assessment and Teaching Resources on the MathVIDS website: Mathematics Dynamic Assessment - Click “Instructional Strategies - Complete List of Strategies” CRA Instruction Sequence -Click “Instructional Strategies - Complete List of Strategies” Explicit Teacher Modeling at CRA Levels - Click “Instructional Strategies - Complete List of Strategies” Teaching Plans for Selected Mathematics Concepts/Standards at CRA Levels that integrate research supported instructional practices - Click “Teaching Plans”

  26. How Does Systematic CRA Instruction Address Needs of Struggling Learners? CRA Provides Students A Tangible Foundation for Conceptual Understanding of Abstract Concepts & Processes • Through Concrete & Representational Learning Experiences • That Are Purposeful; • That Are Systematic in Implementation; • That Are Explicitly Associated with the Abstract; • That Provide Multiple Opportunities To Respond; • Where Students Use Language To Describe What They Understand.

  27. CRA Instruction & Related Instructional Practices Research Support • Allsopp, D.H. (1999). Using modeling, manipulatives, and mnemonics with eighth grade students. Teaching Exceptional Children, 32(2), 74–81. • Allsopp, D.H., Lovin, L., Green, G.W., & Savage-Davis, E. (2003). Why students with high incidence disabilities have difficulty learning mathematics and what teachers can do to help. Mathematics Teaching in the Middle School, 8, 308–314. • Allsopp, D.H., Minskoff, E.H., & Bolt, L. (2005). Individualized course-specific strategy instruction for college students with learning disabilities and ADHD: Lessons from a model demonstration project. Learning Disabilities Research & Practice, 20, 103–118. • Baxter, J.A., Woodward, J., & Olson, D. (2005). Writing in mathematics: An alternative form of communication for academically low-achieving students. Learning Disabilities Research & Practice, 20, 119–135. • Beirne-Smith, M. (1991). Peer tutoring in arithmetic for children with learning disabilities. Exceptional Children, 57, 330–33.

  28. Black, P., & Dylan Williams, D. (1998). Assessment and classroom learning. Assessment in Education, 5(1), 7–74. Bottge, B., Heinrichs, M., Metha, Z., & Hung, Y. (2002). Weighing the benefits of anchored math instruction for students with disabilities in general education classes. The Journal of Special Education, 35, 186–200. Bottge, B.A., Heinrichs, M., Chan, S., Mehta, Z.D., & Watson, E. (2003). Effects of video-based and applied problems on the procedural math skills of average- and low-achieving adolescents. Journal of Special Education Technology, 18(2), 5–22. Bryant, B.R. (1996). Using alternative assessment techniques to plan and evaluate mathematics instruction. LD Forum, 21 (2), 24–33. Burns, M. (1996, April). How to make the most of math manipulatives. Instructor, 45–50. Butler, F.M., Miller, S.P., Crehan, K., Babbitt, B., & Pierce, T. (2003). Learning Disabilities: Research & Practice, 18, 99–111. Calhoon, B., & Fuchs, L. (2003). The effects of peer-assisted learning strategies and curriculum-based measurement on the mathematics performance of secondary students with disabilities. Remedial and Special Education, 24, 235–245. Carnine, D.W., Dixon, R.C., & Silbert, J. (1998). Effective strategies for teaching mathematics. In E. Kameenui & D. Carnine (Eds.), Effective teaching strategies that accommodate diverse learners. Upper Saddle River, NJ: Prentice Hall. Cawley, J. (2002). Mathematics interventions and students with high incidence disabilities. Remedial and Special Education, 23, 2–6.

  29. Chard, D., & Gersten, R. (1999). Number sense: Rethinking arithmetic instruction for students with mathematical disabilities. Journal of Special Education, 33, 18–28. Englert, C.S., Garmon, A., Mariage, T., Rozendal, M., Tarrant, K., & Urba, J. (1995). The early literacy project: Connecting across the literacy curriculum. Learning Disability Quarterly, 18, 253–275. Englert, C.S., & Mariage, T.V. (1991). Shared understandings: Structuring the writing experience through dialogue. Journal of Learning Disabilities, 24, 330–342. Fuchs, L.S., & Fuchs, D. (2001). Principles for the prevention and intervention of mathematics difficulties. Learning Disabilities Research & Practice, 16, 85–95. Fuchs, L.S., Fuchs, D., Hamlett, C.L., Phillips, N.B., & Bentz, J. (1994). Classwide curriculum-based measurement: Helping general educators meet the challenge of student diversity. Exceptional Children, 60, 518–537. Gagnon, J.C., & Maccini, P. (2001). Preparing students with disabilities for algebra. Teaching Exceptional Children, 34(1), 8–15. Geary, D.C. (1993). Mathematical disabilities: Cognitive, neuropsychological, and genetic components. Psychological Bulletin, 114, 345–362. Gersten, R. (1998). Recent advances in instructional research for students with learning disabilities: An overview. Learning Disabilities Research and Practice, 13, 162–170. Ginsburg, H.P. (1987). How to assess number facts, calculation, and understanding. In D.D. Hammill (Ed.), Assessing the abilities and instructional needs of students (pp. 483–503). Austin, TX: PRO-ED.

  30. Harris, C.A., Miller, S.P., & Mercer, C.D. (1995). Teaching initial multiplication skills to students with disabilities in general education classrooms. Learning Disabilities Research & Practice, 10, 180–195. Kennedy, L.M., & Tipps, S. (1998). Guiding children’s learning of mathematics (8th ed.). Belmont, CA: Wadsworth. Kerrigan, J. (2002). Powerful software to enhance the elementary schools mathematics program. Teaching Children Mathematics, 8(6), 364–370. Liedtke, W. (1988, November). Diagnosis in mathematics: The advantages of an interview. Arithmetic Teacher, 181–184. Lock, R.H. (1996, Winter). Adapting mathematics instruction in the general education classroom for students with mathematics disabilities. LD Forum 21(4), 1–9. Maccini, P., & Gagnon, J.C. (2000). Best practices for teaching mathematics to secondary students with special needs. Focus on Exceptional Children, 32, 1–22. Mercer, C., Harris, C., & Miller, P. (1993). First invited response: Reforming reforms in mathematics. Remedial and Special Education, 14, 14–19. Mercer, C.D., Jordan, L., & Miller, S.P. (1996). Constructivistic math instruction for diverse learners. Learning Disabilities Research & Practice, 11, 147–156. Mercer, C.D., Lane, H.B., Jordan, L, Allsopp, D.H., & Eisele, M.R. (1996). Empowering teachers and students with instructional choices in inclusive settings. Remedial and Special Education, 17, 226–236. Mercer, C.D., & Mercer, A.R. (2005). Teaching students with learning problems (7th ed.). Upper Saddle River, NJ: Prentice Hall.

  31. Miller, S., & Mercer, C. (1993). Mnemonics: Enhancing the math performance of students with learning difficulties. Intervention in School and Clinic, 29, 78–82. Miller, S.P., Butler, F.M., & Lee, K. (1998). Validated practices for teaching mathematics to students with learning disabilities: A review of the literature. Focus on Exceptional Children, 31, 1–24. Miller, S.P., & Mercer, C.D. (1993). Using data to learn about concrete-semiconcrete-abstract instruction for students with math disabilities. Learning Disabilities Research & Practice, 8, 89–96. Miller, S.P., & Mercer, C.D. (1997). Educational aspects of mathematics disabilities. Journal of Learning Disabilities, 30, 47–56. Miller, S.P., Mercer, C.D., & Dillon, A.S. (1992). CSA: Acquiring and attaining math skills—A systematic and practical approach to teaching basic math skills at the concrete, semi-concrete, and abstract levels. Intervention in School and Clinic, 28, 105–110. Minskoff, E., & Allsopp, D. (2003). Academic success strategies for adolescents with learning disabilities and ADHD. Baltimore: Paul H. Brookes Publishing Co. Peterson, S.K., Mercer, C.D., & O’Shea, L. (1988). Teaching learning disabled children place value using the concrete to abstract sequence. Learning Disabilities Research, 4, 52–56. Sullivan, P. & Lilburn, P. (2002). Good questions for math teaching: Why ask them and what to ask, Grades K–6. Sausalito, CA: Math Solutions Publications. Sutherland, K.S., & Wehby, J.H. (2001). Exploring the relationship between increased opportunities to respond to academic requests and the academic and behavioral outcomes of students with EBD. Remedial and Special Education, 22, 113–121.

  32. Swanson, H.L. (1999). Instructional components that predict treatment outcomes for students with learning disabilities: Support for a combined strategy and direct instruction model. Learning Disabilities Research & Practice, 14(3), 129–140. Van de Walle, J.A. (2005). Elementary school mathematics: Teaching developmentally (5th ed.). White Plains, NY: Longman Publishing Group. Vaughn, S., Bos, C.S., & Schumm, J.S. (1997). Teaching mainstreamed, diverse, and at-risk students in the general education classroom. Needham Heights, MA: Allyn & Bacon. from research syntheses. Exceptional Children, 67, 98–114. Witzel, B., Mercer, C.D., & Miller, D. (2003). Teaching algebra to students with learning difficulties: An investigation of an explicit instruction model. Learning Disabilities Research & Practice, 18, 121–131. Woodward, J., & Howard, L. (1994). The misconceptions of youth: Errors and their mathematical meaning. Exceptional Children, 61, 126–l36.

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