1. www.FLDOE.org © 2014, Florida Department of Education. All Rights Reserved. Concrete – Representational – Abstract Instructional Model Shelly Miedona, Elementary Math Specialist Bureau of Standards & Instructional Support Shelly.miedona@fldoe.org

2. www.FLDOE.org © 2014, Florida Department of Education. All Rights Reserved. Research Behind the Theory http://padlet.com/shellytimesthree/CRA InstructionalModel

3. www.FLDOE.org © 2014, Florida Department of Education. All Rights Reserved. Jerome Bruner 3 Stages of Intellectual Development 1.Enactive Learning through actions on physical objects 2.Iconic Using models and pictures 3.Symbolic Think abstractly “Leads to more effective learning.” (Bruner 1976) http://www.lifecircles-inc.com/Learningtheories/constructivism/bruner.html

4. www.FLDOE.org © 2014, Florida Department of Education. All Rights Reserved. Jerome Bruner The Culture of Education (1996) Bruner reminds us that, “Education cannot be reduced to mere information processing, sorting knowledge into categories. It’s objective is to help learners construct meanings, not simply to manage information.”

5. www.FLDOE.org © 2014, Florida Department of Education. All Rights Reserved. What is CRA? Research based math intervention – 3 Levels: 1. Concrete Concrete materials 2. Representational/Pictorial 2 dimensional written/pictorial representation 3. Abstract symbols numbers Algorithms http://www.lifecircles-inc.com/Learningtheories/constructivism/bruner.html http://www.ldworldwide.org/educators/strategies-for-successful-learning/1096-deepening-mathematics- teaching-and-learning-through-the-concrete-pictorial-abstract-approach

6. www.FLDOE.org © 2014, Florida Department of Education. All Rights Reserved. Significance of the 3-level progression: Graduated, conceptually supported framework Creates meaningful connections Ensures thorough understanding of math concepts/skills

7. www.FLDOE.org © 2014, Florida Department of Education. All Rights Reserved. Level 1: CONCRETE Tangible, Hands–On Tools Connecting cubes Square tiles Two colored circles Base 10 blocks Measuring tools Coins Buttons Pattern blocks Anything a student can touch and manipulate

8. www.FLDOE.org © 2014, Florida Department of Education. All Rights Reserved. CONCRETE EXAMPLES

9. www.FLDOE.org © 2014, Florida Department of Education. All Rights Reserved.

10. www.FLDOE.org © 2014, Florida Department of Education. All Rights Reserved. Level 2: REPRESENTATIONAL/PICTORIAL Drawings Dots Circles Tally marks Pictures to represent objects used

11. www.FLDOE.org © 2014, Florida Department of Education. All Rights Reserved. REPRESENTATIONAL EXAMPLES

12. www.FLDOE.org © 2014, Florida Department of Education. All Rights Reserved. 4 th Grade NBT.2.5 Area Model

13. www.FLDOE.org © 2014, Florida Department of Education. All Rights Reserved. 2 nd Grade MAFS.2.NBT.2.7

14. www.FLDOE.org © 2014, Florida Department of Education. All Rights Reserved. Level 3: ABSTRACT Numbers Letters Symbols Equations Algorithms

15. www.FLDOE.org © 2014, Florida Department of Education. All Rights Reserved. ABSTRACT EXAMPLES

16. www.FLDOE.org © 2014, Florida Department of Education. All Rights Reserved. K - 2 18 = 10 + 8 23 + 10 = 20 + 10 + 3 13 – 4 = 13 – 3 – 1 8 + ? = 11 15 – 9 = 6 + 23 + 15 = 20 + 10 + 3 + 5 15 < 11 + 5 3 - 5 4 X 3 = (2 X 3) + (2 X 3) 25 x 15 = (20 x 15) + (5 x 15) 215 ÷ 5 = (200 ÷ 5) + (15 ÷ 5) As well as the traditional algorithms for addition, subtraction, multiplication and division.

17. www.FLDOE.org © 2014, Florida Department of Education. All Rights Reserved. Sequencing It is important to follow the CRA sequence and not skip from concrete to abstract. Students need to proceed through all three levels to demonstrate understanding and mastery(University of Kansas). It is important to spend sufficient time in each as well as side by side when students are ready to move on. Have the concrete place value blocks on the desk as the students draw a picture of the concrete. Have the students draw the representational problem of 25 + 49 then teach the “short cut” of the algorithm with regrouping. http://www.specialconnections.ku.edu/?q=instruction/mathematics/teacher_tools/concrete _to_representational_to_abstract_instruction

18. www.FLDOE.org © 2014, Florida Department of Education. All Rights Reserved.

19. www.FLDOE.org © 2014, Florida Department of Education. All Rights Reserved. What does that mean for students? Students learn a focused set of major math concepts and skills that progressively build to more challenging concepts Students participate in real world, hands on, problem solving activities Students develop a greater conceptual understanding of mathematical principles

20. www.FLDOE.org © 2014, Florida Department of Education. All Rights Reserved. Additional Reading Acker, J. V. (2014). Concrete/Representational(Pictorial)Abstract. Retrieved from Prezi: https://prezi.com/1rfj4g22dqua/concreterepresentationalpictorial-abstract/https://prezi.com/1rfj4g22dqua/concreterepresentationalpictorial-abstract/ American Institutes for Research. (2004). Doing What Works. Retrieved from Math Specialist: http://www.mathspecialists.org/dww/Response_to_Intervention/Practice_ Intentional_Teaching/See/1923_it_mat_cra.pdf http://www.mathspecialists.org/dww/Response_to_Intervention/Practice_ Bruner, J. (1996). The Culture of Education. MA: Harvard University Press. Joan Gujarati, E. (2013). Deepening Mathematics Teaching and Learning through the Concrete-Pictorial-Abstract Approach. Retrieved from Learning Disabilities Worldwide: http://www.ldworldwide.org/educators/strategies-for-successful- learning/1096-deepening-mathematics-teaching-and-learning-through-the- concrete-pictorial-abstract-approachhttp://www.ldworldwide.org/educators/strategies-for- Kansas, T. U. (n.d.). Concrete-to-Representational-to-Abstract (C-R-A) Instruction. Retrieved from The University of Kansas: http://www.specialconnections.ku.edu/?q=instruction/mathematics/teach er_tools/concrete_to_representational_to_abstract_instruction http://www.specialconnections.ku.edu/?q=instruction/mathematics/teach Sousa, D. A. (2008). How the brain learns mathematics. Thousand Oaks, CA: Corwin Press.

21. www.FLDOE.org © 2014, Florida Department of Education. All Rights Reserved. References Acker, J. V. (2014). Concrete/Representational(Pictorial)Abstract. Retrieved from Prezi: https://prezi.com/1rfj4g22dqua/concreterepresentationalpictorial-abstract/ https://prezi.com/1rfj4g22dqua/concreterepresentationalpictorial-abstract/ American Institutes for Research. (2004). Doing What Works. Retrieved from Math Specialist: http://www.mathspecialists.org/dww/Response_to_Intervention/Practice_ Intentional_Teaching/See/1923_it_mat_cra.pdf http://www.mathspecialists.org/dww/Response_to_Intervention/Practice_ Balka, H. a. (n.d.). What is Conceptual Understanding? Retrieved from Math Leadership: http://www.mathleadership.com/sitebuildercontent/sitebuilderfiles/conceptualUnderstandin g.pdf http://www.mathleadership.com/sitebuildercontent/sitebuilderfiles/conceptualUnderstandin Ben-Hur, M. (2006). Concept-Rich Mathematics Instruction. Retrieved from ASCD: http://www.ascd.org/publications/books/106008/chapters/Conceptual-Understanding.aspx http://www.ascd.org/publications/books/106008/chapters/Conceptual-Understanding.aspx Brickwedde, J. (2012). Developing Base Ten Understanding: Working with Tens, The Difference Between Numbers, Doubling, Tripling..., Splitting, Sharing and Scaling Up. Retrieved from Project for Elementary Mathematics: http://www.uwosh.edu/coehs/cmagproject/concepts/documents/Developing_Base_Ten_Und erstanding.pdf http://www.uwosh.edu/coehs/cmagproject/concepts/documents/Developing_Base_Ten_Und Bruner, J. (1996). The Culture of Education. Cambridge: Harvard University Press. CPALMS Where Educators Go For Bright Ideas. (2015). Retrieved from CPALMS: http://www.cpalms.org/Public/ http://www.cpalms.org/Public/

22. www.FLDOE.org © 2014, Florida Department of Education. All Rights Reserved. References (continued) Florida Standards. (2015). Retrieved from Florida Department of Education: http://fldoe.org/academics/standards/florida-standards http://fldoe.org/academics/standards/florida-standards Hanner, B. (n.d.). Education. Retrieved from Pinterest: https://www.pinterest.com/barbara_hanner/education/ https://www.pinterest.com/barbara_hanner/education/ Joan Gujarati, E. (2013). Deepening Mathematics Teaching and Learning through the Concrete-Pictorial- Abstract Approach. Retrieved from Learning Disabilities Worldwide: http://www.ldworldwide.org/educators/strategies-for-successful-learning/1096-deepening- mathematics-teaching-and-learning-through-the-concrete-pictorial-abstract-approach http://www.ldworldwide.org/educators/strategies-for-successful-learning/1096-deepening- Press, H. U. (1995). Theories of Learning in Educational Psychology. Retrieved from Life Circles, Inc: http://www.lifecircles-inc.com/Learningtheories/constructivism/bruner.html http://www.lifecircles-inc.com/Learningtheories/constructivism/bruner.html

23. www.FLDOE.org © 2014, Florida Department of Education. All Rights Reserved. Department of Education Updates

24. www.FLDOE.org © 2014, Florida Department of Education. All Rights Reserved. PAEMST Presidential Awards for Excellence in Mathematics & Science Teaching Applications are due May 1 st Reviewers will be assigned by May 9 th Two rounds of reviews Paid stipend to reviewers Contact – Heidi.brennan@fldoe.orgHeidi.brennan@fldoe.org

25. www.FLDOE.org © 2014, Florida Department of Education. All Rights Reserved. Resources http://www.fldoe.org/academics/standards/subject-areas/math-science/mathematics

26. www.FLDOE.org © 2014, Florida Department of Education. All Rights Reserved. Parent Resources

27. www.FLDOE.org © 2014, Florida Department of Education. All Rights Reserved. Computer Science Standards New Computer Science Standards for 16/17 school year Public Review February 16 th Districts may use these standards to submit K- 12 courses Certification to teach courses not yet finalized Possibly a Math, Science and Computer Science Certification

28. www.FLDOE.org © 2014, Florida Department of Education. All Rights Reserved. Contact Information: Rashad Bennett 850-245-0830 Rashad.Bennett@fldoe.org Jonathan Keener 850-245-0808 Jonathan.Keener@fldoe.org Shelly Miedona 850-245-0660 Shelly.Miedona@fldoe.org Heidi Brennan 850-245-7805 Heidi.Brennan@fldoe.org Contact Information:

29. www.FLDOE.org © 2014, Florida Department of Education. All Rights Reserved. www.FLDOE.org