1. Concrete – Representational – Abstract Instructional Model
Shelly Miedona, Elementary Math Specialist
Bureau of Standards & Instructional Support

2. Research Behind the Theory

3. Jerome Bruner 3 Stages of Intellectual Development
Enactive
Learning through actions on physical objects
Iconic
Using models and pictures
Symbolic
Think abstractly
“Leads to more effective learning.” (Bruner 1976)
Bruner was one of the founding fathers of constructivist theory. According to Bruner, learning is facilitated by the teacher through lessons that provide the information and allows the learner to organize it for his/her learning. The learner plays an active role in building understanding and making sense of the information presented.
He postulated three stages of intellectual development.
Using a combination and moving through these stages will lead to more effective learning.

4. 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.”
Using the Concrete-Representational-Abstract (CRA) model of instruction facilitates the meaning of abstract concepts. On the next slide we will look more closely at this model of instruction.

5. 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
teaching-and-learning-through-the-concrete-pictorial-abstract-approach
To meet the needs of all students in twenty-first century classrooms, teachers must incorporate multiple representations of mathematical ideas in their instruction to reach all students with their multiple learning styles. Although this was and is coined as a math intervention, students at all levels, can benefit…..gen ed, ESOL and ESE students. Teachers look to differentiate instruction on a daily basis. Using the CRA or CPA, pictorial as some say, instructional approach meets the diverse needs of students so they can develop true conceptual understanding through interactions and experiences.

6. Significance of the 3-level progression:
Graduated, conceptually supported framework
Creates meaningful connections
Ensures thorough understanding of math concepts/skills
Students experience and discover mathematics rather than simply memorizing processes and procedures. This instructional approach helps to facilitates the mastery of Florida Mathematics Standards and fluency of math facts. As Bruner described, it helps learners to construct meaning and take ownership of their understanding so they can apply learning to other tasks and problems.

7. 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
When introducing a new manipulative, allow 2 minutes of “free play” so that the students can become familiar with the new manipulative. Set an expectation that these first 2 minutes are for you to “play” with them. Once the lesson begins, students must stay with you and “play time” is over. Doing this at the beginning will ensure on task behavior as you teach the concept with the manipulative.
Spend as much time here as needed to ensure students understand the concept at the concrete level. They should be able to manipulate and solve problems using manipulatives without teacher support before moving on to the representational/pictorial stage.

8. CONCRETE EXAMPLES

10. Level 2: REPRESENTATIONAL/PICTORIAL
Drawings
Dots
Circles
Tally marks
Pictures to represent objects used
It is important for teachers to explain the connection between the concrete and the representational/pictorial stage.
Too often the connection is not made or not enough time is spent here.
When first presenting this stage, it is important for students to manipulate the concrete first, then draw the picture. Spend sufficient time here before taking the manipulatives away. Use your judgment and differentiate as necessary based on student need.
With the Florida Math Standards, some standards and math concepts stay here at this stage. Mastery would be to draw a picture or a representation.
Teaching students to draw solutions to solve problems provides an excellent strategy for problem solving in the future.

11. REPRESENTATIONAL EXAMPLES

12. 4th Grade NBT.2.5 Partial Product Area Model
Although numbers are listed in this example, it would be considered representational because the students are not using the algorithm.  This student demonstrated understanding of multi-digit multiplication by using the area model and partial product.  Area model is when the student “breaks apart” one or both factors and shows the distributive property in an array or area of a rectangle.  Partial product does this similarly but the array is no longer used or necessary.  Partial product is getting the student ready for the algorithm and students will begin to see connections.
Partial Product
Area Model

13. This example shows understanding of this standard by using a drawing and place value. The student then explained in a written method that she must take ones from ones, tens from tens and hundreds from hundreds.
2nd Grade
MAFS.2.NBT.2.7

14. Level 3: ABSTRACT Numbers Letters Symbols Equations Algorithms
When moving to the abstract model, it is important for teachers to explain how the symbols provide a shorter more efficient way to solve problems. 
With the Florida Math Standards, dependent upon the grade level and the standard, students may not learn the algorithm/procedure until a later grade.  For instance, students in 2nd grade learn addition with composing ones to make a ten, but the algorithmic procedure occurs in the 4th grade standards. 

15. ABSTRACT EXAMPLES

16. K - 2
3 - 5
18 = = – 4 = 13 – 3 – ? = – 9 = = <
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. 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 then teach the “short cut” of the algorithm with regrouping.
_to_representational_to_abstract_instruction
Students need to have the understanding of why they are doing something before teaching the algorithm for regrouping.

18. When moving from one stage to the next, be sure students begin with the stage they’ve been working in, then introduce the next stage. To go from concrete to representational, first have the students show = 9 with the blocks, then have the students draw what they did as on this slide. When the students move from representational to abstract they first draw the representation then they show the abstract.

19. 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
Students are not only learning new math concepts, they are learning to problem solve and apply what they have learned.

20. Additional Reading
Acker, J. V. (2014). Concrete/Representational(Pictorial)Abstract. Retrieved from Prezi:
American Institutes for Research. (2004). Doing What Works. Retrieved from Math Specialist: Intentional_Teaching/See/1923_it_mat_cra.pdf
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: successful- learning/1096-deepening-mathematics-teaching-and-learning-through-the- concrete-pictorial-abstract-approach
Kansas, T. U. (n.d.). Concrete-to-Representational-to-Abstract (C-R-A) Instruction. Retrieved from The University of Kansas: er_tools/concrete_to_representational_to_abstract_instruction
 Sousa, D. A. (2008). How the brain learns mathematics. Thousand Oaks, CA: Corwin Press.
Additional readings to provide additional information on the CRA model of instruction and how to implement the model in the classroom and at home.

21. References
Acker, J. V. (2014). Concrete/Representational(Pictorial)Abstract. Retrieved from Prezi: American Institutes for Research. (2004). Doing What Works. Retrieved from Math Specialist: Intentional_Teaching/See/1923_it_mat_cra.pdf Balka, H. a. (n.d.). What is Conceptual Understanding? Retrieved from Math Leadership: g.pdf Ben-Hur, M. (2006). Concept-Rich Mathematics Instruction. Retrieved from ASCD: 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: erstanding.pdf Bruner, J. (1996). The Culture of Education. Cambridge: Harvard University Press. CPALMS Where Educators Go For Bright Ideas. (2015). Retrieved from CPALMS:

22. References (continued)
Florida Standards. (2015). Retrieved from Florida Department of Education: Hanner, B. (n.d.). Education. Retrieved from Pinterest: Joan Gujarati, E. (2013). Deepening Mathematics Teaching and Learning through the Concrete-Pictorial- Abstract Approach. Retrieved from Learning Disabilities Worldwide: mathematics-teaching-and-learning-through-the-concrete-pictorial-abstract-approach Press, H. U. (1995). Theories of Learning in Educational Psychology. Retrieved from Life Circles, Inc:

23. Department of Education Updates

24. Presidential Awards for Excellence in Mathematics & Science Teaching
PAEMST
Presidential Awards for Excellence in Mathematics & Science Teaching
Applications are due May 1st
Reviewers will be assigned by May 9th
Two rounds of reviews
Paid stipend to reviewers
Contact –

25. Resources http://www. fldoe
Access these resources under the Standards and Instructional Support tab in Academics
Then click Mathematics, Science & STEM
Finally click Mathematics
There are links in the Science tab as well for Instructional Support with more to come in both areas

26. Parent Resources All content areas are updating the parent resources.
Also encourage parents and students to go here for support with Florida Standards and preparing for the FSA.

27. Computer Science Standards
New Computer Science Standards for 16/17 school year
Public Review February 16th
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. Rashad Bennett 850-245-0830 Jonathan Keener 850-245-0808
Contact Information:
Contact Information:
Rashad Bennett
Jonathan Keener
Shelly Miedona
Heidi Brennan