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V.6 #2 Mathematics - Deepening Mathematics Teaching and Learning through the Concrete-Pictorial-Abst

Brought to you by Learning Disabilities Worldwide (LDW®) through the generosity of Saint Joseph's University.

In order to meet the array of students’ needs in twenty-first century classrooms, it is important for teachers to incorporate multiple representations of mathematical ideas in their instruction because this increases the chance that they reach all students through their diverse learning styles (e.g., auditory, visual, kinesthetic). Students can benefit from mathematics approached in varied forms through hearing, seeing, saying, touching, manipulating, writing or drawing concepts within mathematics (Shih, Speer, & Babbitt, 2011). Students who have difficulties with mathematics can benefit from lessons that include multiple models that comprise a mathematics skill or concept at different cognitive levels (Sousa, 2008) since students with learning difficulties rarely learn from only seeing or hearing mathematics (Shih, Speer, & Babbitt, 2011). Due to the importance of diverse approaches to differentiate instruction, this article explores mathematics teaching and learning through the concrete-pictorial-abstract (CPA) approach. Concrete-Pictorial-Abstract (CPA) Approach Research has shown that the optimal presentation sequence to teach new mathematical content is through the concrete-pictorial-abstract (CPA) approach (Sousa, 2008). This approach also goes by other names: the concrete-representational-abstract approach or the concrete-semiconcrete-abstract approach. Regardless of the terminology used, the instructional approach is similar and is based on the work of Jerome Bruner (Bruner, 1960). I utilize the CPA approach with my preservice elementary teacher candidates in the mathematics methods courses I teach and utilized it when I taught elementary school to foster a deeper understanding of mathematics so that students are gaining greater conceptual knowledge rather than mere procedural knowledge. Through this approach, students are experiencing and discovering mathematics rather than simply regurgitating it. Concrete. At the concrete level, tangible objects, such as manipulatives, are used to approach and solve problems. Examples of concrete tools include: unifix cubes, Cuisenaire rods, fraction circles and strips, base-10 blocks, double-sided foam counters, or measuring tools. Almost anything students can touch and manipulate to help approach and solve a problem is used at the concrete level. Pictorial. At the pictorial level, representations are used to approach and solve problems. These can include drawings (e.g., circles to represent coins, pictures of objects, tally marks, number lines), diagrams, charts, and graphs. These pictures are visual representations of the concrete manipulatives. It is important for the teacher to explain this connection. Abstract. At the abstract level, symbolic representations are used to approach and solve problems. These representations can include numbers or letters. It is important for teachers to explain how symbols can provide a shorter and efficient way to represent numerical operations. The CPA Approach in Action In this section, I present two mathematics problems in which I showcase examples of how students, who are at varied cognitive levels, might approach the problems. Example 1: Sophia’s mother baked some chocolate chip cookies for her daughter’s birthday party. In the first batch, she baked 26 cookies. In the second batch, she baked 35 cookies. How many cookies did Sophia’s mother bake all together?

  • Concrete. To solve the problem, students might use actual cookies (or unifix cubes, base-10 blocks, or other counters to represent the cookies) to join 26 cookies and 35 cookies to reach 61 cookies all together.

  • Pictorial. One approach (pictured below) might be for students to use base-10 blocks to show representationally that 26 cookies plus 35 cookies yield 61 cookies in total.

  • Abstract. Students may approach the problem using the traditional algorithm in which they know to regroup (“carry”) the “1” (which is really one group of 10), as shown below.



+ 35


Example 2: Martin picked 24 apples. He picked four times as many apples as Julia. How many apples did Julia pick?

  • Concrete. To solve this problem, students could use real apples (or unifix cubes, double-sided foam counters, or other counters to represent the apples). Since the problem states that Martin picked four times as many apples as Julia, students may take Martin’s 24 apples (the whole) and split them up into four equal groups. One group of 6 apples would then represent how many apples Julia picked.

  • Pictorial. Scaffolding the concrete level, students may represent the problem using red circles as apples to portray that Julia has 6 apples (in relation to Martin’s 24 apples), as shown below.

  • Abstract. Students may approach the problem as 4 x ?=24. Realizing that this is really a division problem (despite the word “times” in the word problem), they could then write 24 ÷ 4=6. Therefore, Julia picked 6 apples.

Significance of the CPA Approach Ultimately, teachers want students to reach that abstract level by using symbols proficiently which can also lead to greater efficiency in problem solving. However, before they can reach that abstract level of understanding, they need to have had experiences with the concrete and pictorial levels to build a foundation for this abstract understanding; otherwise, they are learning by rote procedures which may mask their true understanding. In example 1 shown above, if students started at the abstract level to solve the problem using the traditional algorithm, yes, they might get the correct answer by following set procedures the teacher may have demonstrated, but they may not necessarily have experiences to understand that the regrouped “1” is really one group of 10. When having my preservice teachers experience mathematics through concrete and pictorial methods before moving toward the abstract, I have had so many students have “ah ha” moments in which they finally truly understood what that regrouped “1” really meant, and how they could now better assist their future students. Many lamented that if they were not taught so abstractly all the time through formulas and procedures in their formative years, they would have understood mathematics better which may have helped shape their mathematics dispositions toward more positive ones earlier in their lives. The CPA approach can benefit all learners, but has been shown to be particularly effective with students who have mathematics difficulties because it shows students a gradual progression from tangible objects to pictorial and finally to symbols where they have experienced mathematics in various forms (Jordan, Miller, & Mercer, 1998). It gives students strategies for tackling mathematics problems rather than simply searching for an answer (Sousa, 2008). This approach is not grade level dependent either. Although a student in fifth grade may be at the abstract level for multi-digit addition and subtraction of whole numbers, she may be at the concrete level for the addition and subtraction of fractions where manipulatives, such as fraction strips/circles or Cuisenaire rods, are needed to build understanding of this concept. Therefore, students can be at various cognitive levels depending on the mathematics concepts at hand. Overall, teachers of all grade levels are encouraged to approach mathematics through diverse modalities in order to meet students’ cognitive mathematics needs.

References Bruner, J. S. (1960). The process of education. Cambridge, MA: Belknap Press. Jordan, L., Miller, M., & Mercer, C. (1998). The effects of concrete to semi-concrete to abstract instruction in acquisition and retention of fraction concepts and skills. Learning Disabilities: A Multidisciplinary Journal, 9(3), 115-122.

Shih, J., Speer, W. R., & Babbitt, B. C. (2011). Instruction: Yesterday, I learned to add; today I forgot. In F. Fennell (Ed.), Achieving fluency: Special education and mathematics (pp. 59-83). Reston, VA: National Council of Teachers of Mathematics.

Sousa, D. A. (2008). How the brain learns mathematics. Thousand Oaks, CA: Corwin Press. Joan Gujarati, Ed.D., is an Assistant Professor in the Department of Curriculum and Instruction at Manhattanville College in Purchase, New York where she teaches the childhood mathematics methods courses. She is a former elementary school teacher and Math Teacher Leader. Dr. Gujarati’s research interests include early childhood and elementary mathematics education, teacher beliefs and identity, teacher quality, effectiveness, and retention, and curriculum development. Dr. Gujarati can be reached at

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