## CPA Model

The CPA approach is a framework that is widely used in classrooms around the world, particularly in Singapore, as a way to teach math through mastery. CPA stands for concrete, pictorial, and abstract -- a framework developed by Jerome Bruner. As per Purwadi, Sudiarata, & Suparta (2019), "There are three stages; (1) concrete (learning through real objects); (2) pictorial/representational (learning through image representation); and (3) abstract (learning through abstract writing)" (p. 1114). Accordingly, the CPA approach starts with concrete materials, moves towards providing students with pictorial representations around the concept, and ends with teaching the abstract, using notations and symbols. More specifically, Schultz (1986) explains, "Examples of concrete models are blocks, sticks, chips, Cuisenaire rods, and Dienes blocks. Examples of pictorial models are pictures of blocks, sticks, chips, Cuisenaire rods, and Dienes blocks. These pictures can be on worksheets, textbook pages, bulletin boards, paper or felt cutouts, or cards. Symbolic models are numerals on worksheets, textbook pages, chalkboards, bulletin boards, or cards" (p. 54).

As Schultz (1986) suggests, "Certain types of representational models are appropriate for certain types of learning and not for others" (p. 54). Therefore, the CPA approach works very well in elementary and middle school years, but may not be feasible for all concepts. That is not say that concrete and pictorial approaches are obsolete in high school, but the content tends to be more abstract in nature, where both concrete and pictorial representations are less relevant or no longer fully applicable.

As Schultz (1986) suggests, "Certain types of representational models are appropriate for certain types of learning and not for others" (p. 54). Therefore, the CPA approach works very well in elementary and middle school years, but may not be feasible for all concepts. That is not say that concrete and pictorial approaches are obsolete in high school, but the content tends to be more abstract in nature, where both concrete and pictorial representations are less relevant or no longer fully applicable.

## Concrete

In the concrete stage, students are able to use manipulatives and objects to solve problems. Here, students are able to “build” or “make” the learning happen. According to the Ontario Ministry of Education, "Manipulatives are a useful tool to help students construct their understanding of concepts through meaningful investigation. They are crucial for visual and kinesthetic learners, and are beneficial for all students as they move from the concrete to the abstract stages of mathematics" (p. 1). Similarly, Larbi & Mavis (2006) suggest, "Instructional materials or manipulatives provide the physical media through which the intents of the curriculum are experienced. These physical media appeal to the senses of the learners which bring things that are far beyond their environment near. In other words, they make imaginations more vivid and accurate" (p. 54). Therefore, teaching concretely seems beneficial to student learning around math.

However, manipulatives have to be used appropriately in order for them to be effective. This means having the teacher think about how to use their knowledge to teach students, and then to think about how to use the manipulative for a given topic. In addition, it includes narrowing the use of a single manipulative for a topic, which Clements & McMillen (1996) explain can support students learning to "gain expertise through using a tool over and over on different projects" (p. 277). This does not, however, mean that other manipulatives cannot be used for the same concept. Other manipulatives are encouraged to teach students how to navigate a concept, but it has to be introduced in a meaningful way rather than what Clements & McMillen call a "trivial" use. Furthermore, Clements & McMillen (1996) suggest using "manipulatives, such as interlocking cubes, can be used for counting, place value, arithmetic, patterning, and many other topics. This versatility allows students to find many different uses. However, a few single-purpose devices, such as mirrors or Miras, make a significant contribution" (p. 277).

As an example, if students are asked to solve 5 + 3, they can use blocks or other manipulatives to solve the problem using concrete materials. In the examples below, I have shown how popsicle sticks can be used to solve 5 + 3 or how blocks can be connected to find the answer. Over time, the consistent use of teaching concepts concretely supports students build knowledge to move towards pictorial and abstract representations.

However, manipulatives have to be used appropriately in order for them to be effective. This means having the teacher think about how to use their knowledge to teach students, and then to think about how to use the manipulative for a given topic. In addition, it includes narrowing the use of a single manipulative for a topic, which Clements & McMillen (1996) explain can support students learning to "gain expertise through using a tool over and over on different projects" (p. 277). This does not, however, mean that other manipulatives cannot be used for the same concept. Other manipulatives are encouraged to teach students how to navigate a concept, but it has to be introduced in a meaningful way rather than what Clements & McMillen call a "trivial" use. Furthermore, Clements & McMillen (1996) suggest using "manipulatives, such as interlocking cubes, can be used for counting, place value, arithmetic, patterning, and many other topics. This versatility allows students to find many different uses. However, a few single-purpose devices, such as mirrors or Miras, make a significant contribution" (p. 277).

As an example, if students are asked to solve 5 + 3, they can use blocks or other manipulatives to solve the problem using concrete materials. In the examples below, I have shown how popsicle sticks can be used to solve 5 + 3 or how blocks can be connected to find the answer. Over time, the consistent use of teaching concepts concretely supports students build knowledge to move towards pictorial and abstract representations.

## Pictorial

In the pictorial stage, students are able to visually represent the problem to solve it. Through a visual representation, students are able to “see” the learning. According to Ruchti & Bennett (2013), "The use of pictorial representations can support students in developing reasoning skills, building conceptual understanding, identifying and correcting misconceptions, and building proficiency. Specifically, using various types of models—area, length, and set—and discussing their usefulness in different contexts help students develop deeper mathematical understandings" (p. 36). This can include, for example, using diagrams and pictures. It can also mean using number charts, number lines, ten frames, and so on.

Accordingly, if solving 5 + 3, students can draw illustrations to add the two numbers. I have shown two such examples here:

## Abstract

Finally, in the abstract stage, students use symbols to solve the problem. At this stage, students are “symbolically” representing and/or solving. This includes using numbers, notations, and mathematical symbols. By this time, students who have built knowledge and understanding through the concrete and pictorial stages should be able to add 5 + 3 abstractly. Here, students can denote 6 + 3 = 8.

## Another Example

The example provided above is a simple one, but the CPA approach is applicable for various other concepts in math. For example, if students are learning about fractions, we can apply the CPA approach. Purwadi, Sudiarata, & Suparta (2019) suggest it enhances students' mathematical conceptual understanding (MCU) and students' mathematical representation (MR) around fractions. Furthermore, it provides a scope to address misconceptions. For example, Purwadi, Sudiarata, & Suparta (2019) explain how students often believe that adding fractions means adding both the numerator and denominator, which is a misconception.

Considering fraction, here is one example, where I show how 2/4 = 1/2:

Considering fraction, here is one example, where I show how 2/4 = 1/2:

Here is another example: what is 1/5 of 10? Alternatively, this can be written as the following: what is one fifth of 10?

The CPA approach is also applicable for more rigorous concepts. As the grade level increase, it does become difficult to use the CPA model for

In conclusion, the CPA approach is helpful in elementary school, middle school, and the early years of secondary school. However, it is important to note that the CPA approach is not a linear one, but rather a cyclical one. This means that students do not necessarily have to go from concrete to pictorial to abstract in order to simply move away from one stage to the next. A student, for example, can solve a problem abstractly, but still be asked to demonstrate how that same problem can be solved concretely or abstractly. Therefore, the scaffold approach moving from concrete to pictorial and then pictorial to abstract is a process that is building block for math mastery, allowing students to work towards math is more abstract in nature.

*all*concepts. It is, therefore, important to figure out what stages of the CPA approach are applicable for a specific concept.In conclusion, the CPA approach is helpful in elementary school, middle school, and the early years of secondary school. However, it is important to note that the CPA approach is not a linear one, but rather a cyclical one. This means that students do not necessarily have to go from concrete to pictorial to abstract in order to simply move away from one stage to the next. A student, for example, can solve a problem abstractly, but still be asked to demonstrate how that same problem can be solved concretely or abstractly. Therefore, the scaffold approach moving from concrete to pictorial and then pictorial to abstract is a process that is building block for math mastery, allowing students to work towards math is more abstract in nature.

## References

- Clements, D.H. & McMillen, S. (1996). Rethinking "concrete" manipulatives.
*Teaching Children Mathematics*, 2(5), p. 270-279. Retrieved from JSTOR database. - Larbi, E. & Mavis, O. (2016). The use of manipulatives in mathematics education.
*Journal of Education and Practice*, 7(36), p. 54-61. - Ontario Ministry of Education (n.d.).
*Professional learning guide: manipulatives*. Retrieved from EduGAINS. - Purwadi, A., Sudiarata, G.P., & Suparta, N. (2019). The effect of concrete-pictorial-abstract strategy toward students' mathematical conceptual understanding and mathematical representation on fractions.
*International Journal of Instruction*, 12(1), p. 1113-1126. Retrieved from Education Source. - Ruchti, W.P. & Bennett, C.A. (2013). Developing reasoning through pictorial representations. Mathematics Teaching in the Middle School, 19(1), p. 30-36. Retrieved from JSTOR database.
- Schultz, K. (1986). Representational models from learners' perspectives.
*The Arithmetic Teacher*, 33(6), p. 52-55. Retrieved from JSTOR database.

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