“Any curriculum for primary mathematics must incorporate the progression from the concrete to the abstract and subsequently a need to appreciate the importance of abstraction in mathematics” (NCERT,2006). To master any mathematical concept students should develop both the conceptual understanding and the procedural fluency. The Concrete-Pictorial-Abstract instructional approach provides meaningful connections between different stages of learning and develops both conceptual and procedural knowledge. In this approach, students learn to enjoy mathematics than seeing it as a subject with formulas and procedures. Kurniawan, H., & Budiyono, S. (2020) found “Motivation to learn in students who are subjected to the CPA approach is better than learning motivation in students who are subjected to conventional approaches. Singapore is one country out-performing others in the international benchmarking studies like PISA and TIMSS. The CPA approach is a key instructional strategy advocated by the Singapore Ministry of Education.

What is the CPA approach?

The Concrete-Pictorial-Abstract (CPA) is also referred to as Concrete-Representational-Abstract (CRA) or Concrete-Semi Concrete-Abstract (CSA). The idea was developed by Jerome Bruner in 1960. It is a three-stage learning process where students learn through physical manipulation of concrete objects, followed by learning through pictorial representations of the concrete manipulations, and then solving problems using abstract notation.

In the Concrete stage, the teacher begins instruction by modelling each mathematical concept with concrete materials (e.g. chips, cubes, base ten blocks, fraction bars etc).

In the Pictorial stage, the teacher transforms the concrete model into a representational (pictorial) level, which may involve drawing pictures; using circles, dots, bars, a number line etc.

The Abstract stage uses abstract symbols to model problems. At this stage, the teacher helps to develop the mathematical concept at a symbolic level, using numbers, notation, and mathematical symbols to represent the number algorithm.

Shown below are some examples of the CPA approach used for teaching addition in grade 1 from NCERT Mathematics textbook and teaching equivalent fractions from Elementary and Middle School Mathematics by Van De walle.

 Addition of numbers Equivalent fractions Concrete  Pictorial  Abstract  It is very important for the teacher to select appropriate manipulatives at the concrete stage and make meaningful connections to the pictorial and the abstract stages.

Is there any research evidence?

Mercer and Miller (1992) reported that students taught using the CSA sequence acquired the target skills, demonstrated an understanding of the respective operation, and were able to maintain the skills over time. In another study, Witzel, Mercer, and Miller (2003) showed that students who learned how to perform algebra transformation equations through a concrete to representational to abstract (CRA) sequence outperformed peers receiving traditional instruction during both post instruction and follow-up tests. In a study by Butler, Miller, Crehan, Babbitt, & Pierce (2003), students with learning disabilities receiving CRA instruction performed at least as well as general education 8th grade students on assessment on fractions. They also outperformed general education students on a subtest on word problems that had fractions and equivalency embedded in the problem.

Can the CPA approach be used in digital learning tools like Mindspark?

The CPA approach can be used in digital learning tools. The concrete objects will be replaced with virtual objects and animations. Here are some such examples where the CPA approach is used in Mindspark.

Example 1: Addition of single-digit numbers:

 CPA Stage Sample Page Concrete (Virtual objects/animations) (By showing concrete objects like marbles, pens, balls etc. and let the student count those objects.) Pictorial (By using dots in a 10-frame) Abstract (Just using the numbers and symbols) Example 2: Solving a linear equation in 1 variable

 CPA Stage Sample Content Concrete (Virtual objects/animations) This animation helps the student experience how the equation can be balanced. Concrete (Virtual objects/animations) Through a series of steps, this animation shows how both sides of an equation can be balanced by adding or removing some weights. Step 1 Step 2 Step 3 Pictorial Here, students will be able to frame equations from the given representation and vice versa. They will use the visual to balance the equations and generalise the procedure to solve abstract equations by balancing both sides of the equation in the next stage. Abstract A student is expected to solve an equation abstractly and the tool evaluates each step entered by the student and gives relevant error messages for wrong steps to remediate the students’ errors. Are there any challenges in using this approach?

The main challenges with CPA are:

• Unable to spend good time in teaching each of these stages: One should spend enough time to teach the concept through these 3 stages by using appropriate examples and giving sufficient practice to the student. If not done well, students may not be able to make the connections between these stages and they will be more confused about the appropriate technique they needed to solve the problem.
• Not having good resources that covers CPA activities for all the 3 stages of a concept: Student can’t develop robust understanding of the concept unless the connections between these CPA stages are more meaningful and well established. There is a need to have more evidence-based CPA activities covering all essential Math concepts. Teachers may skip using some of the stages due to lack of resources or may not be able to make a logical connection between these stages.

One such example is the approach used in teaching addition and subtraction of integers. In most of the textbooks in India, the concept is introduced using the concrete phase by using number line or counters as the manipulative. Then in the abstract stage, some rules like ‘when you have one positive and one negative integer to add, you must subtract but answer will take the sign of the bigger integer’ are given.

From the student interactions we have noticed that the students are just following these rules often without understanding why these rules work. This encourages rote learning of procedures without developing the conceptual understanding of the topic and at times inappropriate recall/application of these rules also lead to wrong answers. We wouldn’t recommend encouraging such kind of rules if learnt without understanding why they work.

The possible reason for a typical teacher promoting these rules might be that he/she couldn’t find any better method or resource where she can extend the understanding from the concrete phase to the abstract phase.

In Mindspark, we have used CPA approach to teach the concept of addition and subtraction of integers that aids connections between the 3 stages without using/stressing the rules like the one shown above. Sharing the approach and sample content below.

 CPA Stage Sample Content Concrete / Virtual manipulatives (A number line is used as the manipulative. In the concrete stage, an animation shows the meaning of addition on a number line by making a frog jump as per the given addition problem.) Pictorial (This question helps in establishing a connection between the concrete stage and the pictorial representation.) Here the pictorial representation helps in representing operations using large numbers. Students will be able to see the result of any operation on the number line as a distance from zero. This establishes the connection to the abstract problems.   Abstract (Students will be able to answer the abstract problems with conceptual understanding.) The concrete stage helps the student understand the meaning of the operation using a number line. The pictorial stage helps in finding the result of the operation on a number line even for large numbers. If students get enough practice in the first two stages, they will develop good number sense and in the abstract stage, they will be able to answer the abstract problems either by just visualizing the number line in their head or by developing a good number sense and hence understand the magnitude of the result. For example: to solve 84 + (-90), a student can think that adding (-90) to 84, which is, actually, moving 90 units towards the left from 84 on a number line. One reaches zero after moving 84 steps. So, the result will be 6 steps to the left of zero which is (-6). Or they can visualize the result as the difference between 90 and 84 but it will be on the left side of zero and answers (-6). Even if they have learnt the rules from other sources, they will be able to understand the logic behind the rules without just rote learning the rules.

We have piloted this content of addition and subtraction of integers using the CPA approach with around 23 students who are at the end of grade 5. A pre-test was given before they started the topic and a post-test were given just after they completed the topic. Both the pre-test and the post-test contains around 19 questions testing integer fundamentals and addition and subtraction of integers. The students have done around 3 to 4 hours of Mindspark on integers topic without any teacher intervention. On average there is an improvement of 15% in the post-test score compared to the pre-test. The median score in the post-test is 84.2%.

The pilot results are encouraging. There is a significant learning gain of 0.7 SD when students learn the topic, Integers, using this approach. The CPA approach can bridge the gap between conceptual understanding and rule-based algorithms. Teachers and educators should consider using more such strategies in their teaching and make students learn Math with understanding.

References

Butler, F.M., Miller, S.P., Crehan, K., Babbitt, B., and Pierce, T. (2003). Fraction Instruction for Students with Mathematics Disabilities: Comparing Two Teaching Sequences. Learning Disabilities Research & Practice. 18(2), 99-111.

Kurniawan, H., & Budiyono, S. (2020). Siswandari. Concrete-Pictorial-Abstract Approach on Student’s Motivation and Problem Solving Performance in Algebra. Universal Journal of Educational Research, 8(7), 3204-3212.

Leong, Y. H., Ho, W. K., & Cheng, L. P. (2015). Concrete-Pictorial-Abstract: Surveying its origins and charting its future.

Mercer, C. D., & Miller, S. P. (1992). Teaching students with learning problems in math to achieve, understand, and apply basic math facts. Remedial and Special Education, 13, 19–35.

Sealander, K. A., Johnson, G. R., Lockwood, A. B., & Medina, C. M. (2012). Concrete–semi concrete–abstract (CSA) instruction: A decision rule for improving instructional efficacy. Assessment for Effective Intervention38(1), 53-65.

Witzel, B., Mercer, C. D., & Miller, M. D. (2003). Teaching algebra to students with learning difficulties: An investigation of an explicit instruction model. Learning Disabilities: Research & Practice, 18, 121–131

### Praveena K

Praveena K is a lead educational specialist working on math pedagogy research