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A Logical Way to Learn Math

There are many schools of thought on how Mathematics should be taught (and learned). Growing up, I was taught to just memorise steps for solving questions by heart. As long as I remember which step leads to what, I did not need to wonder about the why and how of mathematics. In fact, I was discouraged to question it in class.

Back then, the focus on learning mathematics was on speed. As students, we were expected to be able to solve x number of questions in the shortest time possible, regardless of whether or not we understood what we were doing. This has been the basis of many mathematics classes around the world. Students were just supposed to remember the process of solving a problem, and then train themselves to execute it as quickly as possible. 

Common as it is, does this method really help pupils understand (or like) mathematics though? If it is effective, why do so many students get imaginary headaches the moment they look at a math problem?

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In this article, we will share another concept used in the teaching and learning of mathematics: the CPA Approach. This approach was developed by American psychologist Jerome Bruner, who proposed it as a way of scaffolding learning. He believed that learning can be difficult to children if it’s all abstract. 

For example, if you present a child with a textbook full of information and just expect them to remember everything, it will be very difficult for them to do so. It’s like giving directions to a stranger in town, who has no idea where he is or how anything looked like, and expecting him to be able to find his way around with purely verbal instructions. Instead of that, Bruner suggested that abstract concepts in learning should be accompanied by concrete and tangible aids. When students are able to visualise how mathematical concepts work in real life, their understanding of the subject will be strengthened, leading to more effective learning. 

There are three stages in the CPA Approach: concrete, pictorial, and abstract.

Concrete Stage:

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The concrete stage is also known as the “doing” stage. Students are encouraged to use real life objects to model mathematical concepts. For example, let’s say they’re learning about addition. They could be handed a set of Lego blocks, and then given the question: what is the sum of 2+1. In this case, they would be asked if they should add 1 block to 2 existing blocks, or should they remove the blocks. Since it’s addition, the correct answer would be adding a new block to the existing ones. This hands-on approach allows pupils to improve logical reasoning and thinking skills, and helps them form a connection in their brains about the application of addition in math.

The concrete stage is very important, as it helps students build their fundamental understanding on how mathematics works, before proceeding to deeper equations.

Pictorial Stage:

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The next stage is the pictorial stage, which is also called the “seeing” stage. As the name suggests, this stage involves students creating visual representations to model the problems that they’re working on. For example, if a question asks about removing 4 apples from a basket of 10, students are encouraged to draw it out to solve the problem.

One concept that can be introduced in this stage is the Bar Model, where students create visuals using bars. It is a more abstract method for representing numbers, compared to actually drawing out objects that are involved in children’s mathematical questions. This model acts as a bridge between the pictorial stage and the abstract stage, as students slowly transition from literal visuals (like fruit) to more abstract ones (like bars). Because it is very versatile, and only requires pencil and paper, it is very useful for students during classes and exams. The Bar Model can be applied to solve questions related to the four operations (addition, subtraction, multiplication, division), fractions, and algebra.

Abstract Stage:

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The final part of the CPA approach is the abstract stage. This stage can only be taught when students have demonstrated a solid understanding of the first two stages. In the abstract stage, teachers can now introduce actual mathematical symbols (e.g +, -, x, ÷, ≥, ≤ etc) and numerals in problem solving. If students have built a solid foundation from the two earlier stages, they should be able to solve mathematical questions using numbers and symbols easily.

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Tying it all up

While there are many ways that kids can learn math, the CPA approach has been shown to be extremely helpful. In fact, it is so helpful that it was designated as the mandatory teaching method for mathematics in Singapore. The CPA approach does not need to be limited to math; it can be applied to other subjects and topics as well. Since kids have a limited vocabulary and understanding of complex words, visual representations are a great way to introduce new concepts to them. The next time your child is unable to solve a question, try this approach, and see if it works. All the best!

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