Concrete - Pictorial - Abstract

Do you hate Math?

I am sure many people will say yes to that question. We find math to be a difficult and challenging subject, but I believe that's partly because we have been taught math the wrong way. Rote learning and memorization are not the only way to learn math. Remembering the steps is important, but students need to develop number sense and understand why those steps are important.

One way to do that is using the concrete, pictorial, abstract (CPA) approach. Even though I have been using this approach back in Singapore, I learned something new today from Dr. Brian Sharp. He shared with us a 4-step approach for getting to the standard algorithm of adding and subtracting whole numbers that makes use of the CPA approach to help students understand how the standard algorithm works.

Before I get to the 4-step approach, the book Elementary and Middle School Mathematics: Teaching Developmentally by John A. Van de Walle says this about the standard algorithm for addition and subtraction, which is something we should bear in mind always:

Be sure students continue to view it as one possible algorithm that is a good choice in some situations (just as invented strategies are good choices in some situations).

Here are the 4 steps using 54 + 67 as an example.

Step 1: Concrete

In this step, students will use concrete manipulatives (base-ten blocks) to add these two numbers.

Students will then regroup or "trade" 10 ten-blocks to make 1 hundred-block. They can also "trade" 10 one-blocks to make 1 ten-block. So altogether, they would have 1 hundred-block, 2 ten-blocks, and 1 one-block, so that will give them the answer of 121.

Step 2: Semi-concrete

Here, students represent the base-ten blocks with a diagram or model. This allows students to move from the concrete to the pictorial. Regroupings or "trades" are clearly shown in the diagram.

Step 3: Develop the Written Record

This process requires students to record each step as it is done during the concrete and semi-concrete phases. It is interesting to note that it does not matter whether you start from the biggest place value (left to right) or the smallest place value (right to left). Students should follow their natural instincts. What you see in the picture is an example of starting from the biggest place value (left to right).

The written record is important because it is the final bridge to help students move to the abstract.

Step 4: Mapping Written Record to Standard Algorithm

In this step, it is important to show how the different numbers from the written record appear in the standard algorithm, especially how the number 1 is regrouped after adding the ones place.

By mapping the written record to the standard algorithm, students will be able to understand what each digit means in a way that makes a lot of sense to them.

Although it might seem like a lot of work just to get students to learn the standard algorithm, I believe that the effort and time is worth it, as it allows students to understand how to add and subtract numbers conceptually, and not just by pure memorization, which is easily forgotten.

So, what do you think? Is this something you've taught your students before? Will you try this with them the next time you teach them how to add or subtract numbers using the standard algorithm?