The principle of C-P-A sequence is a key instructional strategy for the development of primary mathematics concepts in Singapore, according to Ministry of Education.

C-what?

C stands for Concrete, P stands for Pictorial, A stands for Abstract. These are stages of development in how we learn about quantities.

To understand about C-P-A, let’s try to recall: How were you like when you were at your child’s age? Before you knew fractions, multiplication and proportions?

If you are like me, you probably started learning about numbers by counting fingers. Or counting beads. Or counting something. It’s safe to say that none of us started with algebra, or fractions, number lines, or any such abstract concepts.

Concrete

We probably spent some days or weeks chanting, “One… Two.. Three… Four…” while pointing at physical objects. Sometimes we counted too fast, skipping over some objects while we count. Sometimes we counted too slow, pointing at an object when we went, “Two…Three…”. Gradually we learnt to synchronise the number we uttered with the objects we pointed at.

Pictorial

Then we were introduced to the idea that, objects in pictures can be counted too.

That was when we evolved to being able to say, “There are 6 apples in the picture!”. Two skills were involved in this seemingly simple task (to us): (1) skill to recognize distinctive shapes as “apples” and (2) skill to count these shapes.

In Singapore, primary school children spent a great deal of time learning how to draw a kind of mathematical mental model, called “Bar Model”. The Bar Model is a case of using pictures to represent quantities. Instead of drawing 6 apples, in Bar Model drawing we draw a bar which is 6 times as long as the bar for 1 apple.

Abstract

At some point, we were introduced to numbers. So, instead of having to draw 6 apples, or 6 shapes to represent the apples, we simply write a symbol that looks like an upside-down beansprout, or opened padlock, or an upside-down “g”.

All of a sudden, quantities are unchained from their proportions. 6 apples take up the same space on paper as 1 apple. 100 apples and 999 apples are represented by three digits figure.

Can you remember the confusion and cognitive shock you probably have felt? It probably is greater than when you were first told that the was such a thing called “imaginary number”.

The introduction of numbers or symbols, instead of pictures or icons, represents the abstract stage of mathematics education.

What has C-P-A got to do with my child?

When we have more understanding of how children evolve in their concept of quantities (I avoid using the word “numbers” because “numbers” are symbols – an Abstract concept), we can perhaps appreciate it better when our children struggle with seemingly a simple written statement, with no accompanying pictures, like “Which is more? 9 balls or 5 balls”?

To our trained minds, immediately we can conjure images of 9 balls VS 5 balls. But these images flashed past our minds so quickly that we are probably not even aware of it. Almost instantly we conclude that 9 balls is the answer. But for a child who is just transiting from Pictorial to Abstract stage, her mind first have to conjure the image of 9 balls, then 5 balls, then place them side by side mentally to come to a conclusion.

It is quite a lot of mental maneuverer for a child.

So, how should I teach differently?

Now that we know that our concept of numbers really started from counting concrete objects or pictorial representations, there is plenty of ways to apply some techniques to make our children’s learning easier.

Technique #1: Have some maths manipulatives in your house

Manipulatives are small counters like LEGO bricks, beads, cuisinaire rods, spinners, etc. In short, anything that makes counting quantities more tangible. They are very handy in giving young children a concrete sense of the quantities we are talking about.

For example, there was once my daughter added numbers wrongly, insisting that 2 + 3 = 4. No amount of explanation from me could convince her, so I took out some old buttons and asked her to take 2 white buttons, followed by 3 black buttons. Then I asked her to count how many buttons there were. Only then she was convinced that 2 + 3 = 5.

You can visit https://www.kindergarten-lessons.com/teaching-with-math-manipulatives/, www.fishpond.com.sg and www.singaporemaths.com to understand more of what are available.

Technique #2: draw a bar model

I would only recommend drawing bar model when you are sure that your child is comfortable with the Pictorial stage of understanding quantities.

Let’s consider this question. It’s taken from 2018 Nan Hua Primary School Primary 3 SA2 Paper.

If your child struggles to comprehend this question, one of the reason can be because she do not understand the quantity of “314 bottles”.

Going back all the way to Concrete stage will be challenging – I don’t know about you, but I certainly don’t want to spend all night counting 314 buttons or beads!

Due to the big number involved, even drawing pictures of 314 bottles can be exhausting. A more practical technique is to draw a bar to represent 314 bottles. Like this:

When we have a diagram like this, it still takes some children some imagination to associate the rectangular bars with bottles, but it’s better than only having the number 314.

I tried representing quantities in maths problem sums with many children, from maybe P3 all the way to PSLE, and I usually get affirmation from my students that drawing Bar Models really helped them to conceptualise what was going on in the questions.

Using Model Method effectively requires some training. I have put together a live interactive online course called “Teach Your Own”, in which I personally teach parents the art and science of using Bar Model to solve challenging maths problem sums. You can find out more about it here.

Now that you have two more techniques at your disposal, I hope that you will be more equipped to help your child learn maths.

Share your experience

I’d love to hear about your experience with these techniques. Do share with me at jason@lovetolearnsg.com or leave a comment!