The Concrete – Representational – Abstract Model

What’s the best way to teach maths to under 5s?

The concrete-representational-abstract model.

It sounds complicated but the idea is simple: play with physical objects to get a feel for number.

But before we examine it in more detail, let’s look at how not to do it.

The wrong way to teach maths

Think about all those books on doing maths at home with your preschooler. They are full of all kinds of strange symbols. What does ‘+’ or ‘=‘ mean to a four-year-old?

How about ‘3’? Surely they know what that means?

Hmmm. Try it.

The threeness of three

’3’ is a symbol.

When we see it we say, ‘three’.

But what is three?

  • A symbol
  • A collection of three objects
  • One more than two
  • One less than four
  • Position 3 in a series

Not so straightforward, is it?

And yet, even the very youngest children can count. Even babies.

Don’t believe me? Find a ten-month-old.

Take one biscuit in your left hand and two in your right. Which hand will she choose?

She will take the two biscuits every time. And that’s because she can count, even though she doesn’t yet know any numbers. She has an innate sense of quantity.

If you really want to be mean and enjoy laughing at how easy it is to trick children, you can take this a step further.

Put one biscuit in your left hand and one in your right, but snap the second biscuit into two pieces. Your child will still choose the two pieces, even though the total amount of biscuit is the same.

A child counting with beans, an example of concrete materials

Why maths seems hard (even though it isn’t)

All children have an innate sense of maths. The problem is that we put obstacles in the way of their understanding.

I give you two sweets. You take one more from the packet. How many do you have now?

The maths is easy. You can see that the quantity has increased.

You count how many you have in your hand. It’s something your preschooler can do with confidence

But if I write 2 + 1 = ? on a piece of paper and give it to my daughter, she will look at me blankly.

What’s the solution?

We have to make the abstract concrete.


This is teachers’ big secret.

When we teach new concepts in school, the first thing we do is give children concrete materials to play with. Concrete materials are anything that you can touch and move. That’s why classrooms are full of maths cubes, counters and beads.

Cuisenaire rods help bridge the gap between concrete and abstract

5 x 3 is completely meaningless to a child. It is completely abstract. It bears no relationship to the real world. But show her five groups of three objects and it becomes instantly clear.

But that doesn’t mean that she can go from sharing out objects to formal sums. There is a step in between – representation.

After she has shared out fifteen sweets, she could represent this on paper by drawing five sets of three. The symbols she chooses could look like sweets…

Sharing sweets helps to understand division

…or simply be lines on the page.

Using simple marks to represent numbers

It doesn’t matter.

The point is that she is learning to represent physical objects with a symbol.

This is the first step on the road to discarding concrete materials and working entirely with symbols.

What does this mean for how I play with my child?

Forget about symbols and written calculations. They can wait until school.

Far better to give your child a feel for maths by integrating it into your daily life. Give her meaningful tasks and show her practical ways to use her knowledge.

My preschooler likes to scan the products through the self checkout at the supermarket. Some items don’t have barcodes so she has to count them up and key in the total.

One, two, three, four lemons. Look, Daddy! There’s the number ‘4’. That’s the one I press.

Later that day, back in the kitchen and inspired by the shopping trip, she decides to make a fruit salad. But we don’t have all the ingredients. Make a shopping list, I say.

A pictorial representation of a shopping list

She will be able to read this list independently when we return to the shop.

Three bananas, two apples, four oranges.

When we play football in the garden, the action is so fast we sometimes forget the score. I like to know how much I’m winning by (I have a fragile ego) and so we keep track of the goals with a tally chart.

The game is too rough for the preschooler to join in but she is delighted to be involved by marking each goal with a chalk line on the wall. Like most children her age, she still can’t write numbers perfectly, but the tally lines show that she understands the underlying principle of ‘one more’.

Keep score with a tally chart - an early way to represent quantities without numbers

Once your child understands that ‘IIII’ stands for four goals and that adding another line to the end makes five, you can introduce + and =.

But take it slowly.

Don’t do this:

Using numerals to represent a sum

Your aim is to keep the symbols anchored to the real world.

Drawing pictures to represent quantities makes sums easier

Ideally, combine the representation with concrete materials. Write the sum above and then check using real bananas, or other countables like dried beans or maths cubes.

And if your child ever seems to be struggling with a concept, go back a step, to representation or even concrete materials.

Your aim is to constantly reinforce the association between concrete and abstract.

Keep some pencils and a tray of paper handy. Once your child has understood that she can represent numbers and operations (sums) using pictures, you will notice that she is motivated to record all kinds of mathematical activity.

It’s lots of fun.

Are you ready for school?

Do you have a preschooler? Would you like them to develop some key skills before starting school? How wonderful to go into school on that first day feeling like you belong.

Get Set Five is a year-long course full of fun and free activities to do with your child.