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Beginning Mathematics

How is it at one extreme, some three years can do abstract maths in their heads, while at the other extreme, some five years olds can't reliably real count to five.

What makes the difference?

I'll retell a conversation as best I remember it, just so you realize what is possible... because it stunned me.


One day I was driving down the road and I had my three year old in the back of my car. His older brother had commenced school and was learning his first lot of multiplication tables that being x 2.


Try to imagine a three year old  strapped in his car seat, holding up the correct amount of fingers as he said the following:

'Two lots of two... that's four isn't it Mum?  Two fingers up on each hand.

'Yes,' I replied.

'And two lots of four, that's eight isn't it.'

'Yes.' I again replied and caught a glimpse of him with four fingers on each hand and moving each consecutive finger as he counted them in order. I was impressed by his early understanding and I hadn't ever see him do it before.

'And two lots of eight, that's ...'

I smiled as I thought he's run out of fingers but I caught a glimpse of his face in the rear view mirror. He was initially perplexed but then his eyes rolled up to the left and he imagined.

'... And that's sixteen isn't it Mum,' he said. And he smiled because he knew what he'd done was hard.

So how did this happen?

When I analysed it I realized it was because my son had been exposed to maths concepts incidentally from a very young age and had lots of practice with real maths examples.

"Look how much taller it's grown. It's the same height as your brother. Will it get as tall as me?"

"I wonder which plant will have the most? Let's count and see."

"If Grandma wants three tomatoes from this plant how many will be left for us?"

"Who can find the biggest tomato?"

Then progress real counting to five.

My son could just 'see' the answer in his head in abstract form.

So what can I tell you that may help help your child to understand maths early?


To begin; I'll divide maths concepts into basic areas to help you understand how you can expose your children to maths as part of everyday activities.



Some of this may seem obvious but I've learnt over years never to assume that knowledge and to begin at the beginning, particularly with mathematics. Below I am suggesting an order of the introduction of difficulty as your child learns to count. For activities you will need to go to the link page at the end of this page.

Rote counting is the sequence.

Real counting is the number.

1. Begin with rote counting to three

and reinforce with real counting. Include simple songs, chants, rhymes to reinforce three.

       How many yellow blocks are there?                       How many yellow flowers?                       Let's put five blueberries on each dessert.                                                                                                                                                                   


  N.B. Children will initially have to move items as they count them.

Later they will be able just to touch them. Finally they will be able to count with their eyes.

Some will be able to recognise collections of dots such as on dice faces.


2. ​​Progress rote counting to six, then eight then ten.


Find real examples of counting up to ten.

​N.B: There are often Counting Pitfalls

The most common pitfall people make is thinking if their child can do it faster it must be better; but that's rarely the case.


They may even start saying things like 'What's two plus two.' And the child can answer four which parents think is good but the child may not even know what plus means. They may simply be parroting the answer with no concept of addition at all.


If your child has the chance to understand real or concrete maths before moving them onto visual representations of maths, or abstract maths, they will have a lot better chance of understanding how maths works and feel more positive about it over the long term... because they'll be good at it.


Many children have difficulty keeping the counting sequence correct between 13 and 17 inclusive.

3. Progress rote counting to twenty.

  • Considering that 'th' is one of the last sounds for a child to be able to say, listen to how, many four or five year olds, actually count.

'firdeen, fourdeen, fivdeen, sixdeen, sevendeen...'


  • But if they're younger or still can't accurately produce even the sounds above, counting often sounds more like this:

'birdeen, bordeen, bideen, dideen, dendeen...'

  • The similarities between birdeen (thirteeen,) and bordeen (fourteen) are obvious and children often omit one of them.


  • Also  fifteen and sixteen have the same middle sound so again one of these is frequently omitted.


  • And then having sixteen and seventeen both beginning with 's' adds yet one more confusing factor.

What I did to overcome this problem, was to enunciate my sounds very clearly and see if the child could hear which sound each number started with. I also raised my pitch as I counted through fourteen, fifteen and sixteen  emphasizing each number differently.


And I also reinforced the progression with numerals once they were reasonably reliable with their counting, explaining that thirteen was a bit like three teen, fourteen is obvious, fifteen is like five teen, six teen and seventeen are also obvious and that helped some children who understood the concept but had a speech delay.

4. Progress rote counting to thirty.

  • Another pitfall is twenty nine to thirty, which trips a lot of children up and it often needs to be treated specifically. Many children will go 'Twenty ten, twenty eleven, twenty twelve, etc'

There are a multitude of simple counting songs and activities on

'The Importance of Counting Page.'

Considering we're looking at learning to count over years guided by the child's ability I've included the next two areas as well. But I strongly suggest looking at the Web page The Importance of Proximal Development to help guide you.

5. Progress rote counting to one hundred.

Another Counting Pitfall:

Yet another pitfall while counting to one hundred is omitting the fifties, the sixties or the seventies completely. This can be overcome by introducing a numerals chart and explaining how it works

6. Progress rote counting to one thousand.

Something to be aware of: Going from one hundred and nine to one hundred and ten to one hundred and eleven also seems to trip children up and may need to be taught specifically while most others in the sequence through to one hundred and ninety nine progress naturally.



It is also prudent to include numerals incidentally at first so children realise numerals represent a value. And it's particularly useful once you get over thirty.

The Importance of Counting
Learning Maths Incidentally
Fun Maths Games for 5-6 Y
Show More


Keeping it simple: Beginning with Big and Little

  • Developing the concept that size is relative. For example we are huge to an ant but tiny next to a Blue Whale.

  • Over time developing a vocabulary around size. For example: Microscopic, tiniest, small, middle-sized, large, larger, largest, gigantic, infinite, shortest, short, middle sized, tall, taller, tallest, or mid-length, long, longer, longest, wide, wider, widest, narrow, narrower, narrowest, etc
















​​Activity Ideas

  • Compare two items at a time. Initially which is bigger and which is smaller? Swap items. When they've grasped that concept introduce which is longer? And later again introduce which is wider? Substitute pictures of real items and see if your child can remember some items that are always larger or smaller than the other item.

  • Expand on the above, ordering three to five items according to size. For example: Ordering the people in your family according to height, ordering your saucepans, etc.

  • Building structures out of blocks, or interlocking bricks, or empty cartons and discussing the height as they do so. e.g. You could comment encouraging them to make a part of their building taller, longer, wider, etc?

  • Measuring pieces of wool against one another and sorting them from shortest to longest and then using them for a collage.

  • Comparing volumes of containers while in the bath. Which has more/ less? Which is half full? Which overflows? Rank the containers according to which holds the most to the least.


Keeping it simple: Beginning with Red, Blue and Yellow

  • Developing the concept of colour and identifying basic primary colours: red, blue, yellow, black and white.

  • Showing how to mix primary colours to make green, orange, purple, brown and identify them.

  • Add black and white to palette and identify these shades of colour.

  • Look at what happens to red with the addition of white making pink and shades of colour like light green, paler pink, light blue, apricot, mauve, fawn, lemon, etc.

  • Over time experimenting with colours to make their own shades of colour and name them.

Activity Ideas

  • Sort counting items into red, blue, yellow plastic bowls. When they can manage introduce green, then later orange and purple. Sorting clothes pegs onto coloured cardboard is a favourite. I cut cardboard into different echidna shapes and both of my children loved clipping the pegs onto the correct coloured echidna. For echidna shape go to Craft Links: Echidna

  • Paint a picture using three basic colours. Look at what happens where the paint overlaps. You can use cake colours to create simple dot pictures where colours overlap.

  • Have a specific colour day to highlight a specific colour beginning with their favourite colour, then your favourite colour, then other family member's favourite colours.

  • Sort patty pans or other colourful items.

  • Make chalk rainbows on the outside path and sing 'I Can Sing a Rainbow.' Read the book 'The Chalk Rainbow.'

  • Provide pencils and textas for your child to draw with and discuss the colours used incidentally.

  • Make a rainbow birthday cake and discuss colours. Blow up different coloured balloons for a birthday party and have your child help you make a colourful sign for the party.


Keeping it simple: Beginning with Circle, Triangle, Square and Rectangle



  • Look at a circle. Discuss it has one side that goes the whole way around. Look for environmental examples of a circle. Wheels, pots, bottoms of jars, bottoms of paper tubes.

  • Make a picture out of patty pans

  • Make Lego models and add wheels

  • Discuss how circles make our lives easier



  • I explained that squares had four sides and were the boring ones. No matter how many times they turned themselves over, they were always the same.

  • Make a picture out of squares. Notice how the shapes can tesselate exactly.

  • Look for environmental examples of squares: paper packaging, the sides of stackable crates, sides of dice, some snack biscuits or water crackers


  • Explain that a triangle has three sides. Show that if a bug tries to climb up on the side it can be difficult and they slide back down due to the slope so they try and try again (Tri...angle).

  • Look for environmental examples of triangles: roofs, pennants, bridge supports, cranes, Give Way signs, patterns and tesselations.

  • Cut sandwiches into triangles. Look for triangular shaped foods in the supermarket: spinach and feta triangles, gow geys, lentil triangles

  • Build a tent and discuss the shape of the roof. Why are triangles useful? For strength in buildings. To facilitate run off during rain, etc


  • Explain that a rectangle is a bit like a square but somebody stretched and stretched it and wrecked it. I also added that it was the lazy one of the shapes because even if itstood up tall, it also liked to lie down a lot on its side.

  • Look for environmental examples of rectangles in your home. They're everywhere.

  • Your child might be able to take photos of the rectangles in your home. E.g. Cupboards, walls, books, boxes, folders, sheets of paper, table tops, bricks, phones, computer screens, playing cards, etc

For more Maths Ideas go to the links below:

The Importance of Counting
Learning Maths Incidentally
Fun Maths Games for 3-5 Year Olds
Show More

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