I’ll explain to you how to teach some basics of geometry to a 4 year old. – My grand-son is now 4 years 2 moths, I just a while ago taught him. 4 years is excellent age for these studies, perhaps the very best.

You’ll need either a chalk-board or a white plane with a peg in the middle. Then you’ll need a string and a chalk or a pencil. Then you’ll have to make a tiny groove to the peg, as well as one to the chalk, or the pencil. – Best if the distance of the groove from the plane is about the same in the peg and in the chalk/pencil.

Then you tie one end of the string to the peg, and the other to the chalk or pencil. Make sure the ties hold and that the length of the string is not too long. – If you use a pencil, it should be B 6 (quite soft, the kind artists use).

Then show the child how to keep the string straight, as long as possible, that is. Then instruct him/her to move the chalk/pencil to draw on the plane. – Then, like a miracle, a circle is being drawn.

It is a true miracle to the child. – The child will soon learn that a really fine and even circle will only be formed IF the string is kept tense, stretched to the utmost.

By now you have DEMONSTRATED the nature of a circle to the child. Now he/she understands the SOUL (as topologist say) of a circle.

Then use various lengths of the string – and you get a set of concentric circles. – And the child’s understanding gets deeper.

Children of this age get really enthusiastic. – My grandson Mikko was jumping and shouting: I drew a CIRCLE, I can make a CIRCLE!!! A FINE circle!!!

Now you have a great opportunity to discuss all you have done together. The child will absorb all the information like a sponge absorbs water.

But this is not all. Now the child is ready to understand the functions a pair of compasses. – That it has to be stiff, with a joint in the upper end. And with a sharp spike in the other end, as well as a piece of lead in the other end.
Then take a paper-block, and let the child draw various kinds of circles on the paper. – Some of them overlapping each other. – You’ll get various Venn’s diagrams to discuss.

Make sure that you pace these demonstrations and discussion in accord to the child’s enthusiasm and interest. – With the first signs of slackening attention, stop and tell the child that you continue some other day. (My grandsons are both BDM Baby Dance children, so they both have exceptionally long attention spans & are quite skillful.)

Then take an A4 sheet of paper and draw as big a circle as you can on the sheet. – Then two points of the circle will touch the edge of the paper (if you have measured it’s shorter side & chosen a middle point as the center of the circle).

Then take the pair of compasses, taking care not to change it’s angle (which you point out to the child), and put the spike on any point on the circumference of the circle. Then mark two points on the circumference, one to one direction, one to the other direction.

Then take a ruler and draw lines connecting the two marks + the little hole left by the spike. – Now you have got an equilateral triangle inside the circle.

Then cut the circle out of the sheep of paper. – Then fold the paper according to the lines drawn with the ruler.
– Then you’ll have a triangle AND a circle. – When flat, it is a circle, when folded, it is a triangle with “wings”.

This, too, is for the child a miracle. – My grandson Mikko absolutely wanted to save this miracle. And he spoke for days to come, that granny made a triangle out of a circle.

You may then point out that all the “wings” are of equal size. – By this time you have had lots of opportunities to demonstrate “equal” and “unequal” to the child. – Also “more” and “less”. Etc., etc.

The next phase, where you go from plane geometry to three-dimensional geometry I have not yet done with my grandson.

For that you need these Japanese Origami papers in various colors, in order to get the most of it.

First, choose a selection of the biggest sheets, with a variety of colors. – You also need a pair of compasses, a ruler, scissors and glue. And a pencil, of course.

Then make, according to the rules above, a set of these triangles with “wings”.

Then start gluing them together, two wings at a time first. – Now establish a rule: the two “wings” to be glued together must always be of different colors. – Let the child choose the colors. – Do not, however, take a new color if it is not needed because of the rule.

When the glue holds, show the child that a third triangle with “wings” can be attached to the former two by two of their “wings”. – Remember the rule about colors!

The idea of a curved space begins now to show itself.

You then continue, and in the end you will have a Platonic Solid, the Octahedron. – The “wings” are there to remind the child that you started with circles. – Also, you may point out to the child that an imaginary sphere, a bigger one may be thought – around the one formed out of triangles. This imaginary sphere is determined by the “highest points” of the wings. – There is the idea of concentric spheres demonstrated.

You may also dwell on the idea of determination. (You may need a little practicing to learn to speak so that a child can understand what you mean. One of the benefits for you yourself is that by learning to do so, you simultaneously learn to understand yourself. – Not a small benefit, I assure you!)

Now you can put a lamp inside, so you get a multicolored lamp shade, preferably hanging from the ceiling.
This is ideal for the child (as well as for you) to meditate on. – It works miracles on geometrical imagination!

Note that you have here introduced the basics of the famous map-coloring problem as one side-product (one amongst many).

Then all you need to do is to nourish your child’s geometrical imagination – and mathematical in general – from time to time — and just wait for your little genius to grow and prosper!

If you happen to feel a shortage of geometrical imagination to nourish that of your child’s, then just take up Euclid’s Elements to enliven it.

Cheers,