This my response to a question in Quora: As a mathematician, how would you mentor your child and help her to learn, do and live mathematics in her free time as she is growing up?

I write from the position of a mathematician about what a *mathematician* can do for her child.

First of all, a mathematician understands and can use the fact of life non-mathamticians are not aware of:

Mathematics is done by the subconscious.

Encourage in your child, and help her, to develop all kinds of intuition, guesswork (with subsequent checking, whenever possible, of the correctness of the guess). Help her to train her vision of the world, see relations in the world, identify mathematical structures present in the world.

What follows are a few random examples, chosen from what I did myself with my (grand)children or had seen my colleagues doing with their (young, pre-school or primary school age) children.

- Adult spends time with a child, aged between 3 and 4, in a garden, watching insects and ants, and discussing with the child how the world looks from the ant’s viewpoint: that the tree trunk is a like a street, and pathches of algae and of moss on the bark are like lawns and bushes along the street. Child: ”and this branches are like sidestreets”.
- The same adult uses every opportunity to explain to the child the structure of an actual street in a big city: street signs, house numbers which go in progression and odd on one side of the street and even on another side. A year later, the child is able to confidently guide the adult (and his little sister) across a unknown part of the city using a map. Observing an ant on a tree helped. This is an ecouraging sign of mathematical development.
- Of course, a child’s ability to read is useful. Street names, all kinds of shop signs provide an excellent material for reading and proof that reading gives information about the world.
- An adult and a child (aged 5) send to each other, from opposite corners of a sofa, small strips of paper with messages written in a substitution cipher: each letter is substituted by the next one (cyclically, z is substituted by a). Suddenly the child exclaimes: “And I invented my own cipher — each letter is replaced by the
*previous*one!” IMHO, this will help the child understand algebraic notation where nubers are substituted by letters. [It is worth remembering that Vieta, the inventor of algebraic notation, was the frist cryptographer known to us by name. His deciphering of intercepted diplomatic coresspondence directly infuenced Europian politics of his time.] The child is now 7 years old and can handle variables in Scratch. - Children are invited to guess weight of every household object they can handle by weighing it in hand and check the result by weighing on scales. The same with temperature of water in hte bath, checked by thermometer.
- All kinds of estimates with subsequent checking: how many steps are in this staircase? How many steps are to the end of the street? How to estimate the number of cars in the parking lot without counting them all?
- Playing lego ( with child of 4). An adult encourages a child to pick correct bricks (say, 2 by 3 studs) without looking at them, by touch only. They together follow step-by-step instruction in the manual. Building a symmetric model (say, a plane), the adult builds the left wing, the child builds the right wing by mirroring the adult. Very soon the child starts picking details of correct orientation even before the adult touches his detail.
- Playing snakes and ladders with two dices. A player can pick one of the values or their sum. The catch is that, for winning the game, 100 has to be hit without overshooting — for otherwise the player gets back to the beginning, the path is circular, 97 + 3 = 6 . A fast, furious, and vicious game which trains tactical thinking.
- Actually the rules of snakes and ladders can be changed in variety of ways. Adult encourages the child to invent her own rules. Crucially, the new rules need to be agreed and written down before the start of the game.

I can continue this list, but, I hope, it already gives some idea. Sorry for typos.