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Consider explaining this to a 10 year old.

When we think of mulitplication with the natural numbers, the intuition is essentially "repeated addition". Consider the product $2\times4$. From this "repeated addition" intuition, the answer is simply $2+2+2+2$. Hence a solution of $8$. Great a 10 year old can understand this. However what happens when we now extend multiplication to real numbers, this intuition breaks down. Consider $0.5\times0.8$. What does it mean to add $0.5$, $8$ times. How can I intuitively explain this extension of multiplication to a 10 year old?

Consider, division now. For natural numbers, division can be thought of "splitting things into groups". Great this is easy to understand. However, once again this intuition breaks down as soon as we move to real numbers. $\frac{0.4}{0.2}$. We can still intuitively thing about division with numbers less than 1 by thinking of how many times we must add $0.2$ together to get $0.4$.

We learn fairly early on that dividing by a fraction, is the same as multiplying by its reciprocal. This is rather trivial to prove algebraically. However, the intuition to me is not entirely clear. How can we intuitively think of this process?

Lastly, is it better to understand these concepts through rigorous definitions and proofs? Intuition is different for everyone, and is based on what the individual already knows. How would you explain these fundamental concepts of multiplication and division, even addition and subtraction to a 10 year old?

EDIT: Any exciting or clear intuitions for multiplication and division in the answers would be much appreciated.

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    $\begingroup$ "we now extend multiplication to real numbers"... what intuition have 10 years old boys? Having said that, maybe use fractions. $\endgroup$ Oct 17, 2023 at 7:28

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Conceptually division is the one that should be explained first. If a child can understand that $0.2=\frac{2}{10}$ i.e. two of something else, not $1+1$ but $0.1+0.1$ then it is easier, since we can talk about multiplying smaller parts like $0.1=\frac{1}{10}$ first, since that one is likely most difficult to understand. However, the fact that $\frac{1}{10}\frac{1}{10}=\frac{1}{100}$ is easy to understand if we take the concept of dividing something in tens twice. We get again $100$ parts, as if we multiplied $10$ by $10$, but they are smaller than whole. Another concept is necessary which is commutativity: that things can be arranged any way and that $2\cdot5$ is the same as $5\cdot2$, which means that parts and number of parts are exchangeable. Once these two are in, renaming parts and commutativity it is time to introduce abstract thinking and what is $2x$, $3a$, $5b$, $\frac{c}{2}$, i.e. that we can start dividing abstract things that are not predefined.

And no! By all means do not introduce math to a child through definitions. Let him get to it. Even when you are very mature there is an unsurmountable problem in all mathematical expositions. You can see and learn and understand a proof, but you still need quite some time to understand the limitations that shaped the proof or even a theory. An obstacle that created a part of the proof, that forced a mathematician to circumvent it, may never be revealed, you see the proof not reasoning that brought it to light. But the struggle is absolutely essential for any form of understanding. Polishing, as it is done in the proofs helps having a solution, but just learning the proof gives far less than what one would desire.

Let the kid struggle until everything is clear as a sun.

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