Today’s session was inspired by a different math circle book: that by Anna Burago, which is targeted at slightly older children.
We started with 7 coins, and asked the children to place them so that 4 were heads up and three tails up. The rules of the game are as follows:
- in each turn, you can pick any two coins of your choice
- you must turn both those coins over
Your aim is to get all 7 coins facing heads up.
The task is impossible, but the children didn’t see so immediately. One child even thought he had found a solution, but was unable to repeat it. One child noticed that he could get them to all tails, but not to all heads. Very quickly, children of all ages noticed that “odds” and “evens” had something to do with this difficulty.
One of the older children offered a partial explanation: by turning over two heads, the number of heads-up coins decreases by 2; by turning over two tails, it increases by 2; by turning over one head and one tail, it stays the same. So the only possibilities after one move are to have 2, 4, or 6 heads face up, and all these are even.
I helped the children to generalise this to “an even change in the number of heads up” after any number of moves. Unexpectedly, one child objected to zero being considered an even number, which led me to take a detour. One serious concern I have with typical primary school mathematical education, which I share with Hung-Hsi Wu, is the absence of clear definitions. So I viewed this as an ideal opportunity to get the children to formulate the definition of an even number. I received these suggestions from the children:
- a number that you can divide by two
- a number that is a multiple of two
- a number that ends in 0, 2, 4, 6 or 8
One child objected to the first definition, “because I can divide 1 by 2, to get 0.5”. So after some prompting, the children changed the first definition to “a number that you can divide by two and the answer is a whole number”.
We then worked through each definition in turn, testing it for whether zero was an even number. The oldest child provided a clear verbal proof that the first two definitions were equivalent, and a proof that any number meeting the third criterion also met the other two (but not vice versa).
The child who had objected to zero being even seemed to accept that by these definitions zero was even, but suggested that this would make , “which is not the case because “. I paused this question, writing on the board to return to, because we were in danger of getting too far off track.
For the last part of the session, I tried to get the children to reason algebraically, suggesting that – from their own definition – any even number can be written as and, therefore because , we still have an even number after any number of moves. I think I overestimated the preparedness of the children for this form of reasoning. The older kids were fine with this, one of the middle kids who had obviously seen but not understood algebraic notation before, upon seeing the exclaimed “oh no, those numbers, I hate those numbers!” and the youngest had not met brackets in arithmetic expressions. Trying to cover all this at once was probably too much, but I hope we managed to get the basic ideas across.
Finally, I returned to the question, which I had rephrased, “what is zero divided by zero”. We looked at the equation , and how – if the child who suggested it were correct, we could equally conclude from that the answer was “3” and from that the answer was 4. I suggested that there is no well defined answer, so we refer to it as undefined. Seemingly unsatisfied with this answer, one child left to get a calculator. I was hoping this to be my moment of triumph where the calculator reported “-E-“. Stupid calculator reported “0” (!) However, at least one child obviously could see what was going on, and asked whether 1 divided by zero would therefore be infinity. In the last few moments, I managed to explore this question with a subset of the children by looking at the growth of as increases in size, while the others packed away.
I’d view this as a mixed session. Lots of very interesting questions raised, not all answered satisfactorily, but at least most of the children were thinking mathematically.