Only Part of the Picture
I remember when I was in secondary school, my class learned about conjectures in mathematics class. This section was probably one of the most confusing part of my mathematics class because there was nothing definite about it. I found it strange how we went about doing these problems, and I was happy that we didn’t spend too much time on it (a sign that I probably should have spent more time on it).
Just to give a quick explanation of a conjecture, it’s essentially a statement that does not have a proof, but states a certain property or relationship between things. For example, most of us have seen some form of proof for the Pythagorean theorem, involving squares and geometry. However, suppose you never had seen the proof and was told that the square of the sums of two legs of a triangle is the same as the squared length of the hypotenuse, that would be a conjecture.
If you thought it was false, you’d try and find a counterexample. After trying a couple of different triangles, you’d see that they do indeed fit the conjecture. At one point, you may throw your hands up in frustration and say, “All these stupid triangles fit the conjecture!”
You haven’t actually proved anything, but you now have a stronger suspicion that the conjecture is correct. You couldn’t just keep using triangles with different lengths if you wanted to prove this relation. You would have to take a triangle with arbitrary lengths and do some kind of proof to get to the conjecture. Then you could be sure of the conjecture, and it would become a proved statement.
Instead of learning that though, what I learned was that you needed to either find a counterexample (in which case you were then done), or you had to do three examples of the conjecture and show that it does indeed give the correct answer each time. Basically, it’s a glorified version of arithmetic, with only a bit of thinking involved.
I never got why we were doing this, nor how we were supposed to form a strategy for these conjectures. Was I just supposed to pick random numbers and see if they worked or not? Was I supposed to try and think of a counterexample?
What I think would have made much more sense now was if I was shown that conjectures are like “proofs-in-waiting”. They are ideas looking for proofs. Additionally, I don’t get why we were supposed to just examine the conjectures. After all, they are pretty boring on their own. Instead, we should have had to use those conjectures to learn how to construct proofs. It would have been a great example of the power of proving something.
Unfortunately, the closest I’ve come to proving something as an exercise on my own (read: not through the teacher’s notes on the board) was when I learnt how to do mathematical induction. The idea is that if your premises are true, than this is what you get. It was a powerful way to prove things that happen in series (such as repeated matrix multiplication). It gave me a taste of what mathematics can do.
Therefore, I believe we need to do give more effort to show young students the trajectory of what they will do with the tools they are learning now. Maybe it isn’t always necessary to complicate their lives, but I think it’s important to show them that these ideas will be used in the future. It gives them the sense that mathematics isn’t just a series of hurdles to jump through.