Chances are that if I asked you to add 1756678 and 99810023 together without using a calculator, you would have a way to accomplish that. This is already pretty interesting, if you think about it, in that it is likely you have never before added those two particular numbers in your life. If I asked you to check your result on a calculator, you would know how to do that, and it is quite likely you have additional ways to discover the result. Some of those ways may be to ask an authority figure; but even short of relying on authority figures – whether teachers, parents, older siblings, buddies, librarians, spouses, accountants, you really do have alternative ways to discover the result. In fact, there are many ways even without resorting to counting out 1756678 blocks and then counting out another 99810023 blocks, throwing them all in a big pile and then counting how many blocks are in this joint pile. And this is a good thing because counting that many blocks may feel like a prison sentence rather than anything you’d remotely want to be involved with.
The idea of something being discoverable is fairly simple, though I haven’t heard the word used much in school settings. The idea applies to mathematics but not only to mathematics. If I want to find out if salt water freezes at a lower temperature than plain water, I can discover this; that is, I can try it out. I don’t need to rely on Google or text books or teachers or any other authority. The question about salt water might be resolved as simply as taking an ice cube out of the freezer and sprinkling some salt on it and see if it melts faster than another ice cube you take out at the same time but without salt on it. There might be other and better ways to discover the effect of salt on the freezing point of water, sure – but it isn’t the kind of thing that relies on having the right person with the right credentials and the right magic wand performing the right magic invocation.
There are lots of things in mathematics as well as other fields that are not discoverable. Some authority, early in your life, told you what a “two” looks like, it looks like this: “2”. That is a convention, and this one happens to be a world-wide convention. Somebody in authority, early in your life, told you that to add things we use the “+” symbol. That too is a convention. These are conventions with a long pedigree and very wide-spread adoption. Some conventions are recent, and some have local significance but not global significance and acceptance. In much of the world, unlike the USA, the “,” symbol is used to separate whole from decimal part; and the “.” is used to mark thousands and millions etc. – just the reverse of what is used in the USA, Britain, Australia, and other countries. This means, for example, that the meaning of “7,040” is not discoverable outside of a cultural setting. In the USA, it clearly means seven thousand and forty; where in most of Europe, the same 7,040 would mean seven and 40 thousandths. For “7.040” the same would hold but in reverse.
When you look at a map, you know that the top of the map corresponds to North. This is a convention, and not discoverable. A kid is not stupid for not being able to figure it out. The question “which way (on this map) is North?” is a question that tests knowledge of a convention, of a cultural legacy being handed down.
When you ask a student what the square root of 9 is, things are a little more complicated. It may be that the student has never heard or seen the term “square root” or has forgotten it. In that situation, the answer to the question about the square root of 9 is not discoverable. But if the student knows what the term “square root” refers to, then the job of figuring out what number, when multiplied by itself, gives 9, that part is discoverable. It doesn’t depend on any authority to tell or confirm that the answer is correct.
Discovery, and following conventions, those two are entirely different beasts. Nothing bad about either, but confusing the two can lead to considerable mischief.
When a student discovers something in mathematics, it bolsters something really important. It fosters the idea not only that mathematics can make sense, but that “sense making” is the very essence of mathematical thinking. It raises the confidence of the student that he or she can make sense of the mathematics, that he or she is capable, that he or she can figure this out. (Note that by “discovery” I don’t mean something that’s never been seen before by anybody. For example, noticing for oneself that adding two even numbers together will always produce another even number, that would count as having made a discovery – very different from being able to reproduce something that one has been told to memorize.)
When a student learns something that is part of the mathematical heritage, it opens up participation in a community. A student who learns how to take a pair of numbers and plot it as a point on a graph with axes at ninety degree angles and regularly-spaced markings on each axis (an ordinary Cartesian graph, in other words), this student now has access to a new community and a new world, in essentially the same way that learning to read opens up a new world and a new community. If a child were to learn written language on his own, writing with a toe in the sand, inventing his own private written language, that would be an extraordinary feat, but even so it wouldn’t give this kid access to Dr. Seuss or Harry Potter or any other piece of our joint heritage.
Though not everything in math is either a convention or something to be discovered, it is nevertheless very eye-opening to sort out for oneself whether something I’m teaching is a convention or a discovery. This is not a particularly hard thing to do, and yet so often we leave it befuddled. Am I inducting somebody into a society of people who share a rich common legacy and heritage? Or am I enabling somebody to figure out something powerfully for themselves, so they are left with a real sense of having made a real discovery for themselves? Do I find myself doing only one of these without hardly ever doing the other?