We must teach proofs early on
Otherwise we might as well not teach maths at all
I think the practice which has become common in mathematics curricula from countries like the UK and the USA, that students don’t learn what is a mathematical proof till they reach university, makes maths classes almost completely useless.
To understand why, let’s first think of what are the benefits of students learning maths. Here are some of the most fundamental ones:
it develops reasoning abilities;
it develops precision (and hence clarity) in thinking and writing;
it develops problem-solving skills, including first principles thinking;
and in particular it develops the ability to break down a complex problem into manageable steps;
it develops a certain kind of creativity;
it develops abstraction abilities;
it strengthens the ability to persist through difficult tasks.
Of course, there are also some more narrow benefits, like providing certain prerequisites to learn physics or helping with data literacy, but these are usually deeply conected to what I just listed anyway.
If one thinks about it, almost everything I listed is closely related to the activity of searching for, finding and writing up proofs of mathematical statements — meaning taking such a statement and thinking about it in certain ways until you find a precise step-by-step argument of why it is true.
And if one does not include teaching students how do find and write up proofs, which is what happens in many countries nowadays, then they have no way of reaping these benefits. This is because what people do instead of proofs is training to execute a particular list of procedures that get you the right answer to questions from a particular narrow list. This activity clearly has nothing to do with anything I listed above, and by the way, is something AI already does better than any human, meaning if it helped develop some other skill, that skill is anyway made obsolete by AI.
So I think it’s now clear that a problem with not learning proofs is that it greately decreases the benefits coming with learning maths. But there is another problem.
Because mathematical truths do not depend on things like physical experiments (this is why maths is considered so universal), the way one arrives at such truths is through a proof. So by not learning proofs, students do not learn anything about the nature of mathematical truth, in a way ending up thinking that something is true because it says so in the textbook, not because they understand why it simply must be true. In particular, this means they are never given the possibility to understand well the concepts they are taught — which of course leads to frustration and lack of interest — but it also means that they don’t even understand anything about what is this subject they’re studying, called maths.
And now, what is the use of a class which gives you very few benefits, which you find frustrating and uninteresting, and after which you don’t even actually know anything about the nature of subject it was about? I think I’ve made my point.