# Why is highschool math so unrigorous?

I am an highschool student (I'm starting soon the italian equivalent of 12th grade) and I have many problems with the way hs math education works. I don't understand why everything is explained only at an intuitive level, and there are no clear and precise definitions/proofs.

The biggest problem is the approach to real numbers: we always use them even though we have never defined what they are, which is understandable because it's not really easy to explain Dedekind Cuts/Cauchy sequences to highschool students, but we also have never touched upon on why the real numbers are different from the rational ones, that is also the reason why everything we do works: the least upper bound/completeness property of $$\mathbb{R}$$. As a consequence of this, we get a lot of definitions that are not really definitions but are more like "defining things into existence", for example this year we learned about the square root function and this was the definition: "Let $$x \geq 0$$. The square root of $$x$$ is the unique number $$r \geq 0$$ such that $$r^2 = x$$". And the same thing happened for exponentials and logarithms (by the way it's not so hard to define these functions using only elementary techniques: you can use some simple inequalities + the completeness property and you're done).

The other big problem is the approach to mathematics, in highschool math most of the stuff you do is use some result you never really understood to simplify an arbitrary expression or solve an arbitrary equation, instead of doing abstract thinking/problem solving or analyzing some key examples in the theory to really understand the results.

Why is that? Why doesn't anyone try to change this?

• To me, the way to do things is with three levels: "Concrete - Pictorical - Abstract". In this scheme, you need to "try things first" before state them as properties. Aug 23 at 16:03
• In the U.S. only a small percentage of students (probably under 5%) could be reasonably expected to work at the level you want. As for me, I hardly gave any attention to what was being covered in school classes. For me it was two different worlds, stuff we did in school (which required virtually no thought and whose assignments I usually managed to do on the school bus or during homeroom, at least for math classes) and stuff I did on my own (like I did many other things of interest to me when not in school). Just get a more advanced text and read ahead at your leisure. Aug 23 at 17:14
• It might interest you to read the observations of an American college professor who visited Europe in in 1934 to see what was being taught to mathematics specialists in schools. Of course, at the time, "rigor" was less about Dedekind cuts and more about the depth of coverage of algebra, geometry, calculus and mechanics. See Cairns, Advanced Preparatory Mathematics in England, France and Italy.
– Dave
Aug 24 at 15:56
• If you want a somewhat precise and easily understandable distinction between rational and irrational real numbers it is simply this; rational numbers are those whose decimal expansion is eventually repeating. In contrast, irrational numbers are those for which the decimal expansion does not eventually repeat. If you look at it like this, the existence of irrational numbers is hardly surprising. To prove the rational numbers are like this it just requires the geometric series which can be argued without the full arsenal of series analysis, at least plausibly so. Why do teachers not do this ? .. Aug 24 at 22:11
• Do you recognize that that vast majority of people would become completely lost if math in high school were taught the way you wish, and moreover that such an approach is not helpful for most people to develop an understanding of concepts? As an analogy, should everyone learn how an internal combustion engine works before they are permitted to learn how to drive a car?
– KCd
Aug 28 at 12:19

For the same reason that elementary counting numbers of more than a single digit are explained as ones, tens, hundreds, etc. . The concept of powers and exponents has not yet been developed. Later it is re-cast in terms of powers of ten with a decimal point delineating the positive/negative powers. Even then, it's often never further defined as powers of a Base with a radix point delineation.

The ability to successfully use the mechanics does not necessarily require understanding the abstract concepts, just as the ability to use a smart phone or computer does not require understanding how they work.

It's simply more practical to teach functionality first. Abstract understanding can come later, or not at all in many cases.

As a consequence of this, we get a lot of definitions that are not really definitions but are more like "defining things into existence", for example this year we learned about the square root function and this was the definition: "Let x≥0. The square root of x is the unique number r≥0 such that r2=x"

You're just mistaken about what rigor consists of. It's very common throughout all of mathematics simply to start from a set of axioms and work forward. This is how Euclid worked, and it's also how people created ZFC set theory, which is the most common foundational framework for mathematics today.

More valid criticisms of this definition might be: (1) It hides a claim of existence inside a definition. (2) It makes it unclear whether this existence is intended to be a theorem, or an axiom. (3) If you want to talk about third roots and so forth, then you may need a proliferation of such axioms, which is kind of ugly. But these are all more like stylistic issues, and they could be easily remedied without having to do an explicit construction of the reals from set theory.

I don't understand why everything is explained only at an intuitive level, and there are no clear and precise definitions/proofs.

The only example you give in the question is a definition that is precise. You haven't given any examples of what you think is an unclear or imprecise proof.

• Well, one such example could be the trigonometric functions. We still haven't covered them in class but the way the book presents them is the most imprecise way possible. There is no treatment of arc-lenght, or of pi but the construction of these functions is based on them. Aug 25 at 6:19

Do not confuse rigorous with foundations-first.

Highschool math is neither, but it is perfectly possible to do rigorous math without starting with the foundations. A key aspect here is being utterly clear on what is taken for granted as a prerequisite. Developing mathematics foundations-first is challenging enough when addressing an audience of mathematicians who already know where the journey is leading. Keeping a novice motivated to follow along will be near-impossible, as the investment required of them before stuff starts to make sense is just too much.

Many teachers would not be qualified to rigorously do math.

As future teachers will not only need to take subject-specific courses, but also courses about how to teach the subject to children. In many systems, a teacher isn't single-subject either. Thus, a math teacher will often have taken (far) fewer math courses at university then someone with a BSc in Mathematics. They'll often have seen some level of rigour in math during their lectures, but being able to rigorously do math themselves is a different story.

Foundations-first school teaching was attempted once. It didn't go well.

The New Math movement lead to an attempt for math teaching based on set theory around the 1960s, in a couple of countries. The experiment was ultimately deeemed a failure. Of course, one can debate whether this is because the attempt was handled in an incompetent way (see the previous item) or whether it was a bad idea in the first place. But this failure will be an obstacle for any similar attempt to reform school teaching.

Probably few pupils could cope.

Mathematics is a special mode of thought, and a lot of people really struggle with the kind of abstract reasoning required. Already basic stuff such as processing a definition doesn't fit in with how most people think - they'd expect to learn a concept by generalizing from examples instead. When proving a statement mathematicians essentially think about ways how the statement could be wrong -- non-mathematicians often look for confirmation instead.

Those that would benefit might be better off elsewhere.

Those rare pupils who really want a rigorous math education are probably better off looking for it elsewhere than at highschool. I, for example, stopped attending math at school in the 10th grade and instead took math courses at university. This was a bit tricky to set up, but that would be the route I'd recommend to any pupil with opinions on whether Dedekind cuts or the Cauchy-completion are a better way introduce the reals!

Highschool math teaching definitely needs to be overhauled though, but not with the OP or me-20-years-ago as examples in mind.

• You're right about eveything, especially the qualification of teachers!! I remember asking my highschool teacher before summer if it was possible to generalize a result (i wanted to know if it was possible to generalize a characterization of the lebesgue measure on the borel sigma algebra to the sigma algebra of all the lebesgue measurable sets) and she didn't even know what a sigma algebra was. It was the most awkward thing ever. By the way i would love to follow some university courses on the topics i'm learning on my own, but i don't know how to that in my country Aug 29 at 12:30
• @RickDoesMath Yeah, with questions like that you should be at a university. If there is a local university where you live, email someone in their math department to see whether you can engage with them somehow. If you were in Germany, I'd recommend the FernUniversitaet Hagen to you - see my question here for Italian counterparts: academia.stackexchange.com/questions/173973/…
– Arno
Aug 30 at 17:07
• Obligatory Tom Lehrer song about New Math.
– J.G.
Sep 9 at 21:31