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Comment by trengrj | Hacker News Reader

I think that maths at school it still too focused on creating engineers (the non software kind) and much of the material is aimed at producing people who can do calculus.

I'd prefer for there to be more focus on proofs, logic, geometry, number theory, graph theory, and cryptography etc. I think these concepts would give people a more rounded understanding of mathematics and prevent their experience of mathematics being one of pain.

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I don't know if there's too much calculus, but there's definitely too much of the wrong type of calculus.

Congratulations, you have just learned to integrate. Let us now spend the next three years running through several thousand different examples of analytic integration of one-dimensional continuous functions.

More dimensions? Path integrals? Numerical integration methods that you'll almost certainly have to apply in the real world because most functions don't have analytic integrals? Differential equations? Never heard of them. Now let's all integrate (x^2 - sin(x))/coth(x).

I think there is too much focus on the mechanics and not enough on the big picture behind what you're trying to do. Mostly it's a lot of. Step one recognize the problem. Step two apply the solutionator+ and turn zee crank. Step three receive kibble.

+Solutionator: Bunch of mechanical steps that the student understands in the exact same way a trained monkey understands an organ grinder.

It's kind of funny, most people who major specifically in math or applied math will take something equivalent to a course in Real Analysis. RA is basically starting over from scratch and reteaching you the exact same material as calculus, but in a much more formal and rigorous manner. Mostly it focuses on eliminating the not-so-rigorous concept of an infintesimal quantity and formalizing the notion of limits and convergence (albeit, there is a way to rigorously treat infintesimals called nonstandard analysis, but I digress). It involves a lot more open-ended thought than the endless memorization of earlier calculus classes. It's often dreaded by undergraduates as being very proof-heavy, but it was one of my favorites. It always felt like learning the "essence" of calculus, at least for me (who sucks at/dreads rote memorization).

My real analysis textbook says on the back:

Was plane geometry your favorite math course in high school? Did you like proving theorems? Are you sick of memorizing integrals? If so, real analysis could be your cup of tea. In contrast to calculus and elementary algebra, it involves neither formula manipulation nor applications to other fields of science. None. It is Pure Mathematics, and it is sure to appeal to the budding pure mathematician.

Complex analysis is even neater. (But that's almost algebraic in its niceness.)

I really enjoyed measure-theoretic probability theory myself.

We did that as a precursor before probability theory. Really puts the latter on a sound footing.

What I wonder is whether it's necessary to teach it the wrong way before we teach it the right way. Neither answer to that question would surprise me.

As an example, I remember that as a high school maths student I had real trouble believing that 0.9 repeater equals one.

Nowadays, post university, it seems trivially true.

It depends I think. My experience has been that many people simply don't do well in "real" (i.e., formal and proof-based) math classes. Not to mention, a deep understanding of calculus isn't necessary for doing a lot of engineering, which provides most of the students in a lot of university calculus courses. On top of that, even if you start with an analysis-like approach of teaching calculus, you ultimately will have to spend some time plugging-and-chugging to get used to working with calculus on a practical level.

On the other hand are people like me, who really don't do well with learn by memorization type classes, but does well in classes that are theory first, with a thick layer of application over that.

That was my experience exactly. In the first class the professor told us that everything we learned in calculus was true, but not quite for the reason they taught us. We then re-proved everything withpoint set topology. But it felt like real math.

I concur. What's worse was when they finally did get to an example of how you would use something, it was always so contrived as to be pointless.

For example, you've learned about dealing with rates of change. Now lets talk about the rate at which we pour water into a bucket.

For the love of god! Stop making problems about pouring things into other things! This has never come up in my life! And as an example, it doesn't help me relate to anything!

I remember the semester I was taking Calc II, I was also playing a lot of the game Terminus. (This was in 2000.) The game prided itself on having realistic space flight physics, so you had to spend the same amount of time accelerating and decelerating when traveling between locations. I remember realizing I could actually use the calculus I was learning to figure out how long / how far it would take to decelerate, and so plan my thruster burns accordingly.

Haven't used it since, though... It's a pity.

As a kid I was very gifted at swallowing mechanics, and misled myself into thinking I was good at math. Like an ability to parse musical score very fast, but unable to play any music. Getting the concept is far more important, but far harder to do in class with quantifiable results (unless we measure the average grin on students face, maybe).

I teach these courses. I can tell you that everything in them is asked for by a client department.

Not saying that change wouldn't be good, but these things are there because we are asked to put them there. In fact, we are asked to put many more things there that don't fit in the course.

I would add statistics to this list. Science and pretty much all the numbers quoted out there are based on statistics.

It’s impossible to reasonably understand statistics without studying calculus. Everyone I’ve talked to who took a statistics course without first studying calculus had an overly formulaic, theoretically wishy-washy, and often wrong understanding of the meaning of basic statistical distributions and techniques.

Not necessary at all. You can teach most of the interesting parts of statistics with just the binomial distribution and numerical techniques like bootstrapping. Nothing about that is wishy-washy, it's just a modified curriculum that sidesteps some interesting things that require calculus but in return allows you to focus on some other really interesting things that don't require it.

There is, for example, nothing rigorous about being able to prove and use the Central Limit Theorem – the whole point of the thing is that it's an approximation and not always a very good one at that. And this is the origin of many statistical techniques and tests: computational shortcuts. The fact that these shortcuts require a good understanding of math doesn't suddenly make them any more central to a conceptual understanding of statistics than they are.

Two good examples of good stats intros that are light on the math: Allen Downey's Think Stats http://greenteapress.com/thinkstats2/ or the more traditional/frequentist http://onlinestatbook.com/.

Um... I feel like there is likely to be a lot of selection bias in your experience. Now, I would notice that, having taken calculus before statistics myself, but...

The introductory economics courses at my school didn't require calculus. This would have been less annoying if every single topic hadn't been explicitly introduced as finding the slope of, or area under, particular curves.

Those who don't learn calculus have to reinvent it, poorly. C.f. so-called "AUC" [0], which is apparently an important metric in medicine.

[0] - https://en.wikipedia.org/wiki/Area_under_the_curve_%28pharma...

What exactly bugs you about AUC? What about ROC curves?

On the other hand, perhaps you are biased in favour of the math relevant to computer science, such as number theory, graph theory, and cryptography?

It might be true... but ask an American 17-year-old with collegiate aspirations what he or she will take in college. "Calculus!" What else? Blank stare. What other math exists? This number theory you speak of... as far as an average high schooler is concerned, you're making stuff up.

Haha, I could agree with that. I think though the problem with calculus is that it is so cumulative, so doing badly in one year leads to the next etc.

That's the nature of mathematical knowledge though. If you can't understand basic calculus then there's a helluva lot that's downstream from there that you won't understand either.

For me, this has been about proving that mathematics is about more than just numbers. The world is made up of quantities, relationships, nodes, probabilities, and much more.

If students are expected to use math to make better decisions and understand the mechanisms of the world, a breadth of math needs to be taught earlier on.

I think this was completely lost on all my math teachers growing up. It wasn't until I started teaching myself programming and exploring the patterns of the universe and reading philosophy and science and even too from being a musician and teacher did I start to FEEL and KNOW math and experience it's concepts directly. Not just memorize steps. I almost feel like math and philosophy should be combined in school into a hybrid course that is very experimental (verbal hypotheses, mathematic proofs) and should focus simply on investigating the world of ideas around you. Get them into a mode of thought where your creative and skeptical insights can get you into mental conundrum, and your mathematical skills can help you get out of it. How wonderful would it be to get students to "rediscover" fundamental insights about the world? THEN you crack the book open and teach them who discovered it first 2000 years ago.

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