> taking all the optional extra math courses (including linear algebra), without there ever being a big emphasis on proofs
Sorry to break it to you, but you didn't take math classes. You took classes of the discipline taught in high school under the homonymous name "math". There is a big difference.
It's the same difference as there is between what you get taught in grade school under the name "English" (or whatever is the dominant language where you live): the alphabet, spelling, pronunciation, basic sentence structure... And what gets taught in high school under the name "English": how to write essays, critically analyze pieces of literature, etc. The two sets of skills are almost completely unrelated. The first is a prerequisite for the second (how can you write an essay if you can't write at all?), so somehow the two got the same name. But nobody believes that winning a spelling bee is the same type of skill as writing a novel.
I know it's a shock to everyone who enters a university math course after high school. Many of my 1st year students are confounded about the fact that they'll be graded on their ability to prove things. They expect the equivalent of cooking recipes to invert matrices, compute a GCD, solve a quadratic equation, or whatever, and balk at anything else. I want them to understand logical reasoning, abstract concepts, and the difference between "I'm pretty sure" and "this is an absolute truth". There's a world of difference, and most have to wait a few years to develop enough maturity to finally get it.
> Sorry to break it to you, but you didn't take math classes. You took classes of the discipline taught in high school under the homonymous name "math". There is a big difference.
If you look at the comments below, you’ll see that this can’t be strictly true. At least, not 20+ years ago in Australia when I was a student. Some of the courses I took were in the math faculty with students who were going on to become mathematicians. At that time this would have been a quarter load of a semester, and was titled “Linear Algebra”, but I can’t remember if it was 1st/2nd or even 3rd year subject (it’s been too long).
Perhaps the lack of emphasis on proofs (I am not saying proofs were absent, I made another comment with more explanation), was a combination of these being introductory courses, the universities knowledge that there were more than just math faculty students taking them, or changes with time in how the pedagogy has evolved.
What is more interesting to me, is what do you think a student misses out on, from a capability point of view, with an applications focused learning as opposed to one focused on reading and writing proofs?
Would a student who is not intending to become a mathematician still benefit from this approach? Would a middle aged man who was taught some “Linear Algebra” benefit from picking up a book such as the one referenced here?
> What is more interesting to me, is what do you think a student misses out on, from a capability point of view, with an applications focused learning as opposed to one focused on reading and writing proofs?
The generalizable value is not so much in collecting a bunch of discrete capabilities (they're there, but generally somewhat domain-specific) as it is in developing certain intuitions and habits of thought: what mathematicians call "mathematical maturity". A few examples:
- Correcting trivial errors in otherwise correct arguments on the fly instead of getting hung up on them (as demonstrated all over this comment section).
- Thinking in terms of your domain rather than however you happen to be choosing to represent it at the moment. This is why math papers can be riddled with "syntax errors" and yet still reach the right conclusions for the right reasons. These sorts of errors don't propagate out of control because they're not propagated at all: line N+1 isn't derived from line N: conceptual step N+1 is derived from conceptual step N, and then they're translated into lines N+1 and N independently.
- Tracking, as you reason through something, whether your intuitions and heuristics can be formalized without having to actually do so.
- More generally, being able to fluently move between different levels of formality as needed without suffering too much cognitive load at the transitions.
- Approaching new topics by looking for structures you already understand, instead of trying to build everything up from primitives every time. Good programmers do the same, but often fail to generalize it beyond code.
> Would a student who is not intending to become a mathematician still benefit from this approach?
If they intend to go into a technical field, absolutely.
> Would a middle aged man who was taught some “Linear Algebra” benefit from picking up a book such as the one referenced here?
Depends on what you're looking for. If you want to learn other areas of math, linear algebra is more or less a hard requirement. If you want to be able to semiformally reason about linear algebra faster and more accurately, yes. If you just want better computational tricks, drink deep or not at all: they're out there, but a fair bit further down the road.
The sibling comment answered most of what you wrote. So, I'll just add that I'm talking about the present day, not 20+ years ago. I don't know about your experience in Australia 20+ years ago, but I'm teaching real students, today, who just got out of high school, in Western Europe. Not hypothetical students 20 years ago in Australia. And based on what the Australian colleagues I met at various conferences told me, their teaching experience in Australia today isn't really different from mine.
Sorry to break it to you, but you didn't take math classes. You took classes of the discipline taught in high school under the homonymous name "math". There is a big difference.
It's the same difference as there is between what you get taught in grade school under the name "English" (or whatever is the dominant language where you live): the alphabet, spelling, pronunciation, basic sentence structure... And what gets taught in high school under the name "English": how to write essays, critically analyze pieces of literature, etc. The two sets of skills are almost completely unrelated. The first is a prerequisite for the second (how can you write an essay if you can't write at all?), so somehow the two got the same name. But nobody believes that winning a spelling bee is the same type of skill as writing a novel.
I know it's a shock to everyone who enters a university math course after high school. Many of my 1st year students are confounded about the fact that they'll be graded on their ability to prove things. They expect the equivalent of cooking recipes to invert matrices, compute a GCD, solve a quadratic equation, or whatever, and balk at anything else. I want them to understand logical reasoning, abstract concepts, and the difference between "I'm pretty sure" and "this is an absolute truth". There's a world of difference, and most have to wait a few years to develop enough maturity to finally get it.