6 ms·

I like the approach, even though in practice I would not necessarily follow the suggested presentation. From my experience, most math courses start with the "d

I like the approach, even though in practice I would not necessarily follow the suggested presentation. From my experience, most math courses start with the "dead" book definitions. Which is not the way mathematics are built: people start with intuitive but incorrect ideas, the idea is proven to be useful, and only then made correct. The definitions end up coming from refinements to avoid contradictions arising from edge-cases. This was a common reproach to the Bourbaki group who wanted to formalize mathematics into an almost computer-digestible form; I think it was Grothendieck who said they were "embalming mathematics". This is absolutely not a good way to learn mathematics, although it is the lowest effort to come up with. I generally think it is much better to go through what do we want to achieve, how it came to be, etc. Which you can generally find with presentation of the Seven Bridges of Königsberg problem, some game mathematics (Rubik's cube, etc), or "You could have invented..." type articles (the famous one being for "spectral sequences"). Although there should be different approaches for different learning types. But this is another problem, which is more linked to the (bad) educational structure.

neel_k 5y ago
It's worth understanding the context Bourbaki arose in. An entire generation of French mathematicians was turned to bits of blood and gristle in the trenches of World War I, and so French mathematicians in the 1920s and early 1930s faced an acute shortage of teachers who were current with modern mathematics. The premise of the Bourbaki effort was to write everything down in enough detail that a sufficiently motivated reader could learn it without having to learn it master-apprentice style -- because too many potential masters were dead.
- woolion 5y ago
  In that case, that means they have entirely failed. The very few people I've met who actually bothered with the books only made fun of how horrible they were to read and understand. I never even did open one myself! I always admired the principle though, because you can reduce everything to pure logic (and even in some cases you can "brute-force" formalism to obtain new results). Which makes me think category theory kind of fills this role in a better way?
- n4r9 5y ago
  Even without the context, there's something to be said for a formalised approach. When I was in undergrad there was a lecture course given by a notoriously aloof and formal lecturer. One of the other - more popular - lecturers decided to give an "understandable" alternative to the course at the same time of the day. Myself and a few others were in the 5% that continued with the official schedule. Those notes were hard as hell to work through, but once you understood something, you REALLY understood it. One of the exam questions was conveniently targeted at one of the lectures from the more difficult course. I think it was proving that A5 is simple by considering the rotations of a dodecahedron.
  - jacobolus 5y ago
    I disagree. An overly dry and formal style (definition, definition, lemma, theorem, corollary, definition, lemma, lemma, theorem, ...) does not make students “really understand” the material. It just focuses students on low level details of formal definitions and symbolic manipulation and gives a lot of practice regurgitating/performing those, often at the expense of knowing the purpose or meaning of the subject. Low-level details are certainly essential, but the only way to really understand is to figure out what the formalism is for (what problems does it solve), grapple with the possibility space of definitions and theorems (if we picked this alternate definition, would that also get us where we want?), figure out how topics and structures relate to each-other, spend some time doing personal explorations, and build up mental models of what the definitions mean, not just their formal content. A too-dry mathematics course/book is like a screenwriting course where you focus on snappy dialog and details of the setting but never talk about the plot or themes of the story.
    - n4r9 5y ago
      I don't disagree. What we found was that, starting from the lecture notes and a few examples, getting to the point where we could complete the exercises meant that we had to do most of the above figuring-out for ourselves. And because we did it ourselves rather than have it laid out for us, the learning was more established.
    - necovek 5y ago
- bsedlm 5y ago
  > a sufficiently motivated reader could learn it without having to learn it master-apprentice style if that's the case, I would say they failed. however, what they accomplished would certainly help jog the memory of somebody who knew the material once upon a time. maybe it's a bit like looking at a zip file directly and uncompressing the contents on the fly in your head? (something about 'understanding' as a compression scheme)
jsmith45 5y ago
Yeah, I must say I prefer the math teaching approach of: 1. Explain some problem that is difficult to solve using other techniques. Ideally the problem should be interesting, even if potentially somewhat abstract. 2. Propose a technique to solve it, without requiring full rigor, but that allows for intuitively feeling that the technique is probably valid. 3. Show that the technique also works on other different problems. 4. Show show some flaws (or limits of generality) of the technique. Propose fixes for the flaw, and/or better outline where the technique viable. 5. Now either prove the technique's validity more rigorously (need not be a perfect proof, especially if a full proof requires far more complicated mathematics) or extend this technique to be able to solve more (but eventually coming back to proving the validity). This sort of approach is has been used in educational videos like some of threeblueonebrown's on youtube, and I've also seen this used as a faux historical development of algebra and high school calculus in "Algebra: the easy way", and "Calculus: the easy way", which did a great job of motivating the development of most of the ideas in those courses. I'm not familiar with similar texts for higher level mathematics (which would have a different tone, since they would be targeting adults, not teens/children), but surely some must exist, right? This is infinitely better than the all too common higher level math textbook approach of: 1. Here is some unfamiliar theorem, that you might not even really understand, and certainly have no clue of the relevance. 2. Now here is how to prove the theorem. (Which you might be able to follow, but you probably don't really care about right now). 3. Now we finally explain what the theorem is, and hint at (but may fail to show why) it might be useful.
- thesuitonym 5y ago
  Unfortunately what we typically end up with when learning math is: 1. Explain some problem that is difficult to solve using other techniques. The problem is painfully abstract and completely absurd. 2. Propose a technique to solve it. Never mind how we got to this technique or why it makes sense, just apply this equation and you'll get the answer. Memorize the numbers and symbols, that's all that matters. 3. Show that the technique also works on different (but really the same) problems. 4. Exam time! Here's a problem that looks nothing like the ones you've seen so far, but use the magic technique you memorized! You did memorize it right? Even though you didn't understand it at all? 5. This class is over so you will never use that technique again--even though it's generally useful, but since you never learned how to actually apply it or why it's useful, you'll forget it the day the exam is finished. 6. Math is great! Why don't people like math?!
xg15 5y ago
Fully agreed! I'm also puzzled why so little math courses seems to involve the actual history of an idea. Often, an idea is presented in a vacuum or - even worse - confusing a formal proof with an explanation of purpose. In my opinion, that's about as effective for teaching mathematical concepts as teaching vi by giving you a copy of the source code. I think explaining the historical context of an idea and the problems the authors back then were trying to solve with it would be much more valuable for understanding than just the bare proof.
- Jtsummers 5y ago
  IME, you will get that information if you get a math degree. You are less likely to get it, unless you read the textbook (that is, not from the teacher) from any K-12 or non-major math classes because that's not their focus (for better or worse). But often the motivational cases for the math are present in high school and non-major math course textbooks. Just not part of the class because that's not what you're going to be evaluated on.
- agumonkey 5y ago
  I found a math book aimed at math grads going into official sub-college teaching positions, and was shocked about the historical context provided. Every idea had 2 or 3 variants that differed in origin. To me it would have made math 2x more fun for a lot of pupils. But none of it made it into our classrooms (lack of time ? a dusty pedagogical approach maybe ..). It's probably vital for most people that don't have their neurons naturally properly aligned to a subject to have various informations and point of views about the problems and goals.
  - MonaroVXR 5y ago
    Which book?
    - agumonkey 5y ago
      It's a french book. "mathematiques au concours de professeur des ecoles" (Hachette editions 2006) For instance the numbering system chapter mentions non roman numeral symbols, babylonian, egyptian up to indian of course. Then various ways to multiply numbers. As a kid everything was pruned and we only saw one thing.

neel_k 5y ago

It's worth understanding the context Bourbaki arose in. An entire generation of French mathematicians was turned to bits of blood and gristle in the trenches of World War I, and so French mathematicians in the 1920s and early 1930s faced an acute shortage of teachers who were current with modern mathematics. The premise of the Bourbaki effort was to write everything down in enough detail that a sufficiently motivated reader could learn it without having to learn it master-apprentice style -- because too many potential masters were dead.

woolion 5y ago
In that case, that means they have entirely failed. The very few people I've met who actually bothered with the books only made fun of how horrible they were to read and understand. I never even did open one myself! I always admired the principle though, because you can reduce everything to pure logic (and even in some cases you can "brute-force" formalism to obtain new results). Which makes me think category theory kind of fills this role in a better way?
n4r9 5y ago
Even without the context, there's something to be said for a formalised approach. When I was in undergrad there was a lecture course given by a notoriously aloof and formal lecturer. One of the other - more popular - lecturers decided to give an "understandable" alternative to the course at the same time of the day. Myself and a few others were in the 5% that continued with the official schedule. Those notes were hard as hell to work through, but once you understood something, you REALLY understood it. One of the exam questions was conveniently targeted at one of the lectures from the more difficult course. I think it was proving that A5 is simple by considering the rotations of a dodecahedron.
- jacobolus 5y ago
  I disagree. An overly dry and formal style (definition, definition, lemma, theorem, corollary, definition, lemma, lemma, theorem, ...) does not make students “really understand” the material. It just focuses students on low level details of formal definitions and symbolic manipulation and gives a lot of practice regurgitating/performing those, often at the expense of knowing the purpose or meaning of the subject. Low-level details are certainly essential, but the only way to really understand is to figure out what the formalism is for (what problems does it solve), grapple with the possibility space of definitions and theorems (if we picked this alternate definition, would that also get us where we want?), figure out how topics and structures relate to each-other, spend some time doing personal explorations, and build up mental models of what the definitions mean, not just their formal content. A too-dry mathematics course/book is like a screenwriting course where you focus on snappy dialog and details of the setting but never talk about the plot or themes of the story.
  - n4r9 5y ago
    I don't disagree. What we found was that, starting from the lecture notes and a few examples, getting to the point where we could complete the exercises meant that we had to do most of the above figuring-out for ourselves. And because we did it ourselves rather than have it laid out for us, the learning was more established.
  - necovek 5y ago
bsedlm 5y ago
> a sufficiently motivated reader could learn it without having to learn it master-apprentice style if that's the case, I would say they failed. however, what they accomplished would certainly help jog the memory of somebody who knew the material once upon a time. maybe it's a bit like looking at a zip file directly and uncompressing the contents on the fly in your head? (something about 'understanding' as a compression scheme)

jsmith45 5y ago

Yeah, I must say I prefer the math teaching approach of: 1. Explain some problem that is difficult to solve using other techniques. Ideally the problem should be interesting, even if potentially somewhat abstract. 2. Propose a technique to solve it, without requiring full rigor, but that allows for intuitively feeling that the technique is probably valid. 3. Show that the technique also works on other different problems. 4. Show show some flaws (or limits of generality) of the technique. Propose fixes for the flaw, and/or better outline where the technique viable. 5. Now either prove the technique's validity more rigorously (need not be a perfect proof, especially if a full proof requires far more complicated mathematics) or extend this technique to be able to solve more (but eventually coming back to proving the validity). This sort of approach is has been used in educational videos like some of threeblueonebrown's on youtube, and I've also seen this used as a faux historical development of algebra and high school calculus in "Algebra: the easy way", and "Calculus: the easy way", which did a great job of motivating the development of most of the ideas in those courses. I'm not familiar with similar texts for higher level mathematics (which would have a different tone, since they would be targeting adults, not teens/children), but surely some must exist, right? This is infinitely better than the all too common higher level math textbook approach of: 1. Here is some unfamiliar theorem, that you might not even really understand, and certainly have no clue of the relevance. 2. Now here is how to prove the theorem. (Which you might be able to follow, but you probably don't really care about right now). 3. Now we finally explain what the theorem is, and hint at (but may fail to show why) it might be useful.

thesuitonym 5y ago
Unfortunately what we typically end up with when learning math is: 1. Explain some problem that is difficult to solve using other techniques. The problem is painfully abstract and completely absurd. 2. Propose a technique to solve it. Never mind how we got to this technique or why it makes sense, just apply this equation and you'll get the answer. Memorize the numbers and symbols, that's all that matters. 3. Show that the technique also works on different (but really the same) problems. 4. Exam time! Here's a problem that looks nothing like the ones you've seen so far, but use the magic technique you memorized! You did memorize it right? Even though you didn't understand it at all? 5. This class is over so you will never use that technique again--even though it's generally useful, but since you never learned how to actually apply it or why it's useful, you'll forget it the day the exam is finished. 6. Math is great! Why don't people like math?!

xg15 5y ago

Fully agreed! I'm also puzzled why so little math courses seems to involve the actual history of an idea. Often, an idea is presented in a vacuum or - even worse - confusing a formal proof with an explanation of purpose. In my opinion, that's about as effective for teaching mathematical concepts as teaching vi by giving you a copy of the source code. I think explaining the historical context of an idea and the problems the authors back then were trying to solve with it would be much more valuable for understanding than just the bare proof.

Jtsummers 5y ago
IME, you will get that information if you get a math degree. You are less likely to get it, unless you read the textbook (that is, not from the teacher) from any K-12 or non-major math classes because that's not their focus (for better or worse). But often the motivational cases for the math are present in high school and non-major math course textbooks. Just not part of the class because that's not what you're going to be evaluated on.
agumonkey 5y ago
I found a math book aimed at math grads going into official sub-college teaching positions, and was shocked about the historical context provided. Every idea had 2 or 3 variants that differed in origin. To me it would have made math 2x more fun for a lot of pupils. But none of it made it into our classrooms (lack of time ? a dusty pedagogical approach maybe ..). It's probably vital for most people that don't have their neurons naturally properly aligned to a subject to have various informations and point of views about the problems and goals.
- MonaroVXR 5y ago
  Which book?
  - agumonkey 5y ago
    It's a french book. "mathematiques au concours de professeur des ecoles" (Hachette editions 2006) For instance the numbering system chapter mentions non roman numeral symbols, babylonian, egyptian up to indian of course. Then various ways to multiply numbers. As a kid everything was pruned and we only saw one thing.