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Calculus and linear algebra seem to _totally_ dominate the curriculum in most (all?) countrie. What about meta-mathematics? Topology? Logics? History of mathem

by jll29 5y ago

Calculus and linear algebra seem to _totally_ dominate the curriculum in most (all?) countrie. What about meta-mathematics? Topology? Logics? History of mathematics? Philosophy of mathematics? Combinatorics? Number theory? Discrete mathematics? Graph theory? In the post, the fieds under "electives" are by far the most interesting ones, IMHO. And I fully agree, in-depth knowledge of probability theory as well as descriptive statistics and of course the application to systematic and sound decision making is absolute key, and ought to be taught to anyone from medic to policy makers (scary: Gigerenzer showed that medics tend to be confused about the difference between P(A|B) and P(B|A) - the very people whose job it is to diagnose whether you have cancer or not!).

eftychis 5y ago
Anything besides a cursory layperson's outlook on a lot of these topics (besides basic logic and history/philosophy of math -- although not sure how you would teach the last two/what you have in mind as curriculum) requires calculus and/or linear algebra. There is a reason they say you can never learn too much linear algebra. And yes probability and statistics are fundamental. I was shocked a bit when I learned it was not taught in highschools world wide (i.e. not in the U.S.A.). But then again I had gotten numb with the current average level in the taught topics people arrive at undergrad at. Note there is a lot of interconnectivity. To understand a new concept you might need concepts in another. E.g. number theory and probability.
- jltsiren 5y ago
  In my experience, mathematics can be roughly divided in two: topics that use real numbers all the time, and topics that rarely need them. In the former, almost everything builds on calculus and/or linear algebra. In the latter, they are just two topics among many others. The true foundational classes in the typical undergraduate mathematics curriculum are logic and abstract algebra. People rarely start with them, because the usual way of teaching mathematics is applications before foundations. You learn linear algebra before abstract algebra, proofs before formal logic, and axiomatic probability before measure theory. And there is definitely such thing as too much linear algebra. Once upon a time, I wanted a decent mathematical background for theoretical CS and continued (at least) until the first graduate-level class in most major topics. Graduate linear algebra was "foundations without applications" for me, as I've never worked on anything building on it.
- t-3 5y ago
  A decade+ ago when I was in school, the policy was to push honors and AP classes for "college-bound" students. Stats (not offered as honors or AP in my school) was one of the fun courses that co-op kids got to take while we were stuck in pre-calc.
- caddemon 5y ago
  I think a layperson's outlook on these topics in high school could be very interesting though. Use this outlook to help motivate learning calculus and linear algebra. With a mix of engineering survey and math survey there are plenty of kids out there that would find parts of the course that stimulate their natural interest. Then the "calculus has many practical applications" thing would feel way less contrived. I mean of course I know now that it is true, but in high school I didn't feel it because I didn't have enough broader context.
dwohnitmok 5y ago
> meta-mathematics? Topology? Logics? History of mathematics? Philosophy of mathematics? Combinatorics? Number theory? Discrete mathematics? Graph theory? Honestly all of those feel more niche than calculus. I agree with you and joatmon-snoo on the usefulness of statistics and would probably support bumping calculus in favor of statistics, but meta-mathematics, topology, logic (which bleeds into meta-mathematics), combinatorics (which is kind of covered by stats), number theory, discrete mathematics, and graph theory are all much less useful even in adjacent STEM fields (discrete mathematics and graph theory matter more in CS, but far less for day-to-day programming). History of mathematics is effectively an entirely separate discipline and philosophy of mathematics has meta-mathematics/mathematical logic as a prerequisite. Calculus unlocks much of physics and engineering (and lots of stats!). Large cardinal theory does not unlock any other field to the best of my understanding.
- caddemon 5y ago
  I think the problem is that students don't know these fields exist unless they took an especially strong liking to the standard school math curriculum. Going into college I couldn't have told you what most of the fields mentioned were about even broadly. Truly I had no idea what math was, because we barely talked about proofs through high school. The only reason I swung back around to learning some of it is because my upper level college CS classes introduced me to ideas that then made me want to revisit my math curriculum. There ought to be some way to encourage late high school/early college students to "survey" the field without necessarily taking full courses in these topics. This could also give some earlier understanding in how the different fields relate - for example you could present a toy graph theory problem within linear algebra as a matrix problem, later presenting the same problem in graph theory section and walk through it using graph representation. I think high school courses struggle with memorability precisely because most units are taught basically in a vacuum. Regardless though, the point of the survey course wouldn't be to remember details so much as to find topics of interest for further study/be generally aware of their existence in case a relevant problem comes up in the future.
edflsafoiewq 5y ago
Calculus and linear algebra continue to dominate in applied mathematics. "How can I turn this into a problem in linear algebra?" is probably the most fruitful mathematical technique that has ever existed.
- JadeNB 5y ago
  > "How can I turn this into a problem in linear algebra?" is probably the most fruitful mathematical technique that has ever existed. And, from that perspective (with which I agree), calculus itself is just another instance of trying to turn a non-linear problem into a problem in linear algebra!
  - Koshkin 5y ago
    True: the differential at a point is a linear map; the integral is a linear form (on the vector space of functions).
ghaff 5y ago
Interestingly the ascent of linear algebra is a relatively recent thing. I have an engineering degree but while I was certainly exposed to basic matrix stuff, never took a linear algebra class. When I was an undergrad, you took differential equations in addition to basic calculus for engineering. (You needed for system dynamics among other things but it was very cookbook.) Linear algebra became a lot more interesting once you had cheap computers and Matlab.
- antishatter 5y ago
  Computers made it insanely useful where before it was an interesting view of systematic math.
- bigger_cheese 5y ago
  When did you do your Engineering degree? When I did my BE Degree (2004 to 2008) Linear Algebra was included. We were introduced to the basics of linear Algebra in year 12 of high school (simple matrix manipulation, intro to vectors, dot and cross products that sort of thing). I'm Australian if that makes a difference. The Math in my 4 year engineering degree was structured something like this: First Semester Year 1 two math courses: Calculus 1, Linear Algebra Second Semester Year 1 two math courses: Calculus 2, Sequences and Series (this one was probably least useful all I remember from this 15+ years later is Taylor Series and Binomial Theorem) First Semester year 2 two math courses: Differential Equations, Statistics for Engineers. From second semester year 2 onwards there were no more discrete math classes this was where the degree really specialized into various engineering streams, Mech Eng, Chem Eng, Civil etc. Had their own courses. I studied Materials Engineering some courses were shared with other eng students (For example I shared Thermodynamics with Mechanical Engineering students) but others such as Non-ferrous metallurgy were pretty deeply specialized. A lot of subject used built on earlier math (Fluid Dynamics was backed up by lots of differential equations for example, stuff like Gamma Function would come up in a lot of places. Solid Mechanics had a lot of integrals second moment of area etc.), Linear Algebra I can remember from Fracture mechanics and crack propagation (Stress and Strain tensors etc.)
  - ghaff 5y ago
diffeomorphism 5y ago
These come as a second step. In the first year (non US) you learn linear algebra, some real analysis and how to write proofs as well as important basics. In your second year you can choose all these electives, which then don't have to spend time introducing natural numbers, induction etc.
caddemon 5y ago
I fully agree with your general point, but in the US I've found calculus to be much more dominant than linear algebra. I went to an "elite" high school that still does not offer any course in linear algebra, despite having enough juniors in AP calc to fill a room. Senior year options for those students are AP stats or multivariable calc. At my college, everyone was (and still is) required to take through multivariable calculus, but there's no such requirement for linear algebra. I have some appreciation for calculus now but I really did not enjoy it much in high school or even in college. It turned me away from learning more math for some time which is unfortunate - linear algebra isn't my favorite either but I liked that much more off the bat, so I wish I had some exposure to it in high school. Then again, maybe the high school teaching style is what made me dislike calc to begin with.

by jll29 5y ago

eftychis 5y ago

Anything besides a cursory layperson's outlook on a lot of these topics (besides basic logic and history/philosophy of math -- although not sure how you would teach the last two/what you have in mind as curriculum) requires calculus and/or linear algebra. There is a reason they say you can never learn too much linear algebra. And yes probability and statistics are fundamental. I was shocked a bit when I learned it was not taught in highschools world wide (i.e. not in the U.S.A.). But then again I had gotten numb with the current average level in the taught topics people arrive at undergrad at. Note there is a lot of interconnectivity. To understand a new concept you might need concepts in another. E.g. number theory and probability.

jltsiren 5y ago
In my experience, mathematics can be roughly divided in two: topics that use real numbers all the time, and topics that rarely need them. In the former, almost everything builds on calculus and/or linear algebra. In the latter, they are just two topics among many others. The true foundational classes in the typical undergraduate mathematics curriculum are logic and abstract algebra. People rarely start with them, because the usual way of teaching mathematics is applications before foundations. You learn linear algebra before abstract algebra, proofs before formal logic, and axiomatic probability before measure theory. And there is definitely such thing as too much linear algebra. Once upon a time, I wanted a decent mathematical background for theoretical CS and continued (at least) until the first graduate-level class in most major topics. Graduate linear algebra was "foundations without applications" for me, as I've never worked on anything building on it.
t-3 5y ago
A decade+ ago when I was in school, the policy was to push honors and AP classes for "college-bound" students. Stats (not offered as honors or AP in my school) was one of the fun courses that co-op kids got to take while we were stuck in pre-calc.
caddemon 5y ago
I think a layperson's outlook on these topics in high school could be very interesting though. Use this outlook to help motivate learning calculus and linear algebra. With a mix of engineering survey and math survey there are plenty of kids out there that would find parts of the course that stimulate their natural interest. Then the "calculus has many practical applications" thing would feel way less contrived. I mean of course I know now that it is true, but in high school I didn't feel it because I didn't have enough broader context.

dwohnitmok 5y ago

> meta-mathematics? Topology? Logics? History of mathematics? Philosophy of mathematics? Combinatorics? Number theory? Discrete mathematics? Graph theory? Honestly all of those feel more niche than calculus. I agree with you and joatmon-snoo on the usefulness of statistics and would probably support bumping calculus in favor of statistics, but meta-mathematics, topology, logic (which bleeds into meta-mathematics), combinatorics (which is kind of covered by stats), number theory, discrete mathematics, and graph theory are all much less useful even in adjacent STEM fields (discrete mathematics and graph theory matter more in CS, but far less for day-to-day programming). History of mathematics is effectively an entirely separate discipline and philosophy of mathematics has meta-mathematics/mathematical logic as a prerequisite. Calculus unlocks much of physics and engineering (and lots of stats!). Large cardinal theory does not unlock any other field to the best of my understanding.

caddemon 5y ago
I think the problem is that students don't know these fields exist unless they took an especially strong liking to the standard school math curriculum. Going into college I couldn't have told you what most of the fields mentioned were about even broadly. Truly I had no idea what math was, because we barely talked about proofs through high school. The only reason I swung back around to learning some of it is because my upper level college CS classes introduced me to ideas that then made me want to revisit my math curriculum. There ought to be some way to encourage late high school/early college students to "survey" the field without necessarily taking full courses in these topics. This could also give some earlier understanding in how the different fields relate - for example you could present a toy graph theory problem within linear algebra as a matrix problem, later presenting the same problem in graph theory section and walk through it using graph representation. I think high school courses struggle with memorability precisely because most units are taught basically in a vacuum. Regardless though, the point of the survey course wouldn't be to remember details so much as to find topics of interest for further study/be generally aware of their existence in case a relevant problem comes up in the future.

edflsafoiewq 5y ago

Calculus and linear algebra continue to dominate in applied mathematics. "How can I turn this into a problem in linear algebra?" is probably the most fruitful mathematical technique that has ever existed.

JadeNB 5y ago
> "How can I turn this into a problem in linear algebra?" is probably the most fruitful mathematical technique that has ever existed. And, from that perspective (with which I agree), calculus itself is just another instance of trying to turn a non-linear problem into a problem in linear algebra!
- Koshkin 5y ago
  True: the differential at a point is a linear map; the integral is a linear form (on the vector space of functions).

ghaff 5y ago

Interestingly the ascent of linear algebra is a relatively recent thing. I have an engineering degree but while I was certainly exposed to basic matrix stuff, never took a linear algebra class. When I was an undergrad, you took differential equations in addition to basic calculus for engineering. (You needed for system dynamics among other things but it was very cookbook.) Linear algebra became a lot more interesting once you had cheap computers and Matlab.

antishatter 5y ago
Computers made it insanely useful where before it was an interesting view of systematic math.
bigger_cheese 5y ago
When did you do your Engineering degree? When I did my BE Degree (2004 to 2008) Linear Algebra was included. We were introduced to the basics of linear Algebra in year 12 of high school (simple matrix manipulation, intro to vectors, dot and cross products that sort of thing). I'm Australian if that makes a difference. The Math in my 4 year engineering degree was structured something like this: First Semester Year 1 two math courses: Calculus 1, Linear Algebra Second Semester Year 1 two math courses: Calculus 2, Sequences and Series (this one was probably least useful all I remember from this 15+ years later is Taylor Series and Binomial Theorem) First Semester year 2 two math courses: Differential Equations, Statistics for Engineers. From second semester year 2 onwards there were no more discrete math classes this was where the degree really specialized into various engineering streams, Mech Eng, Chem Eng, Civil etc. Had their own courses. I studied Materials Engineering some courses were shared with other eng students (For example I shared Thermodynamics with Mechanical Engineering students) but others such as Non-ferrous metallurgy were pretty deeply specialized. A lot of subject used built on earlier math (Fluid Dynamics was backed up by lots of differential equations for example, stuff like Gamma Function would come up in a lot of places. Solid Mechanics had a lot of integrals second moment of area etc.), Linear Algebra I can remember from Fracture mechanics and crack propagation (Stress and Strain tensors etc.)
- ghaff 5y ago

diffeomorphism 5y ago

These come as a second step. In the first year (non US) you learn linear algebra, some real analysis and how to write proofs as well as important basics. In your second year you can choose all these electives, which then don't have to spend time introducing natural numbers, induction etc.

caddemon 5y ago

I fully agree with your general point, but in the US I've found calculus to be much more dominant than linear algebra. I went to an "elite" high school that still does not offer any course in linear algebra, despite having enough juniors in AP calc to fill a room. Senior year options for those students are AP stats or multivariable calc. At my college, everyone was (and still is) required to take through multivariable calculus, but there's no such requirement for linear algebra. I have some appreciation for calculus now but I really did not enjoy it much in high school or even in college. It turned me away from learning more math for some time which is unfortunate - linear algebra isn't my favorite either but I liked that much more off the bat, so I wish I had some exposure to it in high school. Then again, maybe the high school teaching style is what made me dislike calc to begin with.