Introductory vs Upper-Level Math Coures

[vcxp]

Did anyone who now does mathematical research feel as if what they did in their upper-division courses was not only more relevant to their career as a researcher, but very much not related to their introductory courses? I'm sure this depends on the school, but at mine all of the 100 and 200 level (and most of the 300 level) mathematics courses are "for engineers", and they suck.

I ask because I'm finishing up all of these "monkey see, monkey compute" courses at my school, and I will be taking point-set Topology and an introduction to Real Analysis ("Advanced Calculus") next semester. Looking over the books and the materials for these courses has me very excited, as it looks like conceptual mastery and the ability to prove is what rules in these courses, not how many algorithms I can memorize!

I decided to take some courses this semester just to see how much I would like proof/concept-oriented stuff. One was a "math for Computer Science"-type class, where you had to do some basic reasoning (not really a formal proof), but all of the work was based on whether or not you could understand the concepts of a few fields (set theory, combinatorics, graphs, relations). I did so well in that course I got to skip the final exam, which is cool. A lot of people bombed/dropped the class, which was a little surprising.

I took another course that started with a proof-based introduction to number theory. Some of the proofs were a little challenging, but I worked through all of them and I think I got almost every single one correct. Honestly, that was a religious experience for me. I have never sat up past when I normally go to sleep to do schoolwork just because I wanted to, but in that class I did. I got a nice taste of what actual intellectual work is like (I think), and I enjoyed it. Sadly, sometime after the midterm, everything dropped off and became computation-based, at which point I stopped caring about the class. Come to find out, everyone else in the class had hated the proofs and complained about having to do them. Apparently, people are against having to prove something on a test, but are okay with mindlessly applying Euclid's algorithm over and over and over and over again.

So, with all this in mind, can someone tell me there's light at the end of the tunnel?

 

[koab1mjr]

disclaimer I am not a math researcher just a senior math/engineering major

well duh

of course the more advance stuff will be more applicable but you need basics. One can have deep understanding of the material without knowing the formal proof. I would say complete mastery one should know the proofs.

Go into pure math there is plenty for you to gnaw on that goes for applied too

oh and I love the modesty and humility!

 

[vcxp]

well duh

We're off to a good start!

of course the more advance stuff will be more applicable but you need basics.

I guess my whole feeling was the so-called "basics" are either useless or would be learned as necessary by someone looking to study the field anyway. I apologize for not making this clear.

As an example, I think knowing how to row reduce matrices is something everyone should know, but I don't think a huge chunk of a course's content should be based around my ability to row reduce them on half of the problems on a test, nor my ability to solve 20 identical homework problems about row reducing matrices, especially when I demonstrate knowledge of the basic algorithm for doing so and the motivation for doing. To me, that is a supreme waste of my time.

oh and I love the modesty and humility!

* blink * What? Is this sarcasm? Where do I sound immodest?

 

[elkement]

Hi vcxp,

your posting is very interesting! I have studied physics 20 years ago, in Europe. At that time there was no different version of the entry mathematics lectures for real mathematicians and physicist / engineers. I have checked the curriculum of my home university recently: As you say, now there is only 'algebra for physicists' etc.

Back then everybody had to go through very theoretical and proof-oriented lectures right from the start. I was probably lucky because I had teachers in high school who emphasized proofs a lot. In my opinion this theoretical foundation is invaluable: I felt that it was easier for people with this 'pure theory' / proof background to catch up with engineering knowledge than vice versa. Students who already had engineering classes in high school had an advantage at the beginning of university engineering classes but they had a much harder time to catch up with mathematical reasoning than I had with engineering.

I have some insights on curriculum design for MSc programmes targeted to professionals (e.g. BScs with some years of industrial experience). I have been told that potential students are 'terrified' by pure mathematics more by anything else.
In Europe all degrees in higher education are currently being standardized in order to enhance students' mobility and to ease transition from one subject to another (BSc is a new concept in many countries. In the past the MSc (5yrs.) was often the first degree).

So curriculum designers need to make the programmes more 'modular' and 'pluggable' than they had been before so that a greater variety of BSc / MSc and PhD programmes can be 'concatonated'. In addition, the number of universities and programmes has been increased. As a consequence, there is more competition for students and universities need to limit the impact of the 'mathematics panic factor' to attract students.

It would be interesting to hear from people who have graduated a while ago in the US.

 

[micromass]

Did anyone who now does mathematical research feel as if what they did in their upper-division courses was not only more relevant to their career as a researcher, but very much not related to their introductory courses? I'm sure this depends on the school, but at mine all of the 100 and 200 level (and most of the 300 level) mathematics courses are "for engineers", and they suck.

I guess this is a USA-thingy. In Europe, they immediately begin with heavy proof-bases classes. Even the engineers see a lot of proofs! So the first year start with analysis, linear algebra and abstract algebra, all proof-based!

I really wouldn't want to attent your school because math without proofs is boring. If you've done a course in math, and it didn't involve proofs, then you didn't do math, but simply mindless computations. In my opinion, every math class should contain proofs, from high school on! That's the way it is here at least.

I ask because I'm finishing up all of these "monkey see, monkey compute" courses at my school, and I will be taking point-set Topology and an introduction to Real Analysis ("Advanced Calculus") next semester. Looking over the books and the materials for these courses has me very excited, as it looks like conceptual mastery and the ability to prove is what rules in these courses, not how many algorithms I can memorize!

Trust me, you'll LOVE topology and real analysis! But nothing stops you from checking these things out right now!

So, with all this in mind, can someone tell me there's light at the end of the tunnel?

Yes! Things will be better tomorrow! It seems like you are a true mathematician, so be a bit patient, things will be more interesting later!

 

[Nicook5]

I'm in high school and I'm also quite scared of this. I was lucky enough to have two fantastic teachers in middle school for algebra and geometry who focused heavily on proofs and such. Now though most of my classes have been more plug and chug... which is very boring (I frequently end up just reading a book or something). I also hear that at many universities you can take classes between schools. For example, there's like a MATH version of calculus, and Applied math version, and an engineering version or something. I was talking to a couple alumni who told me, and recommended taking the Math ones.

 

[vcxp]

Hi vcxp,

your posting is very interesting! I have studied physics 20 years ago, in Europe. At that time there was no different version of the entry mathematics lectures for real mathematicians and physicist / engineers. I have checked the curriculum of my home university recently: As you say, now there is only 'algebra for physicists' etc.

Back then everybody had to go through very theoretical and proof-oriented lectures right from the start. I was probably lucky because I had teachers in high school who emphasized proofs a lot. In my opinion this theoretical foundation is invaluable: I felt that it was easier for people with this 'pure theory' / proof background to catch up with engineering knowledge than vice versa. Students who already had engineering classes in high school had an advantage at the beginning of university engineering classes but they had a much harder time to catch up with mathematical reasoning than I had with engineering.

I have some insights on curriculum design for MSc programmes targeted to professionals (e.g. BScs with some years of industrial experience). I have been told that potential students are 'terrified' by pure mathematics more by anything else.
In Europe all degrees in higher education are currently being standardized in order to enhance students' mobility and to ease transition from one subject to another (BSc is a new concept in many countries. In the past the MSc (5yrs.) was often the first degree).

So curriculum designers need to make the programmes more 'modular' and 'pluggable' than they had been before so that a greater variety of BSc / MSc and PhD programmes can be 'concatonated'. In addition, the number of universities and programmes has been increased. As a consequence, there is more competition for students and universities need to limit the impact of the 'mathematics panic factor' to attract students.

It would be interesting to hear from people who have graduated a while ago in the US.

Interesting. If I'm reading your post correctly about the "modularity" of courses, it seems to line up with http://www.lambdassociates.org/blog/decline.htm" account of British universities, particularly,

It was in 1993 that I experienced these changes as a newly-tenured lecturer. We were summoned to be told that the School of Computer Studies at Leeds was henceforth to adopt a buffet-style form of degree whereby students picked and mixed their degree studies rather than the table d'hote system we had used till then. This new system was called 'modularisation' and it represented the drive towards student choice desired by government.

An immediate casualty were some hard-core traditional CS modules like complexity and compiler design. Why, argued students, elect to study some damned hard subject like compiler design, when you could study something cool like web design and get better marks? So these old hard core subjects began to drop off. Even worse, the School (following the logic of the market), having seen that these hard core subjects were not attracting a following, simply dropped them from the curriculum. So future students who were bright enough to study these areas would never get the chance to do so.

After a few years of this system, the results percolated through to my office. I could see the results in the lecture hall, but the procession of students who walked into my office and said "Dr Tarver, I need to do a final year project but I can't do any programming"... well, they are more than I can remember or even want to remember. And the thing was that the School was not in a position to fail these students because, crudely, we needed the money and if we didn't take it there were others who would. Hence failing students was frowned upon. By pre-1990 standards about 20% of the students should have been failed.

 

[vcxp]

I really wouldn't want to attent your school because math without proofs is boring. If you've done a course in math, and it didn't involve proofs, then you didn't do math, but simply mindless computations. In my opinion, every math class should contain proofs, from high school on! That's the way it is here at least.

I agree. What angers me the most is it waters down the value of the degree. In fact, looking at the courses, I think you can graduate with a BS in Mathematics from my school and not do a single proof if you plan correctly; things like Topology, Algebra and Analysis can be avoided, and you'll still have the same degree.

Trust me, you'll LOVE topology and real analysis! But nothing stops you from checking these things out right now!

I'm reading through a copy of Mendelson's "Introduction to Topology", and I will probably find something related to Analysis to read. My course uses Browder's "Mathematical Analysis: An Introduction", but I don't know much about it. Everyone tells me I should avoid Rudin as a first introduction, but I might at least try it.

Yes! Things will be better tomorrow! It seems like you are a true mathematician, so be a bit patient, things will be more interesting later!

Thanks.

 

[vcxp]

I'm in high school and I'm also quite scared of this. I was lucky enough to have two fantastic teachers in middle school for algebra and geometry who focused heavily on proofs and such. Now though most of my classes have been more plug and chug... which is very boring (I frequently end up just reading a book or something). I also hear that at many universities you can take classes between schools. For example, there's like a MATH version of calculus, and Applied math version, and an engineering version or something. I was talking to a couple alumni who told me, and recommended taking the Math ones.

Some schools do have this. The one I've always seen is "Calculus for Math Majors" vs. "Calculus for Engineers" or something similar. I would recommend the mathematics major courses, however, it's my understanding that mixing and matching classes isn't always possible. For instance, often you can't take "Calculus for Math Majors" and have it count for your Calculus credit for an engineering major. However, I would assume that this depends entirely on the school.

 

[elkement]

Interesting. If I'm reading your post correctly about the "modularity" of courses, it seems to line up with http://www.lambdassociates.org/blog/decline.htm" account of British universities, particularly,

Yes - this is it: 'a buffet-style form of degree whereby students picked and mixed their degree studies'! I am not from the UK, but it seems to be a common trend in all European countries.

 

[Inna]

What angers me the most is it waters down the value of the degree. In fact, looking at the courses, I think you can graduate with a BS in Mathematics from my school and not do a single proof if you plan correctly; things like Topology, Algebra and Analysis can be avoided, and you'll still have the same degree.

I am kind of shocked that you can major in Math and avoid proof-based courses.

I guess this is a USA-thingy. In Europe, they immediately begin with heavy proof-bases classes. Even the engineers see a lot of proofs! So the first year start with analysis, linear algebra and abstract algebra, all proof-based!

I wonder, how would one go about transferring their Europian credits to a US school in this case? I have studied in Russia and taken what we called "Mathematical Analysis", which is Calculus plus Real Analysis in one course. However, it got put on my transcript as "Calculus" when I went on to study in the US. Back then I did not realize the problem, because I was majoring in Physics and did not need advanced Math, but now I can't convince foreign transcript evaluators that I have taken Real Analysis in the first year of college. Does anyone have experience with this sort of thing?