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When did you first encounter proof based mathematics?

  1. Nov 7, 2012 #1
    When did you first encounter "proof based" mathematics?

    I've been reading a few forums and have seen many posters say "methods based" mathematics like calculus is easy. The posters would then state that "proof based" mathematics is so hard and calculus isn't high level.

    So when did you first encounter "proof based" mathematics in a classroom setting? Was it in high school or university/college? What year in high school or university did you encounter it?

    Edit: What grade did first encounter it (if it was in high school)?
     
  2. jcsd
  3. Nov 7, 2012 #2
    Re: When did you first encounter "proof based" mathematics?

    My first encounter with proofs was probably in high school (freshman-sophomore year). The proofs were very easy though, but I didn't quite grasp them. Throughout highschool, the teachers kept emphasizing proofs and I got better at them. But I wouldn't say that the classes in high school were proof-based: the proofs were usually very easy and there weren't a lot).

    My first real proof based course was an analysis course in freshman year. The course was literally filled with proofs. I didn't find the course very hard, since I was acquainted with proofs in high school.

    So all in all, I had a very gradual transition into proof-based mathematics. I think that is actually the best way to introduce proofs.
     
  4. Nov 7, 2012 #3

    symbolipoint

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    Re: When did you first encounter "proof based" mathematics?

    Geometry in high school. Mostly proof-based, very difficult.
     
  5. Nov 7, 2012 #4
    Re: When did you first encounter "proof based" mathematics?

    I'm taking my first real proof based course (linear algebra) and a half-proof based course (complex variables) as a freshman at uni. I understand the proofs and can write any reasonable proof with little trouble (some proofs can take 30min-1hr) but the result is rewarding(either course). Complex variables are partially filled with proofs but not necessary rigorous in the sense that what we assume is already true (even if it may not have already been proven) and often skipping steps.

    Regardless, both courses are fascinating. The thing that fascinates me the most is seeing a collection of axioms slowly build an entire branch of mathematics.

    US highschools do a very poor job at introducing proofs. And I mean zero proofs in anything but geometry. I learned about proofs outside of class about 1-2 years ago (Junior/Senior).
     
  6. Nov 7, 2012 #5
    Re: When did you first encounter "proof based" mathematics?

    Usually, you encounter proofs for the first time in high school geometry. If you take calculus, you might see a few proofs. In linear algebra, maybe a little bit more. Then, you might take some kind of transitional class--I took a set theory class, for example. Then, I took real analysis, which was my first really serious proof-based course.
     
  7. Nov 7, 2012 #6
    Re: When did you first encounter "proof based" mathematics?

    I took my first proof based course as a freshman. At my university, if you study math, physics, computer science or actuarial science, with very few exceptions, all the math courses you'll take are going to be proof based.
     
  8. Nov 7, 2012 #7

    phinds

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    Re: When did you first encounter "proof based" mathematics?

    I think in the US at least everyone's first encounter with proof-based math is high school geometry. I remember well the delight I felt at finding that math wasn't just about algebraic equations and arithmetic. I loved doing the proofs.
     
  9. Nov 7, 2012 #8

    MarneMath

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    Re: When did you first encounter "proof based" mathematics?

    Geometry in high school, but it didn't really click for me that this was a 'proof-based' course. At university, calc I with spivak was my first real proof based course. I was an immature student and basically flunked hard, but lesson learned!
     
  10. Nov 8, 2012 #9
    Re: When did you first encounter "proof based" mathematics?

    i believe that is atypical. at my school only math and CS majors took proof based math. in physics you never touched proofs because math proofs are useless in physics. just a hint: there are indeed functions in Hilbert space that do not vanish at infinity. Are they valid wavefunctions? no of course not they're completely unphysical. But the math works... no here is when you ignore the math and say that simply can't exist.

    First time I ever saw an epsilon sign that wasn't a physical constant was 3 months ago when I started grad level mathematical physics. Hit me like a reinforced concrete wall.

    In other schools they don't teach it like this, but my current professor loves theoretical math, since that was the way he was taught... so...
     
  11. Nov 8, 2012 #10
    Re: When did you first encounter "proof based" mathematics?

    :confused: What is a function in Hilbet space? And what does it mean that it vanishes at infinity???
     
  12. Nov 8, 2012 #11
    Re: When did you first encounter "proof based" mathematics?

    sorry. Wavefunctions which are vectors in Hilbert space.

    However, vectors in Hilbert space need only be square integratable, and not necessarily have zero value at infinity such that psi(x), when lim(x->infinity) psi(x) =/= 0. Physically, we restrict attention to wavefunctions such that lim(x->infinity) psi(x) = 0. Straight off Griffith "Quantum Mechanics" pg. 14 footnote 12.

    Well that wasn't a good example of the uselessness of proofs to start off with...
     
  13. Nov 8, 2012 #12
    Re: When did you first encounter "proof based" mathematics?

    Oh, you're talking about [itex]L^2(\mathbb{R})[/itex] or something? Yeah, then it makes sense. But there are more Hilbert spaces than that :biggrin:
     
  14. Nov 8, 2012 #13
    Re: When did you first encounter "proof based" mathematics?

    yeah that was quite embarrassing to talk about the uselessness of proofs and make that mistake =)

    guess I may have to re-evaluate my position about their usefulness but their difficulty is not affected.
     
  15. Nov 8, 2012 #14
    Re: When did you first encounter "proof based" mathematics?

    Hmm, first encounter with proofs was middle school geometry, but my first encounter with a proof based class is Linear Algebra 1 that i'm taking at the minute.

    It's actually great. It's my first real encounter with mathematics. The lectures are extremely heavy on proofs. Only 1 or 2 corollarys are left as "given" EVERYTHING else is proven and it's great I love it. But the exams are 100% computation so there pretty easy LOL.

    I kinda see it as the best of both worlds.. :)
     
  16. Nov 8, 2012 #15
    Re: When did you first encounter "proof based" mathematics?

    My first proofs were induction proofs during my first year in high school, then direct, contrapositive and ad absurdum proofs came during second year. Looking back, there were quite a few proofs in my high school textbooks, although they were not explicitly announced as proofs. My university classes were all definition-theorem-proof from the get-go.
     
  17. Nov 8, 2012 #16
    Re: When did you first encounter "proof based" mathematics?

    There are various types of proofs in proof theory http://en.wikipedia.org/wiki/Proof_theory#Kinds_of_proof_calculi

    There is a theorem prover using set theory http://en.wikipedia.org/wiki/Zermelo–Fraenkel_set_theory called Metamath. They prove a wide variety of theorems just using set theory.
    eg proof of triangle inequality http://us.metamath.org/mpegif/abstrii.html

    You can even use proofs to construct programing languages. Prolog is based on a type of Horn-clause logic and running a Prolog program is equivalent to generating a proof
    http://en.wikipedia.org/wiki/Prolog
     
  18. Nov 8, 2012 #17
    Spivak for calc I? Wow.

    To the OP: The way proofs are presented in most US high school courses (if presented at all) makes them seem just as banal as straight algebraic manipulation. At least in my experience. . .
     
  19. Nov 8, 2012 #18
    Re: When did you first encounter "proof based" mathematics?

    I remember proofs as far back as geometry and precalculus/trig before university, I don't know remember when exactly, but I'm pretty sure I had precalc/trig in 9th or 10th grade, since I had calculus in 11th (not a US student).

    My university encounters with calculus (also Spivak/Apostol and later Marsden) and linear algebra in my first year of university also did not skimp on the proofs. I don't remember proofs in high school calculus.
     
  20. Nov 8, 2012 #19

    WannabeNewton

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    Re: When did you first encounter "proof based" mathematics?

    This is quite normal if one takes an honors calculus sequence. The usual suspects are either Spivak or Apostol if the course is any good.
     
  21. Nov 8, 2012 #20
    Re: When did you first encounter "proof based" mathematics?

    I also learned my first proofs in high school, first geometrical like proving that two triangles were congruent, but i think my first "serious" proof was that sqrt(2) is irrational.

    I agree that many theoretical math proofs are rather useless for physics and you simply don't have the time to learn the same background for all the math you use as a mathematics student does. But i believe one of the basic things about becoming a physicist and a scientist in general is being critical and question everything. And it's definately useful to know were basic math as l'hôpital's rule or the chain rule come from, knowing why this stuff works so why you can appy it.
     
  22. Nov 8, 2012 #21
    Re: When did you first encounter "proof based" mathematics?


    Well, I don't find proofs useless... I believe that the things you learn in physics make a lot more sense if you know proof based math.

    For example, take Laplace's equation, I'm sure that the first method that comes to your mind is separation of variables, but why can you make the assumption that the solution is a product with the variables separated? How do you know it will work? A lot of physicists just scratch their heads and say "well who cares! it works!", but if you knew the math, you'd know that it has to do with the symmetries of the Laplacian, the method actually makes sense!

    Maybe proofs are useless when calculating stuff, but they make things a lot clearer (at least for me)...
     
  23. Nov 8, 2012 #22
    Re: When did you first encounter "proof based" mathematics?

    I've read that there is a way of doing the standard school geometry problems WITHOUT the standard proof method. To make it more intuitive for kids and stop them hating maths.

    Anybody know anything about this?
     
  24. Nov 8, 2012 #23

    WannabeNewton

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    Re: When did you first encounter "proof based" mathematics?

    I hope to god this isn't an actual thing. The last thing we need is a dumbing down of the already sub - par average high school math curriculum. i agree the way they teach proofs in those geometry classes is atrocious however.
     
  25. Nov 8, 2012 #24
    Re: When did you first encounter "proof based" mathematics?

    So, what do you propose? Just giving out the statements and let the kids memorize that? Yeah, that way they're gonna stop hating math.

    While I agree that high school geometry classes have a lot of problems and could be improved a lot, I don't think proofs is something we want to eliminate. What should be eliminated are stupid things like two-column proofs and memorizing definitions of obvious and useless terms (seriously: high school geometry books seem to get high from defining useless terms that nobody really cares about).


    Furthermore, geometry is a field with a very rich history. But this rarely gets told in the classroom. One can use geometry to make a link to so many exciting subjects: for example, when I was in geometry, we learned as an axiom that through every point there exists a unique line parallel to a given line. I find it a shame that they didn't pick up a sphere and show that this wasn't true there. They also could have made connections to modern physics and they could have told us that our space is not euclidean. All of these things would make geometry class so much more exciting. But no: my entire geometry class was just a collection of dry facts nobody really cared about.
     
  26. Nov 8, 2012 #25
    Re: When did you first encounter "proof based" mathematics?

    I cant find the web page that I was looking for but here is a book with similar ideas:

    https://www.amazon.com/Proofs-without-Words-Exercises-Classroom/dp/0883857006
    "Proofs without Words: Exercises in Visual Thinking (Classroom Resource"Materials) "

    I have education & math degrees & some kids are visual learners. I've marked many high school math papers where kids "just dont get it", and the job of educators is top help the worst as well as best students. If alternative strategies are needed then so be it.
     
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