When did you first encounter proof based mathematics?

[Best Pokemon]

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)?

 

[micromass]

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 high school, 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.

 

[symbolipoint]

Geometry in high school. Mostly proof-based, very difficult.

 

[Klungo]

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 high schools 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).

 

[homeomorphic]

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.

 

[alissca123]

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.

 

[phinds]

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.

 

[MarneMath]

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!

 

[chill_factor]

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.

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...

 

[micromass]

there are indeed functions in Hilbert space that do not vanish at infinity. Are they valid wavefunctions?

What is a function in Hilbet space? And what does it mean that it vanishes at infinity?

 

[chill_factor]

What is a function in Hilbet space? And what does it mean that it vanishes at infinity?

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...

 

[micromass]

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...

Oh, you're talking about $L^2(\mathbb{R})$ or something? Yeah, then it makes sense. But there are more Hilbert spaces than that

 

[chill_factor]

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.

 

[Darth Frodo]

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.. :)

 

[espen180]

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.

 

[Devils]

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

 

[FalseVaccum89]

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!

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. . .

 

[Lavabug]

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.

 

[WannabeNewton]

Spivak for calc I? Wow.

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.

 

[thephystudent]

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 definitely 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.

 

[alissca123]

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...

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)...

 

[Devils]

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?

 

[WannabeNewton]

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?

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.

 

[micromass]

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?

So, what do you propose? Just giving out the statements and let the kids memorize that? Yeah, that way they're going to 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.

 

[Devils]

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

I can't find the web page that I was looking for but here is a book with similar ideas:

https://www.amazon.com/dp/0883857006/?tag=pfamazon01-20
"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 don't 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.

 

[moouers]

Like others, my first experience with proof-based mathematics was a geometry course during my freshman year of high school. I wasn't very good at it then, because I didn't put a lot of effort into learning the material needed to formulate a proof.

 

[moouers]

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.

That is, word for word, exactly what my high school geometry teacher demonstrated for us. I was quite blessed to have him as my teacher for four years.

 

[pk1234]

First year at university now, the whole 1st semester is full of proofs in both real analysis and linear algebra. Quite different from high school - the only proofs I remember from there are things like "square root of 2 is an irrational number."

Surprisingly, I find the real analysis proofs easier compared to the linear algebra ones, unlike most of the other students - while the the proofs in linear algebra are simple and obvious, the technical part of writing them down with all the various indexes makes it very confusing... It's weird to me that writing down some linear algebra proof can be so long and tedious, when the "idea" behind the proof is so straight forward.

Might be because of the more "systematic" nature of the subject? I don't know.

 

[Tsunoyukami]

I first encountered proofs in Grade 11 (I am Canadian and unsure of what this translates to in terms of a US equivalence). The only proofs we did were basic trigonometric proofs.

However, when I entered my first of university I was overwhelmed with the number of proofs in my first-year calculus class. I felt as if I "understood" the proofs (in that I could read one and understand what was happening and was able to follow the logic) but I was unable to prove things myself. The class assumed you were already familiar with proofs and I did poorly in it.

Since then the math classes I have taken have been at an applied level as opposed to a theoretical level for the most part.


I feel as if most high schools do a poor job of preparing students for rigourous proof based mathematics.

 

[Robert1986]

This thread is interesting to me. I am a US student and I've lived here my whole life. I took Calc. I, II, and III as well as linear algebra before encountering what I would consider a real "proof-based" class. Sure, there were proofs mentioned in the calc classes and we did some "baby proofs" in linear algebra (prove this thing isn't or is a subspace, etc). My first real proof-based class was probably the second semester of my second year in college.

I find it interesting that people do proofs in high school (though, I wouldn't call most geometry classes 'proof-based') or as freshmen in college. Those of you who have done that, are you in school in the US? or somewhere else?

 

[homeomorphic]

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?

An example that is somewhat along these lines is Curves and Surfaces: A Practical Geometry Handbook. It does have proofs, but they are somewhat non-rigorous at times. The point is to get a lot of intuition about why things are true, rather than dotting all the i's and crossing all the t's. In particular, the book discusses how. I would tend to suspect that it would work best for "gifted" children, but a really good teach might be able to convey it to a wider audience.

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

I think it makes sense to just introduce kids to the intuition of why things are true first, before doing formal proofs. That will help some of them stop hating math. Non-rigorous proofs are the core of math--those are the things you can actually take away and remember and gain insight from. The formal ones are just to check it all and make sure nothing goes wrong. The formal side is important, too, but it can't take the place of deep understanding.

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).

The proofs aren't the problem. The problem is emphasizing formal proofs over seeing satisfying reasons why things should be true intuitively. But, I think if you started with non-rigorous proofs, that would come across better.

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.

Those things are interesting, but personally, I think ruler and compass constructions are really fun. If I were to teach geometry, I might do it like Euclid does--doing geometric constructions and showing why they work, but a little bit less formally. These kinds of geometric constructions might arise in surveying or drafting, so you can see the usefulness and the beauty of it together.

 

[homeomorphic]

I have education & math degrees & some kids are visual learners. I've marked many high school math papers where kids "just don't 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.

I don't think that visual learners should be considered the "worst" students. They are really the best students. In fact, the main reason I find so much math to be "dumbed down" is because it is targeted for people who have difficulty with visualization. If visual learners seem like they have trouble, it's the math that is wrong and the students who are right. The fact of the matter is that it is often, but not always the case that ONLY the visual thinkers are capable of a really deep understanding because there are many cases where if you can't visualize it, you don't really understand it. Of course, there are some visual learners who aren't that good at math, and some people who excel at non-visual aspects at math who aren't very visual thinkers, so I'm not saying it's an absolute rule that visual thinkers will be "better" students, just that visual thinking is sometimes necessary for a deep understanding of certain topics. Also, it is possible to try to visualize too much. People who aren't that good at visual thinking might think that it's a handicap, and sometimes it can be in a kind of screwed-up way, but it's only because of the WAY the material is presented, rather than something inherent in the material. The visual thinkers will outperform everyone else, given proper explanations of things.

 

[pk1234]

This thread is interesting to me. I am a US student and I've lived here my whole life. I took Calc. I, II, and III as well as linear algebra before encountering what I would consider a real "proof-based" class. Sure, there were proofs mentioned in the calc classes and we did some "baby proofs" in linear algebra (prove this thing isn't or is a subspace, etc). My first real proof-based class was probably the second semester of my second year in college.

I find it interesting that people do proofs in high school (though, I wouldn't call most geometry classes 'proof-based') or as freshmen in college. Those of you who have done that, are you in school in the US? or somewhere else?

I'm in Europe. It's not like at my university everywhere, but generally across Europe, I think real analysis is compulsory for math students in first semester, if the school has any reputation. If you're a financial math student/informatics/physics however, the proofs aren't emphasized as much, with the exception of financial math, where calculus is taught instead of analysis, and the amount of proofs there really is minimal - there's some "important" proofs taught in the first semester (from around arithmetic of limits to l'hopital's rule, but a lot of things are skipped), but after that it's really just counting problems.

I think it's a bit of a tradition really. The attitude of all the professors (not just the analysis professors) is basically "analysis and linear algebra are basic subjects of higher math, upon which advanced math is based, and as such they should be taught as early as possible."

 

[Best Pokemon]

So, what do you propose? Just giving out the statements and let the kids memorize that? Yeah, that way they're going to 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.

Physics schmysics! That would only make it worse, especially for girls.

 

[Best Pokemon]

The last two years in high school I learned topics such as algebra, functions, advanced trigonometry and trigonometric identities, vectors, calculus, sequences, series, approximations, counting (permutations and combinations), matrices and complex numbers. In most of these we had to do proofs. The main proof methods we used were direct proof, proof by contradiction and proof by induction.