When did you first encounter proof based mathematics?

[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.

 

[Patrick Kale]

I first encountered "proof-based mathematics" in high school in a Geometry class. The work could have been more engaging, although, and I think this would have been good for me and the other students too. I plan on learning more math with you all and on this forum. Best Pokemon, your second to last post doesn't seem appropriate but perhaps I have misunderstood your writing.

 

[cochemuacos]

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.

Which uni do you attend?

 

[Best Pokemon]

I first encountered "proof-based mathematics" in high school in a Geometry class. The work could have been more engaging, although, and I think this would have been good for me and the other students too. I plan on learning more math with you all and on this forum. Best Pokemon, your second to last post doesn't seem appropriate but perhaps I have misunderstood your writing.

I'm sorry if I sounded that way. I was just saying that adding physics wouldn't make it more interesting.

 

[MathINTJ]

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)?

What college/major was this person? This is an outright lie. I did many proofs in calculus.

 

[symbolipoint]

What college/major was this person? This is an outright lie. I did many proofs in calculus.

Experiences vary, depending on district, depending on era, depending on the educational fashion of the time and place. Schools (secondary level, or college level) may have courses designed for different student levels and for different major field emphases.

 

[Evo]

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

For girls?

 

[atyy]

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.

I somehow never learned proofs in school in Sinagpore. We learned a bunch of silly rules which worked, like "opposite angles are equal". So I am completely unable to do rigourous American high school geometry, although I can do the physics just fine.

I have no real idea what a proof is. There are only two proofs I have studied. One was Shannon's noisy channel theorem, and the other was Goedel's incompleteness theorem in Hofstadter's book. I read them because they didn't seem intuitive to me, whereas I was able to naively "buy" all the other "maths" I've needed.

 

[PAllen]

For me, first exposure was high school geometry, as for others in US. They were actually experimenting with new curriculum, so the course was probably 90% proofs.

Then, in college, it was linear algebra, then Real Analysis using Dieudonne book.

Funnny, but at least at this level (rather than research math), I found proofs very easy.

 

[alissca123]

Which uni do you attend?

UNAM. In Mexico.
http://en.wikipedia.org/wiki/National_Autonomous_University_of_Mexico

 

[lingualatina]

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.

I'm a freshman taking a geometry class right now, and I couldn't agree with you more! I really enjoy seeing the general theory behind math rather than just crunching numbers all day. Exponents were my nemesis in Algebra I.

 

[cochemuacos]

UNAM. In Mexico.
http://en.wikipedia.org/wiki/National_Autonomous_University_of_Mexico

Lol i was suspecting that.

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.

This just seemed too familiar. Facultad de Ciencias FTW!