How did you learn Mathematics?

[Greg White]

Bit hard to say. Mostly just learned from paying attention in class until halfway through high school and then just started working through how the textbooks said to do it. Cut it a bit close in Geometry because for the life of me I couldn't learn what any of the proofs were supposed to be for. Numbers I understand, but remembering which mathematical law makes 2+2=4 is just pointless.

All told, high-end math is pretty pointless unless you plan on going into a field(engineering, math, or physics since the other sciences are horrible at math). Thankfully I stick with computer networking, so basic figuring or just remembering your patterns is more than enough.

 

[Weaver]

I went to University in the math faculty for 5 years and they made me learn math.

 

[Truman Soutar]

Easy, I just researched The Wheel first then it was full steam ahead to the Classical Era =3


Seriously though I had a math teacher in junior high who I absolutely loathed, and I'm at least fractionally sure that he didn't like me either, not that I gave him any reason to. Somehow I learned quite a lot more from him than any other math teachers I was taught by before or since. That year one of the students in our class aced the final and did quite well on a couple provincial standard exams I think. Basically what I'm getting at is he was a real good teacher; he retired after that year so I guess I actually lucked out.

Also he had this policy of not letting anyone leave until everyone had asked at least one question about something that was covered that day, which I think was a pretty smart idea.

 

[CrazyCapnMorgan]

Was just naturally good at it. That, and I loved to count all of the numerical statistics in my first Final Fantasy strategy guide.

Oh, and I also studied a little bit of Pokemon math way back when. Especially as it concerned shiny Pokemon. That helped a bit.

 

[mrdude2010]

I always did really well in math without trying up through about 9th grade. Even then, as long as I did the homework, I tended to get B's in the advanced/AP math classes my HS offered. Starting from multivariable calc on, I had to study hard and all that to do well in math.

What made me get miles better at math was practicing it regularly in high level physics courses. If you stare at enough tensors and ODE's, you'll eventually get pretty good at using/solving them.

I was lucky to, in general, have pretty engaging and helpful math teachers. The only rough spot was a semester where I had a recent Asian immigrant who had a really thick accent and was obviously there to do research rather than teach.

 

[kurokotetsu]

Hi all

I'm currently writing a book about the teaching and learning of mathematics and I'm interested in hearing any anecdotes you may have about your experiences. Did you have particularly good or bad teachers? At what age did you decide whether or not you enjoyed the subject? Any particular moment at which you suddenly discovered you didn't understand a word the teacher was saying?

Also, I'm interested in specific country-by-country information. In particular, one question interests me, which is that in Britain, young children tend to be taught by a single teacher each year, who is expected to teach all the required subjects, including Maths, Science and English. Is that the case where you are, or did you have subject specialists at your age?

Anything else you may have to say on the subject would be of interest too. For example, what does the word mathematics mean to you? What would you say a mathematician studies? What makes a good mathematician? What makes a good maths teacher?

Thanks for any thoughts you may have!

Hmmmm, terrible idea I have to say. Teaching and learning, anything really, really changes with age. It is not the same to try to teach children than high school students, and the content and necessary abstractions are completely different.

For bad teachers, well, a high school teacher just wrote some problems in the board and then started gossiping, so that year was terrible. He didn't even know how to handle inequalities when multiplying by a negative number. A waste of a year. For best, well college or my high school teacher who was a statistician, all who really understand math and love it.

math has always been easy to me, at least until real math, which is challenging but fun. What a mathematician does is kind of a game. There are rules (your axioms) and a world to explore, following rules and finding new ways the rules are made and bent. A puzzle where you have to find a completely unknown picture. Or create models of the world and see their limits. A good mathematician must be thorough, imaginative and good at logic and abstraction, for that is what you use to make a theorem work.

I was in a half-British, half-Mexican school. During elementary school that meant two teachers, one in English the other in Spanish, doing both,s o each teacher while having to do several subjects they didn't have to do them all. Math in particular was shared. In the last year of elementary school I changed to specialized teachers per subject. In Math my parents where very useful too, as one was a mathematician and the other a physicist and CS Master, so while they didn't have to help me with problems they talked to me about math, even me showing me problem like Königsberg Bridges and so.

My calculus teacher made me love it, at least until it got more complicated. It wasn't until my real analysis class that I realized that even mathematics is not immune to subjectivity, gray areas, and controversy. I believe the axiom of choice is the hot topic these days, although I've always been intrigued by zero factorial.

I would like to understand more how the Taylor series was derived, and perhaps by doing so I can figure out how Euler solved the Basel problem. Wikipedia is a start, but it's tough to work with. Still, I don't have the heart for mathematics that those guys did, so I'll probably never fully understand their techniques. There is no substitute for desire. Desire is much more powerful than mental capability, even though I'm lacking in that area as well.

0!=1, that is it. Nothing more. As a matter of fact Donald Knuth's Concrete Math does a very good job of expanding the factorials and why it works that way.

The axiom of choice though is fascinating and very hot. As it is unprovable, it has to be taken as an axiom, but while very useful it present unintuitive results if kept, like the possible good order of the real numbers. But the usefulness means that most mathematicians I know still o for it, rather than reject it.

Don't go to wikipedia. A book like the Spivak or Apostol of Calculus should, if I remember correctly, have proofs for taylor and maybe a more in depth look at Basel Problem. But the later is just choosing point of expansion where the series is zero in the Taylor series.

I've been reading up on this aspect of the philosophy of maths myself recently and I was surprised how much it's debated. As far as I'm concerned, one of the main joys of Maths is that it's the one subject where everything you learn is true. But people seem to get very bothered by this notion and start worrying about questions like whether it's based on cultural conventions, or whether we can prove the principles we use to prove things, and suchlike philosophical nonsense.

Obviously it's perfectly possible to set up alternative mathematical realities where 2 + 2 =/= 4, just as we already deal with mathematical worlds where there are no parallel lines or where negative numbers have a square root. Maths isn't about whether these things are 'real', it's about examining them and working out how they behave.

It is not only that. While the question of discovery versus invention is huge in Philosophy of Mathematics, there a not so philosophical debate related. There are controversies of proofs, like the Four Colour Theorem proof, suing computers and if ti is a valid proof. There is the Axiom of Choice and transfinite numbers, which are undecidable and whether they should be accepted. The true nature of Math exists, but what rules do we play by and what can be derived is problematic. We can thank Gödel for that.

Also teh 2+2=/=$ has been done to death. Really people modular algebras are a thing. Hell, they can be used in cryptography. That is old news.

 

[DanielBrown]

I'm from Sweden and the way we're taught here has already been covered, so I can skip that!
I've only had one math teacher who actually was able to teach me things. Disliked math ever since I was six or seven years old when we started learning the multiplication table. Around the time when we were taught Ma B(algebra and the likes) I lost my shit and started failing every single test. I couldn't make the graphs right even if I followed the textbook step by step.

Mathematics beyond the basics feels to me like unecessary learning. It has some uses based on your job, but personally I've never had any use for it yet. A good math teacher is one who explains well and has time to help his students. That's what I had with my one good teacher. With the others the classes has always been too big(average of 20-30/class).
Can't really blame it on the teachers though. The core is my problems is most certainly my lack of intrest.

 

[Scarim Coral]

Sometime in Highschool I had a good Math teacher as in she was fun to talked too but I thnik it's cos of that I dropped a grade down resulting me being put with the other Math Teacher.

He looked like Bill Clinton but without the fun aspect of the former president and god he was a bore at teacher Math. He petty much treat it as a University Lecture (I kid you not, there was one time he spend the entire lesson on Math lecture that we didn't do any Math excersie of any sort)! He was also somewhat intimidating because he was also the deputy Head teacher but in saying so it could of been worse (there was another Math teacher who was very strict to non-sixth former pupils).

While I think I wasn't good at Math but by the end of it (GCSE) I did get an C (which I was content with) but that was thanks to my revision time on it but I supposed I have to aknowledge that teacher to some degree (he did called me and my mate mathematicians to this other education authority person during this ceremony).

 

[Flatfrog]

I've always been bad at math, but the best help I had was in a game (in a foreign language no less), where I could take my time and just figure things out by myself. It put me a bit ahead of the class and gave me a confidence boost.
As soon as it went past basic math though, I lost interest completely and concentrated on the things I found important or was naturally good at.

If you don't have examples of practical use for the knowledge you're presenting, you're going to lose a lot of kids. I feel that's the most important lesson. If kids, teenagers and whatnot have no fucking clue of why you're teaching them this, then they'll lose interest and motivation.

Mathematics beyond the basics feels to me like unecessary learning. It has some uses based on your job, but personally I've never had any use for it yet.

This is one of the questions that interests me greatly, and has been a key part of the debate on best practices on Maths teaching for a long time. What it boils down to, essentially, is 'What is maths for?'

There are two main schools of thought. In essence, the first is that we should learn Maths because it's beautiful while the other is that we should learn it because it's useful. I've always been a bit suspicious of both approaches, but for me the second is the most dangerous because of exactly the point you both made, which is that if we're focusing on the 'useful' bits of Maths, we're going to run out rather quickly for most people. It's relatively hard to find *any* individual part of Maths that is genuinely important to most people's day-to-day lives. Even the arithmetic stuff, which was vital a century ago, is relatively unnecessary now we have computers.

So teachers and examiners have to resort to coming up with questions where Ahmed buys seven calculators for £12.56 and wants to know how much he'll have to pay for seventeen calculators, which *pretend* to be about real-life situations but any child can tell that's bullshit.

My opinion is that we should learn Maths because it is useful, but not in any individual part. Maths is important as a *way of thinking* - it's about spotting patterns, extrapolating by analogy to bigger answers, finding algorithms to solve problems, and many other skills which genuinely are useful every day and in a million different contexts. So while very few people will ever need to solve a quadratic equation, the ability to combine two pieces of information together and use them to work out the solution to a problem in a logical way is surely useful to everyone.

 

[odolwa99]

My time at secondary school was 'the' single worst experience of my life and, sadly, I learned very little.
More recently, however, I have gone through the entire course, bar the advanced mathematics end of things which I'm leaving for another day. By going to math forums, pacing myself and asking all the questions that 'I' need answered without fear of being bullied for being a swot, I am now actually quite angry as I have never had a better understanding of the course than I do now.

I mean this - without any hyperbole - I am convinced that higher level secondary school mathematics can be taught to 'anyone' in as little as 6 months, providing the help you get is of a good standard. You don't need to spend 'years' learning this stuff, and anyone who says otherwise is wasting your time.

Needless to say, I am f**king livid at how badly my education has gone, more so now given that I actually see how straightforward a lot of the material actually is.

The following is worth noting, as it 'will' affect your chances:
1) The classrooms have too many people. Too many people means too many questions and not enough time to help everyone.
2) A hostile environment. Ask a question, teacher makes you look like an idiot or you get bullied.
3) Private tuition. A damning indictment that schools can't teach, so now we're paying someone to do the job properly. What's that? You're family is struggling financially week by week and can't afford private tuition? Well then, you can go f**k yourself.
4) You should expect to be left behind (i.e. willfully ignored) if you aren't naturally gifted at rote memorization, securing a decent grade and arse licking.
5) Grind school. See points 1), 2), 3) & 4)
6) No-one in your family has attended university, meaning you don't have well educated family members to help you out.
7) Contacts. Because hey, as we all know, it's not what you know it's who you know. Mommy/ Daddy is an engineer/ CEO/ mathematician? Well doesn't matter if you get a 3 at university, they'll set you up for life. Yaay!! Education works!! xD

Conclusion: The education system needs to go back to school as it clearly doesn't work as well as it should. Multiple studies show time and time again, if you have money, you will go far. The rich get richer, the poor get poorer.

Rant done.

TL;DR: Don't rely on school for your maths education. Do it yourself!!!

 

[Mersadeon]

[...]
Also, I'm interested in specific country-by-country information. In particular, one question interests me, which is that in Britain, young children tend to be taught by a single teacher each year, who is expected to teach all the required subjects, including Maths, Science and English. Is that the case where you are, or did you have subject specialists at your age?
[...]

That is similar to how it is done in most primary schools in Germany (at least when I was there). We had one teacher for everything except stuff that one person can't teach - so swimming (graded as a subset of sport/P.E.) was taught by two different teachers (since you always have to have two teachers during swimming). The turkish children also had a teacher for turkish, and that was the only male teacher in the school.
Other than that, everything was taught by the same teacher.

Well, I don't have many memories of learning math for myself, so here is the one that I have: I learned the "Einmal Eins" during long car rides, during which my mother would quiz me. The Einmal Eins is essentially just all multiplications from 1*1 till 10*10. Problem is, you kinda are expected to not just do them in your head, you're supposed to know them by hard. So that's why I did that.

Other than that, I can only tell stories about the kids I teach (as a private tutor).
One girl I had learned fractions for the first time, and it blew her mind. I eventually explained it with imaginary pieces of cake. I get the feeling that talking about cake helps her with pretty much any topic.

As to specific teachers and things they did wrong: oh, boy.
My math teacher from 5-7th grade was awful. She only taught through extreme repetition with the help of boatloads of worksheets. In the end, it stopped us from doing any of them - you got a bad grade if you even forgot ONE, so why even bother?
After that was done, I was put in the advanced math course (advanced courses in the schoolform I was in are needed to get any further than 10th grade, but german schoolforms are complicated) and had the hardest math teacher in school. Hard, but... actually, not fair at all. He essentially takes the best five people in the class and ignores everyone else. Luckily, I was one of those five, so I had a blast. We blazed through topics at about double the rate the other advanced course did. It was awesome. And he was a dick, in all regards. A very uncomfortable man - but I always felt that a bit of praise from him was worth more than the best grade.
When I started my "Abitur" (11-13th grade), I had a teacher that just didn't give a shit. He only taught those that really, really wanted to AND was talented in it (which was exactly one person), everyone else just got a barely passing grade. I was grateful for that, because mathematics at that point become very... abstract.

As a tutor (and mostly math at that), I have to fix the mistakes of teachers all the time. Teachers who treat their students like university students, expecting them to understand abstract concepts and connect them without help and without introducing them to the methods needed.

But probably the worst one pertains to the "cake" story I had above, and it shows something that is (in my humble opinion) one of the biggest problems in teaching: they act like everything they say, even the extreme oversimplifications are true. In this case, my student was taught that THERE ARE ONLY NATURAL NUMBERS. When she asked if you could divide five by three, she was told it was impossible.
Look, lady, just because they aren't ready to learn about it, reinforcing the idea that everything they can't do yet is impossible isn't the best way to go.
So you come to a point where students have problems understanding new concepts because those very concepts were impossible just last lesson. Those teacher set up rules (with good intent, sure) and them break them. She needed a lot of time to understand that yes, you can have numbers between one and two.
This is also a problem in chemistry, where students essentially walk in the footsteps of science - they start with really old (and thus, simplified) models of atoms and reactions and then essentially follow history with ever more complex versions of the systems they already learned. Problem is, the teacher treats the version they are at as if it is the absolute truth. Students then get taught a new one that pretty much invalidates the previous one, often without explanation as to how we got to that conclusion, because a lot of chemistry teachers don't care about the history of chemistry or teaching the scientific method.

 

[Flatfrog]

This is also a problem in chemistry, where students essentially walk in the footsteps of science - they start with really old (and thus, simplified) models of atoms and reactions and then essentially follow history with ever more complex versions of the systems they already learned. Problem is, the teacher treats the version they are at as if it is the absolute truth. Students then get taught a new one that pretty much invalidates the previous one, often without explanation as to how we got to that conclusion, because a lot of chemistry teachers don't care about the history of chemistry or teaching the scientific method.

My solution to this problem when I'm teaching science is to have a little sign which I am always ready to hold up, which reads 'Actually, it's a bit more complicated than this'.

 

[Mersadeon]

This is also a problem in chemistry, where students essentially walk in the footsteps of science - they start with really old (and thus, simplified) models of atoms and reactions and then essentially follow history with ever more complex versions of the systems they already learned. Problem is, the teacher treats the version they are at as if it is the absolute truth. Students then get taught a new one that pretty much invalidates the previous one, often without explanation as to how we got to that conclusion, because a lot of chemistry teachers don't care about the history of chemistry or teaching the scientific method.

My solution to this problem when I'm teaching science is to have a little sign which I am always ready to hold up, which reads 'Actually, it's a bit more complicated than this'.

I hope you didn't trademark that, BECAUSE I'M STEALING THAT IDEA! ^____^

 

[Blow_Pop]

Southern Californian here.

I haven't actually had more than very basic mathematics stick with me. Part of that has been godawful teachers. I literally had a teacher 2 years in a row that would assign us a chapter and then tell us we had to learn it ourselves because her college homework was more important than teaching us. The first year that happened, the baseball coach actually helped me pass the class. The second year I had a lot of help from my neighbour two doors down, in fact he helped me pass all my maths courses in high school. Geometry I didn't start understanding angles and how they worked until I started playing pool funny enough. Algebra I still don't understand at all and anything higher level goes straight over my head. I've tried but I honestly don't understand it. Nor do I understand electrical systems even though I've been trying to learn to make myself more employable and marketable in the automotive field. But I've also had quite a lot of cashiering type positions to do without registers that have fully put into my head the basics. I'll understand it when it's being explained to me but once you try and have me work out problems and add in letters and irrational numbers my brain shuts down.

Our schooling was that in elementary(or grade)school(depending on where you're from in the US it's called different things)we had one teacher to teach all the subjects. That lasted until 6th grade which was approximately 11 years old. 11-13 years of age was middle school where we started having different teachers for different subjects and that continues into high school from age 14-18 approximately(depending on when your birthday fell during the year).