Completely new to Mathematics - I am an anxious learner

[conure]

Hello all,

A brief history is that as a child (I'm only 26 now) I had performed poorly at Mathematics and it wrecked my confidence. I had no interest in long division, basic algebra etc and focused elsewhere because I never viewed it as a means to an end. Suffice to say I suffered a bit of Math anxiety.

I decided to overcome it about six weeks ago and have been learning at quite a steady pace (as I am returning to education to pursue a degree in Software Engineering early next year, which will require a good level of Mathematical knowledge I am sure - though luckily the first half of year one is an intro to Mathematics!).

My first question is, though I am still very much a beginner, sometimes when I am shown a rule (i.e Prime Factorisation) I can understand how to do it, how to interpret the results...but...I can't always get my head around why it works. I struggle to visualize the way the sum is working. For some things I can, for some I can't. Maybe with time it will come but, it seems like there are a number of facts in Mathematics that just work (i.e A squared + b squared = c squared) - it works, I don't have a complete grasp on why it works, but I know it does...

Will this stop me from ever becoming proficient at Mathematics? That would be a shame as I very much enjoy it! I am not asking if I'll ever be a top Physics researcher or mathematician because I know that will be beyond me, but will I be able to progress to a relatively high level, at least far enough to be a good software engineer and perhaps appreciate some scientific equations...I look forward to any encouragement or advice :)

Sorry for the long post but this stuff has been bothering me for a while!

 

[micromass]

You might be surprised to hear this reaction, but I think you are already quite mathematically advanced if you ask questions like these.

Your statement:

it seems like there are a number of facts in Mathematics that just work (i.e A squared + b squared = c squared)

tells me enough. They have been teaching math the wrong way. Math is far from a collection of facts that work. That is how high schools make it appear, but it's far from the truth. Mathematics is all about why things work. Mathematicians are interested in prime factorization and Pythagoras' theorem to find out why they work the way they do.

You seem to me that you're not anxious of mathematics, but you seem anxious of a mindless collection of useless facts (= high school mathematics).

I remember, when I was still very young. I was thought prime factorization. And I asked the teacher: "But how do you know prime factorization always works". The teacher responded: "Do you know a number for which it doesn't work?" Only recently, I found out that the teacher said that because he didn't know the answer himself. This is sadly what happens in a lot of math classes. The teachers don't know math themselves, so they teach it badly. The textbooks don't think the students can handle math, so they dumb it down. It leaves students frustrated and anxious.

Read this to see what kind of sham the high school math classes are: www.maa.org/devlin/lockhartslament.pdf

 

[homeomorphic]

it seems like there are a number of facts in Mathematics that just work (i.e A squared + b squared = c squared) - it works, I don't have a complete grasp on why it works, but I know it does...

It's hard for me to convey the proof here, but I felt the same way at some point many years ago and came up with a proof. My first proof wasn't very satisfying. However, I eventually came up with one (which I later found out had been discovered by some Indian mathematician hundreds of years ago) where you just have a big square and 4 triangles that you can make out of construction paper (or imaginary construction paper). You just move the 4 triangles between 2 different positions and the Pythagorean theorem then becomes obvious. I had to be clever to come up with this trick, but I taught it to a bunch of non-math grad students once and they thought it was pretty cool. Because many people don't really understand what an enlightening proof of something should look like, they may be able to prove things, but their proofs just establish the truth of something without actually giving you an idea of why they are true. Sometimes, this is a result of people just blindly copying proofs, while forgetting the inspiration behind them.

Anyway, a lot of it is finding the right proof. Not all proofs are equal as far as understanding goes.

 

[nickadams]

However, I eventually came up with one (which I later found out had been discovered by some Indian mathematician hundreds of years ago) where you just have a big square and 4 triangles that you can make out of construction paper (or imaginary construction paper). You just move the 4 triangles between 2 different positions and the Pythagorean theorem then becomes obvious.

Can you please share the proof?

 

[homeomorphic]

Can you please share the proof?

I found a version of it here:

http://www.cut-the-knot.org/pythagoras/index.shtml

The one I came up with is proof number 9 on the list, there.

 

[Fredrik]

This is one of the pictures from proof #9 on the page that homeomorphic linked to:

Call the shortest side of the triangle a, the second shortest b, and the longest c. The first proof I learned goes like this: The area of the blue square is obviously c². But it's also equal to the area of the large square minus the area of the four yellow triangles. Each of those triangles has area ab/2. So we have
$$c^2=(a+b)^2-4\cdot\frac{ab}{2}=a^2+b^2+2ab-2ab=a^2+b^2.$$
There are also many proofs on the Wikipedia page about the pythagorean theorem. This one is very similar to homeomorphic's proof. It's beautiful and almost self-explanatory:

The symbols a,b,c denote the same things here as in the proof I just showed you. The idea is that since we're just moving two of the triangles, the area of the black region before the move (c²) is the same as the area of the black region after the move (a²+b²). The proof of that is that it can be calculated as ##(a+b)^2-4\cdot ab/2##, regardless of where the triangles are. Note that in this proof there's no need to do the algebra to simplify ##(a+b)^2-4\cdot ab/2## to ##a^2+b^2##.

 

[DrewD]

If you dedicate yourself to learning math, I'm sure you'll do well, but I have a warning. In the early days of math it was just a bunch of facts. Then the Greeks formalized it a bit, but that level of formalism wasn't really seen again for a long time. In general, the goal is to find the connections between these facts, but you need to be very patient and get used to using facts that you don't fully understand.

Now, you claim that you know [itex]A^2+B^2=C^2[/itex] but don't know why. Well, this is a fact (if the we assume that you are talking about a right triangle) only if you are in a flat Euclidean space. In fact, this fact depends on an axiom that does not necessarily have to be true (I might be wrong about this precisely, but I'm going to go forward because I think my point is valid)! For about 2000 years the parallel postulate was assumed and people tried to prove it. In the 1800's mathematicians realized that not only could it not be proved, but it didn't even need to be true.

So my point is that, while the connections are what matters, sometimes it is okay to just accept things while you keep learning. If Euclid had spent all of his time trying to prove the parallel postulate, we would not know his name. In software development, you will probably learn a lot of math without the proofs. Don't be bothered by that. Use it; be happy that somebody else figured it out; keep reading/researching until you figure out why, but it may take a long time.

PS
I've always loved math, but I sort of put it down for a while. The book Fermat's Last Theorem was one of the books that got me interested again by drawing connections throughout history. There may be better books, but I like this one. It put math progress in perspective for me.

 

[twofish-quant]

Will this stop me from ever becoming proficient at Mathematics? That would be a shame as I very much enjoy it! I am not asking if I'll ever be a top Physics researcher or mathematician because I know that will be beyond me, but will I be able to progress to a relatively high level, at least far enough to be a good software engineer and perhaps appreciate some scientific equations...I look forward to any encouragement or advice :)

One important thing here is that physicists and mathematicians use math in very different
ways, and there are a lot of different skills involved. One reason I ended up in physics rather than mathematics is that I "feel" equations. When you write down an equation, I try to associate it with an emotion and that let's me "feel" how the equation works intuititively. This works very well in physics research. I'm dreadful with proofs, since often I can "feel" how something works, but I can't write it down.

Conversely, something that works for me sometimes is *not to think about the equation*. For example, when learning group theory, what works for me is sometimes just to treat things like a crossword puzzle. Just think of it as symbols on a page, and then pretend I'm doing a word game.

Something else is that the level of math that you need to be a proficient physicist is something that "hard core" people consider kindergarten work. If you can get to the point where you are proficient with partial differential equations, that will get you in most areas of physics.

 

[twofish-quant]

Only recently, I found out that the teacher said that because he didn't know the answer himself. This is sadly what happens in a lot of math classes. The teachers don't know math themselves, so they teach it badly.

There is a reason for that. Mathematics classes are intended to teach some very basic skills to large numbers of people. There aren't enough skilled mathematicians to do this, so they rely on semi-skilled teachers to teach the classes.

Also one thing that you'll find in college is that a lot of people that know something really well can teach it badly. I've found that often genius mathematicians are terrible teachers because they don't understand how someone can't understand something that's obvious to them.

I was fortunate to have some excellent math teachers in high school. They were excellent because they just gave me some math books to read, and then left me alone.

The textbooks don't think the students can handle math, so they dumb it down. It leaves students frustrated and anxious.

The thing is that many, maybe most students *can't* handle math. If you try to push them then you'll have a confused student and a confused teacher, and they won't learn anything at all. Also the people that are in this forum probably *hate* the way that math is taught in high school. But there are people that *love* this. It's a personality thing.

Read this to see what kind of sham the high school math classes are: www.maa.org/devlin/lockhartslament.pdf

I think people that write things like this are being unfair. If you had infinite amounts of money and could hire people with infinite skill, then we'd teach math differently. Real world educational system have to deal with limited budgets and students with wildly different skill levels and motivations.

 

[homeomorphic]

One important thing here is that physicists and mathematicians use math in very different
ways, and there are a lot of different skills involved. One reason I ended up in physics rather than mathematics is that I "feel" equations. When you write down an equation, I try to associate it with an emotion and that let's me "feel" how the equation works intuititively. This works very well in physics research. I'm dreadful with proofs, since often I can "feel" how something works, but I can't write it down.

I suppose the difference for me is I feel it, and I can translate that feeling into a proof. Sometimes, it takes a lot of effort, though.

Only recently, I found out that the teacher said that because he didn't know the answer himself. This is sadly what happens in a lot of math classes. The teachers don't know math themselves, so they teach it badly.

There is a reason for that. Mathematics classes are intended to teach some very basic skills to large numbers of people. There aren't enough skilled mathematicians to do this, so they rely on semi-skilled teachers to teach the classes.

Also, most mathematicians find high school level teaching to be very uninteresting.

Also one thing that you'll find in college is that a lot of people that know something really well can teach it badly. I've found that often genius mathematicians are terrible teachers because they don't understand how someone can't understand something that's obvious to them.

I have great difficulty lecturing to students in a way that they understand, although I have improved a lot. When I talk to students one on one, it is fairly rare for me to have significant difficulty in getting through to them. Of course, it is hard to be aware of that at times, since students may go through the motions without actually understanding. Never the less, I can usually get students to get through all the problems they need help on if I am talking to them one on one. That is how I have learned to teach a lot of things. It is still hard for me to imagine finding some of these things difficult, but I can usually figure out how to get across to them even so with a little trial and error if I use reasonable methods of teaching (one on one, as opposed to lecture).

Teaching is not a one dimensional thing. You can be a great teacher in more than one way. Graduate students are thrilled by some of my lectures. I believe I am a great teacher from a mathematical point of view, but I struggle to be a decent teacher to most low-level students.

The thing is that many, maybe most students *can't* handle math. If you try to push them then you'll have a confused student and a confused teacher, and they won't learn anything at all. Also the people that are in this forum probably *hate* the way that math is taught in high school. But there are people that *love* this. It's a personality thing.

They can handle math more than you might imagine. The key is to push them only so much as they can handle. Get them just a little bit out of their comfort zone, I think. Very few people love the way math is taught in high school, and even if they love it, it might not be in their best interest. Some people are demonstrating that there is a much better way of doing things.

http://opinionator.blogs.nytimes.com/2011/04/18/a-better-way-to-teach-math/

By the way, JUMP math doesn't always insist on presenting math in a way that conforms 100% to the ideals of the people in this thread discussing it. I think it ultimately is in accord with our point of view, but it doesn't insist that students always need to understand why everything works from the outset. Some people attack it for not being conceptual enough, but I think, in the end, it IS conceptual. Of course, I only have a nodding familiarity with the stuff.

I think people that write things like this are being unfair. If you had infinite amounts of money and could hire people with infinite skill, then we'd teach math differently. Real world educational system have to deal with limited budgets and students with wildly different skill levels and motivations.

I think you misunderstand what Lockheart was saying. It's a lament. He doesn't have an answer as to what needs to be done about it or even whether it is possible to fix it--just that the situation is bad, and I think that's fair to say. As JUMP math has demonstrated, we need not have infinite skill or money in order to bring about drastic improvements. Of course, there may be reasons for the current situation, some of them hard to fight against. But if we underestimate what is possible, we risk too much complacency and missing out on the changes that ARE possible.