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Re: Can anyone be good at math?

mwalsh — Wed, 11 May 2011 19:37:56 GMT

I had a math teacher, he majored in Math and then became a HS math teacher/vice principal until he retired may 2010, that there were several "leaps" in math that people get lost at. These were

1) Going from concrete math to abstract math - ie 1 apple and 2 apples is 3 apples to 1+2=3

2) Adding in "letters" - basic variables

3) the infinite summation of infinetly small things is a number- basically infinite things and calculus

4) Supposedly more- but he never went that far.

To me, I haven't had trouble in math- yeah I have earned 2 Bs in math- Calc 3 and Diff Eq, but that is because I didn't put the effort into memorizing the formulas to simplify a problem so it took 5 minutes not 50, and could have finished the test.

Re: Can anyone be good at math?

Caley McKibbin — Wed, 11 May 2011 19:34:36 GMT

He used singular form. It seemed like a reference to rank.

Re: Can anyone be good at math?

Scrooge McDuck — Wed, 11 May 2011 18:48:25 GMT

College algebra, calculus, differential equations, linear algebra, physical math, etc.

Re: Can anyone be good at math?

Caley McKibbin — Wed, 11 May 2011 14:33:12 GMT

Clayton, to your last paragraph, what level of mathematics is required to understand this?

What is a "level" of math?

Re: Can anyone be good at math?

AJ — Wed, 11 May 2011 09:32:05 GMT

Scrooge McDuck:

AJ, can you elaborate on (1)? As I think more about this topic it seems apparent that individuals wrap math in a sheet of mysticism. I don't know if it is because these individuals are mathematically ignorant and/or that some math is so complex and specialized that it intimidates some and seems unknowable. But given that math is reason and logic, it should be knowable by all.

As far as advice about definitions in math, there's not much to say except simply to examine definitions very carefully. Like if something seems wrong, go back to the definitions and make sure you have them straight. If you don't understand a term in a definition (or theorem), look up its definition.

Then just always assume that you misinterpreted things when you get the wrong answer, not that you can't do it. (2) I've seen what you might call "not the sharpest tools in the shed" who excel at math simply because they are excessively confident that it is all very easy, and if they mess up they just assume they misread something, never that it's beyond them. Since it is just logic - the same logic people use for doing everything in everyday life - it shouldn't be hard as long as the terminology and symbols don't throw you off.

Re: Can anyone be good at math?

Marko — Wed, 11 May 2011 08:18:58 GMT

Yes you can be good at math. It is the same thing as having a high IQ. But you can be good at math and fail the class, or not be that good at math and stil max out your grade. Where being good at math shines through is math competitions where it is a must. We had those and the results were similar year in year out. There was a correlation to grades, but not really. Eg there were honor students who consistently failed to achieve a noteable score etc.

Re: Can anyone be good at math?

Clayton — Wed, 11 May 2011 00:31:29 GMT

@Scrooge: Well, I think everyone is constantly using the "visualization" aspect whenever they speak to one another, especially about physical objects. Consider the surface of a table. Geometrically, it is a "bounded plane". Now, imagine that surface in your mind's eye, without the table, just the surface. Now, remove the bounds of the surface and allow it to extend indefinitely. This is a geometric plane or just "plane". Now, envision two such planes which are not parallel... they must eventually "intersect" or cross through each other. Now, envision the "joint" where there two planes intersect... what could you describe it as? We are now in a position to state a geometric fact about planes: the intersection of two planes is a line.

For a long time, arithmetic and geometry were separate disciplines. Arithmetic was an artisan skill of the merchant class and geometry was mathematics. But by the time of Rene Descartes, the foundations had been laid for the arithmetization of geometry. The key concept is imagining a line as if it were a ruler with regularly spaced numbers ordered from 0 in the middle to infinity and negative infinity in either direction. This is called "the number line." By arranging two such lines together, you can speak of "functions" and their "graphs" which are a visual, quasi-geometric representation or summary of the behavior of the function. A function is simply an equation into which you can plug one or more numbers and get one (or more) numbers out. The discipline of the study of the geometry of algebraic functions is called analytic geometry.

It turns out that an ordinary geometric plane can be thought of as the graph of a function. Imagine a 3-dimensional space with three number lines all intersecting at their zeros and each at 90 degree angles to one another (also called "axes"). Each axis is normally labeled x, y and z. There are actually three planes already defined in this space just from what we've described... the first plane is defined by the function x = 0, the second by y = 0 and the third by z = 0. Now, we can make an arithmetic statement of the geometric fact described above... the intersection of the plane x=0 and the plane y=0 is the line (x+y=0). That may not have made sense to you but my point is simply that geometric facts which you can visualize have a corresponding arithmetization and one of the first tricks you need to learn is how to switch between them.

This video is a not-half-bad introduction to analytic geometry. There are other videos linked which continue explaining more advanced concepts.

Good luck.

Clayton -

Re: Can anyone be good at math?

Scrooge McDuck — Wed, 11 May 2011 00:05:50 GMT

Yes, it absolutely makes sense. I forget intermediate rules of algebra and get intimidated by big equations in calculus, trig functions, etc.

Re: Can anyone be good at math?

Stan — Tue, 10 May 2011 23:54:50 GMT

I think the root cause of so much math "hatred" comes from not having mastered the basic fundamentals in math. Most math concepts starting from calculus and beyond is pretty interesting (and dare I say, fun?). But when it comes to solving problems that inevitably incorporate old concepts like fractions, long division, etc. people seem to struggle and forget and then get frustrated. Does that make sense to anyone? And with things like Khan Academy, it's becoming a lot easier to relearn old concepts and review.

Re: Can anyone be good at math?

Scrooge McDuck — Tue, 10 May 2011 23:32:30 GMT

Clayton, to your last paragraph, what level of mathematics is required to understand this? Perhaps my math knowledge is so poor that I am not understanding your point. I think I understand the gist of your post, and I see math used to explain visuals, but I don't understand math well enough to "get" those equations with regard to visualization. As I mentioned earlier, I have no idea how math explains M-theory or anything like that and I'm not sure what level of math is needed to do so.

Re: Can anyone be good at math?

Clayton — Tue, 10 May 2011 22:38:13 GMT

@Scrooge: I think there are two different sorts of math. The first sort of math is more constructive in nature, it is math that you can easily build a machine to perform. All forms of numerical calculation fall into this category.

The second sort of math is less constructive, it's more definitional and I think this kind of math is more contextual to the human brain, specifically language. When you speak of a "sphere" for example, an image pops into your mind, maybe something like a super-thin, see-through basketball. From this image, you are able to "visualize" the rules of the geometry of spheres. Then, these rules which you began with (which you could visualize in your head) can be generalized to other dimensions which you may not be able to visualize... a two-dimensional sphere (circle), one-dimensional sphere (line segment), zero-dimensional sphere (point), four-dimensional sphere (hard to visualize) and so on. The rules of the geometry of spheres were not chosen because "these are the only possible rules" but, rather, they were chosen because they were the best formalization of the mental picture that we share when we speak of a "sphere in the abstract." I believe this mental sphere is a by-product of the human brain. An alien race intelligent enough to engage in mathematics and geometry would not necessarily visualize the world in the same way we do and may not choose to begin with the same rules of geometry and may derive a different, conflicting set of rules regarding the behavior of geometric spheres.

Where I think people run into trouble is in connecting the formalism (rules of symbol manipulation) with the base visualization. You might learn the rules of symbol manipulation but then it's "empty symbol manipulation" until you connect it to the visualization. But then, you might understand how mathematical objects that you can visualize interact while having difficulty translating this to the rules of symbol manipulation and thereby lose the ability to extend into what cannot be visualized.

Clayton -

Re: Can anyone be good at math?

Scrooge McDuck — Tue, 10 May 2011 21:58:53 GMT

AJ, can you elaborate on (1)? As I think more about this topic it seems apparent that individuals wrap math in a sheet of mysticism. I don't know if it is because these individuals are mathematically ignorant and/or that some math is so complex and specialized that it intimidates some and seems unknowable. But given that math is reason and logic, it should be knowable by all.

Re: Can anyone be good at math?

AJ — Tue, 10 May 2011 18:43:40 GMT

Some people have a higher aptitude for math than others, but I've noticed that anyone can learn any level of math as long as they (1) are very careful with definitions and (2) don't get intimidated no matter how many mistakes they make. People that hate math usually have a problem with one of those two.

Re: Can anyone be good at math?

baxter — Sat, 07 May 2011 00:56:07 GMT

IIRC p-adic numbers have some ambiguity. Here's an example

Since -1/3 = 1/(1-4) =1+4+16+64+...=....101010101xb

Then 2/3 = ...101010101xb + 1 = ...101010110xb

But 2/3 can also be represented as 0.1010101...xb

Re: Can anyone be good at math?

Clayton — Fri, 06 May 2011 23:40:48 GMT

@baxter: Yeah, the p-adic literature is pretty impenetrable. I like p-adic numbers because they actually form a number system and you can do addition, subtraction, etc. on them just like you would decimal or binary numbers. In addition, they are, at least in some sense, more "natural" in that they include negative as well as positive numbers where standard decimal or binary requires the use of an external "minus sign" to indicate the positivity or negativity of a number.

Clayton -