Veering Away from Mechanical Thinking

[thrill3rnit3]

Veering Away from "Mechanical Thinking"

I'm a current high school student aspiring to become a mathematician. I am doing pretty well in my math classes, taking college level math here and there. However, I'm afraid that continuing to do the math that I'm taking right now leads me further and further into the path of "mechanical thinking" - procedural approach to math that pretty much uses plug and chug. I figured that it probably has to do with the way teachers nowadays teach math - teach the "procedure" on how to do it, and then assign routine problems at the end of each chapter.

This approach loses the abstractness of math that I enjoy, and it makes me less and less able to think outside the box - say, doing proofs, solving olympiad problems, etc. I read advanced level math, I study the procedures on "how to do it", but the beauty of the theorem and their proofs in the process.

I want as much as possible to stay away from this procedural approach of doing math. I'm not really sure what kind of "question" I should ask, but I guess general advice would be really helpful.

 

[tiny-tim]

Hi thrill3rnit3!

You need to know all the procedural stuff so that you can apply it when you do think out of the box.

Heisenberg didn't know anything about matrices, and Born had to tell him about them before he got his Nobel prize.

 

[chroot]

This is a pretty old conundrum, and it shows itself in virtually every kind of educational process.

If your goal is to teach students how to solve problems, then you must teach at least some "mechanical thinking." If you provide students only with concepts -- even very well-explained concepts -- they tend to do very poorly on exams, even if they "understand" the material very well.

A famous example of this are the lecture notes prepared by the great physicist Richard Feynman for undergraduates at Caltech. The lecture notes have been celebrated ever since as one of the most beautiful, insightful, and approachable expositions of physics ever written. Unfortunately, Feynman himself admitted that the lectures turned out to be very poor at teaching students to solve problems.

The best approach probably lies somewhere between purely "mechanical" and purely conceptual education. If you feel that you are getting only mechanical education from your educators, I suggest that you spend more time with them, one-on-one, in their office hours. You might also ask them for suggestions on books that might be appropriate, or perhaps just out of reach, to really push your conceptual understanding forward.

Also, when you find yourself slaving away at a particularly mechanical set of problems, try to transcend them and work on a higher level. Instead of simply solving dozens of problems, look for patterns in them, and abstract out higher-level structure. When I used to get boring problem sets, I would write computer programs that would solve the problems for me. I did not simply numerically solve them, though; I applied all sorts of sometimes "unrelated" machinery like sets, lists, and loops. I wrote genetic algorithms, invented tree structures, and used a bazillion hours of CPU time. I looked for the weirdest ways to solve things, and those weird solutions both entertained me and educated me. I was able to learn both computer programming and mathematics in a way that cemented both firmly in my head.

- Warren

 

[qntty]

The only way to get better at solving problems is practice. With practice you get used to thinking about problems in a certain way and learn how to approach problems even if you've never seen a problem similar to it.

 

[thrill3rnit3]

Also, when you find yourself slaving away at a particularly mechanical set of problems, try to transcend them and work on a higher level. Instead of simply solving dozens of problems, look for patterns in them, and abstract out higher-level structure. When I used to get boring problem sets, I would write computer programs that would solve the problems for me. I did not simply numerically solve them, though; I applied all sorts of sometimes "unrelated" machinery like sets, lists, and loops. I wrote genetic algorithms, invented tree structures, and used a bazillion hours of CPU time. I looked for the weirdest ways to solve things, and those weird solutions both entertained me and educated me. I was able to learn both computer programming and mathematics in a way that cemented both firmly in my head.

- Warren

This is probably a good idea. I'm a newbie in programming, but I should have time during the summer to get started.

I guess the root of my problem is the way math is taught nowadays. My current high school math teacher just gives us the equations, rules, and procedures to pretty much memorize on our own, with little to no explanations on how in the world they derived those. I had to dig deeper to answer my own questions.

 

[chroot]

I had to dig deeper to answer my own questions.

If you have access to a program like Mathematica, I would also suggest you being "playing" with it. I'm not suggesting that you use it to solve problems -- you really do need to do some plain old hard work to learn math -- but it can be an amazing exploratory tool.

- Warren

 

[evilpostingmong]

Yeah I remember on a Calc II quiz I messed up on finding the limit of a sequence
(where you need to divide by the highest degree of n) though I know the concept
of diverging and converging. IMO, you can't neglect either in math. As much as I
hate reducing matrices, sometimes it needs to be done. Just be sure to ice your
wrist when finished :). Good luck with your future studies, you'll do excellent!

 

[millitiz]

This is probably a good idea. I'm a newbie in programming, but I should have time during the summer to get started.

I guess the root of my problem is the way math is taught nowadays. My current high school math teacher just gives us the equations, rules, and procedures to pretty much memorize on our own, with little to no explanations on how in the world they derived those. I had to dig deeper to answer my own questions.

Well, maybe I should ask, what else do you expect? From your posts, it seems like you are an extraordinary student. As the word implies, you are more than an average HSer, and classes in HS is, by definition, provided for average HSer.
Notice that not everyone needs to know the why, or how. For example, as a physics student, I don't really need to know all the theorems behind the integration (say, what exactly is Reman intergration, and when to apply and when not too). The important part is to know how to do the integration, and do it fast. Sometimes it is even better since one is fine with some "dirty math" that pure mathematicians would not dare to use.
Also, some of the whys and hows are way too difficult to explain rigorously to HSer. And sometimes there just does not exist even a sloppy explanation.
Also, practicing these problems routinely sometimes could be a good thing. Yes, it is boring, but as one might say, practices make perfect.
Just for instance, let's talk a little bit of Algebra while one is trying to solve for an function that is, say, something very complicated, x to the 5 and what not. At this point, if you have a very solid practices in solving Algebra II routinely, you can apply all the techniques you know, faster than your classmates, and solve the problem faster than others.
Also, notice that all the insight/intuition of math and physics are normally obtained from tons of practices.
And at the end of the day, it is pretty much the same thing even in advance math, or pure math, for that matter. In upper level math you solve the problems (routinely, in some sense). You learn some techniques/ tactics to solve problems, and try them out when you face new problems. And in research, again, you solve some harder problems. Sometimes by those techniques/insight that you developed along your way up to that point.

 

[thrill3rnit3]

Well, maybe I should ask, what else do you expect? From your posts, it seems like you are an extraordinary student. As the word implies, you are more than an average HSer, and classes in HS is, by definition, provided for average HSer.
Notice that not everyone needs to know the why, or how. For example, as a physics student, I don't really need to know all the theorems behind the integration (say, what exactly is Reman intergration, and when to apply and when not too). The important part is to know how to do the integration, and do it fast. Sometimes it is even better since one is fine with some "dirty math" that pure mathematicians would not dare to use.
Also, some of the whys and hows are way too difficult to explain rigorously to HSer. And sometimes there just does not exist even a sloppy explanation.
Also, practicing these problems routinely sometimes could be a good thing. Yes, it is boring, but as one might say, practices make perfect.
Just for instance, let's talk a little bit of Algebra while one is trying to solve for an function that is, say, something very complicated, x to the 5 and what not. At this point, if you have a very solid practices in solving Algebra II routinely, you can apply all the techniques you know, faster than your classmates, and solve the problem faster than others.
Also, notice that all the insight/intuition of math and physics are normally obtained from tons of practices.
And at the end of the day, it is pretty much the same thing even in advance math, or pure math, for that matter. In upper level math you solve the problems (routinely, in some sense). You learn some techniques/ tactics to solve problems, and try them out when you face new problems. And in research, again, you solve some harder problems. Sometimes by those techniques/insight that you developed along your way up to that point.

I understand your point.

I just want to delve deeper into the theory behind the math rather than just doing it. I understand that physicists/engineers generally utilize math as a tool to solve problems, but I'm more into the theory and beauty behind all the formulas and theorems.

Maybe that's why I'm more inclined to be a math major than an engineer

 

[qspeechc]

It's sounds like your maths classes should be no problem for you. So this mean you should have some free time to read some maths books on your own, outside of class. Find some books you think you can tackle, and look interesting to you, and get stuck into them in your own time.