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Math education: It’s not about numbers, it’s about learning how to think

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Re: Math education: It’s not about numbers, it’s about learning how to think

#21

Post by chebureki »

hasenj wrote: post #13
lordnacho wrote: post #4

I think a major issue with math problems in school is that they're obvious. By that I don't mean it's easy. But when you're grappling with some problem, whatever it is, eg find some angle or integrate some function, if you don't find the answer, someone will show you, and you'll think "OMG why didn't I think of that?" And you won't have any excuses for why you didn't think of it. Because math is a bunch of little log…

> if you don't find the answer, someone will show you, and you'll think "OMG why didn't I think of that?" Really? There are many times where I thought "How am I supposed to think of that?" In fact, that's probably what would happen most of the time. The problem with high school math is that it has no context. It's just a bunch of seemingly arbitrary rules. It's as if when learning programming, you would learn fragmen…

My university education experience seems to contradict this notion. Many classes were heavily focused on Java to allow students to learn data structures and object-oriented programming, in general. Other classes, like Operating Systems Programming were focused on C. In the end, you learn that there is the right tool for the right job.

Re: Math education: It’s not about numbers, it’s about learning how to think

#22

Post by sp332 »

netcan wrote: post #15
taneq wrote: post #2

As my old boss once said, "never confuse mathematics with mere arithmetic ."

I don't understand why this point seems so important to people. Seems like a semantic point, if we're talking about math education in schools. I mean, there is theory & practice in every field, technique and discovery. A medical scientist studying the development of aberrant cells in a petry dish has a pretty different job to a pediatrician or radiologist. They generally seem to coexist happily under the "medicine" l…

Arithmetic is theory too. Graph theory has lots of practical applications. I don't think it's that kind of divide. Arithmetic is just a very small part of mathematics.

Re: Math education: It’s not about numbers, it’s about learning how to think

#23

Post by Koshkin »

taneq wrote: post #2

As my old boss once said, "never confuse mathematics with mere arithmetic ."

Yet,

- arithmetics (a.k.a. Number Theory) is a vast branch of modern pure mathematics;

- numeric calculations, i.e. how to do them efficiently, is a very important part of the (applied) mathematics.

Re: Math education: It’s not about numbers, it’s about learning how to think

#24

Post by hasenj »

jdmichal wrote: post #18
Koshkin wrote: post #11

Learning "how to think" is just one part of it. The other part - the one that makes it much more difficult for many, if not most, people to learn math - especially the more abstract branches of it - is learning to think about math specifically . The reason is that mathematics creates its own universe of concepts and ideas, and this universe, all these notions are so different from what we have to deal with every day…

Is that really true before calculus? In my recollection, even in pre-calculus it was pretty easy to accurately tie constructs to physical representations. Limits is the first time I remember learning something that could not be generalized to a physical form, because it specifically dealt with what happened when a function "broke".

I find this sentiment somewhat surprising.

Limits is incredibly easy to link to real physical things: movement, speed, acceleration.

If speed is distance over time, what's the speed of a moving object at a very specific instant?

Re: Math education: It’s not about numbers, it’s about learning how to think

#25

Post by hasenj »

chebureki wrote: post #21
hasenj wrote: post #13

Earlier quoted context omitted.

> if you don't find the answer, someone will show you, and you'll think "OMG why didn't I think of that?" Really? There are many times where I thought "How am I supposed to think of that?" In fact, that's probably what would happen most of the time. The problem with high school math is that it has no context. It's just a bunch of seemingly arbitrary rules. It's as if when learning programming, you would learn fragmen…

My university education experience seems to contradict this notion. Many classes were heavily focused on Java to allow students to learn data structures and object-oriented programming, in general. Other classes, like Operating Systems Programming were focused on C. In the end, you learn that there is the right tool for the right job.

I think you misunderstood my post (or perhaps I wasn't able to articulate the point clearly).

My University computer science was also broken up as you describe. The introductory class was in Pascal, the next course was Java, and then each course used its own language.

That's not what I'm talking about at all.

Re: Math education: It’s not about numbers, it’s about learning how to think

#26

Post by JoshTriplett »

One of the most critical skills I see differentiating people around me (co-workers and otherwise) who succeed and those who don't is an analytical, pattern-recognizing and pattern-applying mindset. Math itself is quite useful, but I really like the way this particular article highlights the mental blocks and misconceptions that seem to particularly crop up around mathematics; those same blocks and misconceptions tend to get applied to other topics as well, just less overtly.

Re: Math education: It’s not about numbers, it’s about learning how to think

#27

Post by rf15 »

Koshkin wrote: post #11

Learning "how to think" is just one part of it. The other part - the one that makes it much more difficult for many, if not most, people to learn math - especially the more abstract branches of it - is learning to think about math specifically . The reason is that mathematics creates its own universe of concepts and ideas, and this universe, all these notions are so different from what we have to deal with every day…

There's also a hurdle in having to work through the wildly varying notational and structural views/tastes of the individuals who created their specific categories/theories (biggest problem at least for me - while programming languages are nice and structured and standardized, traditional math is highly impure/unstandardized and relies too much on how individuals feel like when they write things down)

Re: Math education: It’s not about numbers, it’s about learning how to think

#28

Post by throwawayjava »

crispyambulance wrote: post #17
lordnacho wrote: post #4

I think a major issue with math problems in school is that they're obvious. By that I don't mean it's easy. But when you're grappling with some problem, whatever it is, eg find some angle or integrate some function, if you don't find the answer, someone will show you, and you'll think "OMG why didn't I think of that?" And you won't have any excuses for why you didn't think of it. Because math is a bunch of little log…

I think the more pervasive problem in schools is a lack of mastery in basic foundational skills before moving on to more advanced topics. Too many students lack the ability to perform basic algebra. I mean simple stuff like simplification, factoring, working with exponents and even fractions. Faced with a typical problem in calculus, of course they're going to flounder and make errors when each calculus problem invol…

> No amount of explanation and reasoning will help the students and instructors get around the fact that they just have to spend the time to develop fluency with the basics.

Conversely, no amount of memorization and drilling will provide students with the intuition necessary to solve problems just beyond the grasp of naive symbolically-driven proof search algorithms.

In a typical non-proof-based calculus sequence, tricky u substitution and integration by parts problems are the two places where you really see "I was good at math in high school" students start to struggle because they lack a good conceptual foundation in trig and logarithms. Up until that point the students with trig identities etched into their skulls by practice and drilling do just fine. But from that point onward, they really start to struggle because they need a bit of geometric intuition and some practice with more effective problem solving techniques than "blindly apply equations until something works".

Really, both are needed -- drilling and practicing, as well as a solid (and preferably proof-backed) intuition about elementary properties of trig funcations and logarithms.

Re: Math education: It’s not about numbers, it’s about learning how to think

#29

Post by simias »

I completely agree. I think we start all wrong too, the first memories I have of maths at school was learning how to compute an addition, a subtraction and later a multiplication and division. Then we had to memorize by heart the multiplication tables.

That can be useful of course (especially back then when we didn't carry computers in our pockets at all times) but I think it sends some pupils on a bad path with regards to mathematics.

Maths shouldn't be mainly about memorizing tables and "dumbly" applying algorithms without understanding what they mean. That's how you end up with kids who can answer "what's 36 divided by 4" but not "you have 36 candies that you want to split equally with 3 other people, how many candies do you end up with?"

And that goes beyond pure maths too. In physics if you pay attention to the relationship between the various units you probably won't have to memorize many equations, it'll just make sense. You'll also be much more likely to spot errors. "Wait, I want to compute a speed and I'm multiplying amperes and moles, does that really make sense?".

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