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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

#31

Post by sixo »

d3ckard wrote: post #19

Maybe I'm wrong, but I have always believed that if you want people to be good at math, it's their first years of education which are important, not the last ones. In other worlds, push for STEM should be present in kindergartens and elementary schools. By the time people go to high school it is to late. I never had any problems with math until I went to university, so I was merely a passive observer of everyday stru…

Not sure it's exactly the early years. But I think there's zero point to moving past a topic until you have mastery & can think WITH it (not just, about it, or hack through it on paper). And since the early years are often where that fails, they need to be rethought.

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

#32

Post by Koshkin »

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".

Even simple objects, like vectors, for example, can very quickly become so abstract (higher dimensions; complex vector spaces) that the physical intuition stops being useful and can even lead to mistakes. This is the point when new kinds of mathematical intuition becomes necessary, and building them is very difficult and takes time.

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

#33

Post by keymone »

throwawayjava wrote: post #20
keymone wrote: post #16

i always found munging numbers and memorizing formulas discouraging. i think physics classes teach kids more math than math classes and in more interesting ways (or at least have potential to).

Interestingly, my experience was that your two sentences are totally contradictory. Physics and engineering was all about using memorized formulas to solve word problems, and mathematics was where you learned how those equations systematically fall out of some obvious assumptions. I think the real take-away is that both physics and mathematics can be taught poorly, and both can be taught well.

i get a feeling you're not talking about elementary school. but i agree - both can be taught good or bad. bad is dumb memorization, good is understanding real world phenomena.

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

#34

Post by dahart »

I wonder if a large part of our math problem is our legacy fixation on Greek letters. Would math be more approachable to English speakers if we just used English?

I like to think about math as language, rather than thought or logic or formulas or numbers. The Greek letters are part of that language, and part of why learning math is learning a completely foreign language, even though so many people who say they can't do math practice mathematical concepts without Greek letters. All of the math we do on computers, symbolic and numeric, analytic and approximations, can be done using a Turing machine that starts with only symbols and no built-in concept of a number.

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

#35

Post by DiffEq »

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…

I suppose one can learn how to think by doing math, or they can just learn how to follow rules...and perhaps some may want to think about match specifically. However, I think most people would be encouraged to learn math if they are shown that math is just a way to model the world they live in and it can tell them specifically about some of the assumptions they have about it. For instance, I might assume if you put some fuel in a rocket and point it straight up that it will go into space and be somewhere. Math can tell me where it will be and the path it will take to get there...people identify more with those concrete things.

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

#36

Post by jfaucett »

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…

"all these notions are so different from what we have to deal with every day that learning them takes a lot of training"

I disagree in the general sense. I think most of mathematics with which "normal people" (by that I mean people who don't get a masters or phd in mathematics) have problems is directly correlated to problems and situations in the real world. The problem is that we teach mathematics as if we were computers - which we are not. Its about formulas, recognizing patterns, precise answers and proofs instead of general understanding and intuitions about things - which is where humans excel and how we work and remember things.

For example, being able to solve quadratic equations is nice but IMHO what's much more important is knowing how the coefficients and their signs shape the plot of the lines and how that might effect the trajectory and hang-time of a projectile. Also being able to look at a projectile and give an approximate quadratic equation is again IMO more important than knowing all the steps in the quadratic equation.

But this is just my opinion.

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

#37

Post by Koshkin »

hasenj wrote: post #24
jdmichal wrote: post #18

Earlier quoted context omitted.

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?

"Ma'am, you were driving at ninety miles an hour."

"But, sir, I have only traveled twenty miles!"

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

#38

Post by dahart »

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…

> Learning "how to think" is just one part of it. The other part [...] is learning to think about math specifically.

Do you feel like you were taught how to think about math specifically, or how to think at all? Like probably most people here on HN, I love math and programming, but I'm struggling to recall whether and how I was taught to think in school. We went through a series of problems, and talked about the syntax and mechanics of solving those problems, using various frameworks (geometry, calculus, diff. eqs) So it feels like we learn math by example. Is learning how to think directly part of education, or only indirectly something that happens by seeing what others did?

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

#39

Post by devmunchies »

d3ckard wrote: post #19

Maybe I'm wrong, but I have always believed that if you want people to be good at math, it's their first years of education which are important, not the last ones. In other worlds, push for STEM should be present in kindergartens and elementary schools. By the time people go to high school it is to late. I never had any problems with math until I went to university, so I was merely a passive observer of everyday stru…

school teachers, administrators, and legislators most likely did not do STEM, so that pattern will most likely continue.

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

#40

Post by redial »

dahart wrote: post #34

I wonder if a large part of our math problem is our legacy fixation on Greek letters. Would math be more approachable to English speakers if we just used English? I like to think about math as language, rather than thought or logic or formulas or numbers. The Greek letters are part of that language, and part of why learning math is learning a completely foreign language, even though so many people who say they can't…

Symbols are symbols, are symbols.

Instead of trying to compress everything into a very limited set of "daily language" symbols we should be trying to expand our ideas, including our language(s).

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