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Poll: How much calculation work should be taught in schools?

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Poll: How much calculation work should be taught in schools?

#1
When mathematics is taught in schools (K-12 and college), a lot of the time is spent learning how to do the nitty-gritty calculation work. My question is: how much time should be spent teaching the calculation work versus the higher orders of thinking needed in mathematics?

Should you let the student do calculations because it gives them an understanding of the bigger concepts?

Or should you let the computers do the calculations and not waste the students' time?

To learn more about what I'm talking about, read the debate going on over here: http://news.ycombinator.com/item?id=2213258

Also, please watch this video that includes some of my views: http://www.ted.com/talks/conrad_wolfram_teaching_kids_real_math_with_computers.html

Re: Poll: How much calculation work should be taught in schools?

#4
It's really obvious to me - having dealt with people raised on calculators for several decades now - that if you can't do arithmetic in your head, you don't understand any math.

Calculator kids will type in 10 + 10, get 100, and then assume the answer is 100 because they have no intuitive understanding of the meaning of any of it. It's just abstract operations relating to button pressing.

Where this goes off the rails is the ineffective methods used in many schools, and the frequent methodology changes due to the latest teaching fad, which results in 12th graders practicing addition because they still haven't learned it. This has led some to conclude that they should skip the basics and move into advanced things. No. They should instead spend the very small amount of time it takes to teach the basics properly, and then move into advanced things.

College freshmen are arriving at schools and they can't make change from a dollar after 14 years of classes.

Re: Poll: How much calculation work should be taught in schools?

#5
Depending on the area, both... Some things, like solving simple algebra and simultaneous equations are done to death, get the students to understand the concept, do a few different examples and move on. I hate when students are given say 20 equations to do which are all the same rules with different numbers.

Something like differential equations, if you are plugging these into a calculator at first, you probably won't really get exactly what is going on. It requires more examples and exercises (there are a lot more different cases to give to) before a student could plug it into a calculator rather than going through the repetition of doing it by hand.

I just missed some of the newer calculator heavy course at the end of high school, my brother got it the year after I left I guess, wasn't a big fan of it by the looks of it though.

Re: Poll: How much calculation work should be taught in schools?

#6
It would have saved me a ton of frustration and wasted effort if it were a lot less than currently. In high school I remember hating math - it was obvious some people were far better at grinding through computations than I could ever hope to be, and it made me resentful being forced to go through what seemed a pointless exercise.

In university I made a late switch to computer science, and took real analysis thinking it would be very painful, but would make up for having no calculus. I actually enjoyed so much I switched to math, and so did sort of a math crash course over a year to prepare for grad school. Some of the math was hell. Studying differential equations, complex variables (computation oriented), and bits of differential geometry, etc. would leave me with so frustrated, wondering why I was putting myself through this. On the other hand, measure theory, complex analysis, functional analysis, algebra, and galois theory were so illuminating. I could just sit there for hours working through proofs, and not burn out.

However, this would not work for everyone (or most even). It shocked me that everyone else didn't see it my way, so it was very interesting observing how others studied/thought about math. I had a friend in all my pure math classes whom I worked on assignments with. He was a computational wizard who could flawlessly plow through pages of computations. My rate of errors - flipping a sign, carelessly misapplying a rule, etc. - was so much higher than his I had to conclude our brains were just wired in a totally different way. I noticed, when trying to prove something, we'd proceed very differently. He'd take what he knew to be true, and just begin enumerating some logical consequences of that, and go in the direction which seemed to have the smallest blowup in data. I'd usually assume the statement was false, and think and think and think about why that would be so absurd. Then, when I came up with the abstract explanation in my mind, in my mind I'd shape it into something concrete enough to write down (or even describe).

I identify a lot with the mathematician Alexander Grothendieck (at least before he became a little eccentric), because he's one of the rare examples (I know of) of an outstanding mathematician who seems to have approached math the way I do, and derives value from it for similar reasons. AG was reknowned for thinking about math in an extremely abstract way: many people learn by taking specific examples and then playing around with them mechanically until they get a general 'feel' for what's going on. Instead AG would describe the phenomenon being observed in the most abstract and general way possible, sometimes building an entire new theory of which the solution to the original problem was merely a trivial consequence.

Here is a two part piece biographical essay that, regardless of your interest in math, you'll probably find very interesting. He led an extraordinary life:

http://www.ams.org/notices/200409/fea-grothendieck-part1.pdf

http://www.scribd.com/doc/35435936/As-If-Summoned-from-the-V...

EDIT: Hmm, I sort of went off on a general rant instead of making the point I intended (this tiny text field makes that so easy, haha). I wanted to point out it's a common distinction in math made between the "problem solvers" and the "theory builders" (AG, mentioned above, epitomizes the latter). Most subjects in pure math are populated by people in one field or the other, and I believe that by forcing a computational approach on people early in their development, you're completely turning people off of math who could have fallen in love with the abstract theory. How prevalent this is, I don't know.

Re: Poll: How much calculation work should be taught in schools?

#7
post #4

It's really obvious to me - having dealt with people raised on calculators for several decades now - that if you can't do arithmetic in your head, you don't understand any math. Calculator kids will type in 10 + 10, get 100, and then assume the answer is 100 because they have no intuitive understanding of the meaning of any of it. It's just abstract operations relating to button pressing. Where this goes off the rail…

10 * 10 you mean?

However I do agree with you - I learned math not only by doing it by hand, but also by brute forcing it into my brain. I memorized my times tables from third through fifth grade until I knew them up to 25. I did the quadratic formula by hand until I knew the first five Pythagorean Triples.

These aren't necessarily the "right" ways to learn math - but it certainly made me more curious to find a deeper meaning of why mathematics works the way it does.

Re: Poll: How much calculation work should be taught in schools?

#8
I don't think there's such a dichotomy. I certainly believe more focus should be placed on "higher order" thinking but the calculation should still be covered in depth in order to solve those problems.

Essentially teaching in the form "have a problem, learn some techniques" instead of "learn a technique, get some problems that will use it." The latter is how my math education went and it doesn't teach students how to choose the right techniques to use in the first place.

Re: Poll: How much calculation work should be taught in schools?

#9
post #4

It's really obvious to me - having dealt with people raised on calculators for several decades now - that if you can't do arithmetic in your head, you don't understand any math. Calculator kids will type in 10 + 10, get 100, and then assume the answer is 100 because they have no intuitive understanding of the meaning of any of it. It's just abstract operations relating to button pressing. Where this goes off the rail…

Yeah it's all about getting the understanding of how something is working, then looking to move onto something that makes the repetition easy. It's the same in programming, it's great to learn how your code is compiled to assembly but once you have that understanding you can take it back up to a high level language which helps remove repetition.

Re: Poll: How much calculation work should be taught in schools?

#10
post #7
post #4

It's really obvious to me - having dealt with people raised on calculators for several decades now - that if you can't do arithmetic in your head, you don't understand any math. Calculator kids will type in 10 + 10, get 100, and then assume the answer is 100 because they have no intuitive understanding of the meaning of any of it. It's just abstract operations relating to button pressing. Where this goes off the rail…

10 * 10 you mean? However I do agree with you - I learned math not only by doing it by hand, but also by brute forcing it into my brain. I memorized my times tables from third through fifth grade until I knew them up to 25. I did the quadratic formula by hand until I knew the first five Pythagorean Triples. These aren't necessarily the "right" ways to learn math - but it certainly made me more curious to find a deepe…

I've confirmed my geekiness. I didn't even consider it could have been 10 * 10.. I assumed it was binary! :-)
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