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Stop teaching calculating; Start teaching math

computerbasedmath.org

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Re: Stop teaching calculating; Start teaching math

#51

Post by recursive »

jobu wrote: post #33
magice wrote: post #19

This is sad. Really sad. My personal opinion is that Americans just dislike academic matters so much that they are incapable of forming a coherent solution for their children's schooling. Let me tell you some personal anecdotes. I moved to Minnesota during my High school senior year from Vietnam. It sucked: I couldn't hold on to conversations, and people couldn't understand my pronunciation; so I was bored and very l…

I'm all for giving kids challenges, but doing hand calculation isn't hard, it's just tedious. Computers do these calculations faster and with fewer mistakes, and everyone working in the real world uses them every day to do even simple calculations. If we stop spending as much time on calculations we can move kids more quickly into advanced concepts - with the side benefit of fewer kids being turned off math by how bo…

> everyone working in the real world uses them every day to do even simple calculations

That's not really true. I frequently do calculations manually just to convince myself that I can still think.

Re: Stop teaching calculating; Start teaching math

#52

Post by gtani »

jackpirate wrote: post #42
kiba wrote: post #38

Earlier quoted context omitted.

Back to problem of American math schooling. Here is my prescription: just make the damn students do it. Require them to work on math. Forget about the whole, um, personality and uniqueness and whatnots. You know where these things are from? The toil of work (for students, that's studying) builds personality, values, and one's importance and uniqueness. You know where self esteem is from? It's from holding a piece of…

Learning Haskell is a great back-door way to learn about math, specifically abstract algebra and category theory. You'll learn about why it matters and make tons of connections between subjects you never thought were related. Plus, coming from a programming environment gives you the context you need to understand why people care.

Road to Logic: recommended, and only about $25

http://homepages.cwi.nl/~jve/HR/

Re: Stop teaching calculating; Start teaching math

#53

Post by csense »

brr wrote: post #49

"And, of course, as we all know, with a computer you can take a simple problem like solve 5x2+2x+1=7 but you can make it harder, for example solve 5x4+2x+1=7. The principle of the problem is still the same." This is not, in fact, true for polynomial equations of order greater than four. From Wikipedia, "the Abel–Ruffini theorem states that there is no general algebraic solution... to polynomial equations of degree fi…

For that matter, asking students at the high school level to derive the quadratic formula will likely be quite challenging for even the best students in the class.

Asking students to come up with the formulas for 3rd or 4th degree polynomials on their own is probably going to be beyond anyone below the very top tier of human capability. A teacher would be extremely lucky to see one or two students that can do this in their career.

Then again, most people could easily solve these problems if they know what terms to Google :)

Anyway, as I recall the third- and fourth-degree cases require clever substitutions (whereas the derivation of QF is just completing the square with general coefficients).

Re: Stop teaching calculating; Start teaching math

#54

Post by csense »

Mathematicians are basically just accountants, right? And the IRS has gazillions of accountants who only really work during tax season...so, if our math programs suck, let's move our tax due date to overlap with schools' summer break, fire all the math teachers and replace them with those IRS guys!

It might be better than the existing system and it could hardly be worse. There's already been some work done on a curriculum (warning: PDF link): http://www.cs.amherst.edu/~djv/irs.pdf

Re: Stop teaching calculating; Start teaching math

#55

Post by UK-AL »

anonymoushn wrote: post #37
UK-AL wrote: post #26

Earlier quoted context omitted.

Err. This what he saying, you focus more on manipulating equations in the language of maths such as algebra, calculus, and less time on remembering what's the answer to 7 * 8.

Is this really a win? For most small problems or subproblems, a person who has solved similar problems many times before can solve it mentally much more quickly than by consulting a computer.

But one produces engineers, the other produces shop assistants.

Re: Stop teaching calculating; Start teaching math

#56

Post by anonymoushn »

UK-AL wrote: post #55
anonymoushn wrote: post #37

Earlier quoted context omitted.

Is this really a win? For most small problems or subproblems, a person who has solved similar problems many times before can solve it mentally much more quickly than by consulting a computer.

But one produces engineers, the other produces shop assistants.

Which is which? A person who works in either role will do so more quickly by virtue of not needing to constantly consult a machine over trivial problems.

Re: Stop teaching calculating; Start teaching math

#57

Post by UK-AL »

anonymoushn wrote: post #56
UK-AL wrote: post #55

Earlier quoted context omitted.

But one produces engineers, the other produces shop assistants.

Which is which? A person who works in either role will do so more quickly by virtue of not needing to constantly consult a machine over trivial problems.

Focusing on algebra, calculus, and so forth would produce more engineers/scientists/mathematicians etc . These are things rote teaching the times table can not do. His not saying let computers do all maths, just the repetitive bits.

Symbolic Manipulation is still necessary.

I learned vectors(Mostly for games), eigenvectors far more intuitively by programming them on computer as a kid, then calculating them by hand.

I knew what adding two vectors actually represented, or what the crossproducts was, rather than just calculating it(Which the computer always did).

I get top marks in calculus, linear mathematics and most college level maths classes, but my little secret is that i'm not actually very good knowing my times tables by 'art. If someone asked me to calculate a basic times table question, it would take a few seconds to work it out manually. Yet the general public thinks that's weird, but I actually found it's quite common at this level.

Re: Stop teaching calculating; Start teaching math

#58

Post by dinkumthinkum »

I didn't really see much specific stuff here, to be honest. There was some program that you could use sliders to figure the best life insurance policy ... I mean, ok ... that's great ... What's the big deal again?

What are some specific examples. There's all this "wah schools only teach calculation." Well, what specifically are they teaching that is bad and what specifically should they teach that is better?

So, are we saying, no reason for a student to know how to take a derivative, they should know how to turn the knobs on some program to figure out the best fuel efficiency of some device in some really neat simulator?

The example about limits ... ok yeah that's cool but if you aren't actually teach them how to "calculate" (is the word "calculate" used as a boogey man?), how is that you are not doing anything other than teaching "math appreciation?"

Re: Stop teaching calculating; Start teaching math

#59

Post by jcnnghm »

kiba wrote: post #38
magice wrote: post #19

This is sad. Really sad. My personal opinion is that Americans just dislike academic matters so much that they are incapable of forming a coherent solution for their children's schooling. Let me tell you some personal anecdotes. I moved to Minnesota during my High school senior year from Vietnam. It sucked: I couldn't hold on to conversations, and people couldn't understand my pronunciation; so I was bored and very l…

Back to problem of American math schooling. Here is my prescription: just make the damn students do it. Require them to work on math. Forget about the whole, um, personality and uniqueness and whatnots. You know where these things are from? The toil of work (for students, that's studying) builds personality, values, and one's importance and uniqueness. You know where self esteem is from? It's from holding a piece of…

I agree with what you are saying. I've been contemplating the reasons why I don't understand math the way I understand programming for some time, and opened up my old Calculus book a few days ago and started rereading it from the beginning. I intend to work through the entire book over the next couple of months. I believe because of the way math is taught, I never developed a truly intuitive understanding of what was going on after algebra. When I taught myself how to program, I learned then immediately applied what I was learning, and continued applying it continuously in all the software I wrote after. Algebra and arithmetic are pretty analogous in that regard.

For me, it is necessary to develop an intuitive understanding of something before I can really appreciate it, and more importantly, manipulate and apply it to arbitrary situations. The way math is taught, intuition is never really delivered. In programming, it's possible to look at an algorithm and have trouble understanding what it does. However, I have never implemented and debugged an algorithm or data structure and not developed a thorough understanding of it in the process. In Calculus, the rules were given, but there was never any effort spent to foster an understanding of why they are the way they are, or the bigger picture. To me, this would be like learning merge sort by running through the steps, but never actually implementing the algorithm as a whole to truly understand what is going on. In going back through it, I intend to relearn it the way I learned programming, so I can truly apply and reason around such a powerful tool.

Re: Stop teaching calculating; Start teaching math

#60

Post by dspeyer »

These how-to-apply-math skills are valuable, but they aren't math itself. Math is about building more general abstractions, keeping strict rigor, being careful with axioms, and developing techniques that are really, really broad as a result.

Granted, calculation isn't a very good way to get into this either.

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