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Stop teaching calculating; Start teaching math

computerbasedmath.org

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Re: Stop teaching calculating; Start teaching math

#61

Post by dkarl »

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…

There got to be far more to mathematics than simply learning how to follow steps.

Nope, that's really all there is. Just like computer programming is nothing but learning to string arcane commands together in anal retentive ways. It's just a lot of making variable names and putting data in them and using little symbols like "argc" and "foldl" and "fopen" that have excruciatingly dry definitions that are completely disconnected from the real world. You memorize rules of syntax and look up function definitions and follow all of the rules exactly correctly, and even after you've done what feels like a ton of that you're still just writing a dumb program that you would never use anyway. What's the point?

But I bet you didn't feel that way about programming when you were first learning, right? Things like foldl and dlopen seemed kind of neat, if not mind-blowing. People who become good at programming get there because they take delight in the simple things that make up the basics. Is there anything intrinsically beautiful about loops? There is the first time you encounter one. Oh, man, the possibilities! The feeling passes quickly, but I can't imagine how anyone who never enjoyed loops could endure the boredom long enough to become skilled at programming.

It's the same thing with math. Like programming, it really is just symbol manipulation and nothing more. The pleasure and beauty is in your perception of it, the concepts you develop to give intuitive substance to the abstract rules. Your appreciation gets deeper the more you learn. But to get there you have to enjoy the basics, the simple things. How many times did you have to write

  for (i = 0; i
before it became second nature? You must have derived some satisfaction from your initial fumbling with loops and functions or your interest would have died before you learned to write more complex programs. (The kids who don't enjoy the basics quit programming as soon as they realize they won't be making a state-of-the-art video game in a weekend.) Learning math is the same process; you start by enjoying where you are, even if you're starting at the mathematical equivalent of

  for (int i = 0; i
That may not sound very attractive to an educated person with many sophisticated things to think about. It's entirely possible that there are some people who are not simple-minded enough to enjoy math. However, I think those people do not enjoy programming, either ;-)

Re: Stop teaching calculating; Start teaching math

#62

Post by beambot »

dkarl wrote: post #61
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…

There got to be far more to mathematics than simply learning how to follow steps. Nope, that's really all there is. Just like computer programming is nothing but learning to string arcane commands together in anal retentive ways. It's just a lot of making variable names and putting data in them and using little symbols like "argc" and "foldl" and "fopen" that have excruciatingly dry definitions that are completely di…

Like programming, it really is just symbol manipulation and nothing more. I strongly disagree. Arithmetic, perhaps... but not "math."

Some of the best math proofs require little-to-no direct symbol manipulation. You might be able to reduce a mathematical statement down to the symbolic level (eg. using sets), but so much of the beauty is in the simple intuition. One simple example: the pigeon hole principle -- you can teach it to a 3 year old, and yet its incredibly powerful.

EDIT: If you read about Lockhart's Lament (as mentioned elsewhere in this thread), he specifically writes:

The cultural problem is a self-perpetuating monster: students learn about math from their teachers, and teachers learn about it from their teachers, so this lack of understanding and appreciation for mathematics in our culture replicates itself indefinitely. Worse, the perpetuation of this “pseudo-mathematics,” this emphasis on the accurate yet mindless manipulation of symbols, creates its own culture and its own set of values. Those who have become adept at it derive a great deal of self-esteem from their success. The last thing they want to hear is that math is really about raw creativity and aesthetic sensitivity. Many a graduate student has come to grief when they discover, after a decade of being told they were “good at math,” that in fact they have no real mathematical talent and are just very good at following directions. Math is not about following directions, it’s about making new directions.

Re: Stop teaching calculating; Start teaching math

#63

Post by bostonvaulter2 »

RyanMcGreal wrote: post #17

In high school physics about 22 years ago, we did an experiment to calculate the acceleration due to gravity by repeatedly dropping a weight with a ticker tape attached to it that fed through a hole punch tool and measuring the spacing of the holes in the tape over time. As part of the analysis and write-up, we had to calculate the standard deviation across our sample runs. That seemed really tedious to me, so I wrot…

I think it would be cool if calculators weren't allowed in school unless you programmed them yourself. Now trying to enforce that might be difficult but it is an interesting idea.

Re: Stop teaching calculating; Start teaching math

#64

Post by dkarl »

beambot wrote: post #62
dkarl wrote: post #61

Earlier quoted context omitted.

There got to be far more to mathematics than simply learning how to follow steps. Nope, that's really all there is. Just like computer programming is nothing but learning to string arcane commands together in anal retentive ways. It's just a lot of making variable names and putting data in them and using little symbols like "argc" and "foldl" and "fopen" that have excruciatingly dry definitions that are completely di…

Like programming, it really is just symbol manipulation and nothing more. I strongly disagree. Arithmetic, perhaps... but not "math." Some of the best math proofs require little-to-no direct symbol manipulation. You might be able to reduce a mathematical statement down to the symbolic level (eg. using sets), but so much of the beauty is in the simple intuition. One simple example: the pigeon hole principle -- you can…

Be fair. I do get around to saying there's more to it. Why, it's in the very next sentence after the one you quoted :-)

The pleasure and beauty is in your perception of it, the concepts you develop to give intuitive substance to the abstract rules.

Re: Stop teaching calculating; Start teaching math

#65

Post by ja27 »

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…

For me it had a lot to do with who taught my math classes. Out of all of my high school math teachers, I'd say less than half really understood what they were teaching beyond the most superficial depth. (It was even worse in my science classes.) I love teaching and would have preferred that as a career but the pay scale is really discouraging even at the university level. So now I build educational apps and hopefully can be part of the solution.

Re: Stop teaching calculating; Start teaching math

#66

Post by tkahn6 »

dkarl wrote: post #61
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…

There got to be far more to mathematics than simply learning how to follow steps. Nope, that's really all there is. Just like computer programming is nothing but learning to string arcane commands together in anal retentive ways. It's just a lot of making variable names and putting data in them and using little symbols like "argc" and "foldl" and "fopen" that have excruciatingly dry definitions that are completely di…

Programming is about architecting superior solutions to problems.

How then is it purely symbol manipulation? There is an intelligence required to imbue them with semantics and combine them into the right order to match the semantics of a correct solution.

Loops, variables, constants, etc. These are building blocks. One doesn't have to find them inherently interesting. The real challenge of programming comes from imbuing them with the right semantics.

The same can be said for mathematics. Sigma notation isn't interesting. It's what you use it to represent that is interesting.

Re: Stop teaching calculating; Start teaching math

#67

Post by UK-AL »

UK-AL wrote: post #57
anonymoushn wrote: post #56

Earlier quoted context omitted.

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…

I apologise for my dodgy english, it's 3am in the morning here.

Re: Stop teaching calculating; Start teaching math

#68

Post by wwweston »

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…

>this has nothing to do with hard work, and everything to do with the way mathematic is taught. I am a programmer and I would love to understand mathematics the way I understand programming.

How much of your understanding of programming has to do with:

a) teaching/exercises in a class

vs

b) time you've spent hacking on programs, either for a job, or in your spare time

?

Now, how much time have you spent on mathematics compared to time you've spent on programming?

The grandparent's point seems to be that fretting about not having "royal road"[1] is less productive than just spending time working with the ideas.

Good pedagogy can make a difference in how far you get in the time you spend, but there's just not a substitute for putting time in.

--- [1] http://en.wikipedia.org/wiki/Royal_Road#A_metaphorical_.E2.8...

Re: Stop teaching calculating; Start teaching math

#69

Post by dkarl »

tkahn6 wrote: post #66
dkarl wrote: post #61

Earlier quoted context omitted.

There got to be far more to mathematics than simply learning how to follow steps. Nope, that's really all there is. Just like computer programming is nothing but learning to string arcane commands together in anal retentive ways. It's just a lot of making variable names and putting data in them and using little symbols like "argc" and "foldl" and "fopen" that have excruciatingly dry definitions that are completely di…

Programming is about architecting superior solutions to problems. How then is it purely symbol manipulation? There is an intelligence required to imbue them with semantics and combine them into the right order to match the semantics of a correct solution. Loops, variables, constants, etc. These are building blocks. One doesn't have to find them inherently interesting. The real challenge of programming comes from imbu…

Apparently I fail at irony :-) Since we are mostly programmers here, I meant people to read the second sentence, say, "Wait... what? That doesn't do it justice," and then reread the first sentence in light of their understanding of the difference between what computers do and what programmers do. Symbol manipulation is what we design our computers to do, and the power of mathematical systems can be described by mathematical logic, which reduces mathematics to symbol manipulation, but what goes on in our heads as we work with these systems is very different.

Re: Stop teaching calculating; Start teaching math

#70

Post by dinkumthinkum »

scoith wrote: post #47

His talk is fundamentally wrong. Turning real life problems into a mathematical form (in other words, modelling) is not a subject of mathematics.

To be fair, I disagree with his position, but he is willing to call it something else, but the end of the day it is to teach his proposal as opposed what we now teach.
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