Math education

[shvarz]

Found (through LJ) an interesting essay on math education in school: http://www.maa.org/devlin/LockhartsLament.pdf

25 pages, but worth reading when you have time. I certainly found something there to think about.

[Numsgil]

I'll try to read it tonight.  Coming from an actual math background, I'd say the present mathematical educational system is problematic (at least in my experience) because it hasn't figured out whether the calculator is a good thing or a bad thing.

For instance, I learned how to do long division.  I can still do it.  But I almost never do, because the calculator is faster and less error prone than my pen and paper.  Sometimes when I'm at the grocery store, I have to do some division to figure out cost per item and things like that (though usually it's pre-calculated on the price sticker nowadays), but is that rare use worth the months and months of elementary school we spent learning to long divide?  Especially when it's so distasteful?

On the other hand, learning to do square roots by hand was always something I wanted to do.  It bugged me that I had no idea.  Most people I asked said to use a calculator (or the really old geezers-- a slide rule).  It wasn't in to calculus with Taylor series that I had any idea how to do square roots with anything but guess and check.  Here I had an honest desire that was left unfulfilled for years.

More recently, I got a B in my ODE class.  In theory, I should be able to solve y''(x) = -b * y'(x) for an equation in the form y(x) = ...  But whenever I come across such an instance, I always load up Maple and do it in there.  I don't hardly remember how to do those ODEs by hand anymore, and Maple is way less error prone.

In 100 years when the school system has figured out that having calculators means people don't need to do math by hand well, I think things will improve.  Teach me how humans and computers solve a problem, then give me the computer and let me go on with my life   I don't do square roots by hand now, even though I know how.  I still use the calculator.  What I want to learn is what to actually do with the tools I learned.

[Shasta]

I very much agree with what Numsgil said. I have alg2/trig and AP Chem right now in high school, basically two math classes. On one hand in alg2 for most of our tests we can only use a old four function calculator, and only get to use a graphing when dealing with things like logs. On the other hand in AP Chem we have a class-room set of TI's new Inspire CAS series and can use them how ever we like. (The Inspire CAS can do things such as solve algebric equations, do logs of x root and so forth)

What I would like to see is a lot more self teaching, instead of telling me what each part of y=ax²+bx+c does, have them graph the equation and drag different parts around and see how the equation changes. Learning like this would make the information more meaningful to the students as it is gained first hand instead of just having it told to you.

[EricL]

I'll push back a little.  One the one hand I agree that it's difficult to learn anything you arn't really interested in and that the best teaching occurs when the student has natural curiosity about the subject matter, whatever it is.  Many of us are here soley becuase we have that natural curiosity w.r.t. ALife and DB.

But I do think there are certain basics, in every subject, that every last person should always be taught even though technology makes that method unlikly to be used often in real life.  Should we stop teaching history becuase you can look up anything you want anytime on the web?  You won't always have a calculator in your pocket (or an internet connection - or maybe you will) but more importantly, for math in particular, I'll argue that some understanding of the underlying process being performed by the machine is important to hold onto at least at some level even if you use the machine all the time.  It provides some intuition as to the correctness of what the machine produces for one thing not to mention an understanding of related topics such as the precision of the answer and the execution cost of the underlying algorithim.  Perhaps not critical for everyone, but I want the engineer building the bridges I drive on to know the difference between using a 16-bit calculator and a 32-bit calculator and what that means for his/her results.  Every sci-fi series ever done has some story about meeting the civilization whose inhabitents no longer know how the machines keeping them alive work...

"Anyone who cannot cope with mathematics is not fully human.  At best he is a tolerable subhuman who has learned to wear shoes, bathe, and not make messes in the house."  Robert Heinlein, Time Enough for Love

[abyaly]

Did you read the article? The problem is that math isn't really being taught in schools. People are being trained to preform tasks that were discovered through math, but the math that led to the discovery is left out. Then they are told that's what mathematics is. It's criminal. Even at the university level, many 'applied math' courses strip out the mathematics and only deliver formulas and equations.

[EricL]

I'm on page 10.  My pushback was to Nums and Shasta's posts.

W.r.t. the article, I'll agree with author's point that mathematics in it's pure form is an art and I agree we should perhaps supliment the teaching of formulas and math-derrived facts with the art and freedom to discover for one' self how those things were derrived.  Certainly the joy of discoverry is missing in today's math education.

But for the sake of argument, I'll push back here too.  First, how do you measure this?  Everyone gets an A in art class.  Building one's intutition for the beauty of math is a wonderful and necessary thing for mathimaticians, but few people will become profesional mathimaticians in the same way few people become professional artists or musicians.  Sure, I would love a world where the pusuit of math for personal pleasure was appreciated and respected in the same way as the personal pusuit of music or art.  I'd love to stroll down to the math-walk on Sunday mornings, gazing out at the eligant works of struggling mathimaticians hocking their wares for $20...

The talented artist stands out head and shoulders above most of us in art class but most people end up painting walls not landscapes.  Contray to the author's statements, carpenters really do use triginomitry, people really do need to balance their checkbooks.  Most people don't need or care to pursue pure math principles to do these tasks in the same way I don't need artistic talent to paint my daughter's room.    I'll all for supplimenting math education with a more artistic, discovery-based approach and perhaps we would see more mathimaticians as a result, but I fear such would provide about as much value to the majority of students as art class.

[abyaly]

First, how do you measure this? Everyone gets an A in art class. Building one's intutition for the beauty of math is a wonderful and necessary thing for mathimaticians, but few people will become profesional mathimaticians in the same way few people become professional artists or musicians.

Measuring mathematical ability really is a problem. In order to make schooling 'fair' we need to adopt a quantitative method for measuring results. This usually amounts to testing. However, testing detects side effects of mathematical ability rather than mathematical ability. It is far easier to learn the side effects in order to test higher in the short run even though it is damaging in the long run. I don't know how this can be solved. Some teachers assign grades qualitatively - they use their best judgment. But since we want our teachers to be accountable for their consistency, we can't really allow this.

I'd love to stroll down to the math-walk on Sunday mornings, gazing out at the eligant works of struggling mathimaticians hocking their wares for $20...

It seems you're implying that if math was taught more like art then mathematicians would not get paid as well. Do you really think so?

I'll all for supplimenting math education with a more artistic, discovery-based approach and perhaps we would see more mathimaticians as a result, but I fear such would provide about as much value to the majority of students as art class.

Mathematical ability is useful outside of the field of mathematics. That's the premise under which math is made a required topic. The LSAT is a test that measures reasoning ability - it's no surprise that math students tend to get the highest scores on it. Reasoning ability is something that makes a person not only able to argue better, but also make better life decisions.

[Numsgil]

I'm not saying you shouldn't learn how to long divide.  But we spent months and months and months on it through 3, 4, and 5th grade.  Maybe in to 6th, I don't remember.  A month would have been sufficient.  Yep, hmm, that's how you long divide.  I don't need to practice it.  The computer does it better than I ever will anyway.  Take those months of learning to long divide (which I'm pretty sure is universally despised by teacher and student) and let's spend it doing something applied or fun or something.  Play Monopoly with more complex math (Billy is staying in 2 rooms of your 13 room hotel which costs $485 to maintain.  Determine an appropriate price).  Heck, we could put up dry wall.  Anything is more fun than long dividing!

Anyway, how humans do things by hand is usually totally different than the way machines do things anyway.  Addition involves bit identities, for instance.  Not really the way I long divide.

[asterixx]

First of all, I know I'm "new" here, but I won't let that discredit me.

Second of all, I'm being careful not to generalize and say that all schools aren't teaching mathematics. There are many schools across North America that have used what EricL is referring to as a discovery-based approach, and such schools have seen the benefits of such a style of teaching and learning. The point is, we ALL learn things better if we can find meaning in our own lives through that selfsame learning.

Thank you for sharing the essay

[Moonfisher]

The whole concept of education is relatively new in the west, it used to be reserved for the rich and the church, teaching the whole population hasn't been done long enough for us to find a propper way.
The root of the problem is that you have one teacher and 20-40 students depending on funds... and the 20-40 students all have different skills and interests.
It's obvious that if you allowed the students to spend more time on the areas they like and have the skills for, the time would be better spendt, you could be done with the university before you turn 18 and you would probably be better at what you do, the only downside being that you would have less "basic knowledge" and therefor wouldn't do well in gameshows.
I know I've never used and never plan to use aproximately 80% of what I've learned in school during my life.
When people don't know much they tend to focus a lot on whether other people know the same things, and if they don't they must be realy stupid.
And I'm not saying that if you're good at math and computers you don't need to learn anything about history and geography, since if you have the right to vote you should have some basic knowledge to understand what's going on in the world. Ofcourse personaly I would say history is being taught the wrong way in most places, trying to glorify people or periods or nations through time, and neglecting the history concerning religious powers out of fear.
But a lot of the time it will be painfully obvious that trying to teach certain people certain things is a complete waste of time.
Trying to teach a fanatic kid about biology or history is a complete waste of time if he's already got his mind set that the earth is 10000 years old and all the stuff that goes with it... is a wate of time.
Trying to teach me how to spell right, or form readable letter using a stick of ink with a hole in one end, or the world cappitals, or memorize dates of important events, or anything else that never caught my interest.... also a complete waste of time, for me, for the teacher, and for everybody who ever had to wait while someone explained it to me...
I don't think we realy spendt much time on long division where I come from, but in general there is a tendency to force whatever you where forced to learn against your will on the next generation. Otherwise you would be admitting that all that time was wasted, that several years of your life where spendt making no progress, as if a small chunk of your life is suddenly worthless.

Anyway you can argue forever about whats important to learn and what isn't, but in the end you need to consider if the person you're trying to teach can and wants to learn it.
You can't force people to remember, and if they're not interested they WILL forget eventualy.
You just can't cram 30 people into a room, hold a lecture and then believe you actualy taught people something. If you forget the fact that people only have a 15 min attention span, peple will still have a different pace which varies over time and according to the subject, they will have different skills, they may get different things from what you said, some will misunderstand what you said or just not understand it and you'll never know, because noone wants to make 30 people sit and wait around for you to ask a question.

IMO you need computers, one for each student no less can ever be acceptable, this way you can use software to assist in the teaching.
Lectures should be banned or reserved for freaks that actualy like those. Teachers should be more like tutors, walking around helping peope with their problems.
This way people can learn whatever subject they feel like today (And haven't completed) at their own pace, they can ask questions without disturbing the entire class, they can explore subjects far deeper than normal education would ever allow, they can test their own skills when they feel ready for it, asf asf....
(Ofcourse there would still be other classes without computers, personaly I would want to add one teaching social skills and manners and generaly how to live in the country you're in without upsetting the local population. IMO if in Rome learn to do as the romans or go back where you came from, I can't even believe integration is a real word, what the hell is it suposed to mean ? People don't need help to get integrated, that's something that happens naturaly over half a generation if you don't fight it.)

I know money is a big issue here, but if poeple finish school sooner they will also work for a longer part of their life, so for countries with socialist education systems it would still add up in the long run...
But I'm hoping we'll see this evolution in the near future as there will be more teachers who actualy know how to use a computer...