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Mathematics: Is Understanding Underemphasized in Our Schools?

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okaywhy
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  #11 Old 01-05-2008 Default

Understanding can mean different things to different people.

1. understanding the problem
2. understanding how to apply the theorems to solve problems
3. understanding how to derive the formulae used

Now, is having the no. 1 and 2 kind of understanding qualified enough to be considered as you have attained the goal of "understanding"?

Let's face it, understanding to the level of no.3, that is to be able to derive and prove is not relevant to most people, even most engineers, scientists and mathematicians can't do that. If we see mathematics as a tool to solve problems, understanding to no.3 is simply useless, it may help you to gain a deeper gratification or intellectual joy; but it is still, utterly useless. Do you still use the method of exhaustion to find the area of a circle? You'd rather use πr?, would you not?

As for teaching students how a formula comes about, I reckon most teachers do not have such mathematical aptitude to do so, granted that they might have the desire to do so. Mathematical proof is a very rigorous thing if it were to be taught in depth.
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  #12 Old 01-05-2008 Default

Quote:
Originally Posted by okaywhy View Post
Understanding can mean different things to different people.

1. understanding the problem
2. understanding how to apply the theorems to solve problems
3. understanding how to derive the formulae used

Now, is having the no. 1 and 2 kind of understanding qualified enough to be considered as you have attained the goal of "understanding"?

Let's face it, understanding to the level of no.3, that is to be able to derive and prove is not relevant to most people, even most engineers, scientists and mathematicians can't do that. If we see mathematics as a tool to solve problems, understanding to no.3 is simply useless, it may help you to gain a deeper gratification or intellectual joy; but it is still, utterly useless. Do you still use the method of exhaustion to find the area of a circle? You'd rather use πr?, would you not?

As for teaching students how a formula comes about, I reckon most teachers do not have such mathematical aptitude to do so, granted that they might have the desire to do so. Mathematical proof is a very rigorous thing if it were to be taught in depth.
You have misunderstood me.

I am in no way championing for everyone to be able to get the "level 3" understanding. And I did mention in the first post that people are born with different level and variety of talents and so it's ridiculous to ask for total, genius-mathematician-level of understanding in everyone. Maybe our little digression into the formulae of circle might have distorted what I originally meant in the first post. No, I don't think everyone should know master or memorise the methods of exhaustion in circular area. πr? does sound like a better idea to me.

However, what I am championing, using your terms, is that we improve the "level 1", "level 2" and a "level 2.5" in all our students - which we are not doing well, in my opinion. Let's look at them one by one:

level 1: understanding the problem

For most people, it's easy for SPM level questions. All the problems are easily understandable.

Level 2: understanding how to apply the theorems to solve (in-syllabus / past year / routine) problems

Everyone who has done enough practice are excellent in it.

Level 2.5: being able to use the knowledge on non-routine problems

This is what I am trying to emphasize in this thread - being able to use your mathematical concept (I prefer this word to formula) in real world, where not all questions are routine questions. I have a feeling that many people are able to get a perfect score for whatever standard past-year-like questions you throw to them; but when you give them a new take or a novel problem which requires no additional knowledge, they would stumble and not be able to tackle them.

To do well in level 2.5, you need a REAL understanding instead of simply a familiarity with formula plugging. When you go to more advanced levels like engineering, programming, maths, science etc, formula plugging simply doesn't work well - you will struggle, unless your lecturer happens to be ones who are happy to repeat past year questions (which unfortunately happen quite frequently in melbourne uni engineering ).

An example for what I mean:
Quote:
if you have a precise clock, between 2 to 3 o'clock, what time does the long hand and the short hand on the clock overlap perfectly? Tell me the minute and second.
This requires no more than basic algebra (or even none, actually) everyone learned since form 2. But I bet many people would not be able to solve it, because it's not the routine question you get in past year paper or pelangi practice books.

(And I imagine that if one year SPM suddenly comes up with such a question, on the subsequent years all our SPM students will turn into clock problem experts, as pelangi will start including three hundred such questions )

Yes we can very well argue that a proportion of people simply do not have the intellectual capacity to solve such problems without practise, and that I empathise and understand. However, my main point all along has been that many people, including ones who actually have the intelligence, simply lose their capacity to solve a problem as such after acculturating themselves with the "practise is the only thing you need to do" mentality in our school.

The capacity to solve such a problem is what you need in uni levels, and I sincerely believe that "just practise" mentality does explain why some undergraduates flounder in uni courses even though they did okay in SPM.

Anyway, mathematicians do the "level 3" thing day in day out - they are not called mathematician for no reason.
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  #13 Old 01-05-2008 Default

I wonder why are there so few math enthusiast on recom??
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  #14 Old 01-05-2008 Default

There are actually quite a number, if you look at the number of participants in the "math / iq problems" thread.

Anyway, I found this good page describing the derivation of formulae for spheres, cones, pyramids etc. It doesn't contain greek alphabets at all so it's suitable for everyone - and here's to everyone who doesn't know how the formulae come about! It's very interesting.

http://www.mathguide.com/lessons/Volume.html

If only our teachers taught it when they gave us the formulae.
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Last edited by youngyew; 01-05-2008 at 10:09 PM. Reason: Automerged Doublepost
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  #15 Old 02-05-2008 Default

because teachers themselves have never emphasized on understanding....

they only focus the steps, what i noticed...
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  #16 Old 02-05-2008 Default

Quote:
Originally Posted by youngyew View Post
Hey by the way, just learned Archimedes also proved that for a sphere inscribed in a cylinder, the sphere has two third the volume and surface area of the cylinder! This is new for me, haven't read about that before. This basically gives the formula for surface area and volume of a sphere. He was so proud of this result, he even asked a sphere and a cylinder to be placed at his tomb. I would like to find out how he did it.

http://en.wikipedia.org/wiki/Archimedes

Sorry for the digression.
didn't our teachers taught this in form 2?

Quote:
Originally Posted by ame.zost View Post
because teachers themselves have never emphasized on understanding....

they only focus the steps, what i noticed...
The steps are there to show you how to move around, like how you would read the instructions manual after you've bought a new gadget. On how you understand it is another matter altogether, especially in a spoon-fed society today...
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  #17 Old 02-05-2008 Default

Huh? Your teacher taught you how to derive a sphere's formula using pyramids? You must have had a great teacher.
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  #18 Old 03-05-2008 Default

I still remember my math teacher in form 2 asked the class to draw a circle, cut it into sectors of equal angles, and then paste it into sawtooth diagram as shown by bluez, to show how the formula for the area of a circle comes about. He also talked a lot about the good old days when he, as a student, had to write out every single reasoning in each step of his math homework.
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  #19 Old 03-05-2008 Default

Good math teacher. If I'm not mistaken, I think the sawtooth thing is actually in our maths textbook, and my teacher did mention it. But how about the pyramid for sphere-volume thing? I have no memory of it whatsoever from secondary school.
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  #20 Old 14-10-2009 Default Re: Mathematics: Is Understanding Underemphasized in Our Schools?

Quote:
Originally Posted by bluez_aspic View Post
Kumon is trash
ahem excuse me?

its kumon that helped me understand the area of a pyramid/cone/sphere by means of integration

apparently kumon worksheets have very simple ways for you to derive the formula of the area of regular polygons and their respective volumes as well.

with calculus, of course
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