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July 21st, 2013, 01:07 PM   #1
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Discussion welcomed about philosophy of mathematics

Greetings MMF members and guests. Recently I had an occasion to ponder an interesting question concerning the 'philosophy of teaching and learning mathematics'. So this thread is inspired by the thread below.

viewtopic.php?f=18&t=41745&p=174281#p174281

I reproduce the post that gave birth to this thread. One of our excellent moderators , [color=#BF0000]CRGreathouse[/color] has requested i begin a new thread which (hopefully) may induce productive discussion , so , here we are.

Quote:
Originally Posted by Hoempa
Yes, this is really a warm-up question. I figured the follow up could be too hard instantaniously for a wide audience.
CRG used 7! too in his previous question. I don't know how I could have posed my questions so you could have understood them at once. Any ideas for that?
Quote:
Originally Posted by agentredlum
Yes, i have an idea, it's not etched in stone, just my humble opinion. It's been my observation that mathematicians prefer to convey the maximum amount of information using the minimum amount of language and/or formulas. This is a 13th - 18th century idea when mathematicians challenged each other for fame , glory , money and the justification of their massive ego's, I think it is time for a change in 'attitude'. The attitude should not be 'Let me make this as concise as possible' but it should be 'How do I convey my ideas to a wider audience without having to write a long essay about it'

When the attitude changes , i believe more people will like mathematics.

Now , having said all that , i think you did a fantastic job when you read the post about me not understanding the question. You gave enough information for me to figure it out on my own , without giving away the answer , thank you for that.

Examples should be essential to focus the attention of the reader , otherwise the reader may become confused. This means more work for the mathematician but the reward is worth it IMHO because it makes your field more 'friendly'.

I invite the free exchange of ideas from MMF members. What are your thoughts on the subject? Perhaps you can share an experience or 2? Maybe you can share an idea or relevant info?. I am not happy with the title of this thread so if someone can suggest a better title i will be greatful.

Looking forward to your replies , thank you for your participation.

Best Regards,
-agentmulder.
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July 21st, 2013, 11:48 PM   #2
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Re: Discussion welcomed about philosophy of mathematics

Hm, though question for the title. I don't like "How to teach mathematics" because that makes it seem to that there is one "trick" to it. One way to guarantarely mastering mathematics. Perhaps "teaching mathemetics" where teaching could be teaching yourself or teaching some-one else.

When I teach others, I think from "what do you want". From there we go use what we know to find out what we don't know. Sometimes you want to write down what you know. I'll use an example to illustrate these ideas.
Say, I would like to teach some-one to solve 3x + 7 = 2x + 8.
"what do you want" would be:
find all values of x such that 3x + 7 = 2x + 8
Use what we know would be:
- 3x + 7 = 2x + 8
- we may add numbers to an equation
- we may multiply non-zero numbers
- 2 lineair expressions have 0, 1 or infinitely many values of x such that they are equal
- the capital of the Netherlands is Amsterdam

This is where I expect you to think like (???) what is that last part?
I've come to notice that sometimes, when I let some-one play around a little come up with ideas that might not help with solving the question. Despite they'll write it down and find out themselves it might not help them with solving the answer. They might not get the third line or find it after trying some equations.

To find out what we don't know could be interpreting the problem but also solving the problem. Both should be done IMO.

Now comes the balancing. When tutoring some-one I might mention these things. But when we are solving a problem that after some thinking would give such an equation, I won't be explaining you how to solve that. One might consider it patronizing or even I could be considered a crank for it.
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July 28th, 2013, 10:51 PM   #3
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Re: Discussion welcomed about philosophy of mathematics

That knave [color=#BF0000]CRGreathouse[/color], asking for a discussion but not participating... we'll have to do something about him.

Quote:
Originally Posted by agentredlum
It's been my observation that mathematicians prefer to convey the maximum amount of information using the minimum amount of language and/or formulas. This is a 13th - 18th century idea when mathematicians challenged each other for fame , glory , money and the justification of their massive ego's, I think it is time for a change in 'attitude'. The attitude should not be 'Let me make this as concise as possible' but it should be 'How do I convey my ideas to a wider audience without having to write a long essay about it'
My views differ, not so much on outcome but motivation. Mathematicians spend a great deal of time thinking about how to think about a problem, and they present their results in a fashion which is conducive to working on that problem. This means that extraneous detail are removed, because
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August 25th, 2013, 07:09 PM   #4
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Re: Discussion welcomed about philosophy of mathematics

I have seen 7 year old kids do differential equations, Trigonometry, and geometry presented in word format.
Presented with a formatted problem, they would have no clue how to solve it. It would be literally another language.

My problem with mathematics teaching in the United States is the textbooks. 75% problems, 15% teaching the language, 10% actual theory.

Give a kid a model airplane, tell him to take it apart.
Give a kid a model airplane, tell him to put it together.

What kid is going to learn more about airplanes?

give a kid an equation, explain it with a word problem. ( this is how most teachers explain math after elementary school through most college math)
give a kid a word problem, explain it with an equation. (this is most teachers explain math in elementary school)
Get a problem, make solution (This is how math is applied in the real world)

We should teach for real world application. Give students a real world problem that requires all the skills they will learn during the course, at the start of the course. Give them individual problems in the same fashion at the start of each chapter. After they finish each chapter, always go back to course problem. What did they learn that will make it easier to solve that problem, have them explain it.

If at any point during the course, a student can solve the course problem, give them a test with 3 more of those problems. If they solve them, send them to an independent math study for the remainder of the semester. Where they will be given the next book, and the opportunity to work through it. They already got an A for the course, this will just give them an advance on the next course.
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August 25th, 2013, 08:59 PM   #5
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Re: Discussion welcomed about philosophy of mathematics

I like what you said about giving examples of all relevant problems at the beginning of the course. Give all students a practice final with detailed solutions to every problem. That way the student knows exactly what needs to be learned , then make the final comprehensive. Too often IMHO , especially in college where most students are serious about learning , the students have no clue what will be on the final and are forced to compete against the rest of the class in a 'sink or swim' philosophy enforced by the college. There is a reason why teachers refer to this as a 'cottage industry' , colleges make a ton of money from students repeating the same courses.

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August 25th, 2013, 11:58 PM   #6
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Re: Discussion welcomed about philosophy of mathematics

Quote:
Originally Posted by Justiful
We should teach for real world application
I disagree at this point. In my country, our textbook contains data charts and we need to find the weighted means from those. I have always wondered what the government is trying to make out of us -- a commerce student or something, just working as a cashier?

The main problem with the teaching of mathematics is that the society don't realize that among some average students, brilliant students are often seen.

IMO, the students must be divided in two sections, O level and E level, for ordinaries and extra-ordinaries, respectively after a certain time. The course for these two groups will be also different, as one can imagine from the names.

I also think brilliant students should be given the privileges to have their own course on their own subjects.

Quote:
Originally Posted by Justiful
Presented with a formatted problem, they would have no clue how to solve it.
What do you mean with "formatted problem"? A word problem? Well, I don't think extra-ordinary students need to solve word problem since they are going to take pure math as their own subject.

Whether or not we should teach a certain student real world application of math, we must first try to understand whether or not he likes to learn real world application of math. I agree that one should do it for a certain grade or so, but data-handling for E level students? I hate to say, but that's just nonsense.

Finally, I can say all this because I am a high-school student of this generation and I am not satisfied the way we are being educated, especially mathematics.

Balarka
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