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July 21st, 2013, 01:07 PM  #1  
Math Team Joined: Jul 2011 From: North America, 42nd parallel Posts: 3,372 Thanks: 234  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:
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Looking forward to your replies , thank you for your participation. Best Regards, agentmulder.  
July 21st, 2013, 11:48 PM  #2 
Math Team Joined: Apr 2010 Posts: 2,780 Thanks: 361  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 someone 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 someone 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 nonzero 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 someone 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 someone 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. 
July 28th, 2013, 10:51 PM  #3  
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  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:
 
August 25th, 2013, 07:09 PM  #4 
Newbie Joined: Aug 2013 Posts: 29 Thanks: 0  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. 
August 25th, 2013, 08:59 PM  #5 
Math Team Joined: Jul 2011 From: North America, 42nd parallel Posts: 3,372 Thanks: 234  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. 
August 25th, 2013, 11:58 PM  #6  
Math Team Joined: Mar 2012 From: India, West Bengal Posts: 3,871 Thanks: 86 Math Focus: Number Theory  Re: Discussion welcomed about philosophy of mathematics Quote:
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 extraordinaries, 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:
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 datahandling for E level students? I hate to say, but that's just nonsense. Finally, I can say all this because I am a highschool student of this generation and I am not satisfied the way we are being educated, especially mathematics. Balarka .  

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