We know that there is a problem in the understanding of mathematics topics for students at colleges of universities.
There are those who do not like mathematics science.
How do we reconcile to overcome these problems?
It is a difficult topic. I like Paul's take on this
http://math.cmu.edu/~bwsulliv/halmos-what-is-teaching.pdf
But of course, I have my own opinions too.
``The most effective way to teach mathematics by problem solving is to keep challenging students with problems which are barely within their reach".
The best topics are the topic related to the subject they are studying. I studied mathematics during my first year of pharmacy and It didn't help me in my career because it was pure abstract maths that has no utility.
Trying to give the students homeworks and asking them to solve on the board questions in front of the students
Personally, I like A LOT OF EXAMPLES. Also, class notes should be hand written and not just posted as a PDF file on the course website.
I do not use notes in general. If I have to use notes then I do not even think about issues like these. I think one needs to first work on eliminating that out of the way. I like to think of a magician who is trying to do a trick but he does not remember the steps.
The Halmos article is interesting -- I studied Topology with Halmos using his method. The result? Yes, for people who were engaged it worked really well. I certainly attribute my expertise in the sublect to the class and his teaching. But, for others they got confused, frustrated and dropped out. Sure, they shouldn't have been researchers and figured that out by the time they dropped. If the question is "how to we teach engaged, motivated Math students" that has a completely different answer than "how to we teach bored students who have been forced to take this class?" Halmos' method will pretty much depopulate any class of the second sort. So which question are we addressing here?
Very good point! The answer to the first question is obviously.... do what ever you want. Those students will learn by themslves and throw you out of the class or pretend that they need your help once in a while. For the second question.... I guess this is the question that a lot of people are asking but they do not want to emphasize things this way, for various reasons, ... I am back to square one. I think I have some idea but I would like to first define things even further: "what is it that we want to teach bored students who have been forced to take ou class ?"
Do we want to teach them the same material with the same expectations regardless, or we wnat them to learn at least (1/2) of the material (1/2) of the ideas, (1/2) of the skils, etc..... Once we go that road... we are doomed.
And I think this is the reality....
I tend not to care about my audience. I try to treat them all as ``engaged, motivated Math students", but of course, my jobs becomes a lot more complicated if they are not. I like to do things like a magician: you put the same show.... all the time. A magician does not choose the audience; He Pledges, and the does The Turn and finally he is in his Prestige act. 1:08 http://www.youtube.com/watch?v=o4gHCmTQDVI
I believe it is worth understanding math (or a fraction of it) for its own sake, not just because it is related to a field we are interested in. It is another question whether we have had/will have good teachers - it certainly helps - but not having them is not an apology to remain disinterested and relate math to a bad experience. I often hear people saying that math is terrible and of no value just because they don't need it or they failed (to understand it). I'll never understand how the latter justifies the former just like they'll never understand that what they are saying makes no sense.
I think teachers should spend more time explaining concepts behind certain problems rather than just solving them - explain the various methods, notations (if there is a variety), where they apply, why they are useful, etc. Understanding should matter, not memorization. I am not sure homework always works. Hurrying through lessons as a method of teaching is definitely not helpful to anyone who wants to learn, not to mention a recurring problem - lecturers tend to explain math to themselves, not to the audience. Finally, math anxiety is an issue which has to be dealt with.
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