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I found that most math classes involve are the first third review, the second third review of old concepts but with the addition of new applications, and the last third new concepts altogether. This curriculum style is really hectically paced for me and confusing to boot. If they'd just explain stuff once clearly instead of twice quickly I think more people might get it.
I do okay in math. I like its nonsubjectiveness and its problem solving aspects, but a lot of times, it's like who cares? Nonsensical/illogical word problems do NOT make math seem more practical.
Also I think a lot of the practice makes perfect aspect of math makes it a sort of thing that really benefits from group work. I'd never understand any math if I couldn't interact with classmates/ other people in figuring out the sequencing of steps and such. It's not a lecture class!
My worst math class was with an instructor who never wrote anything down essentially. And used rulers and other props held up in the air and pointed with his fingers at said rulers in order to illustrate the graphs. (And this was class was Pre-Calc... which as its name indicates is the class before begining calculus.) And verbally discussed people's questions about problems. And used PowerPoint for the lectures. It confused the hell out of me because I need the interaction of seeing how the problem is approached and the method and the sequence. And I didn't seem to have enough simulataneous processing power to construct the mental image of the graph at the same time as seeing how it related to the equations. (For this "lack of process" reason, math textbooks are really gibberish to me.)
Another bad teacher was all about the homework and the procedure and organization (and all that other SJ stuff) and memorization of the names of the corollaries needed for geometry proofs. Boo geometry proofs.
My best math teacher (I'm excluding elementary school teachers here because I can barely remember anything about that) was my Calc 1B teacher. He hated graphing calculators (yah!) and taught the whole class by the overhead projector. Explaining the concepts and lots and lots and lots of doing problems! It wasn't rehearsed. If he had the wrong approach the first time around, he'd adjust it later. The bad thing about this was that if you were lost it was really hard to "latch" back on (not really any texts, etc) (and a LOT of people dropped the class). I hate graphing calculators because complex programming (I can never remember how to summon a certain type of graph, etc, etc). And graphing things I wouldn't even dream of being able to graph on my own (with a paper and pencil).
I think the worst thing about math is that I'm never sure if I "get" the concepts. If I'm lucky and work hard at it, I can use them, but how do you know the philosophy of math? I mean, obviously it would be quite impossible for me to be led by the hand to rediscover how people arrived at all the math milestones one by one, but I do want a little more of this bottom-up approach! (Like in Chemistry, for example: "X-person saw that certain elements had similar qualities and started organizing them.... Leading to the creation of the periodic table... and the similar qualities can be linked to their outer shell electron #.... reactivity, nonreactivity of noble gases... pi bonding, sigma bonding.)
It depends a lot from the teacher, really. My results have ranged from exceptional to rather bad depending on the teacher. I had great results with ENTJ, ISTP, ISFJ, ENTP teachers. Bad results especially with INTPs and ISTJs, these two types always focus on minutia that I am unable to pay attention to. Obviously this is my own subjective opinion - they may say that what I call minutia is essential for learning.
However, fortunately math is a subject that can be sufficiently easily learnt from textbooks.
ENTj 7-3-8 sx/sp
I tend to be suspicious of voodoo math reform, to be honest. I think most of the problems are in education in general and aren't specific to mathematical pedagogy, although perhaps the rigor of the subject exaggerates some of those problems.
You can't wait for inspiration. You have to go after it with a club. - Jack London
As far as the ability to learn math goes the only type(s) that really seem predisposed toward it are thinkers. Furthermore I think this has more to do with interest than ability. The jobs that require a lot of math are generally not jobs that would attract feelers (engineer, physicist, etc...). On the other hand among doctors there are probably more feelers than thinkers in the field, and pre-med students have to do very well in math and science to be accepted to med school. Feelers can learn math just as well as thinkers if they have the motivation. Overall I think type affects interest more than ability.
My wife and I made a game to teach kids about nutrition. Please try our game and vote for us to win. (Voting period: July 14 - August 14)
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Generally, pay teachers a better salary, and you will get more people wanting to be teachers.
Give them more respect, and again, we will attract more people wanting to be teachers. Abolish the "those who can't do, teach" mentality.
Specifically for Math, I like the "magic trick" approach:
1) Give students a fairly involved problem that they will struggle to solve without knowledge of a particular concept (like integration, for instance), and have them attempt to solve it. Some may even succeed, but either way they will have a deep insight into the concept about to be presented.
2) Then place them in groups, and have them try to solve the problem as groups. Encourage them to be creative and to transform the problem in to equivelent ones, or approximate ones, or smaller ones, etc. Have them go at it as if they were trying to solve a puzzle. Many more will likely solve the problem, without the aid of the concept being taught. Keep track of the approaches the students take that are most promosing, or interesting. Have them discuss/present the most interesting approaches.
3) Develop the concept from its founding principles. This is the usual exposition. Then try and relate the theoretical foundation to the approches students took on the problem. The teacher will likely trip up in several spots trying to do this. That's OK, and actually prefered; it shows that even proffesionals have to struggle with math sometimes, and shows how they struggle with it. Point out that there "maybe some connection here or there." But then go back and summarize the core theoretical concept concisely. Don't show them the solution to the orignal problem, yet.
4) Place them back into groups, and have them have one last go at the orignal problem, now armed with the concept just presented. For those who did just understand the concept may marvel at the "magic" of using the new concept in making the problem easier.
5) Present theoretical solution to the problem. Take questions for clarification, etc.
6) Quiz/Test the individuals on many similar problems in different contexts to the one solved. If a review of the concept is needed, then repeat steps 1-6 and a different appropriately chosen problem.
7) Quiz/Test individual on extremely diverse sets of problems requiring the same concept. If a review of the concept is needed, then repeat steps 1-7 and a different appropriately chosen problem.
To have time for this approach (which I estimate to be 3x-4x normal exposition), give students a short time to prepare for a pretest of review information, and only allow those who "pass" to take the class.
Sorry, but without the foundations, going further is often futile.
Accept the past. Live for the present. Look forward to the future.
Robot Fusion
"As our island of knowledge grows, so does the shore of our ignorance." John Wheeler
"[A] scientist looking at nonscientific problems is just as dumb as the next guy." Richard Feynman
"[P]etabytes of [] data is not the same thing as understanding emergent mechanisms and structures." Jim Crutchfield
Good stuff. Especially the trying to solve a problem before learning the concept. They just need to remember to give lots of time for that, or else the excercise is perfunctory and frustrating.
You might of mentioned it, but in calculus they should stop designing tests that tell you exactly what you'll have to do. What I think is great about the AP calculus exam is it didn't tell you what kind of problem you were looking at, so you had to use what you knew (and you needed conceptual knowledge to know how to solve the problem). Which actually matches real life, and science, where you will never be told beforehand what kind of math you will need to model something and solve a problem.
I agree with this. Perhaps there should be a supplementary history of math (or math principles, whatever you want to call it) to go along with regular math. This way you could understand exactly why and how the concepts that are so familiar were thought up and proven.
At my university they had a special set of math classes for elementary education majors where the instructor would make them go back through the principles of the basic elements of math (+,-,x, etc...) so that they could understand them thoroughly. It was amazing to see people who had taken advanced calculus courses struggle to understand why the basic division algorithm works (not the procedure, but the concept). Of course, they may have just been a dumb class.
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