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Dan Meyer dissects the flaws of math textbooks (video)

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Re: Dan Meyer dissects the flaws of math textbooks (video)

#41

Post by yequalsx »

jimbokun wrote: post #38
yequalsx wrote: post #17

Earlier quoted context omitted.

Yeah, Wolframalpha is a disruptive technology. Is it still relevant to learn algebraic manipulations? I don't know. There are compelling arguments for both points of view. When you say problem solving do you really mean so called practical applications? If so then I disagree. If you mean solving problems to understand the essence of mathematical ideas then I agree. For instance, linear equations are the easiest equat…

"Yeah, Wolframalpha is a disruptive technology. Is it still relevant to learn algebraic manipulations?" We've had calculators for decades now, and my kids are still learning how to do arithmetic.

But is it worthwhile or important?

Re: Dan Meyer dissects the flaws of math textbooks (video)

#42

Post by yequalsx »

olliesaunders wrote: post #40
yequalsx wrote: post #26

Earlier quoted context omitted.

Great generalization and oversimplification. How about this oversimplification? Or you can teach a generation of students that if isn't 'practical' then it shouldn't be done and certainly isn't worth any effort or struggle. Hopefully then, we can eliminate poetry, music, art, literature, and p.e. We have practical problems to solve!

I consider engaging and consuming forms of art akin to play. At first sight to spend lots of time playing might be seen as a waste of time but the fact that a child prevented from playing would fail to develop properly says something about its purpose. Basically I don't agree that those subjects are purposeless.

That's great that you consider this. Many do not consider poetry, music, art classes worthwhile because they consider those classes impractical.

I don't agree that learning how to abstractly apply rules and manipulate objects is useless. It's particularly important in programming, for instance.

Re: Dan Meyer dissects the flaws of math textbooks (video)

#43

Post by yequalsx »

glyn wrote: post #39
yequalsx wrote: post #6

I'm a teacher of mathematics at a community college. It appears to me that the crux of the problem is that people, including the guy in the video, confuse problem solving with mathematics. The utility of mathematics comes in the remarkable fact that a great deal of phenomena can be adequately modeled mathematically. The focus of an algebra class ought to be in learning the language of algebra. That is, in manipulatin…

While I'd agree that students often approach word problems by basically memorizing mappings between a problem class and a small set of equations, I'm not sure that merely teaching them the mechanics of mathematics in the abstract is sufficient for imparting a thorough, practical understanding of the subject. My experience with math-related courses was that I had the greatest difficulty generalizing the word-problem a…

I agree with you. The problem I've encountered is that students who lack intellectual curiosity aren't motivated by the practical applications found in algebra textbooks. They aren't motivated by any sort of problem. So there is a reduction to the lowest denominator. That is, we give word problems that are reducible to mappings to a small set of equations.

Re: Dan Meyer dissects the flaws of math textbooks (video)

#44

Post by yequalsx »

yaks_hairbrush wrote: post #36
dhimes wrote: post #33

Earlier quoted context omitted.

I taught physics in a community college as a full-time prof for 10 years, and had the occasion to teach math there a few times as well. After that stint, I started work with an ed software company developing math content for algebra students in the cc. In my experience, the "word problems" can be used to motivate the students as to the interesting part of what they are studying. My focus wasn't on making them learn h…

In response to yequalsx: I see 2 applications of rational expressions off the top of my head: 1) From an abstract viewpoint, the message is "We can do the same stuff with polynomials that we do with integers. The only issue is that factoring is harder." This message gets very garbled 2) Solving ratio problems A/B = C/D come up very often. For example: I've won 43/87 freecell games. How many in a row do I have to win…

I agree with both of these examples. And here is my point. The sort of practical applications of rational functions involve very simple rational functions. So if our motivation comes solely from word problems then the natural question is why are we learning to manipulate

( x^2 + 2xy + y^2)/(x - y) + 1/(x^2 - y^2)

Has there ever occurred a practical application in which this expression has come up? If our motivation is expanding our understanding of manipulating algebraic objects then this expression is very much a practical application. It's an extension of an existing idea - rational numbers - to a new class of objects. Be able to do this sort of abstraction is useful in mathematics, and in fact is the essence of mathematical thought. It's also useful in things like programming.

Re: Dan Meyer dissects the flaws of math textbooks (video)

#45

Post by yequalsx »

kscaldef wrote: post #35
yequalsx wrote: post #20

Earlier quoted context omitted.

Do they really learn such motivation from solving word problems? Especially when many word problems are not practical or even correct? How about this for motivation, we solve such equations because we can. Solving equations is useful and the more equations we can solve the better. It's not really motivating when we give a word problem whose model is a radical equation and then say solve the equation. Most equations c…

Your second paragraph contradicts itself. You say solving equations is useful. So, why not show your students the use?

Having the ability to solve equations is useful. Randomly write down an equation. That particular equation isn't likely to have practical applications. Is it worthwhile to explore whether or not it can be solved by algebraic methods?

Why is it that first, second, third, and fourth degree polynomials can be solved by algebraic methods but not arbitrary polynomials of higher degree? Is this worth studying only if it has practical applications? It does have applications today but it didn't for the first thousand years this problem was tackled. The requirement that something be 'useful' before it is worthy to be studied is too great a burden. It's a detriment to intellectualism.

Re: Dan Meyer dissects the flaws of math textbooks (video)

#46

Post by nitrogen »

yequalsx wrote: post #42
olliesaunders wrote: post #40

Earlier quoted context omitted.

I consider engaging and consuming forms of art akin to play. At first sight to spend lots of time playing might be seen as a waste of time but the fact that a child prevented from playing would fail to develop properly says something about its purpose. Basically I don't agree that those subjects are purposeless.

That's great that you consider this. Many do not consider poetry, music, art classes worthwhile because they consider those classes impractical. I don't agree that learning how to abstractly apply rules and manipulate objects is useless. It's particularly important in programming, for instance.

I think the other poster was suggesting that students will perceive abstract symbolic manipulation as useless unless they're taught why it isn't useless, probably via application problems. Considering that the majority of students will never have a genuine interest in math for its own sake, I think that textbooks and curricula designed for public education should include high doses of application problems, with enough theoretical problems to interest the true mathematicians and verify the students' understanding of the abstract concepts.

Re: Dan Meyer dissects the flaws of math textbooks (video)

#47

Post by nitrogen »

argv_empty wrote: post #37
wortiz wrote: post #24

Earlier quoted context omitted.

Then perhaps you and him are assigning the wrong word problems or not giving proper introductions to said word problems.

If all students are taught to do is "memorization of problem types" (to borrow GP's term), then they will only be able to handle actual problems that match one of the necessarily finite set of types of problems they have memorized. A different selection of problem types to memorize will not provide general coverage -- that requires learning to construct a solution to a new type of problem.

Nowhere in the thread has it been demonstrated that it would be impossible to design a word problem that does require the construction of a solution.

One of the problems I had with my dif. eq. class was that the grad student teaching the class just focused on "tank problems" and other "x problems" sets. So I see why story problems alone are not helpful. But, if they're judiciously used to encourage the understanding of abstract mathematical concepts, I think they are invaluable.

Re: Dan Meyer dissects the flaws of math textbooks (video)

#50

Post by stan_rogers »

yequalsx wrote: post #6

I'm a teacher of mathematics at a community college. It appears to me that the crux of the problem is that people, including the guy in the video, confuse problem solving with mathematics. The utility of mathematics comes in the remarkable fact that a great deal of phenomena can be adequately modeled mathematically. The focus of an algebra class ought to be in learning the language of algebra. That is, in manipulatin…

I wrote a bit about this a few years ago, before my stroke:

http://stanrogers.blogspot.com/2005/03/girls-math-stuff.html

Warning: the site in general is a personal blog, so don't expect to see much else of interest to anyone.

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