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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)

#51

Post by dhimes »

yequalsx wrote: post #34
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…

"I think where math goes wrong is displayed beautifully in the rational expressions units (I didn't go to the original link, so forgive me if I'm entirely redundant here). When I was building the math content for the ed s/w company I was struck that nowhere in the chapter where students learn to simplify fractions involving a polynomial numerator and a polynomial denominator (tough stuff when you are first learning i…

One thing that was quite easy to do is to set the rational expression equal to something and solve for one of the variables. We did a lot of problems where there was only one variable: (x^2+9x+20)/(x^2-25)

In motivating them you don't necessarily have to give every reason to learn something, just reason enough to buy into what you are trying to teach them.

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

#52

Post by dhimes »

yequalsx wrote: post #44
yaks_hairbrush wrote: post #36

Earlier quoted context omitted.

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 i…

The motivation for learning to manipulate rational expressions has to come well before you get to this.

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

#53

Post by jimbokun »

yequalsx wrote: post #41
jimbokun wrote: post #38

Earlier quoted context omitted.

"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?

I think it is.

If you can't perform the algorithms yourself, I don't think you really understand what the symbols on the machine really mean.

When I worked at a bank, I remember complaints about an employee whose job involved running certain calculations in a spreadsheet. Sometimes the numbers would be off orders of magnitude due to some error somewhere, but the employee would just carry on as normal. If you have some clue as to what the calculations are supposed to be doing, there should be some alarms going off if the numbers are not at least reasonable.

Or what if you just wrote a program that includes some arithmetic calculations, and need to verify that you are getting the right answers for your test data?

I remember another time at that same bank, where a developer did not know that multiplication had precedence over addition, which led to errors in a financial calculation.

I'm sure you could easily think up many more examples like this.

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

#54

Post by yequalsx »

jimbokun wrote: post #53
yequalsx wrote: post #41

Earlier quoted context omitted.

But is it worthwhile or important?

I think it is. If you can't perform the algorithms yourself, I don't think you really understand what the symbols on the machine really mean. When I worked at a bank, I remember complaints about an employee whose job involved running certain calculations in a spreadsheet. Sometimes the numbers would be off orders of magnitude due to some error somewhere, but the employee would just carry on as normal. If you have som…

There was a time when it was important to know how to use interpolating polynomials. Rationalizing the denominator was only useful before calculator became ubiquitous. Being able to make fire was without use of matches or flint used to be essential.

There are plenty of ways to teach algorithms without using arithmetic. Maybe a knowledge of doing basic arithmetic by hand is essential to understanding higher math. I don't know. It's worth exploring though.

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

#55

Post by yequalsx »

dhimes wrote: post #51
yequalsx wrote: post #34

Earlier quoted context omitted.

"I think where math goes wrong is displayed beautifully in the rational expressions units (I didn't go to the original link, so forgive me if I'm entirely redundant here). When I was building the math content for the ed s/w company I was struck that nowhere in the chapter where students learn to simplify fractions involving a polynomial numerator and a polynomial denominator (tough stuff when you are first learning i…

One thing that was quite easy to do is to set the rational expression equal to something and solve for one of the variables. We did a lot of problems where there was only one variable: (x^2+9x+20)/(x^2-25) In motivating them you don't necessarily have to give every reason to learn something, just reason enough to buy into what you are trying to teach them.

But the motivation for solving complicated rational equations does not come from word problems. The word problems for this topic involve very simple rational functions. My point has been that the motivation for studying and doing much of mathematics ought not come, solely, from practical word problems.

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

#56

Post by dhimes »

yequalsx wrote: post #55
dhimes wrote: post #51

Earlier quoted context omitted.

One thing that was quite easy to do is to set the rational expression equal to something and solve for one of the variables. We did a lot of problems where there was only one variable: (x^2+9x+20)/(x^2-25) In motivating them you don't necessarily have to give every reason to learn something, just reason enough to buy into what you are trying to teach them.

But the motivation for solving complicated rational equations does not come from word problems. The word problems for this topic involve very simple rational functions. My point has been that the motivation for studying and doing much of mathematics ought not come, solely, from practical word problems.

I agree with you that you (we) shouldn't feel the need to have a one-to-one mapping between problems we solve and word problems. Even in elementary physics you could argue that the word problems in many cases aren't practical, but we teach it to teach the flavor of the approach; the attitude of how we attack problems (or, less poetically, what exactly we mean when we say "cause and effect").

That said, the motivation is still greater when the students see some hope that what they are learning is meaningful.

What that motivation is depends on the level of the class. For algebra 2, it might be "we learn to manipulate rational functions because it gives us a tool to understand (or solve) a certain class of problems. Someday you may own a business where you have to worry about the average cost of something that depends on things that change alot, or variables as we call them. And you know what an average is: it's the amount of something divided by how many there are. Well if your amounts are represented by an algebraic expression, and your total is represented by an algebraic expression, then the quantity you will be interested in will look something like this (writes a rational function on the board). Now, what the fuck do we do with this?"

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

#57

Post by jey »

flatline wrote: post #14
anuleczka wrote: post #13

Earlier quoted context omitted.

In a world where Wolfram Alpha can do all the algebraic calculations for you, isn't learning to solve problems more relevant? I'd say that problem solving is the core of mathematics. Am I wrong?

The conceptual understanding is certainly essential, but if you want to use a computer to do symbolic/algebraic manipulation, how could you be sure to understand the results or even properly phrase the question without a thorough understanding of the symbols and the rules for their manipulation? The best I think that something like Alpha could do is to help elucidate the underlying principles or give you a quick answ…

You contradict yourself. As you said, one does not need to be skilled in combinatorial search, er I mean, algebraic manipulations to be able to solve problems; one only needs a solid conceptual grounding and understand the symbols and the rules for their manipulation. I wouldn't be surprised if most students who pass college calculus courses with good grades are entirely unable to recognize the situations where calculus is applicable.
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