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Ed Wall
Posts:
845
Registered:
12/3/04


Musings about Question 2
Posted:
Mar 15, 2001 11:22 PM


Professor Zhang
Before I reply to this question, I notice logical thinking has been put forward again as a reason for doing something. I am not convinced that pure mathematics develops logical thinking; however, perhaps I misunderstand your use of the word logical. Logical covers a bit of territory  there is for example, mathematical logic or symbolic logic (Hegel is known for his Science of Logic (Wissenschaft der Logik)  which is quite different from either of these) and so forth. I would even venture to say that if what you indicate is true, then practitioners of pure mathematics must be among the most logical since they have had the most practice. Unfortunately, taking logical in the larger sense, there is substantial evidence that this is not true. So, and I am open to argument, perhaps one could say justly that pure mathematics is useful in developing a logic useful for doing pure mathematics. Then a question arises: What else is this particular logic useful for? Let us say, for example, it is useful for physics. Could one argue that all the logic a top notch physicist needs is the logic of a pure mathematician? Let us say, for example, it is useful for biology? Could one argue that all the logic a top notch biologist needs is the logic of a pure mathematician? Let us say, for example, it is useful for carpentry? Could one argue that all the logic a top notch carpenter needs is the logic of a pure mathematician? So, and I am open to argument, perhaps a basic course, one in which one only experiences pure mathematics, is insufficient logically (as taken in a larger sense). Perhaps something that incorporates 'real life mathematics' might offset that insufficiency. Again there are tradeoffs that one might carefully think about.
Now 'real life mathematics.' If one wants to provide a certain kind of motivation to students at a certain point in their studies this might make sense. However, it is unclear that a problem must be real life in order for students to be mathematically interested and curious. And it is very unclear what makes a problem 'real life'  sometimes what is passed off as this is anything but 'real life.' It might, for example, a fairly standard piece of basic mathematics dressed up in the vocabulary of 'real life.' Some of the best 'real life' mathematics problems that I have seen were done in physics classrooms because often only there can you take the time to make it real. That is not to say that there aren't some possibilities and that takes me to the next topic.
Again, keep in mind that I do know some about the Chinese curriculum, but I know little about the detailed facts of enactment. What are needed, I think, are 'real interesting' mathematical problems (and by this I mean something that addresses substantial mathematics). From what I can tell, your curriculum does not always provide such (and ours doesn't either!)  however, how your teachers use the curriculum may very well be another matter (there are good math teachers no matter what the curriculum). If a 'real interesting' math problem looks 'real life' that is great. If it doesn't, that is fine also. Again, this depends on what you want mathematics to open up for students and, probably, some of what is called applied mathematics  although it is not strictly pure mathematics  has all the rigor one might desire (actually, at one time many meant by pure mathematics number theory).
Finally, I agree that 'real life' mathematics is not important at the abstract level if you mean that in the sense of a tautology; that is a play on abstract and real. However, it seems fairly clear, that human children, at often very different rates, tend to roughly move through stages of abstraction. Students in college may be at very different places than they were in high school. It seems, hence, that a teacher (and a curriculum writer) might want to pragmatically think about 'real' in a relative sense. That is, by 'real' I mean an indication of one's mathematical experience at the moment of teaching. So for a small child 'real' might be fingers and toes, and several years later the number line. And, it seems to me, going from one's fingers to the the number line is a substantial feat of abstraction.
Oh, V. A. Krutetskii (1976) The Psychology of Mathematical Abilities in Schoolchildren might provide some useful information as regards abstraction.
Ed Wall
>Question 2: What role does "real life mathematics" have in the classroom? Some would say that Chinese mathematics education is too pure. Others would say that basic mathematics must be pure. [Background] Chinese mathematics teaching emphasizes knowledge of procedures and algorithms. Real life mathematics is important, but not at the abstract level. Pure mathematics consists of algorithms. Developing logical thinking is important too.



