Quote:
Originally posted by al bundy:
Why should limit-based operations such as differentiation and integration be considered and treated in the same spirit as are the basic arithmetic operations? Many learning-theory specialists today argue that the learning of foundation level 'finger-counting' math concepts in no way resembles the learning of the concepts of higher-order mathematics branches, either in the concept assimilation stage or in workplace practic
In this sense they should be treated the same simply because they are unfamiliar operations, and the learner will not come to understand the operation's mechanism simply by seeing an input and an output. A monkey, could be taught to push the buttons of a calculator and communicate it's output, but that doesn't let him understand what multiplication is. The same thing happens when you teach a student to hit the integrate key on his calculator and write down the answer in a test blank. He needs to focus on the concept of what integration is, we agree on that, but I think he also needs to do it by hand at least a few times, so that he knows where his calculator got the number.
Quote:
Originally posted by al bundy:
Why specifically should a lower-division calculus student be made to learn a manual technique such as 'integration by parts' or 'trigonometric substitution'? That is, what purpose exactly does that serve for a student at that level, either toward their learning-development or toward their possible future professional practice?
I suppose that depends on what the student's future career is, and their personal level of curiosity. Many people can obviously get by without theese skills. I believe, however, that there is more to math than reaching an answer. Learning different ways to approach a problem, and seeing the innovation of those who have come before us, without a TI, helps to foster problem solving skillsadn a way of thinking which I believe are necesary for anyone who wishes to innovate themselves.
Quote:
Originally posted by al bundy:
Why shouldn't a modern tool, such as a computer-algebra-system on a TI-89, be used effectively to provide a student with better access to the concepts of calculus? After all, isn't it precisely the concepts underlying the techniques - rather than the techniques themselves - that should be the point of the mathematical education at that level?
My whole point is that calculators don't teach concepts, they give answers. If those answers help someone learn a more complex issue faster that's great, but I have seen many students learn which button to push so that they can pass the test, and then they completely ignore the concepts. How do you design a test that allows a TI89 and that will accuratly guage a student's knowledge of concepts? The only questions like that I've seen have been ones where the calculator isn't necesary anyway. Where calculators are important is when numbers don't come out nice and neat, like all real world data, but when you give a student a real world problem he can push buttons and give you a real world answer without the teacher knowing whether he really understands what's going on.
Quote:
Originally posted by al bundy:
Edit: At this point, a mathematics education reformer will most often take a moment to hear the replies to these questions, before continuing...
[This message has been edited by al bundy (edited August 14, 2001).]
Is 'Mathematics Education Reformer' a political party or something? It sounds like you're saying anyone who wants to reform mathematics has the same opinions you do. I'm certainly not a member of a conservative old guard or anything. I just don't believe a student should give a teacher an answer he obtained by pushing a key, when he doesn't know what the key really means.
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