> I am currently taking a math methods class here at Concordia > College in Moorhead, MN. We have talked considerably about how > technology should and is used in the classroom. . . > > Another question I pose to this group is: At what point do you > use "machines" to do the work for you and your students? I have heard > a horror story about how a college student does not know his/her > multiplication table. One of the main reasons comes from the fact that > the teachers in his/her elementary school used calculators. This example > was to clarify my question at the start.
I can only give you my somewhat poorly defined thoughts on the matter. I choose "machine" to include any method or device which produces a result by methods unknown to user but when operated under appropriate conditions produces the "correct" result. For example, I consider trigonometric tables, calculators, and most mensuration formulae to be machines.
Condition 1) A machine should be used when it is impossible to complete the task without it.
Use of a table of trig values or a calculator is certainly necessary to complete this unless you have a solid understanding of Maclaurin series and wish to use it to construct a power series, analyze the remainder term, and then approximate the proper value. I suppose that you could give the students a correct Taylor polynomial, but that's substituting one black box for another.
Condition 2) A machine should be used as a time management tool.
I, personally, am perfectly capable of dividing 192.46 into 28932.2389; I understand what division means and how this particular one fits into the context at hand. It is of more value to me to do something else besides the division, like interpret and solve another exercise/problem. The key here is understanding what needs to be done first *and* consciously making the decision to save time.
Condition 3) A machine should be used to enhance breadth in problem solving.
Perhaps this is really a combination of conditions 1 and 2. What is the volume of a sphere of radius 3? The formula we all use is 4 pi r^3 / 3. How many of us have seen it derived without calculus? Is is necessary to derive every volume formula if we understand volume and have derived _some_ volumes? I, personally, don't think so.
What I mean here is that millions of people have worked on various facets and special cases of common, useful, central concepts. Many have created results which are available to me. While replicating their work would be valuable, if we all spent our time replicting previously done work, it would seem to me that little if any new ground would ever get broken. Especially if we do this at the expense of tackling new problems; new problems often lend insight to old ones. After all, how many proofs of the pythagorean theorem do you know?
Some quick comments: I confess that under the notion of impossible I don't mean impossible for any human; rather I mean impossible for the majority of people even if they had completely mastered all techniques, concepts, and facts previously exposed to them (at least what I think they probably were exposed to) when a task is most likely first encountered. I would restate that in a less confusing way if I could think of one.
I am not afraid to blindly use a machine (electronic, printed, algorithmic) if I understand when it will give a valid result and I have a reasonable expectation that no errors were made in construction. So much for using many pentium based computers.
I think that this machine question ties in with the division algortihm thread which ran the week before last. I want to make it clear that I am not advocating pure teaching of black box machinery. If you read those earlier postings, you will recall that I defended in depth understanding of that algorithm, as did several others.
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