D H said:
Stephen: You are arguing with Hurkyl over the correct way to say "tomato." Both his approach and yours will yield the same answers to any and all questions of this form. It's a bit silly to argue over which approach is the "right" approach since the two are ultimately equivalent.
If you are grading a student's paper and he gets the correct answer, but doesn't show work that derives it deductively, does he get the same mark as a student who shows a deductive process? I'm not saying the answer to this question is necessarily "no". But it points out how the two approaches are not equivalent.
A similar debate could be had over whether to teach students to understand the distributive law or just teach them "the FOIL method". On the one hand, given that there are many achievement tests in secondary education nowadays and that teaching precise reasoning is a burdensome task for teachers, there are pragmatic reasons for teaching students how to get the right answers quick at the expense of how to get right answers in some deductive manner. The deductive manner isn't likely to be the subject of a multiple choice question.
Suppose a person is writing a proof and must state the consequences of |3x - \delta| < |4 \epsilon + 2|. What will the correct approach be? To use the "method of opposites" and then try to "check your answer"? Granted that not all students of secondary mathematics are going to proceed into such a situation. And perhaps as students proceed in mathematics they must periodically suffer disappointments when the way they learned some things in high school is not recognized as valid.
The "method of opposites" can be fixed so it uses valid deductions. The procedure on the Mainland site could be solved by introducing a theorem: If |a] =b then a = b if and only if a >= 0 and a = -b if and only if a < 0. Then the cases to work the example would be
Case 1: 2x - 3 = x - 5 and 2x - 3 >= 0
Case 2: 2x -3 = -(x-5) and 2x -3 < 0
I don't see that's any simpler than the cases:
Case 1: 2x - 3 >= 0 and 2x-3 = x-5
Case 2: 2x -3 < 0 and -(2x-3) = x -5
I don't see that introducting the theorem and avoiding the use of the definition of the absolute value function has any pedagogical advantage. In subsequent math courses, the definition of the absolute value function will be the more important concept.
As for why they won't fix it, perhaps you are being a bit too argumentative in your messages or too adamant in telling them that the very approach they are using is wrong. That's just a perhaps; I don't know the history of your communications with them. Do you really care whether they use your approach, or Hurkyl's, or someone else's, so long as the approach they do use does yield the correct answers?
I'm not involved in secondary education now, but when I had contact with it, it always amused me to hear mathematics instructors enjoin students to "think logically" and contrast this with the way that the material is presented. It isn't crucial to me whether or how Mainland HS fixes anything. I do find it an amusing task to see if I can provoke them to do something. It's somewhat like buying a cheap product, having it break and then, just for the "fun" of it, trying to jump through all the hoops to get the manufacturer to honor a warranty. I'm a retired guy. I have time for such inane adventures.
It's silly to argue with a closed mind, but reasonable people can argue profitably.
It is not difficult to conclude "x=y or x=-y" using "proper deductive reasoning" from the equation "|x|=y".
