| 8:21a |
That Tortoise and that Hare; more math-numbness; part four...
When you fully understand some chunk of factual information, it's natural to be convinced that it has to be possible to put that understanding into words in such a way that a highly-motivated, hard-working, reasonably intelligent student will be able to understand it too. This is especially true when the information seems to you to be overwhelmingly obvious. And the usual course of events is that the more times you try to explain that chunk to the student, the more obvious it seems to you to be, and the more improbable it seems that the student could keep saying "Sorry; I still don't get it." For any good teacher, this is one of the most frustrating of all teaching scenarios; once in a blessed while it leads to the writing and publication of an excellent new textbook.
Several of you have asked me how I would deal with it if I had a student who was convinced that the terms "phoneme" and "sound" and "letter" meant the same thing, and who seemed unable to understand the differences among them despite genuinely wanting to understand and working hard to do so. I've had students like that, and I can say honestly that I've never had one for whom I wasn't able to clear up the difficulty. But I can explain why that is the case.
Always, my attempts to help these students took place offline, face to face. That meant that the instant the student started perceiving what I was saying as boring or baffling or full of jargon or scary or cross -- or in any other way the opposite of helpful -- I was able to tell that that was happening. I was able to stop talking and ask the student to help me understand what I was doing wrong, so that I could correct whatever aspect of my communication was standing in the way of the student's learning. If that meant re-adjusting my explanation dozens of times in the course of half an hour, if it meant working together through multiple lengthy sessions, that was fine with me; it took as long as it took, and if the student was willing to be patient with me I was willing to be patient in return. The two of us were always able to work it out, finally, so that we both understood. And I always learned at least as much from the experience as my student learned.
All of you who have been trying to help me with the Tortoise/Hare math puzzle, and with the various other math puzzles that grew out of it as we went along, have had to get along without that advantage. You've been trying to clarify those chunks of information for me without having any face-to-face access to my reactions and responses. It may be that that's an impossible teaching situation. [Consider the so-called "documentation" that comes with software programs, which is intended to explain chunks of information under the same conditions.] I suspect that it's likely to work only in the rare and special case of a teacher who has worked out a near-perfect explanation over the course of years of offline face-to-face experience explaining a certain chunk of information, has already encountered every one of the possible difficulties a student could have with it, and has already worked out reliable and effective ways to handle each of those difficulties.
I am grateful to you for your patience and persistence in your efforts to help me with this problem. I don't think it is a failure on the part of any one of you that I still don't understand how it can be true that "Though Hare is only twice as likely to win as Tortoise, his odds of winning are four times better." I understand the parameters of the puzzle far better than I did at the beginning of the discussion. I understand that the problem I'm having revolves around a technical term from mathematics -- at least I understand that that's the group consensus on the matter. I understand a number of other statistical terms and concepts better than I did when we started. I am aware of dangers I was no aware of when we started. Progress has unquestionably been made, despite the fact that I still perceive the original Tortoise/Hare statement as a Zen koan.
The fact that there was progress, even though the only teaching mechanism available to you was written language, means that you should, one and all, be congratulated. And there are all the other benefits -- the positive side effects -- that have been mentioned.
You have my most sincere thanks, and my most sincere admiration. |