On 28 Jan 2007 at 5:42, Allen Esterson wrote: > There are so many fallacies in "contrarian" Stephen Black´s response on > TERC that I scarcely know where to begin. (Sorry Stephen!):
One of the consequences of taking a contrarian position is that people are going to dump on you for taking it. I would have been disappointed if no one did. Fortunately, Allen and some others have spared me that ignominious fate. Yet if people are going to dump, it's only fair that they do so on the basis of what I said, not on what they would have liked me to have said, so that, in truthiness, they can indignantly refute it. And in this evidence-free environment, one person's assertions all too readily can be labeled "fallacies" by another. Allen's fallacies-- er, I mean assertions--can be similarly challenged. I never said that I'm opposed to all rote learning in mathematics education. Basic facility with, for example, the multiplication and division tables is essential. What I argued is that the particular algorithms we were taught for long multiplication and division have had their day. As we regretfully retired the vinyl record for the DVD, and the slide rule for the calculator, so must we retire these two ancient algorithms. I still play my vinyl records on occasion but the generation now being born will find them as quain and useless as we find the gramaphone. Tellingly, no one answered my question concerning how many times they carry out long division and multiplication by hand. But I know the answer: rarely or never. It might be different if these algorithms had some value in illuminating the structure of our number system. But they don't. We can teach students to see how these methods depend on this organization, but that's not the same thing. If the reason for teaching a particular algorithm is that it helps understand how numbers work, I'm confident that there are better methods to do this than these two. Their main benefit was efficiency, not illumination, and that efficiency has had its day and been replaced by more efficient technology. Similarly, if understanding is our aim, there are undoubtedly better ways to make students understand about remainders (Jim Clark's concern) than by teaching them the algorithm for long division. > It is nonsense to say that the "algorithm" approach is "just a set of > instructions to be followed by rote." Any half-way decent teacher will > go through the procedure with simple examples, showing at each stage > the mathematical sense of the procedure, then moving on to more > difficult examples, until the process becomes automatic. Ideally there > doesn't need to be *any* learning by rote, the algorithm is picked up by > practice. Well, the Random House Dictionary defines an algorithm as "a set of rules for solving a problem in a finite number of steps", which is pretty close to my definition. I don't see the point of the distinction that Allen makes between "learning by rote" and "picked up by practice". The end result is the same. The student follows a series of steps to an answer. The student can be made to understand why it works, but doesn't have to in order to get the answer. As I said above, if the reason for teaching the algorithm is because it helps the student to understand, there are better methods available to achieve enlightenment. > It is simply not possible to acquire a reasonable mastery of algebra > unless there is some facility in using numbers *without calculators*. Absolutely. But that doesn't preclude abandoning an obsolete set of instructions which are followed automatically to an answer. > To reiterate: There is absolutely no reason why using an algorithm > should be "learning by rote". And for one more iteration, however you get there, the end result is that the student now has a set of rules which are automatically applied to produce a result. That hard-won illumination imparted by the teacher in the process is likely to fade pretty fast. How many of us can now explain why they work? Does this impair our ability to use mathematics? > First, using calculators is not a "reliable" way of "getting the right > answers". I can´t tell you how many times I had students getting wrong > answers (occasionally absurd answers) because they have done something > wrong, and they don´t have sufficient *understanding* of the problem > they are doing to realise they have got it wrong. Using calculators is a skill which students should have. I see no reason why they can't be taught to perform this skill with high reliability. Second, I don't see how knowing the old algorithms for long division and multiplication will help them recognize when they've gone wrong. Good estimating skills would, but that's something different. No, I think the problem is that the old generation has trouble letting go. I can see the Egyptian scribes in the shadow of the just-erected pyramids telling their young students, "before you can make scratch marks on papyrus (or on clay tablets), you first have to learn how to do it with stones. How else are you going to be able to understand counting?" Stephen ----------------------------------------------------------------- Stephen L. Black, Ph.D. Department of Psychology Bishop's University e-mail: [EMAIL PROTECTED] 2600 College St. Sherbrooke QC J1M 0C8 Canada Dept web page at http://www.ubishops.ca/ccc/div/soc/psy TIPS discussion list for psychology teachers at http://faculty.frostburg.edu/psyc/southerly/tips/index.htm ----------------------------------------------------------------------- --- To make changes to your subscription go to: http://acsun.frostburg.edu/cgi-bin/lyris.pl?enter=tips&text_mode=0&lang=english
