martyn@garfield.UUCP (Martyn Quigley) (02/20/88)
In article <660@cresswell.quintus.UUCP> ok@quintus.UUCP (Richard A. O'Keefe) writes: [Stuff about the desirability of not teaching mathematics via long drills] >I don't know how it's done here, but back home we picked up the >"New Mathematics" where children are taught all about sets and converting >to different bases and how Egyptians wrote numbers and what a commutative >operator is. Talk about *boring*. Talk about *remote* from the interests >of the children. This is, of course, one of the main reasons why the New Math failed; it taught somewhat different things, BUT IT TAUGHT THEM IN THE SAME WAY as the old "boring" stuff. >My experience of University-level mathematics was >that I was constantly appealing to my understanding of ordinary arithmetic >for analogies. >My mathematical >and computational "intuitions" are rooted in the *experience* of arithmetic. Of course, ALL your mathematical intuitions are rooted in your experiences. Read Piaget, Bruner, Dienes, Gagne, Skemp etc. One reason so many people are weak in mathematics is that as it was taught to them, it was NOT related to their prior experiences. Hands up everyone whose first class in group theory began "A group is a set G, with a binary operation * st (i) for all g, h in G, g*h is in G etc etc etc". My experience of teaching mathematics majors is that none of them taught in this way can tell me where to find a group in "the real world", what a group IS, or what it is used for. Ask the nearest mathematics student what an eigenvector has to do with his/her/its reflection in a mirror. To paraphrase Piaget, teaching a child something it has not already met in the course of its spontaneous development is a waste of time. You cannot teach a concept from a definition. >I think we need three things in elementary arithmetic teaching: > principles: "This is WHY the addition algorithm works." So why does it? My experience (as a teacher trainer) is that relatively few elementary teachers know how it works. As for multiplication or division... > relevance: Accounting/shopping/planning examples. > "How to Lie with Statisitics" No argument from me here. Huff's classic ought to be compulsory reading for all stats students. > drills! In computation. In comparison. In estimation. Drills are appropriate if what you want is high performance in a specific task such as in piano playing, figure skating or mental arithmetic. However if what you want is the ability to solve problems, then what you give is a lot of problems, which, by definition, are not amenable to solution by routine methods. As pointed out by a previous poster, 50 years ago facility with arithmetic was a marketable commodity, therefore teaching arithmetic was a useful thing to do and since speed and accuracy were at a premium, drills were relevant. Today, arithmetic has largely vanished from the workplace, it is just not cost effective to pay humans to do arithmetic. Pay them to do the hard stuff. However, the education machine has such huge momentum that getting it to change course is a little hard. Perhaps this discussion IS relevant to comp.ai since it is AI that has taught us just how hard it is to pin down what happens when learning takes place, or indeed, when almost any cognitive process takes place. Martyn ------ martyn@garfield.uucp mquigley@mun.bitnet