In particular he sees mathematics very much in visual terms, in a way that goes further than the mere use of "manipulatives" - practical materials - that has been taken for granted so much since the "I do and I understand" philosophy promulgated by projects such as Nuffield in the 1960s.
I sat in on a session in which Gene presented some algebraic ideas with the use of plastic tiles. Among other similar problems, we were invited to "grow" the following sequence of shapes (see figure 1). It looks like a familiar sort of activity, but the way it is usually dealt with is to write down the sequence of numbers of tiles used, 2, 5, 10, 17, 16, ...
and look at rules for continuing the numbers before evolving a generalisation and, with luck, referring a formula back to the shapes. Gene asked us instead to look straight away at a general case, trying to see it in various ways. Suggestions therefore varied, and in terms of the number of squares in the bottom row, which we could call n, we could construct an algebraic description of what we saw (see figure 2).
The resulting expressions should all be equivalent, and in order to prove this one had to manipulate such things as "clearing of brackets", but with far more purpose than such exercises usually have.
I was also impressed by a video of a lesson given to fifth-graders by Mike Arcidiacono, in which he invited them to share ideas about a shape on a geoboard (see figure 3). The children came up in turn to the overhead projector to explain how they saw that the two triangles were congruent. The question sounds a little trivial, but the explanations were not. I was struck not only by the ease with which the children presented their ideas, but also by the attention given to the presenters by their peers. Children listened to each other, asked intelligent questions, and appeared to value alternative suggestions.
I was also impressed by two aspects of the way Mike behaved. First, he made absolutely no value judgments about the ideas, neither praising nor even thanking each child as is common practice, particularly with American teachers.
Second, he asked questions which were unusual ones for a teacher. Mike either asked questions because there was something he did not know about what the pupil was presenting, or occasionally I felt he asked for some clarification because he felt that the class would not understand something.
The videotape was being shown as a seminar for workshop leaders, teachers who were going to be running teachers' courses based on one of the Center's projects for middle schools. I was surprised no one commented on Mike's behaviour. On reflection, I can see that perhaps for those particular teachers there was nothing on which to comment: this was a normal and acceptable way of working in a classroom.
This acceptance is in a way most remarkable. I have used similar ways of working in this country in the past and actually made teachers angry by it. In the United States, where teaching styles have often been even more didactic than ours, it is a still greater measure of the change in attitudes that, at least in one project, has taken place.
I do not know how widely spread these ideas are. The Math Learning Center, now a very large organisation, works for school districts scattered around the country, including in Alaska and Hawaii.
It may be that I am experiencing a biased sample. But there have been signs of a nationwide move in the direction of reform. The National Council of Teachers of Mathematics published not so long ago two books, usually referred to as "the Standards". These are quite revolutionary for the US, which certainly in the 1970s I generally found to be dominated by textbooks, the curriculum and tests. Now, NCTM is encouraging use of manipulative materials, co-operative working, a problem-solving approach, investigational methods, integration of content. They recommended less stress on complex paper-and-pencil algorithms like that for long division, on rote learning, memorisation and practice, on "one answer one method", on worksheets, on "teaching by telling", on written problems and "the use of clue words to determine which operation to use". And testing is to be avoided when its sole purpose is to assign grades.
It is difficult, reading the Standards, not to feel that the US is moving to where we were at the time of Cockcroft, and we have been obliged to adopt some of the features NCTM is leaving behind.
My last engagement was at a seminar for high school maths chairpersons, the equivalent of our secondary heads of department, except that high schools start at about age 14. Some of the maths was very interesting. But the concerns of the teachers intrigued me. True, the organisation of high schools is quite different from that of our secondary schools, but I was surprised by the talk, in and out of sessions, of discipline, attendance, rules and grades or tests.
One teacher produced the sheet of rules she handed out to her pre-Algebra class, beginning: "ATTENDANCE:Mathematics is a difficult subject. One topic builds on another - daily practice is a must! Be in class and keep up your course work. If you miss more than ten days of class in any nine week grading period, YOU WILL FAIL!" There followed percentages of points to be awarded for tests, quizzes, assignments, class participation group work; details of tests and credits awarded; and rules for class behaviour and the sanctions to be operated for misdemeanours, ending with "Tardies": Lunch detention with the teacher!
This document had also to be signed by the student's parent. Parents in America have always been more involved with schools than ours have here, though not always to good effect. One teacher, who taught in a small rural town in eastern Oregon, told us of the time she described the misdeeds of a troublesome student to his father on parents' interview night. The large, burly man, listened to the complaints for some time, then turned to his son and said, "Well, why don't you humour the little lady?" o References: Curriculum and Evaluation Standards for School Mathematics, 1989, 0 87353 273 2. Professional Standards for Teaching Mathematics, 1989, 0 87353 307 0. From the National Council of Teachers of Mathematics, 1906 Association Drive, Reston, VA 22091, US, $25 each, MastercardVisa accepted David Fielker has retired from in-service work and is now a freelance lecturer and writer.