I discuss some of the ways that knowledge and understanding of mathematics are brought to high school students.
Organization of material:
My opinion is that `Core-Plus' and `Interactive Mathematics Program' are organized by context. I think that this is a poor style of organization for teaching mathematics, because the context rather than the mathematics becomes the focus. [I remark that I have taught (most recently in Spring 2001) a certain mathematics course organized mostly by context (Math 1001). However, it is not the intention of this course to teach a significant amount of new mathematics in the sections organized by context, but rather to teach the contexts themselves while: (i) helping students develop skill in moving back and forth between verbal statements and mathematics; and (ii) reinforcing previously learned mathematics.]A case can be made that mathematics should be taught in an integrated fashion. For me that means, in mathematics for grades 9-12, integration from among the eleven topics mentioned in Mathematics areas in grades 9-12. `Core-plus' integrates story problems with organization of data and to some extent with statistics. I do not think it successfully integrates the other topics. This is my judgement of the situation quite apart from any view as to whether integrated treatments are desirable. From my viewpoint, the major integration issue is whether algebra and geometry should be integrated and, if so, to what extent. If they are highly integrated, problems could develop for students switching from one school to another.
To teach a subject in a spiral means in part, for me, that the topic is split into pieces, with elementary aspects taught at an early stage and more advanced aspects taught at a later stage. This is a good idea if the pieces are not so small that they lack coherency and if there is sufficient coordination so that in the later stages the earlier material is not repeated but only reviewed. Trigonometry, for instance, can first be done for right triangles, later for general triangles; still later the emphasis can be on trigonometric identities and equations, and then finally trigonometric functions can be treated. The spiral method seems to be used some in `Core-Plus' and `Interactive Mathematics Program', but I do not find the various pieces of areas sufficiently well defined.
Organization by area is not quite the opposite of an
integrated approach. For instance in my high school, trigonometry was a half year course for which two years of algebra was a prerequisite. Even though the organization was by area, I solved as many quadratic and cubic equations in trigonometry as I had in algebra, and I gave more attention there to the issue of extraneous solutions than I had in algebra.
Classroom structure: I have often taught in a group learning setting. One advantage is that students can learn good study habits from one another; for instance, the student who has previously acted as if he or she could either solve a problem in 30 seconds or else would have not chance of solving it might learn that if you think long enough about a problem a path towards a solution might appear. A disadvantage is that many incorrect mathematics statements are made, sometimes with considerable authority.
Straight lecturing in mathematics runs the danger that if a student doesn't understand a certain particular step, then he or she might have trouble following the rest of the lecture. Two partial solutions: (i) incorporate redundancy into lectures; (ii) intersperse some short discussions within lectures.
The feed-back aspect of evaluation is an important teaching technique which should receive high priority, regardless of which textbook is being used.
Labor division:
The most common activity in learning mathematics is problem-solving. However, `problem' has come to mean something longer or more difficult than something called `exercise'. Thus, if one is teaching with an emphasis on problem-solving, the implication is that more work is left for the student and the teacher is doing less that can be directly mimicked than would be the case were `exercise sets' consisting of a large number of routine problems the focus. It seems obvious to me that students should work on an array of problems (or exercises) ranging from those that are easy confidence builders to those that are very challenging. A point should be made: typically a medium difficult problem which a student is able to solve on her or his own, possibly after a struggle, is better than a very difficult problem for which the student is given a sequence of hints.There is a slightly different way in which `problem-solving' is used---namely for an approach where the student, thorough a sequence of problems some of which might be quite easy, is supposed to discover for herself or himself an important mathematical theorem or concept. This constructivist approach is asking students to do in a short time interval what the very best mathematical minds took years to do. Of course, one can remove much of the difficulty by choosing the sequence of problems with care. `Core-Plus' and `Interactive Mathematics Program' seem to lean in the direction of this approach, but it seems to me that, in many instances, closure is not achieved. I think that even an occasional use of the constructivist approach requires extreme care.
At the other extreme, it is important that the student have challenges and that the teacher not rush in to help at the first sign of a student struggling.
In the 1980's I taught in summer programs for in-service high school and junior high school teachers. The problems tended to be on the difficult side and fewer in number than in a standard course. I had three advantages in making this `few routine exercises' approach work: (i) My goal was primarily to aid the teachers in making connections among previously learned topics and increase their depth of understanding rather than add new areas of knowledge; (ii) Because of the maturity of the in-service teachers I was able to have various of them working on various problems at various levels; (iii) I wrote the problems on a day-to-day basis, so each evening as I created problems for the next day I could take account of the progress that I had seen a few hours earlier.
Messengers:
The organization of the `Core-Plus' and `Interactive Mathematics Program' is such that adults other than the mathematics teacher are removed as good sources of information for the student. Even parents who are themselves weak in mathematics can often help their children in traditional programs by guiding them in an organized way to definitions and worked examples.With all that is asked of high school teachers, they definitely need teachers guides, but they would not normally need to make constant use of them. On many topics (different ones for different teachers) they should be able to design lessons and class work by matching their own knowledge with the presentation in the students' textbook. Nevertheless I feel that high school teachers who have received an `A' from me in geometry or probability would, when treating these topics in `Core-Plus', have to make considerable use of the teachers guide in order to know at which places various important concepts are starting to appear. While this in itself is no disaster, I think it does give students confidence when the teacher can be spontaneous and still coordinated with the material the students' textbook; and this is most likely to happen when the teacher is working from her or his knowledge base, without reference to the teachers guide.
Motivation:
Uses (or applications) of mathematics can be a motivating factor for learning mathematics. However its value in this respect is easily overestimated, because it is natural when thinking of applications to highlight one's own favorites, but one person's favorite area for an application of mathematics is an area to which someone else is indifferent. The coherency and systematic character of mathematics is itself a motivating factor for many. `Core-Plus' and `Interactive Mathematics Program' stress applications to the extent that the coherency of mathematics itself is largely concealed (and much important mathematics is also crowded out).It might be that evaluation, combined with parent support, is the most important motivational tool.
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