A Discovery lesson on Matrix Inverses

In the Madison Project text Explorations in Mathematics there are several chapters dealing with the system of 2 × 2 matrices. 1 When investigating some of these chapters with fifth or sixth graders, I have always been amazed to see how quickly they can figure out the identity matrix for multiplication; on the other hand, it seems very difficult for these students to figure out the inverse of a 2 × 2 matrix. The technique proposed in the text of just presenting two matrices whose product is the identity seems too arbitrary for my pedagogical taste, so I have preferred to go through another procedure of looking for the inverse to particular matrices and then trying to find relationships between the elements of the two matrices.

Forum on teacher preparation: Topics in geometry for teachers—a new experience in mathematics education

Can teachers capture by themselves the excited enthusiasm shown by children in classes sponsored by such curriculum groups as the Madison Project or Nuffield Project? Can a teacher reared on lecturedrill-homework classes feel and show the drama inherent in “I do and I understand” activities, in peer-group discussions, and in concepts such as the concrete-ikonic foundation of abstraction? A new course at UCLA is focusing on these dynamic factors on that teachers will know their value from their own personal experiences and feelings.

Signed Numbers: A “Product” of Misconceptions

One of the first encounters with a mathematical idea that jars a student's intuition takes place when he is asked to accept that the product of two negative numbers is positive. There are, of course, many different explanations at various levels of abstraction that have been used to persuade the neophyte of the reasonableness of the assertion. One can offer an argument based upon a model, and this is done in such programs as UICSM and Madison Project.1 One can offer a plausibility argument based upon the behavior of positive numbers and faith in some sort of continuity principle, as suggested by Dubisch.2 One can (quite ironically, given the context that “modern” mathematics is an attempt to replace rules by explanations and justifications) assert, as Courant and Robbins do, that, though appeal may be made to various channels for psychological reasons, ultimately the conclusion is a matter of definition.3 The prime psychological motivation would be the desire to preserve principles already existing in the set of positive reals, as is done by SMSG.4

Graphing linear equations— discovery lesson

A discovery lesson sometimes turns out to be a discovery for the teacher as well. This was my recent experience. In my three seventh-grade classes and one eighth-grade class, I devote one lesson each week to teaching modern mathematics using Madison Project materials.

A rationale in working with signed numbers

In the teaching of signed numbers to children in Grades 4-8, I have used the “Postman Stories” model devised by the Madison Project.