Robert B. Davis

Using Videotapes to Study the Construction of Mathematical Knowledge by Individual Children Working in Groups

Videotaping small groups of students in a regular classroom environment makes it possible to study individual student cognitive growth in a social setting. The present report deals with student development of some new mathematical ideas over an extended period of time.

Chapter 6: Building Representations of Children's Meanings

In Chapter 5 (Davis & Maher, this volume) we suggested the importance of paying attention to the fine detail of childrens ideas. We have given instances of videotaped episodes of children engaged in thinking about mathematics as they worked to construct a solution to a problem. An analysis of the childrens mathematical behavior has given us some insight into how they built up their representation of the problem. We have observed their attempts to connect this mental representation to the physical model that they also built, the picture that they drew, and their written symbolic statement of the problem situation. Having looked at children in Chapter 5, in the present chapter we shall look at how difficult is the teachers task of recognizing the actual ideas of students. Background We consider in this chapter a classroom episode in which a teachers representation of a problem situation is in conflict with that of her students, Brian. The teacher, a second-year participant in a teacher development project in mathematics, was attempting to implement her growing knowledge of content, childrens learning, and pedagogy in her classroom (Maher & Alston,1988). She began to include small cooperative group problem-solving explorations as a regular classroom activity, and she was integrating in her lessons problem tasks in which children were encouraged to build physical models to represent their solutions. The Brian and Scott episode described and analyzed in Chapter 5 was representative of her instruction at this stage of her participation in the project. The two fifth-grade students whom we saw in Chapter 5, Brian and Scott, worked together regularly as partners doing mathematics. Having considered the work and thought processes of these two students in the previous chapter, we now attempt to see their thinking from the perspective of their teacher. In fact, because we have the advantage of videotapes of Brian and Scott working together in earlier lessons, and we also have the notes written by the teacher after each of these lessons, we are able to study their thinking in close detail. From this we can gain added insight into their mathematical thinking precisely because we can watch it develop. (Of course, we also have the advantage of hindsight, and the opportunity to look at tapes over and over again, discuss them, look some more, discuss some more, and so on. This is very different from the situation that confronted the teacher when she was actually

Computers and Curriculum Change in Mathematics

Computers have entered the fabric of modern society so thoroughly that they have become pervasive in everyday affairs. Their entry into schools, however, has been slow, and even slower has been their entry into mathematics classrooms. “While electronic computation has been in the hands of mathematicians for four decades, it has been in the hands of teachers and learners for at most two decades, mostly in the form of time-shared facilities. But the real breakthrough of decentralized and personalized microcomputer-based computing has been widely available for less than one decade” (Kaput 1992, p. 515).

What mathematics should students learn

With this Special Issue, entitled “Visions of School Mathematics,” the Journal of Mathematical Behuvior begins the most exciting project that it has undertaken, an international discussion of what mathematics students should be learning. This includes both the selection of “content” or “topics,” and also the more subtle, but even more important question of whut kind of knowledge students need to develop about each topic. This distinction may require explanation. Deciding whether the curriculum should include the topic of “adding fractions” falls into the first category, as a “content” decision. But in what way do we want students to be able to deal with “adding fractions”‘? Know from memory a rote rule’? Be able to show how this rule can be arrived at by thinking of certain heuristics? Be able to make numerical replacements for the variables in a formula such as

What classroom role should the PLATO computer system play

Inserting computers into the ecology of an elementary school classroom involves a combination of promise and uncertainty that parallels similar technological innovations in other areas, whether heart pacers, artificial kidneys, tranquilizers, atomic power plants, transportation, food production or virtually any other area one can think of. In each instance we lack a complete description of the original ecology, and we cannot be, a priori, fully aware of new possibilities.

The problem of relating mathematics to the possibilities and needs of schools and children

I am clearly the worst geometer — perhaps one should say the only non- geometer — here. Conceding that I can add nothing of a geometric nature to this conference, I nonetheless do suggest there is one important matter to which I can speak.

How computers help us understand people

I would answer an unequivocal “yes!“, for at least two reasons. First, as van Oers (this issue) points out, theories do determine curricula and teaching practices, although it is important to recognize that the theories that shape actual school practice are nearly always informal, naive, unexamined, implicit theories that often bear only modest resemblance to the more formal theories presented in the literature. Even people who may claim that they have no theory whatsoever undoubtedly do; they are merely confessing to an unexamined implicit theory (Davis, 1987). Second, in addition to pointing us in correct directions, a good theory is one that gives us tools to think with. In this regard, not all theories are equal. We can reason better with certain kinds of symbols than we can with others, a point I shall return to later when I consider the problem about mixing wine and water. In general, the more tangible and definite a symbol is, the more easily and powerfully we can reason with it. I am not sure that the experience of different nations is exactly the same. At the level of schools and teachers (which is the level that really counts) there seem to be two dominant theories in the United States today. The first is the theory of ‘direct teaching’ (sometimes attributed to Madeleine Hunter), which argues for a highly-explicit identification of what you want students to learn, a very clear exposition of this information, considerable drill