Constructivist Views on the Teaching and Learning of Mathematics || Chapter 7: Discovery Learning and Constructivism

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  • Chapter 7: Discovery Learning and ConstructivismAuthor(s): Robert B. DavisSource: Journal for Research in Mathematics Education. Monograph, Vol. 4, ConstructivistViews on the Teaching and Learning of Mathematics (1990), pp. 93-106+195-210Published by: National Council of Teachers of MathematicsStable URL: .Accessed: 03/10/2013 01:52

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  • Chapter 7: Discovery Learning and Constructivism

    Robert B. Davis Rutgers University

    Recent years have seen two large-scale efforts at improving the curricular goals and pedagogical methods of school mathematics by placing greater emphasis on student experience, on good analytical thinking, and on creativity. The first of these was proclaimed (incorrectly) to have been a failure. Will our present-day sophistication, as represented by today's constructivist perspective, mean that the second attempt will prove any more successful? What, precisely does constructivism mean for a classroom teacher? It is said that "those who do not study history are doomed to repeat it." This is all too likely to be true in the case of reforming school mathematics. Consequently, in the present chapter, we take a more careful look at the previous effort, and try to understand how things may be different the second time around.

    In the four decades since World War II there have been two major efforts to modify school science and mathematics so as to put greater emphasis on the thoughtfulness and creativity that are often seen as the hallmarks of true science. The first of these occurred in the 1950's, and bore the names of projects such as P.S.S.C., SCIS, E.S.S., and the Madison Project, and of individuals such as Jerrold Zacharias, Francis Friedman, Marion Walter, Caleb Gattegno, Frances and David Hawkins, David Page, Geoffrey Matthews, Leonard Sealey, and Robert Karplus. The mathematics parts of these various projects were lumped together with other efforts of quite different sorts and this unlikely combination was given a single label: "the new mathematics." If one looked at the totality of these projects it is probably true that the only thing that they all had in common was that every one of them proposed major changes in the then-typical versions of school mathematics. Some of these projects used manipulatable physical materials, others did not. Some sought to build up mathematical ideas gradually in students' minds, whereas others attempted to "get it right from the very beginning." Some focussed on various uses of mathematics, whereas others dealt only with what might be called "pure mathematics." 1 Putting such different approaches together, and trying to treat them as one single thing, made useful analysis nearly impossible, and in fact the world seems to have learned very little from this quite large investment of time, effort, and money. Even worse, most of what has been "learned" is in fact wrong.

    Most readers, if they have heard of the "new math" at all, have heard that it was installed in a large number of US schools in the 50's and 60's, and turned out to be a failure. Both claims are false, and so is the implied third claim that there ever was such a thing as "the


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    new mathematics ." Given the diversity,"even the irreconcilable differences-that separated the different efforts, there clearly was no identifiable curriculum or set of goals or pedagogical approach that could be thought of as well-defined and testable. But if, as this chapter does, I select a few of these projects that did have something in common, then the approach represented by these projects was never tried in most US schools. It was, however, adopted in a small percent of our schools, and in every reported case where it was conscientiously employed and carefully evaluated, it proved remarkably successful. Far from proving that the "New Math" was not successful in classrooms, the data from the 50's, 60's, and 70's show quite convincingly that there is a better way to help students learn mathematics, and in fact we actually know-at least roughly-what it is.2

    Something did, indeed, fail-but it was not the best of the school programs, it was the analysis of this important episode in American educational history. The present short chapter cannot hope to rectify all of the errors and misconceptions that we seem to have "learned" from this experience, but perhaps it can begin the process of rethinking what really happened, and why it did.

    Why the Analysis Failed I argue that the analysis of these curriculum improvement ventures failed for at least four reasons.3

    1. the lumping--together of disparate interventions (as discussed above); 2. traditional expectations that were far different from the goals and methods of the new


    3. the lack of an adequate theory for discussing these differences;

    4. the need to give more prominence to actual classroom episodes.

    A More Appropriate Focus We can avoid the error of inappropriate lumping-together of dissimilar interventions by choosing carefully the projects that are considered, and making sure that the chosen group of "curriculum improvement projects" do, in fact, have much in common. For a selection of projects that were quite similar in their basic assumptions and goals, I choose: the Madison Project (see, e.g., Davis, 1988a), David Page's Illinois Arithmetic Project, the EDC-based Elementary School Science Project ("ESS"), Robert Karplus' SCIS project at UC Berkeley, and the California state-wide Miller Math Program .4 Interventions in this same spirit

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    occurred in Great Britain, in the work of Leonard Sealey, Edith Biggs, and the Nuffield Science Project, among others. The same fundamental approach was agreed to by most of those who worked on these ventures, and in fact there was considerable sharing of resources and even personnel. Probably the best over-all description of this work is that given in Howson, Keitel, and Kilpatrick (1981; see also Biggs, 1987; Davis, 1988a; and McNeill, 1988).

    What did these projects claim to have in common? From things written and said at the time, their goals included these:

    1. To get each student to see mathematics as a reasonable response to a reasonable challenge.

    2. To get each student to see mathematics as worthwhile and rewarding.

    3. To get each student to see mathematics as a subject where it was appropriate to think creatively about what you were doing, and to try to understand what you were doing.

    4. To get each student to see mathematics as a subject where, in fact, it was possible to understand what you were doing. (There is abundant evidence that most U.S. students do not usually see mathematics in this light, nor is it taught in such a way that understanding is really possible.)

    5. To give students a wider notion of what sorts of things make up the subject of "mathematics." (There is overwhelming evidence that most students think that "mathematics" refers only to meaningless rote arithmetic.5) For example, these projects included science activities that use mathematics, several approaches to geometry, algebra, the use of computers, probability, and mathematical logic.

    6. To let students see that mathematics is discovered by human beings, that their own classmates and they themselves can discover ways to solve problems if they take the trouble to think about the matter and if they work to understand it.

    7. To give students a chance to learn the main "big ideas" of mathematics, such as the concept of function and the use of graphs.

    8. To have students see mathematics as a useful way of describing the real world. (It is quite different to see mathematics as a description of the real world, instead of seeing it as the process of following a set of meaningless rules, which, unfortunately, is how most students view mathematics.)

    Pedagogical approach With some variations between projects, these efforts mainly tended to use the pedagogical approach of creating an appropriate assimilation paradigm for each new idea they sought to teach. Thus, for example, the Madison Project introduced positive and negative numbers by using an activity called Pebbles in the Bag, where a bag, initially containing an unknown

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    number of pebbles, has pebbles added to it or removed from it. The question is never "How

    many pebbles in all are there in the bag?"-that remains unknown-but rather "How many more pebbles are now in the bag?" or "How many fewer pebbles are there in the bag?" Thus,

    6-5 would correspond to "putting 6 pebbles into the bag, and then removing 5."

    We would not know how many pebbles were in the bag at that point, because we did not know how many were in the bag at the beginning, but we would know that there was one more pebble in the bag than there had been when we started. Hence we would write

    6 - 5 = +1, where the "positive one," +1, means that there is one more pebble in the bag as a result of these actions. It does not mean that the total number of pebbles in the bag is one. Suppose, instead, we had put 5 pebbles into the bag, and then removed seven. We would describe this action (note this instance of the theme "mathematics as a description of reality") by writing

    5 - 7, and we know that the result would be having two fewer pebbles in the bag than we had when we started, so we would write

    5 - 7 = -2, where the symbol negative two (-2) means two less than we had before.

    What does this accomplish? Because the children are readily able to visualize the action of "putting five pebbles into the bag, and taking seven out" (remember that the bag did not start empty, but had a goodly collection of pebbles in it at the outset), they have this visualization available to them as a tool that lets them think about the mathematics. The Madison Project called this an assimilation paradigm, and the strategy of basing teaching on this approach was called the paradigm teaching. strategy. (For a more extended discussion, see Davis, 1984, Chapter 21.)

    This same approach-the creation of an assimilation paradigm by providing appropriate experience-is used also by an important present-day program, the middle school mathematics

    program being created in Atlanta, Georgia, by Robert Moses, where the Atlanta subway system is taken as the basis for the fundamental idea of direction implicit in positive and

    negative numbers. If one did not give the children something like this "assimilation paradigm," they

    would have no way of thinking about the mathematics, and any expectation that they would invent methods of solution would be unreasonable. The children are, however, perfectly capable of thinking about bags and pebbles and "putting pebbles into a bag" and "taking pebbles out of a bag." If we use this as a basis for thinking about mathematics-which is

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    perfectly reasonable if one takes the position that mathematics is a description of reality-then it is entirely possible for the children to carry out their own analyses of problems, and to invent their own methods of solution.6 We have given them tools to think with!

    Expectations It should be clear from the preceding discussion that the expectations of these curriculum improvement projects were quite different from the common expectations of most parents, teachers, or even students. Even when the projects believed that they were explaining themselves reasonably clearly, their words probably meant something different to most hearers. An example may make this clearer

    Most of these projects described themselves as trying to have the teacher focus on the task or problem, and to do this at a fundamental level. But this phrase was probably often misunderstood. How were these projects different from usual school practice, which might also be described by this same phrase? Consider the introduction of base three numerals, as used by the Madison Project Two small groups of children are asked to communicate messages back and forth, but they must pretend that nobody can count above three. One group of children-at the front of the room, say-is then given a pile of tongue depressors (let's say that you and I know that there are twenty-two tongue depressors in the pile). The children at the front of the room must send messages to the other group of children (at the back of the room) so that the second group can assemble exactly the same number of tongue depressors.

    How can one tell if the job has been done correctly? That part is easy; after the second group has assembled what they believe is the correct number of depressors, the two collections can be brought together and a one-to-one matching can be attempted.

    But what kind of messages can the first group of children send? Remember, nobody can count beyond three. The children, however, are given some things they may use: rubber bands, plastic sandwich bags, and shoe boxes, among other things. The task of solving the problem is left for the children to work out.

    Sooner or later they do this, by counting three depressors and putting a rubber band around them to make a "bunch," continuing until as many bunches have been made as possible. (If there really were twenty-two depressors, the children will make seven bunches, and there will be one separate tongue depressor left over that is not part of a bunch, as shown in Figure 1.)

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    Figure 1 - Twenty -one Tongue Depressors Tied in Bunches of Three

    But there are too many bunches for the children to be able to tell the others how many there are-nobody can count beyond three, remember?

    However, what worked once can be tried again: count three bunches and put the...


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