CEPA eprint 3923

Discovery learning and constructivism

Davis R. B. (1990) Discovery learning and constructivism. In: Maher C., Davis R. & Noddings N. (eds.) Constructivist views on the teaching and learning of mathematics. National Council of Teachers of Mathematics, Reston VA: 93–106. Available at http://cepa.info/3923
Table of Contents
Why the Analysis Failed
A More Appropriate Focus
Pedagogical approach
Expectations
Constructivism
References
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 focused on various uses of mathematics, whereas others dealt only with what might be called “pure mathematics.”[Note 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 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.[Note 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.[Note 3]
the lumping – together of disparate interventions (as discussed above);traditional expectations that were far different from the goals and methods of the new programs;the lack of an adequate theory for discussing these differences;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.[Note 4] Interventions in this same spirit 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:
To get each student to see mathematics as a reasonable response to a reasonable challenge.To get each student to see mathematics as worthwhile and rewarding.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.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 ja really possible.)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.[Note 5] ) For example, these projects included science activities that use mathematics, several approaches to geometry, algebra, the use of computers, probability, and mathematical logic.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.To give students a chance to learn the main “big ideas” of mathematics, such as the concept of function and the use of graphs.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 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 inventmethods 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 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.[Note 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.)
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 them together into one plastic sandwich bag. Continue until you have filled as many bags as possible. (Under our assumption that there were twenty-two depressors, you should now have two filled bags, one bunch that is not in a bag, and one loose tongue depressor, as shown in Figure 2.)
Figure 2: Bags of Bunches of Tongue Depressors
We have finally arrived at a message that can be sent to the group at the back of the classroom, without anyone needing to count beyond three. While the children can in fact be depended upon to invent a solution to the basic problem, they cannot be expected to invent history. After the children have solved the problem, the teacher needs to intervene at the stage of interpreting what they have accomplished, and helping them to devise a succinct notation (which, of course, will look very much like standard base-three numeral notation; see Figure 3 for the message that the children might send).
Figure 3: Representing Bunches of Tongue Depressors with Numerals
In the language used by the projects, this was a case of focusing on the basic task, and leaving it up to the children to invent a way to solve it. This, in fact, is where place-value numerals come from – they are an elegant solution to an important problem: how can you name a very large number of different numbers by using only a small number of different words? Furthermore, how can you do this so that the names that you will make up for the numbers will reflect the nature of the numbers so accurately that one can use the names themselves in order to work out actions that really involve the numbers.
Let’s contrast this with a more typical classroom lesson that might also seem to satisfy that same description of “focussing on the task at a fundamental level.” Among other things we will see how it was possible for teachers and parents to be confused by what they read and heard. Let us once again use a task involving place-value numerals, perhaps the task of subtracting
1,002–25.
A typical school approach might say: “I can’t take 5 from two, so I regroup” (or “borrow,” or whatever the local language might be). The teacher might say: “Cross out the 1 and write a small 1 next to the zero, so we’ll think of it as ten.” (See Figure 4.)
Figure 4: Initial Step in Regrouping or Borrowing
The teacher might continue: “Now cross out the ten and write a nine over it, and write a small one next to the next zero.” (See Figure 5.)
Figure 5: Consecutive Applications of Regrouping or Borrowing
Might one not describe this approach, also, as focusing on the basic task? What could be more “basic” than focusing on exactly the things that you need to write down on the paper?
But of course these written marks are not what is really basic. They are merely a way of keeping track of what is really basic – the number of things that you have, or the number of things that are being taken away, and so on. These manipulations of written symbols must seem arbitrary, because the meaning is not present right there in the symbols – not in the way that it is present in the physical tongue depressors. When you put 21 tongue depressors into bunches of three each, you must (if you do it correctly) end up with seven bunches. Nobody needs to tell you that – you cannot do it any other way. The logic is right there in the tongue depressors themselves, and it is compelling.
By focusing on meaningless manipulations of symbols, the typical school curriculum gives a student no effective mental symbol system that carries the basic logic of the real situation. The “logic” of rote manipulations has the appearance of being arbitrary – indeed, as far as its intrinsic internal structure is concerned, it is arbitrary. Nothing in the child’s everyday experience has built up a “symbol system of necessary implications” that can function in the way that the child’s symbols for pebbles and for taking pebbles out of a bag can. Each student knows that if you take some pebbles out of the bag, there will be fewer pebbles remaining within the bag. The student does not need to make a special effort to remember
this. There is no need to keep repeating to oneself “Remember – when you put pebbles in there will be more pebbles in the bag. When you take pebbles out there will be fewer pebbles in the bag. When you put pebbles in.” The student’s mental symbol system for pebbles and bags and putting into and removing does have a fully developed intrinsic logic that compels certain outcomes and prohibits others – just as reality does, because this symbol system was drawn from experience with reality.
How shall we describe the difference between the curriculum improvement projects and typical school practice? Clearly, they are built on different assumptions – but it might no be far-fetched to claim that the difference is in epistemological assumptions. The intervention projects made different assumptions about the nature of knowledge. In the view of the projects, you know something when you have powerful mental representations, not merely for “surface level” aspects, but also for the deeper level constraints and possibilities, in much the way that each child knows that the act of putting mom pebbles into the bag will have the result of increasing the number of pebbles in the bag; the child’s mental symbol system makes this clear to him or to her.
The value of such a mental symbol system becomes evident when one tries to think about a problem for which one has no such powerful system. Consider, for example, the physical apparatus shown in Figure 6, consisting of a spool of thread that can roll on a table top.
Figure 6: Spool of Thread on a Table
If the thread at Point A is pulled to the right, what will the spool do? Most people find this problem hard to think about, because their mental symbol system for representing situations like this is not sufficiently well developed.
Traditional school practice viewed “learning mathematics” as a matter of learning, usually by rote, certain meaningless rules for writing meaningless symbols on paper in some very specific ways. This kind of knowledge could only be acquired by being told and by practicing it. The projects viewed “learning mathematics” as a matter of building up, in your mind, certain powerful symbol systems that allow you to represent certain kinds of situations, and a matter of acquiring skill in creating such mental representations and in using them. This kind of skill is not easily acquired by being told; here, too, you have to practice, but it is not the tedious practice of rote arithmetic. It might better be described as practice in thinking.
That these are quite different assumptions about the nature of knowledge becomes clear to anyone who will consider a few examples. They also imply differences in how one would test to see what knowledge the child had acquired. Traditional school practice tests mainly the ability to repeat back what has been told or demonstrated. For the innovation projects, this was not a satisfactory method of determining whether or not a student had acquired appropriate mental symbol systems, and could use them in a powerful way. The emphasis that the projects placed on studying how a student attacks a novel type of problem for which he or she has not been given specific advance preparation arose because only novel problems were seen as testing the power of one’s mental representation systems. Clearly, since such situations are new, they cannot be dealt with by merely “doing what you were told to do.” Your mental representations must give you the power to see new possibilities and new constraints in new situations.
The difference between these alternative views of “knowledge” has been revealed in stark terms in some recent studies of testing practices, and studies of how some school programs prepare children to take tests. Koretz (1988) reports the case of a school mathematics supervisor who noticed that the state’s minimum competency test presented shaded figures to accompany questions asking that one find the area, and presented unshaded figures for questions asking about perimeter. Koretz reports that, based on this observation, the supervisor instructed the teachers to tell children to multiply the numbers in problems where the figure was shaded, and to add the numbers in problems where the figure was not shaded. This is typical of a kind of strategy that raises test scores without actually teaching the relevant concepts, skills, or understandings.
This approach is nearly the antithesis of what the intervention projects intended. Contrast this strategy with the work of Edith Biggs in helping children learn the concept of area (Biggs, 1987). What mental symbols do students need, if they are to think about what “area” really is? They need to be able to visualize some appropriate square units (some projects used square pieces of paper for this); they need to be able to visualize placing these squares carefully in place (much as one puts down square tiles on a floor), they need to be able to visualize cutting tiles into smaller pieces when necessary (because things do not always come out even!, and – if they are to have a more complete idea of area – they need to be able to visualize some process of “taking limits” as one does in calculus (because sometimes even smaller pieces cannot be made to fit exactly). They also need to have a mental symbol system that lets them distinguish two-dimensional problems from three-dimensional problems, and they need to know that “area” refers to a two-dimensional attribute. None of this, of course, was developed by the “multiply if shaded, add if not” rule that was told to those children.
Developing a suitable kind of mental symbol system is so critical for the effective learning of mathematics, and has proved so elusive in efforts to describe what is needed, that a second example may be in order. Suppose this time that the task were to solve an equation of the form:
x/a = b/c.
The typical classroom lesson usually seeks to teach the method of “cross-multiplying.” The teacher carefully shows the children that the number in position “c” is to be multiplied by the number (perhaps an “unknown”) in position “x,” and the number in position “a” is to be multiplied by the number in position “b,” producing the result shown in Figure 7.
Figure 7: Illustration of the Method of Cross-Multiplying
Because of the pattern in which the symbols appear, this is often called the method of cross-multiplying. Is this lesson helping the students to develop an appropriate mental symbol system? We would argue that it is not. To be sure, the students probably are developing a mental representation for where symbols can be written on the paper – but this, again, is merely a surface-level phenomenon, and the relationships on this level are not representative of the true constraints and the true possibilities in this problem.
Here, as in the previous example, one wants mental representations for the deeper-level structure. In this case, what is really involved is the equality of two numbers, or the request that some number be chosen so that two numbers will in fact be equal. But whenever two numbers are equal, the double of one would equal the double of the other – that is, if you doubled each number, the results would again be equal. There is nothing gratuitous or arbitrary about this – it is basic to how numbers themselves actually behave. Nor is this limited to doubling. If you multiplied each number by three, the results would be equal. Or if you multiplied each number by ten. Or if you multiplied by seven and one half. What the students need to develop, if they are to deal with such situations in a powerful way, is a set of mental symbols that show such operations as “multiplying each side of an equation by the same number,” “adding the same number to each side of an equation,” and so on.[Note 7] A student who has developed mental representations for this aspect of how numbers behave can easily invent for himself or herself methods such as the “cross-multiply” method. They are a simple consequence of the way numbers work. But the converse is not true; a student who has learned “cross-multiplying” will not necessarily see why this method works, nor how numbers themselves behave.
Constructivism
In the 1950’s and 1960’s, when the so-called “curriculum improvement projects” were most active, and were being poorly implemented and incorrectly analyzed, one never heard of “constructivism.” The dominant psychology was “stimulus-response” theory, which held that a concern for what was going on in someone’s mind was unscientific, because it speculated about matters that were essentially unknowable. The dominant teaching strategy was to show or tell students what to do, and then to supervise their practice while they attempted to repeat what they had been shown. Knowledge was seen as the ability to regurgitate facts and to imitate rituals. “Testing” was a matter of confirming the accuracy of this regurgitation or imitation.
The scientists, mathematicians, and teachers who created the curriculum improvement projects knew, from their intuitive analysis of their own personal experience, that this misrepresented the true state of affairs. They were, in fact, able to devise interventions that produced far more effective learning in their students (see, for example, Dilworth, 1973). What was not readily available, however, was conceptualization of the process of learning mathematics – a formalized conceptualization, that went beyond the intuitive conceptualizations that many of the individuals did possess, based on their own personal experiences. Anyone who observes mathematics education has to be impressed by the quite sudden eruption of “constructivism” as a central concern of so many researchers. I would argue that while its origins may be somewhat obscure and uncertain, the reason for it is perhaps clear. It is a strong response to the very great need for a better way to think about how human beings deal with the subject called “mathematics.”
Figure 8: Gilpin Diagram for a Control Class. Each arrow represents one child.
Figure 9: Gilpin Diagram for a PLATO Class. Each arrow represents one child.
References
Biggs E. (1987) Understanding area. Journal of Mathematical Behavior 6(3): 197–199.
Davis R. B. (1988a) The world according to McNeill. Journal of Mathematical Behavior 7(1): 51–78.
Davis R. B. (1988b) The interplay of algebra, geometry, and logic. Journal of Mathematical Behavior 7(1): 9–28.
Davis R. B. (1984) Learning mathematics: The cognitive approach to mathematics education. Norwood NJ: Ablex Publishing Company.
Dilworth R. P. (1973) The changing face of mathematics education (Final report of the Specialized Teacher Project: 1971–72) Sacramento, California: California State Department of Education.
Howson G. C. J., Keitel C. & Kilpatrick J. (1981) Curriculum development in mathematics. Cambridge, England: Cambridge University Press.
Koretz D. (1988) Arriving in Lake Wobegon: Are standardized tests exaggerating achievement and distorting instruction? American Educator 12(2): 8–52.
McNeill R. (1988) A reflection on when I loved math, and how I stopped. Journal of Mathematical Behavior 7(1): 45–50.
Nacome (1975) Overview and analysis of school mathematics, Grades K-12. Washington Dc: Conference Board of the Mathematical Sciences.
Papert S. (1980) Mindstorms: Children, computers, and powerful ideas. New York: Basic Books.
Endnotes
1
For a more detailed discussion of the differences among these various efforts, see Davis, 1988a, or Howson, Keitel, and Kilpatrick, 1981.
2
One version of the kind of program described here was evaluated by Robert P. Dilworth (see Dilworth, 1973; NACOME, 1975, pp. 93-94). This very careful evaluation showed that children taught by teachers who had studied in this program did perform better, both on tests of conceptual understanding and on tests of computational skill. Furthermore, students taught by teachers who had studied in the program for two years did better than students taught by teachers who had studied only one year (so the second year of study by the teachers did pay off in improved performance of their students). Dilworth went even further, he tested children a year later – when at least one year had elapsed after the teacher had studied in the program – and found that the gains were still there. The teachers were continuing to have a superior effect on children whom they taught; the improvement in teaching effectiveness was not temporary. A program of this sort – an extension of the Madison Project program – was put on the computer-assisted system PLATO at the University of Illinois, and was evaluated by ETS. In Figures 8 and 9 I reproduce, in a form developed by John Gilpin, the results of one year of student experience with the PLATO computer-delivered lessons. In the Gilpin diagrams, each arrow represents one child; the tail of the arrow is the child’s performance on the Comprehensive Test of Basic Skills at the beginning of the school year (reported in so-called “grade level equivalents”); the point of the arrow is that same child’s performance at the end of the school year. Hence, each arrow represents the progress made by one child. Figure 8 shows the Gilpin report for a control class. Figure 9 shows the report for one of the PLATO classes. The difference is dramatic, and strongly in favor of the new curriculum.
3
Remember, it was the analysis that failed; the interventions themselves did not fail.
4
The interventions listed were most visible in the 1950’s and 1960’s although some of them continue up to the present time. Very active and effective versions of this approach are still available, especially through programs offered by Marilyn Burns Education Associates (21 Gordon Street, Sausalito, California 94965, Tel. (415) 332-4181). The work of Seymour Papert on LOGO environments is in this same spirit, augmented by the use of computers (see, e.g., Papert 1980).
5
One excellent, but not generally available, piece of evidence is the collection of videotaped interviews with children assembled by Eve Hall, Elizabeth Debold, and Edward Estey, of Children’s Television Workshop. The impact on the viewer of hearing these children describe what they are learning as “mathematics” is both striking and painful; few viewers can escape wondering why our society finds it appropriate to subject young people to this sequence of experiences.
6
Indeed, one of the most controversial aspects of these projects in the 1950’s and 1960’s was their claim that children could invent their own methods for solving problems. But the basis for the popular skepticism may lie not so much in questions about the nature of children as in questions about the nature of mathematics. Clearly, if you see mathematics as the process of following some meaningless rules that you have been taught to imitate, then there is no possible way that you could “invent” these rules, no more than you could “invent” the English language. Arbitrary historical accidents cannot be “invented.” But if, instead, you see mathematics as a process of working out reasonable responses to reasonable challenges, then it becomes entirely possible to invent your own methods – indeed, nothing else would really make sense. This possibility becomes all the more real if the instructional program is careful to give you tools for thinking about the mathematics, in just the way that the “pebbles-in the-bag” activity gives a child some tools for thinking about positive and negative numbers.
7
To have a truly effective way of dealing with such problems, the students also need a good representation system for the truth values of mathematical statements. For details, see Davis (1988b).
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