杜宾斯基Apos理论.doc

APOS: A Constructivist Theory of Learning in Undergraduate Mathematics Education Research Ed Dubinsky, Georgia State University, USA and Michael A. McDonald, Occidental College, USA The work reported in this paper is based on the principle that research in mathematics education is strengthened in several ways when based on a theoretical perspective. Development of a theory or model in mathematics education should be, in our view, part of an attempt to understand how mathematics can be learned and what an educational program can do to help in this learning. We do not think that a theory of learning is a statement of truth and although it may or may not be an approximation to what is really happening when an individual tries to learn one or another concept in mathematics, this is not our focus. Rather we concentrate on how a theory of learning mathematics can help us understand the learning process by providing explanations of phenomena that we can observe in students who are trying to construct their understandings of mathematical concepts and by suggesting directions for pedagogy that can help in this learning process. Models and theories in mathematics education can ? ? ? ? ? ? support prediction, have explanatory power, be applicable to a broad range of phenomena, help organize one’s thinking about complex, interrelated phenomena, serve as a tool for analyzing data, and provide a language for communication of ideas about learning that go beyond superficial descriptions. We would like to offer these six features, the first three of which are given by Alan Schoenfeld in “Toward a theory of teaching-in-context,” Issues in Education, both as ways in which a theory can contribute to research and as criteria for evaluating a theory. 1 In this paper, we describe one such perspective, APOS Theory, in the context of undergraduate mathematics education. We explain the extent to which it has the above characteristics, discuss the role that this theory plays in a research and curriculum development program and how such a program can contribute to the development of the theory, describe briefly how working with this particular theory has provided a vehicle for building a community of researchers in undergraduate mathematics education, and indicate the use of APOS Theory in specific research studies, both by researchers who are developing it as well as others not connected with its development. We provide, in connection with this paper, an annotated bibliography of research reports which involve this theory. APOS Theory The theory we present begins with the hypothesis that mathematical knowledge consists in an individual’s tendency to deal with perceived mathematical problem situations by constructing mental actions, processes, and objects and organizing them in schemas to make sense of the situations and solve the problems. In reference to these mental constructions we call it APOS Theory. The ideas arise from our attempts to extend to the level of collegiate mathematics learning the work of J. Piaget on reflective abstraction in children’s learning. APOS Theory is discussed in detail in Asiala, et. al. (1996). We will argue that this theoretical perspective possesses, at least to some extent, the characteristics listed above and, moreover, has been very useful in attempting to understand students’ learning of a broad range of topics in calculus, abstract algebra, statistics, discrete mathematics, and other areas of undergraduate mathematics. Here is a brief summary of the essential components of the theory. An action is a transformation of objects perceived by the individual as essentially external and as requiring, either explicitly or from memory, step-by-step instructions on how to perform the operation. For example, an individual with an action conception of left coset would be restricted to working with a concrete group such as Z20 and he or she could construct subgroups, such as H={0,4,8,12,16} by forming the multiples of 4. Then the individual could write the left coset of 5 as the set 5+H={1,5,9,13,17} consisting of the elements of Z20 which have remainders of 1 when divided by 4. 2 When an action is repeated and the individual reflects upon it, he or she can make an internal mental construction called a process which the individual can think of as performing the same kind of action, but no longer with the need of external stimuli. An individual can think of performing a process without actually doing it, and therefore can think about reversing it and composing it with other processes. An individual cannot use the action conception of left coset described above very effectively for groups such as S4, the group of permutations of four objects and the subgroup H corresponding to the 8 rigid motions of a square, and not at all for groups Sn for large values of n. In such cases, the individual must think of the left coset of a permutation p as the set of all products ph, where h is an element of H. Thinking about forming this set is a process conception of coset. An object is constructed from a process when the individual becomes aware of the process as a totality and realizes that transformations can act on it. For example, an individual understands cosets as objects when he or she can think about the number of cosets of a particular subgroup, can imagine comparing two cosets for equality or for their cardinalities, or can apply a binary operation to the set of all cosets of a subgroup. Finally, a schema for a certain mathematical concept is an individual’s collection of actions, processes, objects, and other schemas which are linked by some general principles to form a framework in the individual’s mind that may be brought to bear upon a problem situation involving that concept. This framework must be coherent in the sense that it gives, explicitly or implicitly, means of determining which phenomena are in the scope of the schema and which are not. Because this theory considers that all mathematical entities can be represented in terms of actions, processes, objects, and schemas, the idea of schema is very similar to the concept image which Tall and Vinner introduce in “Concept image and concept definition in mathematics with particular reference to limits and continuity,” Educational Studies in Mathematics, 12, 151-169 (1981). Our requirement of coherence, however, distinguishes the two notions. The four components, action, process, object, and schema have been presented here in a hierarchical, ordered list. This is a useful way of talking about these constructions and, in some sense, each conception in the list must be constructed before the next step is possible. In reality, however, when an individual is developing her or his understanding of a concept, the constructions are not 3 actually made in such a linear manner. With an action conception of function, for example, an individual may be limited to thinking about formulas involving letters which can be manipulated or replaced by numbers and with which calculations can be done. We think of this notion as preceding a process conception, in which a function is thought of as an input-output machine. What actually happens, however, is that an individual will begin by being restricted to certain specific kinds of formulas, reflect on calculations and start thinking about a process, go back to an action interpretation, perhaps with more sophisticated formulas, further develop a process conception and so on. In other words, the construction of these various conceptions of a particular mathematical idea is more of a dialectic than a linear sequence. APOS Theory can be used directly in the analysis of data by a researcher. In very fine grained analyses, the researcher can compare the success or failure of students on a mathematical task with the specific mental constructions they may or may not have made. If there appear two students who agree in their performance up to a very specific mathematical point and then one student can take a further step while the other cannot, the researcher tries to explain the difference by pointing to mental constructions of actions, processes, objects and/or schemas that the former student appears to have made but the other has not. The theory then makes testable predictions that if a particular collection of actions, processes, objects and schemas are constructed in a certain manner by a student, then this individual will likely be successful using certain mathematical concepts and in certain problem situations. Detailed descriptions, referred to as genetic decompositions, of schemas in terms of these mental constructions are a way of organizing hypotheses about how learning mathematical concepts can take place. These descriptions also provide a language for talking about such hypotheses. Development of APOS Theory APOS Theory arose out of an attempt to understand the mechanism of reflective abstraction, introduced by Piaget to describe the development of logical thinking in children, and extend this idea to more advanced mathematical concepts (Dubinsky, 1991a). This work has been carried on by a small group of researchers called a Research in Undergraduate Mathematics Education Community (RUMEC) who have been collaborating on specific research projects using APOS Theory within a 4 broader research and curriculum development framework. The framework consists of essentially three components: a theoretical analysis of a certain mathematical concept, the development and implementation of instructional treatments (using several non-standard pedagogical strategies such as cooperative learning and constructing mathematical concepts on a computer) based on this theoretical analysis, and the collection and analysis of data to test and refine both the initial theoretical analysis and the instruction. This cycle is repeated as often as necessary to understand the epistemology of the concept and to obtain effective pedagogical strategies for helping students learn it. The theoretical analysis is based initially on the general APOS theory and the researcher’s understanding of the mathematical concept in question. After one or more repetitions of the cycle and revisions, it is also based on the fine-grained analyses described above of data obtained from students who are trying to learn or who have learned the concept. The theoretical analysis proposes, in the form of a genetic decomposition, a set of mental constructions that a student might make in order to understand the mathematical concept being studied. Thus, in the case of the concept of cosets as described above, the analysis proposes that the student should work with very explicit examples to construct an action conception of coset; then he or she can interiorize these actions to form processes in which a (left) coset gH of an element g of a group G is imagined as being formed by the process of iterating through the elements h of H, forming the products gh, and collecting them in a set called gH; and finally, as a result of applying actions and processes to examples of cosets, the student encapsulates the process of coset formation to think of cosets as objects. For a more detailed description of the application of this approach to cosets and related concepts, see Asiala, Dubinsky, et. al. (1997). Pedagogy is then designed to help the students make these mental constructions and relate them to the mathematical concept of coset. In our work, we have used cooperative learning and implementing mathematical concepts on the computer in a programming language which supports many mathematical constructs in a syntax very similar to standard mathematical notation Thus students, working in groups, will express simple examples of cosets on the computer as follows. Z20 := {0..19}; op := |(x,y) -> x+y (mod 20)|; 5 H := {0,4,8,12,16}; 5H := {1,5,9,13,17}; To interiorize the actions represented by this computer code, the students will construct more complicated examples of cosets, such as those appearing in groups of symmetries. Sn := {[a,b,c,d] : a,b,c,d in {1,2,3,4} | #{a,b,c,d} = 4}; op := |(p,q) -> [p(q(i)) : i in [1..4]]|; H := {[1,2,3,4], [2,1,3,4], [3,4,1,2], [4,3,2,1]}; p := [4,3,2,1]; pH := {p .op q : q in H}; The last step, to encapsulate this process conception of cosets to think of them as objects, can be very difficult for many students. Computer activities to help them may include forming the set of all cosets of a subgroup, counting them, and picking two cosets to compare their cardinalities and find their intersections. These actions are done with code such as the following. SnModH := {{p .op q : q in H} : p in Sn}; #SnModH; L := arb(SnModH); K := arb(SnModH); #L = #K; L inter K; Finally, the students write a computer program that converts the binary operation op from an operation on elements of the group to subsets of the group This structure allows them to construct a binary operation (coset product) on the set of all cosets of a subgroup and begin to investigate quotient groups. It is important to note that in this pedagogical approach, almost all of the programs are written by the students. One hypothesis that the research investigates is that, whether completely successful or not, the task of writing appropriate code leads students to make the mental constructions of actions, processes, objects, and schemas proposed by the theory. The computer work is accompanied by classroom discussions that give the students an opportunity to reflect on what they have done in the computer lab and relate them to mathematical concepts and their properties and relationships. Once the concepts are in place in their minds, the students are assigned (in class, homework and examinations) many standard exercises and problems related to cosets. 6 After the students have been through such an instructional treatment, quantitative and qualitative instruments are designed to determine the mental concepts they may have constructed and the mathematics they may have learned. The theoretical analysis points to questions researchers may ask in the process of data analysis and the results of this data analysis indicates both the extent to which the instruction has been effective and possible revisions in the genetic decomposition. This way of doing research and curriculum development simultaneously emphasizes both theory and applications to teaching practice. Refining the theory As noted above, the theory helps us analyze data and our attempt to use the theory to explain the data can lead to changes in the theory. These changes can be of two kinds. Usually, the genetic decomposition in the original theoretical analysis is revised and refined as a result of the data. In rare cases, it may be necessary to enhance the overall theory. An important example of such a revision is the incorporation of the triad concept of Piaget and Garcia (1989) which is leading to a better understanding of the construction of schemas. Th