msimon@psu.edu
Abstract: We
articulate and explicate a theoretical framework for mathematics teaching in
which we specify mechanisms of students’ conceptual learning and the role of
teachers in promoting that learning. We consider the notion of the learning
paradox as defining the problem that our framework addresses. We explicate how
conceptualizing learning in terms of reflection on activity-effect
relationships can address the learning paradox and provide a basis for
specifying an approach to mathematics pedagogy that can foster conceptual
advance in students.
The current mathematics education reform has promoted a large-scale movement away from direct instruction, leaving the field of mathematics education without well-articulated theories of teaching. In this paper, we elaborate and examine constructs that can contribute to re-conceptualizing mathematics teaching. We approach the need for re-conceptualizing mathematics teaching not only from our role as mathematics educators (teachers), but also from our role as mathematics teacher educators and researchers of mathematics teacher development. It is only with clearly articulated conceptions of mathematics teaching that the goals of mathematics teacher development can be defined and the approaches to and results of mathematics teacher education effectively analyzed and evaluated (Simon, 1997).
We work from the notion
that theories of teaching must build on and be integrated with theories of
learning, but that theories of learning do not, in themselves, prescribe
approaches to teaching. We have asked ourselves: what aspects of our
understanding of learning might provide a solid basis for re-conceptualizing
teaching? We identified ways of understanding mathematics learning that exist
in the literature, synthesized and further elaborated these understandings in
order to articulate a framework for teaching based on those understandings of
learning. Examples of teaching that are consistent with these ideas can be
observed occasionally in practice. However, it is our experience that the
theoretical basis for such teaching is generally unformulated. The consequence
of the lack of a well-formulated theory of teaching is that effective
mathematics pedagogy is not produced consistently and that the pedagogical
ideas involved are not accessible subjects of discourse.
In our theoretical and empirical work, we employ a social constructivist perspective in which we coordinate social and cognitive (constructivist) perspectives. In this article, we focus on the cognitive; that is we articulate theoretical constructs with respect to teaching that are built on radical constructivist interpretations of learning. We classify cognitive perspectives based on constructivism as “conception-based.” We use this term to emphasize that the researcher postulates conceptions as a way to characterize learners’ current organizations of their experiential realities. Conception-based perspectives are based on the following principles:
1. Mathematics is created through human activity. Humans have no access to a mathematics that is independent of their ways of experiencing/knowing.
2. What individuals currently know (i.e., current conceptions) affords and constrains what they can assimilate—perceive, understand, and learn.
3. Learning mathematics is a process of transforming one's ways of knowing (conceptions) and acting. This has two implications. New understandings are the result of change in current understandings, and current understandings afford and constrain what understandings can be developed at any point in time.
The idea of learning as transformation of current understandings is in contrast with perspectives that assume that learners can acquire new concepts by perceiving relationships that exist in the world around them. In the latter case, the notion of how to foster conceptual learning is relatively straightforward. However, from a conception-based perspective, promoting the transformation of current understandings toward the development of more advanced understandings is problematic. In particular, the most important and challenging aspect of fostering cognitive growth is promoting the development of new cognitive entities (e.g., number, ratio, function).
Bereiter (1985) made a distinction that is at the core of the framework that we are presenting: “The distinction is between kinds of learning that can be accounted for on the basis of knowledge schemas that the learner already possesses and learning that involves new cognitive structure to which already existing schemas are subordinated” (p. 217). Fostering the latter type of learning is the theoretical challenge that we are attempting to meet.
In grounding this
exploration in constructivism, we continue to eschew the notion that more
powerful concepts can be infused into learners and embrace the idea of promoting
an internal process of construction. This quest however, puts us face-to-face
with what has been called "the learning paradox" (Pascual-Leone,
1976), the need to explain how learners "get from a conceptually
impoverished to a conceptually richer system by anything like a process of
learning" (Fodor, 1980, p. 149 cited in Bereiter, 1985). This is conceived
of as a paradox for the following reason. Piaget’s (1970) idea of assimilation,
a core idea of constructivism, suggests that one needs to have concept X in
order to make sense of one’s experience in terms of concept X. Thus, it seems
impossible for one’s experience to lead to a more advanced concept, because
that concept would need to be already available for assimilation. We stress
that this paradox is a function of adopting a constructivist understanding of
knowing. It does not exist for those who view learning as taking in relationships from the outside
world (See Simon, Tzur, Heinz, Kinzel, & Smith, in press). Our challenge, therefore, is to find a theoretical
explanation for how humans construct more powerful concepts out of less
powerful ones that can serve as a basis for articulating a role for pedagogy in
promoting such learning processes.
In this section, we introduce a way of conceptualizing learning that addresses the learning paradox and provides a powerful basis for re-conceptualizing teaching. We introduce these ideas with a non-mathematical example (mathematical examples would have required more space than we had available) and then consider the theoretical issues in greater detail.
This is a thought experiment involving children’s learning of a game strategy. As such it is not an example of mathematics learning, and therefore should not be considered as evidence to support our framework for conceptualizing mathematics learning and teaching. Rather, we use it as an easily understandable context for presenting a set of ideas.
Consider two children who have learned the rules of checkers. They understand the goal of the game and can play a game according to the rules and determine a winner. However, at this point, they have developed no strategy, that is, they have no basis for choosing one move over another. If these two children were to play checkers regularly over several weeks, without any coaching or contact with more advanced players, would they develop some strategy? We imagine that you responded as we do, “Yes.”
How is this example related to the challenge that we have articulated? The children’s progression from having no strategy to having strategy can be viewed as an example of developing more powerful conceptions from subordinate ones, the issue raised by the learning paradox. Our explanation is structured as follows:
1. The children had a goal (to win the game) and an activity sequence that they could use in service of the goal (moving pieces in accordance with the rules). The activity sequence that led to the development of the new conception was one that the children could carry out without and before the cognitive advance.
2. From the outset, the children were able to distinguish effects of their activity that advanced their goal from those that did not (e.g., taking an opponent’s piece versus losing a piece).
3. As they played more games, they began to reflect (we are not claiming conscious thought) on the relationship between their activity (particular moves) and the effects of that activity (taking or losing a piece).
4. Their reflection resulted in abstracting an activity-effect relationship (between certain moves and changes in the number of pieces), the rudimentary form of a new conception (Tzur, 1996).
Based on the description of conceptual development as a process of reflection on activity-effect relationships, we now articulate the teacher’s role in promoting the development of a new cognitive structure. The teacher's role is to specify and engage students in an activity sequence that the students are capable of carrying out, independent of the teacher, that can lead to the students’ identification of regularities in activity-effect relationships contributing to an intended cognitive advance. Let us look at some of the entailments of such a formulation of teaching. Note that although we are focusing on the teacher’s role and the activity of teaching, much of what we are describing could be carried out by curriculum developers; the same principles would apply.
Specifying students’ current knowledge. Central to the conception-based perspective that is the basis for our framework on mathematics teaching is Piaget’s (1985) notion of assimilation. That is, what individuals know affords and constrains what they can learn. In our formulation of teaching, the teacher endeavors to understand the students’ conceptions in order to anticipate interpretations students can make of proposed tasks, goals that they can set, and activity sequences in which they can engage to work towards their goals.
Specifying the pedagogical goal. In addition to understanding the students’ knowledge, the teacher must be able to specify the conceptual advance intended. This is a difficult undertaking. Not only is it insufficient to specify what the student will be able to do (the traditionally employed behavioral objective), it is insufficient to specify the mathematics to be learned (e.g., “The student will understand the distributive property.”). It is essential to specify the understanding and how it differs from the identified prior state. Thus, in the example of the distributive property, the teacher would likely describe and contrast two stages of understanding multiplication.
Identifying an activity. Once the teacher has a useful specification of the conceptual advance that she wants to promote and a useful understanding of the students’ relevant knowledge, she is ready to consider the key activity involved (e.g., counting as the basis of developing number). The teacher conceives of the new understanding as an abstracted activity-effect relationship. Once the key activity has been identified, the teacher considers whether the students are currently able to generate an activity sequence embodying that activity. In some cases, the students may not have the requisite activity and the goal for instruction will be modified. In identifying an appropriate activity sequence, the teacher must hypothesize about the types of distinctions students could make among the effects of their activity and how these distinctions could lead to the new conception.
Selecting a task. Once the activity sequence is identified, a task can be selected. The task is one that, based on the teacher’s conjectures, will result in the students setting a particular goal and engaging in a particular activity sequence.
We see steps 2-4 as further elaboration of Simon’s (1995) hypothetical learning trajectory.
During the instructional phase, the teacher may need to
negotiate with the students a shared interpretation of the task (including the
goal). Once this is in place, the teacher is engaged in monitoring each aspect
of her conjecture (students’ selection of an activity sequence, sorting of
effects, and reflection on activity-effect relationships). As a result of
monitoring students’ activity, the teacher may revise her understanding of the
students’ conceptions, the task employed, or both. Task modification can range
from asking the students a question related to the task to changing the entire
task (cf. Tzur and Simon, 1999, for specification of tasks).
The theoretical work that we describe in this paper is part of an ongoing effort to understand and explain mathematics teaching and learning in powerful ways. Theoretical advances in this area can contribute to mathematics teaching, curriculum design, teacher education, and research on mathematics teaching and teacher development. Our efforts have been guided by two assumptions: that the mechanisms of learning and teaching underlying successful lessons can be understood, and that this understanding can lead to a more methodical approach to teaching resulting in more consistent generation of successful lessons and informed modification of unsuccessful ones.
Our position is not that this framework is indicated for every instructional situation involving conceptual learning. Indeed, students learn some things spontaneously and other things through relatively unstructured inquiry lessons. Rather, we endeavor to understand basic mechanisms of teaching and learning for approaching the more intractable pedagogical problems. Towards this end, we have described organizing instruction to foster reflection on particular activity-effect relationships. As Bereiter (1985) pointed out, "certain kinds of learning really are problematic . . . the learning paradox helps us see into the heart of the problem" (p. 221). We have described an approach to teaching that addresses the learning paradox.
Note
This research is supported by the National Science Foundation under grant RED-9600023. The opinions expressed do not necessarily reflect the views of the Foundation.
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