(2004). Learning Trajectories in Mathematics Education. Mathematical Thinking and Learning: Vol. 6, No. 2, pp. 81-89.

Learning Trajectories in Mathematics Education

Simon's (1995) development of the construct of hypothetical learning trajectory (HLT) offered a description of key aspects of planning mathematics lessons. An HLT consists of the goal for the students' learning, the mathematical tasks that will be used to promote student learning, and hypotheses about the process of the students' learning. However, the construct of HLT provided no framework for thinking about the learning process, the selection of mathematical task, or the role of the mathematical tasks in the learning process. Such a framework could contribute significantly to the generation of useful HLTs. In this article we demonstrate how an elaboration of reflective abstraction (i.e., reflection on activity-effect relationships), postulated by Simon, Tzur, Heinz, and Kinzel (in press), can provide such a framework and thus a theoretical elaboration of the HLT construct.

Explicating the Role of Mathematical Tasks in Conceptual Learning: An Elaboration of the Hypothetical Learning Trajectory

Efforts to explain learning as a constructive process run into the paradox of having to attribute to the learner prior knowledge that is at least as complex as the new learning to be explained. Although no full solution of this paradox is in sight, it is argued that progress is possible through examination of the wide range of mental resources available to human learners, only a limited range of which are taken account of in current theories. This paper considers 10 relatively neglected resources for the “bootstrapping” of cognitive growth, including chance plus selection, the affective boosting of relevant schemas, the operation of innate biases, and use of spare mental capacity. Implications for educational research are illustrated with reference to recent work on the development of complex composition strategies.

Toward a Solution of the Learning Paradox

The paper criticizes some epistemological presuppositions of Piaget's and of neo-Piagetian's work, in particular, the psycho-Logical principle. This principle is contrasted with a more valid psycho-dialectical one. It is suggested that a dialectical-constructivist (i.e., causal-dynamic) perspective offers a causal theoretical framework for cognitive development that is superior to that of Piaget and many neo-Piagetians. I outline criteria for evaluating causal developmental theories, and point out deficiencies in Piaget's and neepiagetian's stage theories vis-a-vis the criteria. An organismic theory of constructive operators - a dialectical/causal theory - is introduced as a remedy for these deficiencies. I focus on a modular model of mental attention that is constituted by four dynamically interacting functional systems. These systems together explain the ‘beam’ of mental attention and its phenomenological/behavioural effects. I claim that the stages of cognitive development are caused by growth...

Leland D. Van den Daele

Papers

C. A view of cognition from a formalist's perspective