by Norbert M. Seel
Albert-Ludwigs-University of Freiburg, Germany
Schooling, especially in mathematics and science, has been criticized for decades. Actually, despite the increase of budgets, the implementation of new programs, the lengthening of school days and years, and the addition of new subjects and the deletion of others, the results of these efforts have been disappointing. Thus, with regard to mathematics education, Peterson (1988) criticized the elementary school mathematics curriculum “as based on the assumption that computational skills must be learned before children are taught to solve even simple word problems” (p. 7). The curriculum also reflects this sequence at the secondary level: Students generally take arithmetic
, then algebra, then geometry, and it is common to view higher order objectives as more appropriate later in the sequence. Thus, proofs are expected in geometry but not in algebra, and mathematical reasoning is more appropriate in algebra than arithmetic. As students progress through the grades, only the more capable students are able to keep up with the academic mathematics curriculum, so that most fall away before they encounter higher order instructional objectives, such as mathematical problem solving. Raudenbush, Rowan, and Cheong (1993) pointed out that although current conceptions of learning encourage the pursuit of higher order objectives, many teachers lack adequate preparation for teaching higher order thinking, and, additionally, organizational conditions at schools often discourage the pursuit of higher order objectives. From the perspective of Raudenbush et al., the pervasive influence of behaviorism in curriculum and instruction provides a potential explanation for the link between academic tracks (especially of math and science education) and the pursuit of instructional objectives. Shepard (1991) demonstrated that behaviorist theories imply that students learn best when complex learning tasks are broken down into smaller parts that are learned sequentially. Only when the earlier, simple steps are mastered is the learner ready for more complex tasks requiring analysis, hypothesis testing, and evaluation.
An alternative conception, the approach of model-centered learning and instruction, is described in this chapter. This approach is based on theories of mental models (cf. Johnson-Laird, 1983; Seel, 1991) and focuses on self-organized discovery learning. Therefore, the learner has to search continuously for information in a learning environment in order to complete or stabilize a mental model that corresponds to an a priori understanding of the material to be learned. Discovery learning is guided by exploratory models designed with a specific endpoint in mind. Students explore these models by developing hypotheses and then varying input parameters to investigate how well their conjectures align with the models.
With regard to mathematics education, the National Council of Teachers of Mathematics (1994) has formulated the precept that the primary role of algebra at the school level should be to develop confidence and facility in using variables and functions to model numerical patterns and quantitative relationships. For science education, Schauble (1996) formulated such an approach in the following manner: “The goal of scientific reasoning is not primarily the formulation of inductive generalizations, but rather the construction of explanatory models, that . . . account for the observed phenomena” (p. 103). Lesh and Doerr (2000) argued that helping students to develop explanatory models in order to make sense of their experiences of light, gravity, electricity, and magnetism should be among the most important goals of science instruction; otherwise, students invent models of their own that are often incomplete and incorrect (cf. D. E. Brown & Clement, 1989). Clement and Steinberg (2002) described how conventional approaches to electricity instruction in physics start with electrostatics and quickly introduce the concept of “potential difference,” which is defined mathematically. As the concept of electric potential generally remains unlearned when this instructional approach is implemented, Clement and Steinberg advocated the construction of a mental model that helps to develop a qualitative conception of electric potential, based on an analogy with pressure in compressed air that is compelling to most students, before introducing distant action and mathematical representation. Actually, students often do not (and cannot) develop appropriate symbol systems to make sense of mathematical entities such as directed quantities (negatives), multivalued quantities (vectors), ratios of quantities, changing or accumulating quantities, or locations in space (coordinates). However, they can invent significant mathematical solutions when confronted with the need to create meaningful models of real situations.
Model-centered learning and instruction is not simply an add-on activity to the curriculum. Rather, it involves a reformulation of the curriculum that gives primacy to students’ constructions of content knowledge through an inquiry process of experimentation, simulation, and analysis (cf. Doerr, 1996). In the following sections of this chapter, several aspects of model-centered learning and instruction are discussed: The psychological foundations of model-centered learning are described, then important conclusions for instructional design are derived. Following this, research on model-centered instruction is summarized, and, finally, problems of the assessment of mental models are described.
Albert-Ludwigs-University of Freiburg, Germany
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Schooling, especially in mathematics and science, has been criticized for decades. Actually, despite the increase of budgets, the implementation of new programs, the lengthening of school days and years, and the addition of new subjects and the deletion of others, the results of these efforts have been disappointing. Thus, with regard to mathematics education, Peterson (1988) criticized the elementary school mathematics curriculum “as based on the assumption that computational skills must be learned before children are taught to solve even simple word problems” (p. 7). The curriculum also reflects this sequence at the secondary level: Students generally take arithmetic
, then algebra, then geometry, and it is common to view higher order objectives as more appropriate later in the sequence. Thus, proofs are expected in geometry but not in algebra, and mathematical reasoning is more appropriate in algebra than arithmetic. As students progress through the grades, only the more capable students are able to keep up with the academic mathematics curriculum, so that most fall away before they encounter higher order instructional objectives, such as mathematical problem solving. Raudenbush, Rowan, and Cheong (1993) pointed out that although current conceptions of learning encourage the pursuit of higher order objectives, many teachers lack adequate preparation for teaching higher order thinking, and, additionally, organizational conditions at schools often discourage the pursuit of higher order objectives. From the perspective of Raudenbush et al., the pervasive influence of behaviorism in curriculum and instruction provides a potential explanation for the link between academic tracks (especially of math and science education) and the pursuit of instructional objectives. Shepard (1991) demonstrated that behaviorist theories imply that students learn best when complex learning tasks are broken down into smaller parts that are learned sequentially. Only when the earlier, simple steps are mastered is the learner ready for more complex tasks requiring analysis, hypothesis testing, and evaluation.
An alternative conception, the approach of model-centered learning and instruction, is described in this chapter. This approach is based on theories of mental models (cf. Johnson-Laird, 1983; Seel, 1991) and focuses on self-organized discovery learning. Therefore, the learner has to search continuously for information in a learning environment in order to complete or stabilize a mental model that corresponds to an a priori understanding of the material to be learned. Discovery learning is guided by exploratory models designed with a specific endpoint in mind. Students explore these models by developing hypotheses and then varying input parameters to investigate how well their conjectures align with the models.
With regard to mathematics education, the National Council of Teachers of Mathematics (1994) has formulated the precept that the primary role of algebra at the school level should be to develop confidence and facility in using variables and functions to model numerical patterns and quantitative relationships. For science education, Schauble (1996) formulated such an approach in the following manner: “The goal of scientific reasoning is not primarily the formulation of inductive generalizations, but rather the construction of explanatory models, that . . . account for the observed phenomena” (p. 103). Lesh and Doerr (2000) argued that helping students to develop explanatory models in order to make sense of their experiences of light, gravity, electricity, and magnetism should be among the most important goals of science instruction; otherwise, students invent models of their own that are often incomplete and incorrect (cf. D. E. Brown & Clement, 1989). Clement and Steinberg (2002) described how conventional approaches to electricity instruction in physics start with electrostatics and quickly introduce the concept of “potential difference,” which is defined mathematically. As the concept of electric potential generally remains unlearned when this instructional approach is implemented, Clement and Steinberg advocated the construction of a mental model that helps to develop a qualitative conception of electric potential, based on an analogy with pressure in compressed air that is compelling to most students, before introducing distant action and mathematical representation. Actually, students often do not (and cannot) develop appropriate symbol systems to make sense of mathematical entities such as directed quantities (negatives), multivalued quantities (vectors), ratios of quantities, changing or accumulating quantities, or locations in space (coordinates). However, they can invent significant mathematical solutions when confronted with the need to create meaningful models of real situations.
Model-centered learning and instruction is not simply an add-on activity to the curriculum. Rather, it involves a reformulation of the curriculum that gives primacy to students’ constructions of content knowledge through an inquiry process of experimentation, simulation, and analysis (cf. Doerr, 1996). In the following sections of this chapter, several aspects of model-centered learning and instruction are discussed: The psychological foundations of model-centered learning are described, then important conclusions for instructional design are derived. Following this, research on model-centered instruction is summarized, and, finally, problems of the assessment of mental models are described.
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