Abstract
This paper describes how the notion of instructional situation can serve as a cornerstone for a subject-specific theory of mathematics teaching . The high school geometry course in the U.S. (and some of its instructional situations—constructing a figure, exploring a figure, and doing proofs ) is used to identify elements of a subject-specific language of description of the work of teaching . We use these examples to analyze records of a geometry lesson and demonstrate that, if one describes the actions of a teacher using descriptors that are independent of the specific knowledge being transacted, one might miss important elements of the instruction being described. However, if the notion of instructional situations is used to frame how one observes mathematics teaching , then one can not only track how teacher and students transact mathematical meanings but also identify alternative instructional moves that might better support those transactions.
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Notes
- 1.
Brousseau (1997) defines the milieu as the system counterpart to the learner in a learning task; the milieu is the recipient of the learner’s actions and a source of feedback to the learner. In saying teaching milieu, we are using milieu analogously and in reference to the teacher’s work. The teaching milieu would therefore be the system counterpart to the teacher that contains the teacher’s actions and provides feedback to the teacher. Crucially, this teaching milieu contains the students’ actions, which, inasmuch as they concern mathematical work, are subject-specific. Margolinas’ (1995) studies of the work of the teacher have given a basis for this use of milieu in describing the teacher’s role.
- 2.
The word task is used as a general concept here, and the emphasis is on a task as a particular chunk of work (task as a proper subset of work). The task might be to do a problem, to discuss a solution to a problem, or to compare solutions to a problem, but the point is that students’ engagement is through the particular work called forth by a task (see Brousseau, 1997, p. 22).
- 3.
Doyle also included a fourth component, the accountability of a task, or the relative importance of the task when compared to the other work (e.g., other tasks ) that the class might do (Doyle, 1988, p. 169). We incorporate this notion of the role the task plays in the class’s accountability system in our conception of instructional situation and prefer to describe tasks using the three components of goal, resources, and operations.
- 4.
By special quadrilaterals we mean parallelograms, rhombi, rectangles, squares, etc.
- 5.
While the instructional goal was to learn about special quadrilaterals, the work assignment was often stated in ways that kept those quadrilaterals hidden. The definition of M-Quad and questions about M-Quad were mere instruments to organize students’ work, not what was at stake in the unit (as, obviously, M-Quad is a made-up concept with no status in the curriculum or in the discipline).
- 6.
Note, however, that our description of the situation of exploration , in which the teacher and students reify concrete artifacts as mathematical objects, does not entail our personal endorsement of such relationship to geometric knowledge. Our descriptive attention to them owes to the fact that such practices exist in intact teaching .
- 7.
This is, of course, true, and known in mathematics as Varignon’s Theorem (see Coxeter & Greitzer, 1967, p. 51; also http://mathworld.wolfram.com/VarignonsTheorem.html).
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Acknowledgements
Data used in this paper was collected with resources from a grant from the National Science Foundation, REC 0133619 to P. Herbst. All opinions are those of the authors and do not necessarily represent the views of the Foundation.
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Herbst, P., Boileau, N., Gürsel, U. (2018). Examining the Work of Teaching Geometry as a Subject-Specific Phenomenon. In: Herbst, P., Cheah, U., Richard, P., Jones, K. (eds) International Perspectives on the Teaching and Learning of Geometry in Secondary Schools. ICME-13 Monographs. Springer, Cham. https://doi.org/10.1007/978-3-319-77476-3_6
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