The Role of Instructional Engineering in Reducing the Uncertainties of Ambitious Teaching

Abstract

Ambitious teaching is a form of teaching that requires a high level of teacher responsiveness to what students do as they actively engage with the subject matter. Thus, a teacher enacting ambitious teaching is often confronted with uncertainties about how to advance students’ learning while also building on students’ contributions. In this article we propose a framework that aims to deepen understanding about the role of instructional engineering in helping reduce the uncertainties of ambitious teaching, particularly with regard to the design and implementation of task sequences that target academically important but difficult-to-achieve learning goals. To illustrate the framework, we consider how instructional engineering helped reduce the uncertainties in enacting ambitious teaching to advance university and secondary students’ understanding of what counts as “proof” in mathematics.

ACKNOWLEDGMENTS

The two authors contributed equally to the preparation of this article. The authors are grateful to Paul Cobb, Rogers Hall, and anonymous reviewers for useful comments on previous versions of the article.

Notes

We acknowledge that this scenario assumes certain classroom norms or other social relations in which students feel comfortable expressing disagreement.

The name “Monstrous Counterexample” was not mentioned to the students in the class. We use it here for easy reference to the illustration.

All student names in the article are pseudonyms.

The question in the Circle and Spots Problem asked whether there is an easy way to tell for sure the maximum number of nonoverlapping regions into which the circle can be divided when n = 15 (Slide 4). Although the students’ inability to generate 32 regions for n = 6 did not guarantee that it was impossible to generate this number of regions with six points, it did indicate that the emerging pattern offered an insecure way to find the maximum number of regions for n = 15. The teacher's illustration of the counterexample was basically a confirmation of the emerging view in the class that the pattern they identified for n ≤ 5 offered an untrustworthy means to answer the question for n = 15. The class did not have the means to discover or prove the formula for the correct pattern (1 + _nC₂ +_nC₄), but this was not an expectation given the way the problem was phrased.

Additional information

Funding

The article is based on research supported by funds from the Spencer Foundation to both authors (grant numbers: 200700100, 200800104) and the Economic and Social Research Council (ESRC) in England to the second author (grant reference: RES-000-22-2536). The opinions expressed in the article are those of the authors and do not necessarily reflect the position, policies, or endorsement of either funding agency.