Using Student Reasoning to Inform the Development of Conceptual Learning Goals: The Case of Quadratic Functions

Abstract

Despite the proliferation of mathematics standards internationally and despite general agreement on the importance of teaching for conceptual understanding, conceptual learning goals for many K-12 mathematics topics have not been well-articulated. This article presents a coherent set of five conceptual learning goals for a complex mathematical domain, generated via a method of systematic empirical analysis of students' reasoning. Specifically, we compared the reasoning of pairs of students who performed differentially on the same task and inferred the pivotal intermediate conceptions that afforded one student deeper engagement with the task than another student. In turn, each pivotal intermediate conception framed an associated conceptual learning goal. While the empirical analysis of student reasoning is typically used to understand how students learn, we argue that such analysis should also play an important role in determining what concepts students should learn.

ACKNOWLEDGMENTS

This research is supported by the NSF under grant REC-0529502. The views expressed do not necessarily reflect official positions of the NSF. Drafts of a subset of the findings reported here were presented at the Research Presession of the 2009 meeting of the National Council of Teachers of Mathematics held in Washington DC, USA. We thank Marty Simon, Pat Thompson, Lyn English, and the reviewers for their helpful feedback.

Notes

¹By quadratic motion situation, we mean any scenario involving the motion of objects in which a relationship between two of the quantities in the situation can be represented by a quadratic function.

²All subject names are pseudonyms with gender and ethnicity preserved.

³We did not base our distance-time data on earth's gravity (i.e., 9.8 m/s² or 32 ft/s²) because we wanted distance and time values that would be manageable for middle school students. We initially created a context for the task in which the acceleration reflected in our data would be realistic (e.g., updraft winds or gravity on other planets). However, we felt that these details might distract students from the purpose of the task and so we decided to ask students to make sense of the data as is.

⁴Most students seemed unable to differentiate between the rate of change of the altitude and the speed in the Airplane Task and consequently treated the former as the latter. While this is not correct, we suspended our focus on the rate of change of altitude and analyzed students' use of speed, as if it was appropriate.

⁵This “6 feet” refers to a constant difference between the speeds in Mark's table (i.e., 3ft/s, 9ft/s, 15ft/s, 21ft/s, 27ft/s). Rather than reporting the change in speed as another speed, as might be indicated if Mark had said “I found that you add 6 feet per second every second,” Mark reported it as a distance (6 feet).