Middle School Students' and Mathematicians' Judgments of Mathematical Typicality
Authors
Candace Walkington
Department of Teaching and Learning, Southern Methodist University, Dallas, TX, USA
Jennifer Cooper
Department of Psychology, Stonehill College, Easton, MA, USA
Olubukola Leonard
Department of Curriculum and Instruction, University of Wisconsin – Madison, Madison, WI, USA
Caroline Williams-Pierce
Department of Educational Theory & Practice, University of Albany, Albany, NY, USA
Chuck Kalish
Department of Psychology, University of Wisconsin-Madison, Madison, WI, USA
Abstract
K-12 students often rely on testing examples to explore and determine the truth of mathematical conjectures. However, little is known about how K-12 students choose examples and what elements are important when considering example choice. In other domains, experts give explicit consideration to the typicality of examples – how representative a given item is of a general class. In a pilot study, we interviewed 20 middle school students who classified examples as typical or unusual and justified their classification. We then gave middle school students and mathematicians a survey where they rated the typicality of mathematical objects in two contexts – an everyday context (commonness in everyday life) and a mathematical context (how likely conjectures that hold for the object are to hold for other objects). Mathematicians had distinct notions of everyday and mathematical typicality – they recognized that the objects often seen in everyday life can have mathematical properties that can limit inductive generalization. Middle school students largely did not differentiate between everyday and mathematical typicality – they did not view special mathematical properties as limiting generalization, and rated items similarly regardless of context. These results suggest directions for learning mathematical argumentation and represent an important step towards understanding the nature of typicality in math.
