Unpacking Pedagogical Content Knowledge: Conceptualizing and Measuring Teachers' Topic-Specific Knowledge of Students
Heather C. Hill, Deborah Loewenberg Ball, Steven G. Schilling
There is widespread agreement that effective teachers have unique knowledge of students' mathematical ideas and thinking. However, few scholars have focused on conceptualizing this domain, and even fewer have focused on measuring this knowledge. In this article, we describe an effort to conceptualize and develop measures of teachers' combined knowledge of content and students by writing, piloting, and analyzing results from multiple-choice items.
Josh's Operational Conjectures: Abductions of a Splitting Operation and the Construction of New Fractional Schemes
This article reports on students' learning through conjecturing, by drawing on a semester-long teaching experiment with 6 sixth-grade students. It focuses on 1 of the students, Josh, who developed especially powerful ways of operating over the course of the teaching experiment. Through a fine-grained analysis of Josh’s actions, this article integrates Piaget's scheme theory (1950/2001) and Peirce’s logic of abduction (1998) into a new theory about conjecturing that explains Josh’s learning.
How Mathematicians Determine if an Argument Is a Valid Proof
The purpose of this article is to investigate the mathematical practice of proof validation—that is, the act of determining whether an argument constitutes a valid proof. The results of a study with 8 mathematicians are reported. The mathematicians were observed as they read purported mathematical proofs and made judgments about their validity; they were then asked reflective interview questions about their validation processes and their views on proving. The results suggest that mathematicians use several different modes of reasoning in proof validation, including formal reasoning and the construction of rigorous proofs, informal deductive reasoning, and example-based reasoning.