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 examplebased reasoning. Conceptual knowledge plays an important role in the validation of proofs. The practice of validating a proof depends upon whether a student or mathematician wrote the proof and in what mathematical domain the proof was situated. Pedagogical and epistemological consequences of these results are discussed.
This article examines the mathematical sense-making of children as they use physical devices to learn about linear functions. The study consisted of videotaped problem-solving sessions in which pairs of 8th graders worked on linear function tasks using a winch apparatus, a device with springs, and a computerized input-output machine. The following questions are addressed: How do children make sense of physical devices designed by experts to foster mathematical learning? How does the use of such devices enable learners to access selected aspects of a mathematical domain? The concept of transparency is suggested as an index of access to knowledge and activities rather than as an inherent feature of objects. The analysis shows that transparency is a process mediated by unfolding activities and users' participation in ongoing sociocultural practices.
A large number of college students exhibit a common misconception while solving certain algebra word problems. The error appears in writing equations where a multiplying factor is placed on the wrong side of the equation: writing 6S=P instead of S=6P, for example. Protocol analysis can allow us to investigate the cognitive processes producing the error as well as those leading to the correct solution. The findings have also led us to view the nature of the processes underlying the correct use of algebraic symbolization in a new way. The description of these basic processes should make it easier to design more effective strategies for teaching algebraic symbolization skills.
who were able to give reviews and advice. Also, qualitative analyses often use protocol analysis to present evidence. This led to a demand for longer articles. Expanding the size of the journal was not an option, so fewer articles could be published
. ( 2009 ). Coaching: Approaches and perspectives . In J. Knight (Ed.), Thousand Oaks, CA: Corwin Press . 10. Kuusela , H. , & Paul , P. ( 2000 ). A comparison of concurrent and retrospective verbal protocol analysis