Ph.D. Candidate Honored for Paper on Math Education
Aron Samkoff, a Ph.D. student in Mathematics Education at Rutgers Graduate School of Education, was selected to receive the 2012 Janet Duffin Award for his paper “On the different ways that mathematicians use diagrams in proof construction.” The award was shared with the paper's co-authors, Dr. Keith Weber also from the GSE and Dr. Yvonne Lai, University of Michigan.
The award is made annually to the author (or authors) of what is judged to be the most outstanding research paper published in Research in Mathematics and Education (RME) during the preceding calendar year. It is funded through a generous gift made to the British Society for Research into Learning Mathematics (BSRLM) by the late Bill Duffin, in memory of his wife, Janet, a long-standing and active member of the Society.
Samkoff’s research is part of a larger agenda with his advisor, Dr. Weber, looking at how mathematicians and students write and read proofs, in hopes of learning from the mathematicians about how to more effectively teach students. He investigated “1) the ways in which students’ understanding of proofs could be improved, 2) how instruction might be designed so as to improve students’ comprehension of proofs, and 3) whether and how this proposed instruction is effective.” The use of proofs in upper-level mathematics is a necessity, however, research suggested considerable difficulty reading and understanding proofs by students. Often, students base their proofs “on deductive logic, not the appearance of the diagram,” and “students are often reluctant to use diagrams, even for problems where their use might be of highly productive.” Samkoff examined how mathematicians use diagrams to construct proofs, to investigate ways in which students’ understanding of proofs can be improved with the use of such diagrams.
With data collected by co-author Dr. Lai, Samkoff discovered there were a number of different successful ways to accomplish the task of solving proof. The research consisted of interviewing eight mathematicians, who completed a task of writing a full proof to a question that invited the use of a diagram. Results showed variance in the use of pictures and diagrams, which indicated the dissimilarity in the ways mathematicians think. Samkoff commented, “This painted a more complex picture about how mathematicians construct proofs.”
Samkoff found four purposes that participants’ diagrams served; noticing properties and generating conjectures, estimating the truth of an assertion, suggesting a proof approach, and instantiating an idea or assertion. Previous studies suggested that students generally did not use diagrams for the same reasons the research participants did, and these aspects of proof construction were observed to be absent from some mathematics lectures.
From the findings above, Samkoff suggested “The specific ways in which mathematicians used diagrams in this study may be highlighted for students during instruction.” As the use of diagrams by the research participants differed from the common use of diagrams by students, adapting this research in education may offer students another way of understanding proofs. Samkoff noted that “direct instruction is not guaranteed to improve students’ ability to use diagrams to construct proofs, but we argue that diagrammatic reasoning deserves explicit attention in the undergraduate mathematics classroom.”
Samkoff will attend BSRLM’s day conference in June to receive the award, and to deliver the Janet Duffin Lecture.
Aron Samkoff, Yvonne Lai & Keith Weber (2012): On the different ways that mathematicians use diagrams in proof construction, Research in Mathematics Education, 14:1, 49-67.