Teachers’ perspectives on “authentic mathematics” and the two-column proof form

Abstract

We investigate experienced high school geometry teachers' perspectives on "authentic mathematics" and the much-criticized two-column proof form. A videotaped episode was shown to 26 teachers gathered in five focus groups. In the episode, a teacher allows a student doing a proof to assume a statement is true without immediately justifying it, provided he return to complete the argument later. Prompted by this episode, the teachers in our focus groups revealed two apparently contradictory dispositions regarding the use of the two-column proof form in the classroom. For some, the two-column form is understood to prohibit a move like that shown in the video. But for others, the form is seen as a resource enabling such a move. These contradictory responses are warranted in competing but complementary notions, grounded on the corpus of teacher responses, that teachers hold about the nature of authentic mathematical activity when proving. This paper explores geometry teachers' perspectives on how the two-column proof form can be both a resource and a constraint in engaging students in proving. The two-column form has been used in the USA to teach students how to do proofs since early in the twentieth century (Herbst 2002a). But the two-column form has been criticized for perpetrating a formalistic image of proving, and calls for reform in the 1980s (NCTM 1989) had recommended that teachers decrease attention to two-column proving. In this article we examine a counterproposal that emerges from listening to teachers: namely, that the twocolumn form can actually be productive when engaging students in proving, by acting not only a constraint, but also a resource enabling complex and non-linear reasoning.

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“…And yet despite this, the geometry course has been criticized for years as a caricature of authentic mathematics, characterized by a proliferation of unnecessary postulates and imprecise definitions, claims that are accepted without proof, and proofs that valorize form over substance (Christofferson, 1930;Thurston, 1994;Usiskin, 1980;Weiss, Herbst, & Chen, 2009). …”

Understanding on Strategies of Teaching Mathematical Proof for Undergraduate Students

“…This method can play a dual role, so student can connect some mathematical concept easily. While in many cases teachers do use the form in a way that limits students ability to think flexibly when formulating an argument, in other cases teachers can use the form in a way that enables greater flexibility in reasoning and proving (Weiss, Herbst, & Chen, 2009).…”

What’s Involved in the Work of Dissertation Advising? An Interview with Jeremy Kilpatrick and Some Personal Reflections

“…I am especially proud of the fi ve dissertations that, as of November 2013, have come out of the GRIP (Aaron 2011 ;Chen 2012 ;González 2009 ;Hamlin 2007 ;Weiss 2009 ), and the many opportunities for co-authorship and fi rst authorship my students have had as they worked in the GRIP. Additionally, papers that most of those students wrote to transition to candidacy used data and ideas worked out in the GRIP and became published articles (Aaron and Herbst 2012 ;Chen and Herbst 2013 ;González and Herbst 2009 ;Weiss et al 2009 ). I am only regretful that the numbers have not been higher.…”