Abstract
Prior research showed that many secondary students fail to construct arguments that meet the standard of proof in mathematics. However, this research tended to use survey methods and only consider students presenting their perceived proofs in written form. The limited use of observation methods and the lack of consideration of students presenting their perceived proofs orally—in tandem with their written proofs for the same claims—might have resulted in a skewed picture of the potential of students’ constructed proofs, and this raises concern about the validity of research findings. The research reported in this article substantiates this concern. Using data from a design experiment in two secondary mathematics classrooms (14–15‐year‐olds), I explored the role of the written versus the oral mode of argument representation in students’ proof constructions. Findings from the comparison between the written arguments (perceived proofs) that the students produced during small group work and the oral arguments that the students presented in front of the class for the same claims showed that the oral mode of representation is more likely than the written mode to be associated with the construction of arguments that meet the standard of proof. Thus if a study had analysed students’ written arguments only (as in survey research), it would have reported a less favourable picture of the potential of students’ constructed proofs than another study that would focus only on students’ oral arguments (as in observational research). Implications for methodology, research and practice are discussed in light of these findings.
Highlights
The concept of ‘proof’ is fundamental to deep learning in mathematics and in various countries it is considered to be important for students’ mathematical experiences across all levels of education, as early as the primary school (e.g. Ball & Bass, 2003; Yackel & Hanna, 2003; National Governors Association Center for Best Practices & Council of Chief State School Officers [NGA & CCSSO], 2010; Department for Education, 2013)
A main research strand has focused on secondary students’ constructions of mathematical arguments, showing that many secondary students fail to produce arguments that meet the standard of proof (e.g. Senk, 1989; Healy & Hoyles, 2000; Ku€chemann & Hoyles, 2001–03; Knuth et al, 2009)
The construction of arguments that meet the standard of proof is often the last step in a complex and multifaceted activity typically referred to as proving that includes other processes such as the following: work with examples or exploration of particular cases, identification of patterns and generation or refinement of conjectures or other kind of mathematical claims, and attempts to develop informal arguments for these mathematical claims that may offer insight, or translate, into a proof (e.g. Mason, 1982; Boero et al, 1996; Weber & Alcock, 2004; Mariotti, 2006; Stylianides, 2007, 2008; Alcock & Inglis, 2008; Zazkis et al, 2008, 2016)
Summary
A main research strand has focused on secondary (i.e. post-primary) students’ constructions of mathematical arguments, showing that many secondary students fail to produce arguments that meet the standard of proof The construction of arguments that meet the standard of proof is often the last step in a complex and multifaceted activity typically referred to as proving that includes other processes such as the following: work with examples or exploration of particular cases, identification of patterns and generation or refinement of conjectures or other kind of mathematical claims, and attempts to develop informal arguments for these mathematical claims that may offer insight, or translate, into a proof I focus on the presentation of mathematical arguments whose constructors perceive they meet the standard of proof and who might have engaged previously in some other processes within the broader activity of proving. The important point here seems to be that researchers specify clearly the definition of proof that has underpinned their research, as well as their reasons for their choice, so as to facilitate understanding of their findings and comparison of findings across studies (Balacheff, 2002)
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