How Mathematicians Obtain Conviction: Implications for Mathematics Instruction and Research on Epistemic Cognition

Abstract

Abstract. The received view of mathematical practice is that mathematicians gain certainty in mathematical assertions by deductive evidence rather than empirical or authoritarian evidence. This assumption has influenced mathematics instruction where students are expected to justify assertions with deductive arguments rather than by checking the assertion with specific examples or appealing to authorities. In this paper, we argue that the received view about mathematical practice is too simplistic; some mathematicians sometimes gain high levels of conviction with empirical or authoritarian evidence and sometimes do not gain full conviction from the proofs that they read. We discuss what implications this might have, both for for mathematics instruction and theories of epistemic cognition

Mathematicians’ Example-Related Activity when Exploring and Proving Conjectures

“…It is important to note that studies of expert thinking should not be applied uncritically to novice instruction. However, we agree with Weber et al (2014) that even if there are inconsistencies between mathematicians' practices and what is taught in classrooms, Recognizing such inconsistencies requires having an accurate understanding of mathematical practice. Even if it is not problematic for students to be ignorant of mathematical practice, it does not follow that it is acceptable for teachers, researchers, or curriculum designers to be similarly unaware (p. 38).…”

“…Epstein and Levy (1995) contend that BMost mathematicians spend a lot of time thinking about and analyzing particular examples,^and they go on to note that BIt is probably the case that most significant advances in mathematics have arisen from experimentation with examples^(p. 6). Several researchers have accordingly examined various aspects of the interplay between example-based reasoning activities and proof activities among both mathematicians and mathematics students (e.g., Alcock and Inglis 2008;Antonini 2006;Buchbinder and Zaslavsky 2009;Iannone et al 2011;Harel 2008;Knuth et al 2009;Weber 2008Weber , 2010Weber et al 2014). For instance, Alcock and Inglis (2008) constructed case studies of two participants engaged in example use while evaluating conjectures and constructing proofs.…”

Accessibility of reformulated mathematical content

“…The limitations regarding proofs mentioned here need not be problematic, as is argued in Weber et al (2014). Because my argument only has to do with reasoning about the existence of proofs, there is no need to worry about these complications for empirical research regarding general mathematical reasoning abilities.…”

Challenges in mathematical cognition: A collaboratively-derived research agenda

“…Elucidating the Nature of Mathematical Thinking These questions reflect the ambitious long-term goal of understanding mathematical cognition from infancy to mature expertise. Collectively they address the contribution of nativist theories in proposing foundational "number sense" capacities seen in nearly all human infants (e.g., Feigenson, Dehaene, & Spelke, 2004;Geary, 2007), the mature state of refined formal cognitive capabilities associated with highly successful mathematicians (Weber, Inglis, & Mejía-Ramos, 2014), and the as-yet unexplained mechanisms that link the two. Although individual research studies necessarily focus on a relatively narrow part of this spectrum, participants in the exercise recognised the importance of working towards an empirically evidenced understanding of mathematical cognition that accurately captures effective mathematical reasoning, that provides reliable ways of diagnosing atypical development, and that is specific enough to be leveraged for the design of interventions.…”

“…et al, 2013), individual differences in typical mathematics learning (Geary, 2011;Raghubar, Barnes, & Hecht, 2010), and the nature and variety of mathematical expertise (Weber, Inglis, & Mejía-Ramos, 2014). Studies on the effects of the environment, including home (LeFevre et al, 2009) school (Beilock, Gunderson, Ramirez, & Levine, 2010Clements & Sarama, 2004), and culture (e.g., Jones, Inglis, Gilmore, & Dowens, 2012), have also influenced theoretical conceptualisations of mathematical learning and contributed to the design of interventions to support that learning (Cohen Kadosh, Dowker, Heine, Kaufmann, & Kucian, 2013).…”