Nonstandard Student Conceptions About Infinitesimals

Abstract

This is a case study of an undergraduate calculus student's nonstandard conceptions of the real number line. Interviews with the student reveal robust conceptions of the real number line that include infinitesimal and infinite quantities and distances. Similarities between these conceptions and those of G. W. Leibniz are discussed and illuminated by the formalization of infinitesimals in A. Robinson's nonstandard analysis. These similarities suggest that these student conceptions are not mere misconceptions, but are nonstandard conceptions, pieces of knowledge that could be built into a system of real numbers proven to be as mathematically consistent and powerful as the standard system. This provides a new perspective on students' “struggles” with the real numbers, and adds to the discussion about the relationship between student conceptions and historical conceptions by focusing on mechanisms for maintaining cognitive and mathematical consistency.

Four conceptions of infinity

“…In the first case the horizon is still fixed, but located at an improper point (omega-epsilon position). This is described, for example, by Sierpinska (1994) when she speaks about the static conception of infinity, or by Ely (2010) in an interview with a student who, with regard to infinity and infinitesimal numbers, works with equally rigorous notions of non-standard analysis. In the second case the individual still works with the horizon, but its position is indeterminate and can always be altered appropriately.…”

“…The research activities concentrated on several contexts, such as the structure of number domains (Bauer, 2011;Ely, 2010;Katz & Katz, 2010;Singer & Voica, 2008), functions and limit processes (Juter, 2006;Liu & Niess, 2006;Monaghan, 2001), comparisons of infinite sets (Jahnke, 2001;Tsamir, 2001) and a geometrical context (Fischbein, 2001;Jirotkova & Littler, 2004), mostly draw our attention to the difficulties students experience while passing from 'the finite' to 'the infinite' and they all agree upon the fundamental role played by the preceding intuitive ideas (Alcock & Simpson, 2004;Bauer, 2011;Dubinsky et al, 2005;Fischbein, 2001;Jahnke, 2001;Marx, 2013;Tirosh, 2005).…”

“…On the other hand, Fernández-Plaza and Simpson (2016) in their research in the understanding of three basic uses of the notion of limit (the limit of a sequence, the limit of a function at a point, and the limit of a function at infinity) point to a group of undergraduate students who have a static conception of infinity. Similarly, analogous to Sierpinska (1994) or Ely (2010), this 'appears to contradict the assumption that understanding of limit of function at a point necessarily involves thinking about sequences or using dynamic imagery' (Fernández-Plaza & Simpson, 2016, p. 317).…”

“…Fernández-Plaza and Simpson (2016) describe in the above-quoted research that some students are able to carry the idea of neighbourhood natural numbers to the set of real numbers or to the set of points of a line. Ely (2010) and Katz and Katz (2010) show how students manage to work with the idea of infinitely small quantities. For instance, 28 % of the university students in the Kidron and Tall study perceived the infinite sum of functions as a legitimate object but not clearly as the formal limit definition.…”

A new ontology of reasons for inferentialism: redefining the notion of conceptualization and proposing an observer effect on assessment

“…Hußmann et al 2018;Meyer 2018), our stance is more radically consistent with conceptual pragmatism. Ely (2010) correctly recognizes the observer effect and describes his interviewee's concept formation within the interviews as being emergent. We can only make conjectures as to the influence of asking questions about concepts on students' formation of the concepts.…”