Abstract:
Generalisation is a key component of mathematical activity. Mathematicians often seek general formulae or wonder if a rule that holds in a particular dimension also holds in dimension n. However, generalisation is not limited to professional mathematicians; kindergarteners engage in generalisation when they seek patterns and algebra is sometimes described as generalised arithmetic. Because generalisation is so critical to mathematical thought, research that investigates how people generalise is an important part of supporting student learning. In particular, students often struggle to form normatively correct generalisations (e.g., Dorko & Weber, 2014; Jones & Dorko, 2015; Kabael, 2011; Martínez-Planell & Gaisman, 2013, 2012; Martínez-Planell & Trigueros, 2012). Research can help us better understand what ways of thinking are productive for generalising and help us understand the nature of generalisation as a practise. To that end, this dissertation study focuses on how students generalise their notion of function from the single- to multivariable setting. I focus on function because functions are a fundamental mathematical concept and are relevant in everyday life.
The first manuscript describes results from a longitudinal study regarding how five calculus students generalised what it means for a relation to be a function from the single- to multivariable setting. In keeping with the use of the term ‘generalisation’ to refer to both a product (e.g., a theorem) and a process (c.f. Harel & Tall, 1991; Mitchelmore, 2002), I describe both the mathematical ideas students generalised and the nature of their generalising activity. Findings indicate that students generalised mathematical ideas such as function-as-equation, function-as-pattern, the vertical line test (VLT), function notation, inputs and outputs, and univalence. Focusing on equations seemed to prevent students from developing a normative understanding of what it means to be a function in R3. In contrast, some students considered generalisations of the VLT (e.g., applying the VLT to traces, lines parallel to the y axis on an R3 graph) that, while not normatively correct, led to correct generalisations. Similarly, thinking about function notation and inputs and outputs supported students in forming correct generalisations. Findings about students’ generalising activity indicate that students engaged in what Ellis (2007) terms relating, searching, and extending. For example, some students made sense of the notation f(x, y) by relating it to the notation f(x). One student engaged in searching as she sought to generalise the vertical line test. She searched for a way to draw a line on an R3 graph that would intersect the graph exactly once. Other students extended the range of applicability of prior ideas (e.g., input and output) and their definitions of function.
The second manuscript is more theoretical in nature. In this manuscript, I argue that Piaget’s constructs of assimilation and accommodation align with Harel and Tall’s (1991) framework for generalisation in advanced mathematics. Based on what they imagined to be the cognitive processes underlying generalisation, Harel and Tall proposed that generalisation might be expansive (occurring when a student expands the applicability range of an existing schema without reconstructing it), reconstructive (occurring when a student reconstructs a schema to widen its range of applicability), or disjunctive (occurring when a student constructs a new, disjoint schema to deal with a new context). I employ this framework to interpret data about how students generalise their notion of function from R2 to R3 and how they first think about graphing in R3. I then interpret the same instances through the lens of assimilation and accommodation, arguing that they provide explanatory mechanisms for generalisation. I conclude by discussing how assimilation and accommodation explain other empirical findings regarding students’ generalisation of function and graphing.
Taken together, the two manuscripts provide different, complementary insight into the generalisation phenomenon. The first manuscript employs a framework that describes generalisation at a small grain size, while the second focuses on describing generalisation from a much larger grain size. While the first manuscript describes generalisation empirically, the second seeks to contribute to a theoretical explanation for how generalisation occurs. Notably, these investigations occur among a population (undergraduate students) and content area (single and multivariable calculus) in which generalisation has not typically been studied. As a whole, the dissertation extends the existing body of literature by both empirically and theoretically examining the phenomenon of generalisation with undergraduate students in a relatively advanced mathematical setting.