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University Students' Grasp of Inflection Points

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Educ Stud Math DOI 10.1007/s10649-012-9463-1 University students’ grasp of inflection points Pessia Tsamir & Regina Ovodenko # Springer Science+Business Media Dordrecht 2013 Abstract This paper describes university students’ grasp of inflection points. The participants were asked what inflection points are, to mark inflection points on graphs, to judge the validity of related statements, and to find inflection points by investigating (1) a function, (2) the derivative, and (3) the graph of the derivative. We found four erroneous images of inflection points: (1) f ′ (x)=0 as a necessary condition, (2) f ′ (x)≠0 as a necessary condition, (3) f ″ (x)=0 as a sufficient condition, and (4) the location of “a peak point, where the graph bends” as an inflection point. We use the lenses of Fischbein, Tall, and Vinner and Duval’s frameworks to analyze students’ errors that were rooted in mathematical and in real-life contexts. Keywords Inflection point . Concept image . Representation . Definition . Intuition 1 Introduction The notion of inflection points is frequently discussed when dealing with investigations of functions in calculus. Calculus is an important domain in mathematics and a central subject within high school and post-high school mathematics curricula. In the literature and in our preliminary studies, we found a few indications of learners’ erroneous conceptions of the notion (e.g., Ovodenko & Tsamir, 2005; Tall, 1987; Vinner, 1982). While the findings shed some light on students’ grasp of inflection points, it seems important to further investigate students’ related conceptions and to examine potential sources for their common errors. In this paper, we examine university students’ conceptions of the notion of inflection points, and we also use the context of inflection points to examine students’ proofs (validating and refuting) when addressing inflection-point-related statements. In the field of mathematics education, there are several theoretical frameworks proposing ways to analyze students’ mathematical reasoning; yet usually, research data are interpreted in light of a single theory. We believe that the use of different lenses may contribute to our interpretational examinations of the data and may offer us rich terminologies to address and to analyze the findings (e.g., Tsamir, 2007, 2008). Thus, we offer a range of interpretations of students’ conceptions based on three theoretical frameworks which are widely used to highlight possible sources of students’ difficulties in mathematics: Fischbein’s(e.g.,1993a) analyses of P. Tsamir (*) : R. Ovodenko Tel Aviv University, Tel Aviv, Israel e-mail: [email protected] P. Tsamir, R. Ovodenko learners’ intuitive algorithmic and formal knowledge; Tall and Vinner’s(e.g.,1981) review of concept image and concept definition; and Duval’s(e.g.,2006) investigations of the role of representation and visualization in students’ (in)comprehension of mathematics. 2 Theoretical framework In this section, we first survey the literature regarding: “What does research tell us about students’ conceptions of inflection points?” Then, we attend to the question: “What are possible sources of students’ mathematical errors?” 2.1 What does research tell us about students’ conceptions of inflection points? In the literature, there are some indications of difficulties that students encounter when using the notion of inflection points. For example, studies on students’ performances on connec- tions between functions and their derivatives within realistic contexts reported that students tend to err when identifying or when representing inflection points on graphs (e.g., Monk, 1992; Nemirovsky & Rubin, 1992; Carlson, Jacobs, Coe, Larsen & Hsu, 2002). Students also tend to use fragments of phrases taken from earlier-learnt theorems, such as: “if the second derivative equals zero [then] inflection point” even when solving problems in the context of “dynamic real-world situation” (Carlson et al., 2002, p. 355). On this matter, Nardi reported in her book: Amongst mathematicians: Teaching and learning mathematics at university level that “there is the classic example from school mathematics: how the second derivative being zero at a point implying the point being an inflection point” (Nardi, 2008,p.66).Aninteresting,relatedpieceofevidencewasfoundinMason’s(2001) reflection on his past engagement (as an undergraduate) with the task: “Do y=x5 and y=x6 have points of inflection? How do you know?” Mason recalled being familiar with the shapes of y=x5 and y=x6, thus knowing immediately which does and which does not have an inflection point. Still, he clearly remembered being perplexed when reaching in both cases, f ″ (x)=0 at x=0, and wondering why is it that one has an inflection and the other not? These data indicate that even future mathematicians may experience (as undergraduates) intuitive unease when encountering the insufficiency of f ″ (x)=0 for an inflection point. Gomez and Carulla (2001) reported on students’ grasp of connections between the location of inflection points and the location of the graph related to the axes. Students claimed that, if an inflection point of y=f (x) is on the y-axis or “close enough” to it, then the graph crosses that axis; if an inflection point is “not close, yet not too far from the y-axis,” then the y-axis is an asymptote of the graph; if an inflection point is “far enough from the y- axis,” then the graph has other asymptotes; and if the inflection point “is far enough from x- axis,” then it does not cross that axis. Another line of research on students’ conceptions of inflection points addressed issues related to the tangent at such points (e.g., Artigue, 1992; Vinner, 1982; 1991; Tall, 1987). For instance, Vinner (1982) reported that early experiences of the tangent of circles led learners to believe that “the tangent is a line that touches the graph at one point and does not cross the graph” (see also Artigue, 1992; Tall, 1987). In an early study, we examined university students’ conceptions of inflection points. We came across a novel tendency to regard a “peak or bending point” (i.e., a point where the graph keeps increasing or decreasing but dramatically changes the rate of change) as an inflection point (Tsamir & Ovodenko, 2004). We also found tendencies to regard f ′ (x0)=0 necessary for the existence of an inflection point at x=x0 (Ovodenko & Tsamir, 2005); and we observed that different erroneous conceptions of inflection points evolve when students University students’ grasp of inflection points address a variety of tasks. Thus, we instigated another study to examine university students’ conceptions when solving problems that offer rich opportunities to address inflection points. We report here on the findings. 2.2 What are possible sources of students’ mathematical errors? The analysis of students’ common errors calls for the use of related theoretical frameworks. Usually, studies on students’ conceptions use one interpretational framework to shed light on the data. We enriched our scope of analysis by using the perspectives offered by Fischbein (e.g., 1987), Tall and Vinner (e.g., 1981), and Duval (e.g., 2006) that were widely used by mathematics education researchers for analyzing students’ common errors and for examin- ing possible related sources. Fischbein (e.g., 1987, 1993b) claimed that students’ mathematical performances include three basic aspects: the algorithmic aspect, i.e., knowledge of rules, processes, and ways to apply them in a solution, and knowledge of “why” each of the steps in the algorithm is correct; the formal aspect, i.e., knowledge of axioms, definitions, theorems, proofs, and knowledge of how the mathematical realm works (e.g., consistency); and the intuitive aspect that was characterized as immediate, confident, and obviously grasped as correct although it is not necessarily so. Fischbein explained that, while neither formal knowledge nor algo- rithmic knowledge are spontaneously acquired, intuitive knowledge develops as an effect of learners’ personal experience, independent of any systematic instruction. Sometimes, intu- itive ideas hinder formal interpretations or algorithmic procedures and cause erroneous, rigid algorithmic methods, which were labeled algorithmic models. For example, students’ tendencies to claim that (a+b)5=a5+b5 or sin(α+β)=sin α+sin β were interpreted as evolving from the application of the distributive law (Fischbein, 1993a). Fischbein’s analysis of students’ intuitive grasp of geometrical and graphs-related notions led him to coin the term figural concepts, i.e., mental, spatial images, handled by geometrical or functions-based reasoning. Figural concepts may become autonomous, free of formal control, and thus erroneous (Fischbein, 1993a). Fischbein noted that a certain interpretation of a concept may initially be useful in the teaching process due to its intuitive qualities and its local concreteness. But, as a result of the primacy effect, this initial model may become rigidly attached to the concept and generate obstacles to advanced interpretations of the concept. Fischbein’s framework was widely used to analyze students’ mathematical conceptions of various notions, such as functions, infinity, limit, and sets (e.g., Fischbein, 1987). Two other researchers who examined learners’ grasp of mathematical notions are Tall and Vinner (e.g., 1981) who coined the terms concept-image and concept-definition. Concept image includes all the mental pictures and the properties that a person associates with
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