University students grasp of inflection points - Kansas State University

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: pessia@post.tau.ac.il

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 connections 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). An interesting, related piece of evidence was found in Mason'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 yaxis," then the graph has other asymptotes; and if the inflection point "is far enough from xaxis," 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 examining 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 algorithmic knowledge are spontaneously acquired, intuitive knowledge develops as an effect of learners' personal experience, independent of any systematic instruction. Sometimes, intuitive 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(+)=sin +sin 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 the concept name. When solving a certain task, specific aspects of one's concept image are activated, the evoked concept image. Concept definition is a term used to specify the concept in a way that is accepted by the mathematical community, but learners often hold a personal concept definition that may differ from the formal one. Moreover, the concept image which is frequently shaped by some examples that do not fit the concept definition has a crucial impact on the reconstruction of the concept definition when the latter is called for (Vinner, 1990). Occasionally, one part of the concept image becomes a potential conflict factor by implicitly conflicting with another part of the concept image or with the concept definition. For instance, the concept of tangent is usually introduced with reference to circles, implicitly insinuating that a tangent should only meet the curve at one point and should not cross the curve (e.g., Vinner, 1991). This often becomes part of the students' tangent-image (generic tangent c.f. Tall, 1987) that may cause problems later, for instance, at inflection points. Potential conflict factors contain the seeds of

P. Tsamir, R. Ovodenko

future conflict, but a key condition for one to actually face cognitive conflict is awareness. That is, two incompatible images may obliviously coexist and be interchangeably used due to compartmentalization in one's mind (i.e., different ideas are placed in "separate drawers"). However, when conflicting aspects are evoked simultaneously, they could cause an actual sense of conflict or confusion, which can serve as a starting point in instruction. Tall and Vinner's framework proved to be useful in analyzing students' conceptions of various mathematical notions and specifically those related to advanced mathematics (e.g., limits, continuity, tangent: Tall, 1987; Tall & Vinner, 1981; Vinner, 1982).

Duval offered a different perspective that "representation and visualization are at the core of understanding mathematics" (1999, p. 3). He explained that a major difficulty in mathematics comprehension is rooted in the nature of mathematical objects that can be accessed only through signs while, in other sciences (e.g., biology), objects may also be accessed in a direct manner. Duval analyzed representations of mathematical objects and processes and suggested different routes of their mobilizations in learners' minds (e.g., Duval, 2000, 2002; see also an extensive discussion of Duval's contribution in Hesselbart, 2007). He pointed to three types of representation: (a) mental representations (individuals' conceptions and misconceptions about realistic, concrete objects); (b) computational representations (types of information-codification used in mathematical algorithmic performances); and (c) semiotic representations (include signs, relationships, rules of production and transformation, related with particular sign systems such as language, algebra, and graphs). A semiotic representation provides an "organization of relations between representational units" and since visualization allows an immediate and complete capture of any organization of relations, "there is no understanding without visualization" (Duval, 1999, p. 13). Duval emphasized the role of semiotic representations and of related transformational functions in mathematics learning and called attention to essential differences between treatment, i.e., a transformation within a single semiotic system and conversion, i.e., a transformation from one semiotic system to another. For example, transforming x(x+1) into x2+x within the algebraic system is treatment, while transforming x(x+1) into a graph of the parabola, i.e., from the algebraic system into the Cartesian system, is conversion. Duval (e.g., 2006) stressed that conversions are crucial in mathematical activities and that students' difficulties with mathematical reasoning lie in the cognitive complexity of conversions that entail recognition of a mathematical object in different representations and discrimination between "what is mathematically relevant and what is not," when examining a mathematical object. Another major source of difficulties is rooted in the direction of conversion. "When roles of source register and target register are inverted within a semiotic representation conversion task, the problem is radically changed for students. It can be obvious in one case, while in the inverted task most students systematically fail" (ibid., p. 122). Furthermore, conversion of representations requires the cognitive dissociation of the represented object and the content of the specific representation in which the object was first introduced; on the other hand, there is a cognitive impossibility of dissociation of any semiotic representation content and the first representation of the object, because the only access to mathematical objects is semiotic. Consequently, students erroneously perceive two representations of the same object as being two mathematical objects, i.e., the registers of the representations remain fragmented and compartmentalized. Duval's framework has been valuable in analyzing students' engagement in a wide range of mathematical topics such as geometry, functions, algebra, vectors, and number systems (e.g., Duval, 1999, 2000, 2006).

We use the ideas and the terminology offered by Fischbein, by Tall and Vinner, and by Duval to examine university students' conceptions of inflection points, by focusing on the questions: What are students' common errors, and what are possible sources of these errors?

University students' grasp of inflection points

3 Methodology

3.1 Participants

We report on two studies, investigating two groups of university students. In one study, 53 students were asked to solve tasks 1?3; in the second study, 52 students were asked to solve tasks 4?6. All participants had studied mathematics at mathematics faculties or at mathematics-oriented faculties, i.e., computer science, computer engineering, and electronic engineering. All had successfully completed at least two calculus courses, two linear algebra courses, and a course of differential equations. The participants' ages ranged between 25 and 35 years, and all expressed interest and enthusiasm in their studies.

3.2 Tools and procedure

The research tools were questionnaires and individual oral interviews. We present six tasks from the questionnaires that included reference to inflection points.

Task 1: True or false?

Statement 1: f: R R is a continuous, differentiable function.

If A(x0,f (x0)) is an inflection point, then f (x0)=0. True/false, prove:

Statement 2: f: R R is a continuous, (at least twice) differentiable function.

If f (x0)=0, then A(x0, f (x0)) is an inflection point. True/false, prove:

Task 2: Investigate the graphs

Figure 1 presents five graphs. On each graph mark Zi all points of intersection with the axes; Xi maximum-points; Ni minimum-points; and Pi inflection points. Task 3: Define

What is an inflection point?

Task 4: Investigate the function

Investigate the function

f

?x?

?

1 4

x4 ? x3

. Are there:

(a) (b) (c) (d) Task 5

Points of intersection with the axes? Yes/no, if "yes" what are the points? Explain. Maximum/minimum points? Yes/no; if "yes," what are the points? Explain. Inflection points? Yes/no; if "yes," what are the points? Explain. Asymptotes? Yes/no; if "yes," what are they? Explain. Investigate f (x)

Note: in the following task, f (x) is given, but the questions are about f (x). f 0?x? ? 15x2 ? 5x3 . Does f (x) have:

Fig. 1 The graphs presented in Task 2

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