University Students' Grasp of Inflection Points

Home , Inflection, Inflection point, Who (pronoun)

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 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).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 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(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 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 reconstruc- tion 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 ð Þ¼1 4 3 Investigate the function f x 4 x x . Are there: (a) Points of intersection with the axes? Yes/no, if “yes” what are the points? Explain. (b) Maximum/minimum points? Yes/no; if “yes,” what are the points? Explain. (c) Inflection points? Yes/no; if “yes,” what are the points? Explain. (d) Asymptotes? Yes/no; if “yes,” what are they? Explain. Task 5 Investigate f ′ (x) Note: in the following task, f ′ (x) is given, but the questions are about f (x). f 0ðxÞ¼15x2 5x3 . Does f (x) have:

Fig. 1 The graphs presented in Task 2 P. Tsamir, R. Ovodenko

(a) Maximum/minimum points? Yes/no; if “yes,” find the related Xs. Explain. (b) Inflection points? Yes/no, if “yes,” find the related Xs. Explain. Task 6: Investigate the graph of f ′ (x) Note: in the following task, the graph of f ′ (x) is given, but the questions are about f (x). Figure 2 presents the graph of f ′ (x). Does f (x)have: (a) Maximum/minimum points? Yes/no; if “yes,” find the related Xs. Explain. (b) Inflection points? Yes/no, if “yes,” find the related Xs. Explain.

The tasks were formulated with a number of underlying deliberations (Fig. 3). First, since the concept of inflection point has both graphical and non-graphical aspects which play significant roles in students’ figural and conceptual knowledge and influence their concept images (see, for instance, Duval, 2006; Fischbein, 1993a; Vinner & Dreyfus, 1989), the tasks were given in three representations: verbal (tasks 1, 3), graphical (tasks 2, 6), and algebraic (tasks 4, 5). The solutions could be formulated either by treatment within the given representations or through conversion. We learnt from the teachers of these courses that, most commonly, students analyzed algebraic expressions of functions (task 4) and seldom analyzed algebraic expressions of the derivative (task 5). In the case of the graphic representation, students analyzed given graphs quite rarely (task 2) and even more rarely analyzed graphs of the derivative (task 6). Students’ modest experience with such tasks and the impossibility of applying routine algorithms in the related solutions led us to believe that students’ knowledge might be challenged, especially in task 6. Finally, tasks 1 and 3 were in a way familiar to students because definitions and theorems were customarily presented in their classes. However, students were rarely asked to determine whether a statement is valid (task 1). The questionnaires offered a number of opportunities to examine students’ tendencies to erroneously view (a) f ′ (x)=0 as necessary for inflection (in task 1 judging statement 1 as being true,intasks2and6identifying“only horizontal inflection points,” in task 3 adding f ′=0 to their definitions, and in tasks 4 and 5 adding to the algorithm f ′ (x)=0 as a necessary condition); and (b) f ″ (x0)=0 as sufficient for inflection (in task 1 judging statement 2 as being true; in task 6 identifying “only f ″=0 points,” in task 3 not going beyond the condition f ″ (x)=0 in their definitions, and in tasks 4 and 5 stopping the algorithm after the stage of f ″ (x)=0). Moreover, in the tasks presented in the algebraic and in the graphic representations the inflection points

Fig. 2 The graph of y=f ′ (x) University students’ grasp of inflection points

The Task No 1 / No 3 No 4 / No 5 No 2 / No 6 Task Representation Verbal Symbolic - Algebraic Graphical Instruction Judge and Prove / Define Investigate Expression Investigate Graph

The Solution Possible Representation *Verbal / or *Symbolic / and *Graphic / and *Symbolic / or Verbal / or Symbolic / or Graphic / or Graphic / or Verbal / or a combination a combination a combination Connecting Inflection point f '= 0, f "= 0 / open f (x) / f ' (x) f (x) / f ' (x)

Fig. 3 Task analysis * the most expected representations included horizontal points (in task 2, task 4, task 5, and task 6), oblique points (in task 2, task 4, task 5, and task 6), and a vertical point (task 2). The different perspectives were devised to provide students with rich opportunities to evoke images of inflection points (see previous section, Tall & Vinner, 1981). Based on the analysis of their written solutions, 15 participants were invited to individual follow-up interviews. We interviewed students whose written solution we found to be interesting, unclear, or puzzling. During their interviews, students were shown their original solutions and were asked to provide an elaborate explanation of their written ideas. The interviews lasted 30–45 min and were audio-taped and transcribed.

4 Discussion of the results

In this section, we discuss our data about students’ conceptions of inflection points and about proof-related difficulties that students encountered when performing on the inflection- point tasks. We use theoretical lenses to discuss the identified difficulties.

4.1 Students’ grasp of inflection points

Our data indicate four main erroneous images of inflection points: (1) slope zero, f ′ (x)=0 is necessary for an inflection point, (2) non-zero slope, f ′ (x)≠0 is necessary for an inflection point, (3) f ″ (x)=0 is sufficient for an inflection point, and (4) peak points, “where the graph bends” are inflection points. We discuss each of these concept images, first by reporting on relevant findings, and then by analyzing the data presented in light of Fischbein’s, Tall and Vinner’s, and Duval’s theoretical frameworks. 1. Slope zero, f ′ (x)=0 is necessary for an inflection point: This concept image is evident in students’ solutions to all six tasks. In task 1: True or False? about 60 % of the students incorrectly claimed that Statement 1, “if P is an inflection point, then f ′ (P)=0” is true (Table 1). Their explanations usually (45 %) related to the definition of inflection points—“the f ′ (x)=0 condition is in the definition of an inflection point” or “f ′ (x)=0 and f ″ (x)=0 (and perhaps additional conditions) are the conditions for an inflection point.” P. Tsamir, R. Ovodenko

Table 1 Distribution of students’ reactions to the statements (in %) Judgment Inflection f ″=0⇒Inflection point ⇒ f ′=0 Justification N=53 N=53

Falsea 38 55 There is another condition 30 13 Counter-example 8 42 True 59 40 “It’s the definition …” 45 17 Algorithmic considerations 4 23 Irrelevant/no justification 10 – No answer 3 5 aCorrect judgment

In task 2: Investigate the Graphs students were asked to mark on given graphs points of intersection with the axes (Zi), extreme points (Xi–Max and Ni–Min), and inflection points (Pi). Students who marked Zi, Xi,orNi points, but no Pi points, were regarded as students who identified no inflection points. Commonly, students correctly marked the horizontal inflection points P3 on Graph 2 (95 %) and P10 on Graph 5 (75 %); the percentages of identifications of the other inflection points were much lower (0 to 17 % for each of the points, see Table 2). To get an overall perspective of students’ solutions to task 2, we created profiles of their identifications of inflection points in all five graphs (Table 3). We found three main profiles of solutions: (a) partial, (b) correct–incorrect mix, and (c) incorrect. No student correctly identified all inflection points on the five graphs. Students’ tendencies to identify only horizontal inflection points were evident in the partial selection of 19 % of the students who marked only the horizontal points P3, P10. In Task 3: What is an inflection point? Students’ tendencies (15 %) to regard f ′=0 as “a must” at inflection points were manifested in their slope-zero definitions that included f ′ (x)= 0 or slope-zero consideration (Table 4): “A point where f ′ (x)=0 and the graph keeps increasing (decreasing) before the point and after it”;or“an inflection is a point where f ″ and f ′ are zero”. In reaction to Task 4: Investigate the function f (x)=1/4 x4−x3, half of the students used algorithms that erroneously included f ′(0)=0 as a critical attribute and reached a single inflection point at x=0 (Table 5). Most of them (40 %) examined f ′ (x)=0, f ″ (x)=0, f ‴ (x)≠ 0, and wrote (Fig. 4):

Table 2 Distribution of students’ identifications of inflection point in each graph (N=52) Graph 1 Graph 2 Graph 3 Graph 4 Graph 5

none P1 P2 all none P3 P4 P5 all none P6 P7 all none P8 none P10 83 17 17 17 4 95 12 -- -- 42 17 12 8 35 17 -- 75 University students’ grasp of inflection points

Table 3 Distribution of students’ identifications of inflection point Profile N=52 in all five graphs (%) A partial identification 27

Horizontal inflection points (P3/P10)19

Inflection points of different types (except for P4/P5)8 A mix correct and incorrect identificationa 70 Horizontal inflection points+peak curve 41 Inflection points of different types+peak curve 29 a — Incorrect identification where Only incorrect identificationa 3 the graph bends

f ðxÞ¼1 x4 x3; f 0ðxÞ¼x3 3x2; 0 4 f ðxÞ¼0 ¼¼> x ¼ 0; x ¼ 3 ÀÁ µð Þ¼ 2 ¼¼> µð Þ > ¼¼> 3 µð Þ¼ f x 3x 6x f 3 0 A 3; 6 4 Minimum; f 0 0 f ØðxÞ¼6x 6; for x ¼ 0; f ØðxÞ 6¼0 ¼¼> Bð0; 0Þ inflection point

The others (10 %) examined f 0ðxÞ¼0:µf 0ðxÞ¼x3 3x2 ¼ 0 ¼¼> x ¼ 0; x ¼ 3µ: Then they used a table of signs (Fig. 5) and concluded that (0:0) is an inflection point. Many students (42 %) based their solutions to Task 5, Investigate f ′ (x)=15x2−5x3,ona wrong assumption that f′ (P)=0 is a critical attribute. They incorrectly reached only the result of x=0, either by examining f ′ (x)=0 and a table of signs for f ′ (x) (6 %), or by solving f ′ (x)= 0, f ″ (x)=0, f ‴ (x)≠0 (36 %):

f 0ðxÞ¼15x25x3; f 0ðxÞ¼0 ¼> x ¼ 0; x ¼ 3 f µðxÞ¼30x 15x2 ¼> f µð3Þ < 0 ¼> at x ¼ 3 maximum point; f µð0Þ¼0 f ØðxÞ¼3030x; f Øð0Þ¼30 6¼ 0 ¼> at x ¼ 0 inflection point Finally, in reactions to Task 6, Investigate the Graph of f ′ (x), most participants (62 %) incorrectly found an inflection point only at x=0, exhibiting an erroneous assumption that f ′ (x)=0 is a must. Many (35 %) examined f ′ (x)=0 and f ″ (x)=0: “We see that at x=0,f ′ (x)=0, that is, the slope of the tangent of y=f ′ (x)atx=0 is zero, so, f ″ (0)=0==>at x=0 there is an inflection point.” The others (27 %) explained that f ′ (0)=0 and in the neighborhood of x=0 f ′ (x) is positive—“f ′ (0)=0 and before and after x=0 f ′ is positive, so f increases before and after x=0, thus x=0 is an inflection point.”

Table 4 Distribution of students’ solutions to task 3: What is an What is an inflection point? N=52 inflection point? (%) Definition: correct or non-minimal 68 Slope zero definition 15 f ′ (x)=0/slope zero and increase–increase 9 Convex–concave and slope zero 2 f ′ (x)=0 and f ″(x)=0 4 Insufficient (missing) definition 10 f ″ (x)=0 4 Increase–increase 6 No answer 7 P. Tsamir, R. Ovodenko

Table 5 Distributions of students’ solutions to tasks 4, 5, Result Task 4 Task 5 Task 6 and 6 (%) Algorithm f (x)=1/4 x4−x3 f ′(x)=15x2−5x3 Graph of f ′ N=52 N=52 N=52

A (0; 0) B (2; −4)a/ 50 48 27 x=0, x=2a f ″=0; f ″≠0 12 12 – f ″=0 38 36 27 A (0: 0)/x=0 50 42 62 f ′=0;f″=0;f″≠040 36 – f ′=0; table of signs 10 6 27 f ′=0;f″=0 –– 35 Other – 10 11 aCorrect solution

In sum, many participants expressed f ′ (x)=0 inflection point images that were evident in the different types of tasks and in the three (verbal, graphic, and algebraic) representations (Tall & Vinner, 1981). For example, participants exhibited slope-zero figural concepts when marking only horizontal inflection points in their reactions to all five graphs (Fischbein, 1987) and slope-zero definitions, in response to “what is an inflection point?” when including the condition f ′ (x)=0 or slope zero in their personal definitions (Tall & Vinner, 1981). In tasks 4, 5, and 6, 50 %, 42 %, and 62 % of the students, respectively, found an inflection point only at x=0, since f ′ (x)=0 was used as a critical step in their algorithmic models (Fischbein & Barash, 1993, Fischbein, 1993b). Many students explained their f ′ (x)=0 solutions by highlighting that “this is how I calculate (find) an inflection point when I investigate a function.” Here, like in many other cases, students learn to recognize concepts “by experience and usage in appropriate contexts” (Tall & Vinner, 1981, p. 151; authors’ emphasis). According to the Israeli mathematics curriculum for secondary schools, when students start investigating functions, they solve f ′ (x)=0 to find possible Xs of extreme points. In these routine investigations, they accidentally encounter cases where f ′ (x)=0, but there is no extreme point “because the function keeps increasing (or decreasing) in the interval that includes this point.” As Tall and Vinner (ibid) explained, “usually in this process the concept is given a symbol or name which enables it to be communicated” and differentiated from the other, related concepts. Indeed, in cases of f ′ (x)=0, yet non-extreme points, students are first guided to label the points—inflection points because these are inflection points and for purposes of communication and differentiation from extreme points that are discussed at that time. These processes are students’ prime, inflection-points-

Fig. 4 Points marked by the students as inflection points (correct Pi; incorrect Ti) University students’ grasp of inflection points

x x < 0 x = 0 0 < x < 3 x = 3 x > 3 f '(x) negative zero negative zero positive f (x) decreases Inflection-point decreases minimum-point increases

Fig. 5 The table of signs related, mathematical experiences, which intuitively shape their images of the notion, and set a cornerstone for a robust primacy-effect that would be expressed in future experiences with inflection points (Fischbein, 1987). In other words, students are first introduced to inflection points arbitrarily, while implementing rules of calculus to search for extreme points during treatment transformations on algebraic representations of functions, often concluding by conversions, transforming the symbolically represented solutions to a graphic Cartesian representation (e.g., Duval, 2002). Consequently, over a substantial period of time, only horizontal inflection points are repeatedly encountered and labeled in students’ primary experiences that are approved and authorize in class. Thus, students’ recognition of inflection points, both in symbolic–algebraic representations and in Cartesian–graphic representations, naturally becomes limited to horizontal inflection points. In future studies, it becomes quite demanding (or impossible) for students to cognitively dissociate the content and the specific representation in which inflection was first introduced and used in their early schoolwork (Duval, 2006). 2. Non-zero slope, f ′ (x)≠0 is necessary for an inflection point—In reaction to Statement 1 if A is an inflection point then f ′ (xA)=0 in Task 1, several students (15 %) exhibited awareness of their primary tendencies to view f ′ (x)=0 as “a must” and knew that this requirement is wrong. While they correctly answered that the statement is false; they incorrectly added the condition that f ′(x) should not be zero. In his interview, one of them explained: “I used to believe that f ′ (x) is always zero at inflection points, but I realized it isn’t. Quite the opposite, like in cos(x), f ′ (x) should not be zero”. This student made an over-correction, from f ′ (x)mustbezero(error)tof ′ (x) must be non-zero (error). In an attempt to correct their initial error, students excluded the condition f ′ (x)=0 from the definition of inflection points, by shifting to a reverse, erroneous personal definition that included f ′ (x)≠0as“a must.” This error might be rooted in intuitive ideas that interfere with students’ formal knowledge, incorrectly leading them to believe that, logically, the comple- mentary set of “A—always” (f ′ (x)=0 is always true at inflection points), is “A—never” (f ′ (x)=0 is never true at inflection points); or “not A—always” (f ′ (x)≠0 is always true at inflection points), rather than “A not-always” (f ′ (x) is not always zero at inflection points, sometimes it is and sometimes not) (Fischbein, 1987, 1993a). That is to say that this error has roots in the specific content of inflection points, by reflecting on such points and doubting whether “f ′ (x)=0 is a necessary condition for the existence of such points.” At first, the notion of inflection points is placed at the heart of the investigation. However, in our study, following the participants’ conclusion that the condition is not a necessity for inflection point, we surmise that their focus-of-reasoning might have been reallocated to the wide mathematical realm of logic, proofs, and refutations. Nardi (2008, p. 70) reported on a university student who was “confusing not always with always not…”, and this might be the case here too. 3. f ″ (x)=0 is sufficient for an inflection point—Our data indicate this belief mostly in students’ solutions to Tasks 1, 4, 5, and 6. In reaction to Task 1, True or False? 40 % of the students incorrectly claimed that Statement 2 (f ″ (x)=0 ⇒ inflection point) is true P. Tsamir, R. Ovodenko

(Table 1). Many (17 %) justified that “That’s the definition: P is an inflection point if and only if f ″ (XP)=0”. Many others (23%) provided algorithmic considerations, mentioning solutions to investigate-a-function and stopping after solving f ″ (x)=0, e.g., “We find inflection points when looking for extreme points. At extreme points f ′ (x)=0andf ″ (x)≠0, if f ″ (x)=0, it isn’t an extreme point, it’s an inflection point.” In Task 4, Investigate the function, f (x)=1/4 x4−x3 and in Task 5: Investigate f ′ (x)=15x2− 5x3, about 40 % of the students used in their algorithms f ″ (x)=0 considerations as sufficient for finding inflection points (Table 5). In Task 4, half of the students reached the correct result, A(0; 0) and B(2; −4) are inflection points, but 38 % insufficient examined only f ″ (x)=0. They wrote:

4 3 0 3 2 2 f ðxÞ¼1=4 x x ; f ðxÞ¼x 3x ; f µðxÞ¼3x 6x f µðxÞ¼0 ¼¼> x ¼ 0; x ¼ 2 ¼¼> AðÞ0; 0 ; BðÞ2; 4 Similarly, in Task 5, about half of the participants correctly reached inflection points at x= 0 and at x=2, but 36 % of the students used an incorrect algorithm, examining only f ″=0:

If f 0ðxÞ¼15x25x3then f µðxÞ¼30x 15x2 f µðxÞ¼0 ¼> 30x 15x2 ¼ 0 ¼> x ¼ 0; x ¼ 2; Inflection points are at x ¼ 0; x ¼ 2 In Task 6, Investigate the Graph of f ′(x), students (27 %), who correctly found x=0 andx=2 as the Xs of inflection points did not go beyond solving f ″ (x)=0 (Table 5), for example: At x ¼ 0andatx ¼ 2 the function y ¼ f 0ðxÞhas minimum and maximum So; y0 ¼ 0; ¼¼> f 00ðxÞ¼0; ¼¼> x ¼ 0andx ¼ 2 are inflection points Many others (35 %) who found only x=0 as a solution, using the f ′ (x)=0 and f ″ (x)=0 algorithm, stopped their explanations after finding f ″ (x)=0, indicating their grasp of the sufficiency of f ″ (x)=0 for inflection points, for example, “At x=0, the slope of the tangent of y=f ′(x) is zero, meaning that, f ″ (0)=0, thus, x=0 is an inflection point.” So, in task 6, 62 % of the students exhibited this concept image. Generally, students exhibited three f ″=0 related algorithmic models for finding, and for defining inflection points (Fischbein & Barash, 1993; Fischbein, 1993a): (a) solve f ″ (x)=0, it is the only condition needed for inflection points; (b) solve f ′ (x)=0, then, for these Xs, solve f ″ (x)=0 to find inflection points. Clearly, these two algorithmic models are problematic. They do not necessarily lead to inflection points. For example, f (x)=x4 has both f ′ (0)= 0 and f ″ (0)=0, but there is no inflection point at zero (see also Mason, 2001). Then, several students presented an erroneously revised grasp of inflection points: (c) calculate the Xsof points that fulfill both conditions: f ″=0 and f ′≠0. The three algorithmic models that erroneously included f ″ (x)=0 as sufficient for inflection points, intuitively served as algorithmic bases for students’ concept images of the notion (see also, Carlson et al., 2002; Fischbein, 1993b; Tall & Vinner, 1981). 4. Peak points: points where the graph bends—We found evidence of students’ tendencies to view inflection points at peak points “where the graphs bend,” in their answers to the graphic representation of f (x) in Task 2. Students’ reactions to Task 2, Investigate the Graphs, included three types of solutions: (a) correct (Pi); (b) incorrect (always where the graph bends—Ti); (c) no identification. Figure 4 illustrates students’ correct (Pi) and incorrect (Ti) solutions. Many participants (73 %) marked, at least once, an inflection point at a peak-point. Usually (70 %), they marked a correct–incorrect mix of Pi and Ti. Others (3 %) provided University students’ grasp of inflection points only incorrect identifications of inflection points, exclusively where the graph bends (Table 3). One might expect students majoring in mathematics at university level to draw on the definition and to convert the data to a graphical representation and to visual considerations such as, “a point where the graph shifts from concave up to concave down (or vice versa)” or to examine points (as a first step) “where the tangent intersects the graph.” However, students did not justify these choices of inflection points by referring to a definition (see also Nardi, 2008). In their interviews, students provided two types of explanations for T being an inflection point: mathematically based explanations and reality-based explanations. The mathematical explanations were, for example, “at this point the graph keeps increasing, yet at a different pace” or “the graph keeps increasing, but the slope changes dramatically.” These students examined the graphic representation and converted their solution to natural language while addressing the mathematical realm (Duval, 2006). Such explanations are commonly given in Israeli mathematics secondary-school classes, at the early stages of investigations of functions, with good intentions to encourage students’ recognition of extreme points (where the graph does not continue increasing/decreasing) and students’ discrimination between what is mathematically relevant and what is not when addressing extreme points in different representations. At these early stages, the description (e.g., “at that point the graph keeps increasing, but the slope changes dramatically”) and the labeling of inflection points is merely a superfluous byproduct in the studies of extreme points, but with time this knowledge becomes a critical attribute of inflection points (Duval, 2006). Another interpretation of students’ confused ideas may be rooted in their vast mathematical engagement with constructing graphs (converting a verbal representation or a symbolic representation to a graphic representation) and their scarce experience with analyzing given graphs (converting a graphic representation to a verbal representation). Commonly, when working on inverted task, most students systematically fail (Duval, 1999). The reality-based explanations were explanations embedded in daily context, such as, “if I drive north on a curved road, the point where I must turn the wheel, but still keep north, is an inflection point”;or“when a plane takes off, it first goes up moderately, and at a certain point, at an inflection point, suddenly it changes the slope and keeps going up, yet in a steeper manner”;or“it’s like climbing a mountain that at the bottom has a gentle upward slope, and at a distinct point becomes a challenging, bold upward slope. That point is an inflection point.” Duval stated that “Cartesian graphs do not work visually for most students except for giving the naïve holistic information: the line goes up or down, like a mountain road” (Duval, 1999, p. 16). That is, one source of difficulties might be found in the nature of visualization that “consists in grasping directly the whole configuration of relations and in discriminating what is relevant in it; most frequently, students go no further than to a local apprehension and do not see the relevant, global organization but an iconic representation” (ibid., p. 14). Consequently, what might have been expected to be a most straightforward visual representation was actually most problematic for students. So problematic that students’ explanations and reasoning intuitively “crossed the conventional borders of the mathematical realm” to irrelevant realistic contexts. The graphic representation was the only one that evoked “peak points” concept images (the way students envisioned the points), primary intuitions (the way students primarily encountered such points in realistic settings), and figural concepts (the way they grasped the graphical presentation of the points) of inflection points in the solutions of advanced mathematics learners (Fischbein, 1987; Tall & Vinner, 1981; Tsamir & Ovodenko, 2004). In daily communication in Hebrew, the realistic explanations that were commonly given to inflection points are consistent with what we commonly label ‘inflection points’ in daily P. Tsamir, R. Ovodenko context. Thus, it is possibly as a result of the primacy effect that this initial model became rigidly attached to the concept (Fischbein, 1987), and due to processes of compartmentalization (e.g., Duval, 2006;Vinner,1990), incompatible images (e.g., inflection points are horizontal, and inflection points are at peak-points) were interchangeably used. Vinner (1991) stated that particularly with such students, “one of the goals of teaching mathematics should be changing the thought habits from the everyday life mode to the technical mode” (p. 80). Concisely, our analyses of students’ common errors in this study yielded that students’ intuitive knowledge and concept images (as expressed, for instance, in their solutions to Tasks 1, 2, and 3), and their algorithmic models (as expressed, for instance, in their solutions to Tasks 4, 5) of inflection points resulted from their past experiences with the concept in mathematics or in realistic occurrences. Students’ solution methods and their justifications addressed the calculus-outline in the Israeli curriculum for high school student (e.g., first, teach extreme points in graphs investigation, afterwards, inflection points), and realistic– linguistic considerations (in Hebrew the realistic peak-points that were described in reaction to Task 2 are labeled inflection points). Nardi (2008, p. 41) found that university students frequently experienced major tensions between the familiar (concrete, numerical) and the unfamiliar (rigorous, abstract) and between the general and the particular, when trying to employ formal, mathematical reasoning. Our data indicate matching phenomena. The tension between the familiar and unfamiliar revolved around the tension between students’ intuitive, familiar grasp of inflection points either being horizontal (familiar from class) or at peak points on graphs (familiar from reality), and from students’ (usually non- or mal) employment of definitions. The tension between the general and the particular was found in students’ overgeneralization of the necessity that inflection points be (or not be) horizontal.

4.2 Students’ proofs—validations and refutations

In task 1, students had to refute two “for all” statements, for instance, by providing related counterexamples, with which they were familiar from their mathematics classes (e.g., to refute statement 1, f ′ (x)=0 as a necessary condition: f (x)=sin(x); to refute statement 2, f ″ (x)=0 as a sufficient condition: f (x)=x4). Still, only less than half of the participants correctly answered that statement 1 is false, and 55 % of the participants correctly judged that statement 2 is false. Moreover, only 8 % of the students correctly refuted statement 1 by a counter example, and 42 % presented counter examples to refute the statement 2 (Table 1). In both cases, a substantial number of students based their correct “false” judgments on incorrect “there is another condition” considerations. In reaction to statement 1, 15 % of the students erroneously added the “f ′(x) should not be zero” condition, confusing the logical term not necessarily zero with necessarily not zero (see also, Nard, 2008). In reaction to statement 2, 13 % of the students added the unnecessary condition f ′ (x0)=0. Many students wrongly accepted the statement 1 (59 %) and statement 2 (40 %) as true (Table 1). One may wonder why? Are there possible sources for such difficulties that go beyond the specific content of “inflection”? We would like to address these erroneous solutions by relating to wide perspectives of logic and proof. Perhaps, students, who are accustomed to be presented in class with (valid) theorems, or rarely to be informed of the falsity of a statement, but are never asked to decide themselves true-or-false, might have had similar expectations (see, Nardi, 2008). Possibly, their implicit assumptions here were that, as usual, they are asked to address theorems, and thus, they did not even try to look for a counterexample; they were only trying to provide us “validating” justifications to satisfy our request for explanations. University students’ grasp of inflection points

Another reason might be embedded in students’ poor “proving skills” and their lack of strategic knowledge of proofs and refutations (i.e., skills of identification of the available range of counterexamples or theorems, knowledge of ways to select them appropriately, and to draw on them efficiently when proving). Weber (2001) reported on a study where PhD students had strategic knowledge in proving mathematical statements, but undergraduates had difficulties. It might be that such skills ripen with students’ mathematical, academic experience over time. Statement 2, regarding the sufficiency of f ″ (x)=0, is false, yet the converse statement is true for functions which are differentiable at least twice on the range. It was found that when addressing statements like statement 2, university students tended to be “unable to distin- guish between a main and a subordinate clause,” for example, confusing statements like “if you are in the running shower then you are wet” with the converse statement, so that “if you are wet, it doesn’t follow that you are in a shower” (Nardi, 2008, p. 59). On the other hand, students might have been extremely, yet mistakenly, taken by the mathematical content of the statement, so that they were not open to engage in the logical aspects of the statements and of the proofs (see also, Selden & Selden, 1995). Moreover, students “proved” their mistaken “true” judgments to both statements in two ways: (a) by stating that this condition is “part of the definition” of inflection points and (b) by addressing algorithmic considerations related to investigate-the-function tasks. The two erroneous proving methods carried a sense of generality, and neither had the form of empirical proofs. That is, we found no tendencies to base the “proofs” of “the statement is true” answers, on specific examples. These findings indicate that, unlike widely reported, previous data that pointed to extensive tendencies of students of various ages to provide specific examples as proof of for-all statements (e.g., Balacheff, 1987; Bell, 1976; Harel & Sowder, 2007), the participants of this study exhibited no such tendencies. They provided erroneous, seemingly general, pseudo-cover proofs (e.g., Vinner, 1997).

5 Summing up and looking ahead

This study listed four common, inflection-point related errors that were identified in students’ responses to our questionnaire and additional difficulties, related to students’ proofs and refutations. Our investigation and the analysis of the data in light of Fischbein’s, Tall and Vinner’s, and Duval’s frameworks regarding students’ comprehension, intuitions, and images indicate that these errors and difficulties were not light cases of mere confusion but deep, “logical” problems, rooted, among others, in the nature of mathematics, in mathematics knowledge acquisition, and in learners’ primary engagement with the notion in mathematical and daily contexts. Clearly, one may choose other theoretical models to shed light on the findings. Another approach, suggested by Vinner (1997) himself, indicates that students’ erroneous solutions may be rooted in pseudo-conceptual or pseudo-analytical thought processes, which are “based on the belief that a certain act will lead to an answer that will be accepted …or will impress society.” That is, rather than dealing with students’“belief that statement X is true [we deal with their belief] that statement X will be credited by person Y who is supposed to evaluate it” (ibid., p. 115). The desire to get some credit in a certain social setting may guide students to apply fuzzy memory and superficial generalizations, in the way that they may have used the conditions f ′ (x)=0, or f ′ (x)≠0″, and the f ″(0)=0 for inflection points. We found that university students have erroneous concept images of inflection points. Two of the images were that f ′ (x)=0 is necessary and that f ″ (x)=0 is sufficient for inflection points. When we shared these data with colleagues, their reactions were that these P. Tsamir, R. Ovodenko findings are self-evident. It seems that, while many mathematics educators and high school teachers have repeatedly yet randomly encountered such errors, in this study, we provided for the first time, to the best of our knowledge, extensive research-based evidence, systematically collecting these data from students’ reactions to various representations of tasks and their clear declarations about these beliefs. Thus, we regard these findings as valuable research-based evidence for researchers and for teachers. The two other common errors that we found in this study, i.e., students’ tendencies to regard f ′ (x)≠0 as necessary, or “peak points” on graphs as inflection points were neither reported in the literature nor familiar to our colleagues. Again, the data seem to extend the knowledge base regarding students’ grasp of inflection points. However, more research is needed to study students’ intuitions, concept images, visualization, definitions, proofs, and comprehension of conversions from certain semiotic representations to other. Moreover, when considering instructional implications, One must remember that a concept is not acquired in one step. Several stages precede the complete acquisition and mastery of a complex concept. In these intermediate stages, some peculiar behaviors are likely to occur. Several cognitive schemes, some even conflicting with each other, may act in the same person in different situations that are closely related in time… knowledge of these particular cognitive schemes may make the teacher more sensitive to students reactions’ and thus improve communication. (Vinner & Dreyfus, 1989, p. 365) A first step in student-sensitive instruction is an examination of “what do the students that I am going to teach know”? One suggestion would be to present students who are about to learn the notion inflection points with a questionnaire, for instance, like the one we used here to get acquainted with their initial views and common errors. This study provides teachers with ideas regarding errors that should be expected. Since classes vary, and a certain error might be more evident in one class and less in another, a teaching plan should take these data into consideration. Fischbein (1987) and Vinner (1991; 1997) stated that it is important to promote students’ (especially those majoring in mathematics) awareness of their mathematical, intuitive, and pseudo-mathematical ways of reasoning. “It seems that the formation of control mechanisms in aperson’s mind might help a lot … this can be done by encouraging the person to reflective thinking” (Vinner, 1997, p. 127). Conflicts between the concept images and the formal definition should be identified and thoroughly discussed, and “if the students are candidates for advanced mathematics then, not only that definitions should be given and discussed, the students should be trained to use them as an ultimate criterion in mathematics tasks” (Vinner, 1991, pp. 80). Taking into account these recommendations and our findings regarding students’ tendencies to grasp f ′ (x)=0, or f ′ (x)≠0 as necessary conditions, we suggest tasks like the following: A True or false? f: R ⇒ R is a continuous, differentiable function.

Statement 1: If A(x0, f (x0)) is an inflection point, then f ′ (x0)=0. True/false; prove. Statement 2: If A(x0, f (x0)) is an inflection point, then f ′ (x0)≠0. True/false; prove. B Investigate the functions a. For each function, indicate: Are there inflection points? If yes, what are they? For each inflection point, is f ′ (x)=0 or f ′ (x)≠0?

1 ½a f ðxÞ¼sin x; ½a f ðxÞ¼ x4 x3½a f ðxÞ¼12x5 1 2 4 3 University students’ grasp of inflection points

C Reexamination and reflection Look at statement 1 and statement 2 once more; did you change your solutions? Why? Duval claimed that “the use of visualization requires a specific training, specific to visualize each register, [and] the student can succeed in constructing a graph and being unable to look at the final configurations other than as iconic representations” (Duval, 1999, p. 14; p. 17). One way to go about it is to present students with tasks like task 2 (identify inflection points on given graphs) and like task 6 identify inflection points of f (x)by examining the graph of f ′ (x), the latter being more complex and demanding. Duval also referred to the importance of learners’ flexible mathematical performances when roles of source register and target register are inverted within a semiotic representation conversion task. For such purposes, we would suggest, for instance, tasks that present a graph (e.g., graph 3 in task 2 in the questionnaire), asking to mark on it inflection points and to explain the solutions (conversion from graphic to verbal representation). Then, to present the students with a verbal (or symbolic) representation of a related function (e.g., a function that has an identical graph), asking the students to investigate the function, to find inflection points, to complete the investigation and draw the graph, and finally to reflect on the entire activity. Another example for such engagement is task 1, presented in this section. In the mathematics education community, there is wide agreement that common errors should play a significant role in teaching–learning processes, for example, in the design of pre-teaching, diagnostic tools for identifying students’ difficulties and in the design of related instruction (e.g., Fischbein, 1987; Greeno, Collins & Resnick, 1996). Several researchers encourage teachers to challenge students’ mathematical reasoning by asking them to justify the correctness of given solutions or to identify the bugs in incorrect ones (Borasi, 1996). Here is a task (c.f. Mason & Watson, 2001, p. 128), addressing the error “f ″ (x)=0 is sufficient for inflection points.” A common method for finding inflection points of a curve which is at least twice differentiable is to differentiate twice and set equal to zero to find the abscissa. Sometimes this gives a correct answer for a correct reason, sometimes it gives a correct answer for a wrong reason, and sometimes it gives an incorrect answer. Construct examples which exemplify these three situations, and also a family of examples which include all three in each member. Must a function be twice differentiable to have an inflection point? We illustrated some ways to implement our data in designing instruction. Clearly, there are additional instructional ideas, and it would be wise to examine what instructional approaches are efficient in promoting students’ knowledge and in increasing their awareness of and abilities to control their pseudo-mathematical, intuitive ideas.

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