university students grasp of inflection pointsbennett/onlinehw/qcenter/inflectionpoint… ·...

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 participantswere asked what inflection points are, to mark inflection points on graphs, to judge the validityof related statements, and to find inflection points by investigating (1) a function, (2) thederivative, and (3) the graph of the derivative. We found four erroneous images of inflectionpoints: (1) f ′ (x)=0 as a necessary condition, (2) f ′ (x)≠0 as a necessary condition, (3) f ″ (x)=0as a sufficient condition, and (4) the location of “a peak point, where the graph bends” as aninflection point. We use the lenses of Fischbein, Tall, and Vinner and Duval’s frameworks toanalyze 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 offunctions in calculus. Calculus is an important domain in mathematics and a central subjectwithin high school and post-high school mathematics curricula. In the literature and in ourpreliminary studies, we found a few indications of learners’ erroneous conceptions of thenotion (e.g., Ovodenko & Tsamir, 2005; Tall, 1987; Vinner, 1982). While the findings shedsome light on students’ grasp of inflection points, it seems important to further investigatestudents’ related conceptions and to examine potential sources for their common errors.

In this paper, we examine university students’ conceptions of the notion of inflectionpoints, and we also use the context of inflection points to examine students’ proofs(validating and refuting) when addressing inflection-point-related statements. In the fieldof mathematics education, there are several theoretical frameworks proposing ways toanalyze students’ mathematical reasoning; yet usually, research data are interpreted in lightof a single theory. We believe that the use of different lenses may contribute to ourinterpretational examinations of the data and may offer us rich terminologies to address andto analyze the findings (e.g., Tsamir, 2007, 2008). Thus, we offer a range of interpretations ofstudents’ conceptions based on three theoretical frameworks which are widely used to highlightpossible sources of students’ difficulties in mathematics: Fischbein’s (e.g., 1993a) analyses of

Educ Stud MathDOI 10.1007/s10649-012-9463-1

P. Tsamir (*) : R. OvodenkoTel Aviv University, Tel Aviv, Israele-mail: [email protected]

learners’ intuitive algorithmic and formal knowledge; Tall and Vinner’s (e.g., 1981) review ofconcept image and concept definition; and Duval’s (e.g., 2006) investigations of the role ofrepresentation 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 aboutstudents’ conceptions of inflection points?” Then, we attend to the question: “What arepossible 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 usingthe notion of inflection points. For example, studies on students’ performances on connec-tions between functions and their derivatives within realistic contexts reported that studentstend to err when identifying or when representing inflection points on graphs (e.g., Monk,1992; Nemirovsky & Rubin, 1992; Carlson, Jacobs, Coe, Larsen & Hsu, 2002). Studentsalso tend to use fragments of phrases taken from earlier-learnt theorems, such as: “if thesecond derivative equals zero [then] inflection point” even when solving problems in thecontext of “dynamic real-world situation” (Carlson et al., 2002, p. 355). On this matter,Nardi reported in her book: Amongst mathematicians: Teaching and learning mathematicsat university level that “there is the classic example from school mathematics: how thesecond 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 theshapes of y=x5 and y=x6, thus knowing immediately which does and which does not have aninflection 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? Thesedata indicate that even future mathematicians may experience (as undergraduates) intuitiveunease when encountering the insufficiency of f ″ (x)=0 for an inflection point.

Gomez and Carulla (2001) reported on students’ grasp of connections between thelocation of inflection points and the location of the graph related to the axes. Studentsclaimed that, if an inflection point of y=f (x) is on the y-axis or “close enough” to it, then thegraph 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 ofinflection 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 experiencesof the tangent of circles led learners to believe that “the tangent is a line that touches thegraph 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. Wecame across a novel tendency to regard a “peak or bending point” (i.e., a point where thegraph keeps increasing or decreasing but dramatically changes the rate of change) as aninflection point (Tsamir & Ovodenko, 2004). We also found tendencies to regard f ′ (x0)=0necessary for the existence of an inflection point at x=x0 (Ovodenko & Tsamir, 2005); andwe observed that different erroneous conceptions of inflection points evolve when students

P. Tsamir, R. Ovodenko

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 onthe 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 bymathematics education researchers for analyzing students’ common errors and for examin-ing possible related sources.

Fischbein (e.g., 1987, 1993b) claimed that students’ mathematical performances includethree basic aspects: the algorithmic aspect, i.e., knowledge of rules, processes, and ways toapply them in a solution, and knowledge of “why” each of the steps in the algorithm iscorrect; the formal aspect, i.e., knowledge of axioms, definitions, theorems, proofs, andknowledge of how the mathematical realm works (e.g., consistency); and the intuitive aspectthat was characterized as immediate, confident, and obviously grasped as correct although itis not necessarily so. Fischbein explained that, while neither formal knowledge nor algo-rithmic knowledge are spontaneously acquired, intuitive knowledge develops as an effect oflearners’ personal experience, independent of any systematic instruction. Sometimes, intu-itive ideas hinder formal interpretations or algorithmic procedures and cause erroneous, rigidalgorithmic 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 asevolving from the application of the distributive law (Fischbein, 1993a). Fischbein’s analysisof students’ intuitive grasp of geometrical and graphs-related notions led him to coin theterm figural concepts, i.e., mental, spatial images, handled by geometrical or functions-basedreasoning. Figural concepts may become autonomous, free of formal control, and thuserroneous (Fischbein, 1993a). Fischbein noted that a certain interpretation of a conceptmay initially be useful in the teaching process due to its intuitive qualities and its localconcreteness. But, as a result of the primacy effect, this initial model may become rigidlyattached to the concept and generate obstacles to advanced interpretations of the concept.Fischbein’s framework was widely used to analyze students’ mathematical conceptions ofvarious notions, such as functions, infinity, limit, and sets (e.g., Fischbein, 1987).

Two other researchers who examined learners’ grasp of mathematical notions are Tall andVinner (e.g., 1981) who coined the terms concept-image and concept-definition.Concept imageincludes all the mental pictures and the properties that a person associates with the conceptname. When solving a certain task, specific aspects of one’s concept image are activated, theevoked concept image. Concept definition is a term used to specify the concept in a way that isaccepted by the mathematical community, but learners often hold a personal concept definitionthat may differ from the formal one. Moreover, the concept image which is frequently shapedby 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 partof the concept image becomes a potential conflict factor by implicitly conflicting with anotherpart of the concept image or with the concept definition. For instance, the concept of tangent isusually introduced with reference to circles, implicitly insinuating that a tangent should onlymeet the curve at one point and should not cross the curve (e.g., Vinner, 1991). This oftenbecomes part of the students’ tangent-image (generic tangent c.f. Tall, 1987) that may causeproblems later, for instance, at inflection points. Potential conflict factors contain the seeds of


future conflict, but a key condition for one to actually face cognitive conflict is awareness. Thatis, two incompatible images may obliviously coexist and be interchangeably used due tocompartmentalization 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 senseof conflict or confusion, which can serve as a starting point in instruction. Tall and Vinner’sframework proved to be useful in analyzing students’ conceptions of various mathematicalnotions 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 coreof understanding mathematics” (1999, p. 3). He explained that a major difficulty in math-ematics comprehension is rooted in the nature of mathematical objects that can be accessedonly through signs while, in other sciences (e.g., biology), objects may also be accessed in adirect manner. Duval analyzed representations of mathematical objects and processes andsuggested 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 tothree types of representation: (a) mental representations (individuals’ conceptions andmisconceptions about realistic, concrete objects); (b) computational representations (typesof information-codification used in mathematical algorithmic performances); and (c) semi-otic representations (include signs, relationships, rules of production and transformation,related with particular sign systems such as language, algebra, and graphs). A semioticrepresentation provides an “organization of relations between representational units” andsince visualization allows an immediate and complete capture of any organization ofrelations, “there is no understanding without visualization” (Duval, 1999, p. 13). Duvalemphasized the role of semiotic representations and of related transformational functions inmathematics learning and called attention to essential differences between treatment, i.e., atransformation within a single semiotic system and conversion, i.e., a transformation fromone semiotic system to another. For example, transforming x(x+1) into x2+x within thealgebraic 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’ difficultieswith mathematical reasoning lie in the cognitive complexity of conversions that entailrecognition 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 ofsource register and target register are inverted within a semiotic representation conversiontask, the problem is radically changed for students. It can be obvious in one case, while in theinverted task most students systematically fail” (ibid., p. 122). Furthermore, conversion ofrepresentations requires the cognitive dissociation of the represented object and the contentof 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 andthe first representation of the object, because the only access to mathematical objects issemiotic. Consequently, students erroneously perceive two representations of the sameobject as being two mathematical objects, i.e., the registers of the representations remainfragmented and compartmentalized. Duval’s framework has been valuable in analyzingstudents’ 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 byDuval to examine university students’ conceptions of inflection points, by focusing on thequestions: What are students’ common errors, and what are possible sources of these errors?


3 Methodology

3.1 Participants

We report on two studies, investigating two groups of university students. In one study, 53students were asked to solve tasks 1–3; in the second study, 52 students were asked to solvetasks 4–6. All participants had studied mathematics at mathematics faculties or atmathematics-oriented faculties, i.e., computer science, computer engineering, and electronicengineering. All had successfully completed at least two calculus courses, two linear algebracourses, and a course of differential equations. The participants’ ages ranged between 25 and35 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 tasksfrom 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 graphsFigure 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 ðxÞ ¼ 14 x4 � x3 . 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 0ðxÞ ¼ 15x2 � 5x3 . Does f (x) have:

Fig. 1 The graphs presented in Task 2


(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, sincethe concept of inflection point has both graphical and non-graphical aspects which playsignificant roles in students’ figural and conceptual knowledge and influence their conceptimages (see, for instance, Duval, 2006; Fischbein, 1993a; Vinner & Dreyfus, 1989), thetasks were given in three representations: verbal (tasks 1, 3), graphical (tasks 2, 6), andalgebraic (tasks 4, 5). The solutions could be formulated either by treatment within the givenrepresentations or through conversion.

We learnt from the teachers of these courses that, most commonly, students analyzedalgebraic expressions of functions (task 4) and seldom analyzed algebraic expressions of thederivative (task 5). In the case of the graphic representation, students analyzed given graphsquite 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 routinealgorithms in the related solutions led us to believe that students’ knowledge might bechallenged, especially in task 6. Finally, tasks 1 and 3 were in a way familiar to studentsbecause 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 toerroneously view (a) f ′ (x)=0 as necessary for inflection (in task 1 judging statement 1 as beingtrue, in tasks 2 and 6 identifying “only horizontal inflection points,” in task 3 adding f ′=0 totheir 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 task6 identifying “only f ″=0 points,” in task 3 not going beyond the condition f ″ (x)=0 in theirdefinitions, and in tasks 4 and 5 stopping the algorithm after the stage of f ″ (x)=0). Moreover, inthe tasks presented in the algebraic and in the graphic representations the inflection points

Fig. 2 The graph of y=f ′ (x)


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 toprovide students with rich opportunities to evoke images of inflection points (see previoussection, Tall & Vinner, 1981).

Based on the analysis of their written solutions, 15 participants were invited toindividual follow-up interviews. We interviewed students whose written solution wefound to be interesting, unclear, or puzzling. During their interviews, students wereshown their original solutions and were asked to provide an elaborate explanation oftheir written ideas. The interviews lasted 30–45 min and were audio-taped andtranscribed.

4 Discussion of the results

In this section, we discuss our data about students’ conceptions of inflection points andabout 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)=0is necessary for an inflection point, (2) non-zero slope, f ′ (x)≠0 is necessary for aninflection 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 ofFischbein’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 instudents’ solutions to all six tasks. In task 1: True or False? about 60 % of the studentsincorrectly 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)=0condition is in the definition of an inflection point” or “f ′ (x)=0 and f ″ (x)=0 (and perhapsadditional conditions) are the conditions for an inflection point.”

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


In task 2: Investigate the Graphs students were asked to mark on given graphs points ofintersection with the axes (Zi), extreme points (Xi–Max and Ni–Min), and inflection points(Pi). Students who marked Zi, Xi, or Ni points, but no Pi points, were regarded as studentswho identified no inflection points. Commonly, students correctly marked the horizontalinflection points P3 on Graph 2 (95 %) and P10 on Graph 5 (75 %); the percentages ofidentifications of the other inflection points were much lower (0 to 17 % for each of thepoints, see Table 2).

To get an overall perspective of students’ solutions to task 2, we created profiles of theiridentifications of inflection points in all five graphs (Table 3). We found three main profilesof solutions: (a) partial, (b) correct–incorrect mix, and (c) incorrect. No student correctlyidentified all inflection points on the five graphs. Students’ tendencies to identify onlyhorizontal inflection points were evident in the partial selection of 19 % of the studentswho marked only the horizontal points P3, P10.

In Task 3: What is an inflection point? Students’ tendencies (15 %) to regard f ′=0 as “amust” 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 keepsincreasing (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 usedalgorithms that erroneously included f ′(0)=0 as a critical attribute and reached a singleinflection 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 1 Distribution of students’reactions to the statements (in %)

aCorrect judgment

Judgment Inflectionpoint ⇒ f ′=0

f ″=0⇒Inflection

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

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


f ðxÞ ¼ 14 x

4 � x3; f 0ðxÞ ¼ x3 � 3x2;f 0ðxÞ ¼ 0 ¼¼> x ¼ 0; x ¼ 3f µðxÞ ¼ 3x2 � 6x ¼¼> f µð3Þ > 0 ¼¼> A 3;�6 3

4

� �Minimum; f µð0Þ ¼ 0

f ØðxÞ ¼ 6x� 6; for x ¼ 0; f ØðxÞ 6¼0 ¼¼> Bð0; 0Þ inflection pointThe others (10 %) examined f 0ðxÞ ¼ 0 : µf 0ðxÞ ¼ x3 � 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, on a

wrong assumption that f′ (P)=0 is a critical attribute. They incorrectly reached only the resultof 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 0ðxÞ ¼ 15x2�5x3; f 0ðxÞ ¼ 0 ¼> x ¼ 0; x ¼ 3f µðxÞ ¼ 30x� 15x2 ¼> f µð3Þ < 0 ¼> at x ¼ 3 maximum point; f µð0Þ ¼ 0f ØðxÞ ¼ 30�30x; f Øð0Þ ¼ 30 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) at x=0 is zero, so, f ″ (0)=0==>at x=0 there is aninflection point.” The others (27 %) explained that f ′ (0)=0 and in the neighborhood of x=0f ′ (x) is positive—“f ′ (0)=0 and before and after x=0 f ′ is positive, so f increases before andafter x=0, thus x=0 is an inflection point.”

Table 3 Distribution of students’identifications of inflection pointin all five graphs (%)

aIncorrect identification—wherethe graph bends

Profile N=52

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

Only incorrect identificationa 3

Table 4 Distribution of students’solutions to task 3: What is aninflection point? (%)

What is an inflection point? N=52

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


In sum, many participants expressed f ′ (x)=0 inflection point images that were evident inthe different types of tasks and in the three (verbal, graphic, and algebraic) representations(Tall & Vinner, 1981). For example, participants exhibited slope-zero figural concepts whenmarking 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?” whenincluding 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 aninflection point only at x=0, since f ′ (x)=0 was used as a critical step in their algorithmicmodels (Fischbein & Barash, 1993, Fischbein, 1993b).

Many students explained their f ′ (x)=0 solutions by highlighting that “this is how Icalculate (find) an inflection point when I investigate a function.” Here, like in many othercases, students learn to recognize concepts “by experience and usage in appropriate con-texts” (Tall & Vinner, 1981, p. 151; authors’ emphasis). According to the Israeli mathematicscurriculum 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 accidentallyencounter cases where f ′ (x)=0, but there is no extreme point “because the function keepsincreasing (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 becommunicated” 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 becausethese are inflection points and for purposes of communication and differentiation from extremepoints that are discussed at that time. These processes are students’ prime, inflection-points-

Table 5 Distributions of stu-dents’ solutions to tasks 4, 5,and 6 (%)

aCorrect solution

Result Task 4 Task 5 Task 6

Algorithm f (x)=1/4 x4−x3 f ′(x)=15x2−5x3 Graphof f ′

N=52 N=52 N=52

A (0; 0) B (2; −4)a/x=0, x=2a

50 48 27

f ″=0; f ″≠0 12 12 –

f ″=0 38 36 27

A (0: 0)/x=0 50 42 62

f ′=0; f ″=0; f ″≠0 40 36 –

f ′=0; table of signs 10 6 27

f ′=0; f ″=0 – – 35

Other – 10 11

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


related, mathematical experiences, which intuitively shape their images of the notion, and set acornerstone for a robust primacy-effect that would be expressed in future experiences withinflection points (Fischbein, 1987). In other words, students are first introduced to inflectionpoints arbitrarily, while implementing rules of calculus to search for extreme points duringtreatment transformations on algebraic representations of functions, often concluding by con-versions, transforming the symbolically represented solutions to a graphic Cartesian representa-tion (e.g., Duval, 2002). Consequently, over a substantial period of time, only horizontal inflectionpoints are repeatedly encountered and labeled in students’ primary experiences that are approvedand authorize in class. Thus, students’ recognition of inflection points, both in symbolic–algebraicrepresentations and in Cartesian–graphic representations, naturally becomes limited to horizontalinflection points. In future studies, it becomes quite demanding (or impossible) for students tocognitively dissociate the content and the specific representation in which inflection was firstintroduced 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 1if A is an inflection point then f ′ (xA)=0 in Task 1, several students (15 %) exhibitedawareness of their primary tendencies to view f ′ (x)=0 as “a must” and knew that thisrequirement is wrong. While they correctly answered that the statement is false; theyincorrectly added the condition that f ′(x) should not be zero. In his interview, one of themexplained: “I used to believe that f ′ (x) is always zero at inflection points, but I realized itisn’t. Quite the opposite, like in cos(x), f ′ (x) should not be zero”. This student made anover-correction, from f ′ (x) must be zero (error) to f ′ (x) must be non-zero (error).

In an attempt to correct their initial error, students excluded the condition f ′ (x)=0 fromthe definition of inflection points, by shifting to a reverse, erroneous personal definition thatincluded f ′ (x)≠0 as “a must.” This error might be rooted in intuitive ideas that interfere withstudents’ 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 atinflection 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 errorhas roots in the specific content of inflection points, by reflecting on such points anddoubting whether “f ′ (x)=0 is a necessary condition for the existence of such points.” Atfirst, the notion of inflection points is placed at the heart of the investigation. However, inour study, following the participants’ conclusion that the condition is not a necessity forinflection point, we surmise that their focus-of-reasoning might have been reallocated to thewide mathematical realm of logic, proofs, and refutations. Nardi (2008, p. 70) reported on auniversity student who was “confusing not always with always not…”, and this might be thecase here too.

3. f ″ (x)=0 is sufficient for an inflection point—Our data indicate this belief mostly instudents’ solutions to Tasks 1, 4, 5, and 6. In reaction to Task 1, True or False? 40 % ofthe students incorrectly claimed that Statement 2 (f ″ (x)=0 ⇒ inflection point) is true

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


(Table 1). Many (17 %) justified that “That’s the definition: P is an inflection point if andonly if f ″ (XP)=0”. Many others (23%) provided algorithmic considerations, mentioningsolutions to investigate-a-function and stopping after solving f ″ (x)=0, e.g., “We findinflection points when looking for extreme points. At extreme points f ′ (x)=0 and f ″ (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 forfinding inflection points (Table 5). In Task 4, half of the students reached the correct result, A(0;0) andB(2; −4) are inflection points, but 38% insufficient examined only f ″ (x)=0. They wrote:

f ðxÞ ¼ 1=4 x4 � x3; f 0ðxÞ ¼ x3 � 3x2; f µðxÞ ¼ 3x2 � 6xf µðxÞ ¼ 0 ¼¼> x ¼ 0; x ¼ 2 ¼¼> A 0; 0ð Þ;B 2;�4ð Þ

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 0ðxÞ ¼ 15x2�5x3then f µðxÞ ¼ 30x� 15x2

f µðxÞ ¼ 0 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 and x=2as the Xs of inflection points did not go beyond solving f ″ (x)=0 (Table 5), for example:

At x ¼ 0 and at x ¼ 2 the function y ¼ f 0ðxÞhas minimum and maximum

So; y0 ¼ 0;¼¼> f 00ðxÞ ¼ 0;¼¼> x ¼ 0 and x ¼ 2 are inflection points

Many others (35 %) who found only x=0 as a solution, using the f ′ (x)=0 and f ″ (x)=0algorithm, stopped their explanations after finding f ″ (x)=0, indicating their grasp of thesufficiency of f ″ (x)=0 for inflection points, for example, “At x=0, the slope of the tangentof 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 fordefining 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 problem-atic. 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, severalstudents presented an erroneously revised grasp of inflection points: (c) calculate the Xs ofpoints that fulfill both conditions: f ″=0 and f ′≠0. The three algorithmic models thaterroneously included f ″ (x)=0 as sufficient for inflection points, intuitively served asalgorithmic 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’ tendenciesto view inflection points at peak points “where the graphs bend,” in their answers to thegraphic representation of f (x) in Task 2. Students’ reactions to Task 2, Investigate theGraphs, included three types of solutions: (a) correct (Pi); (b) incorrect (always wherethe graph bends—Ti); (c) no identification. Figure 4 illustrates students’ correct (Pi) andincorrect (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


only incorrect identifications of inflection points, exclusively where the graph bends(Table 3). One might expect students majoring in mathematics at university level to drawon the definition and to convert the data to a graphical representation and to visualconsiderations 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 adefinition (see also Nardi, 2008). In their interviews, students provided two types ofexplanations for T being an inflection point: mathematically based explanations andreality-based explanations.

The mathematical explanations were, for example, “at this point the graph keeps increas-ing, yet at a different pace” or “the graph keeps increasing, but the slope changes dramat-ically.” These students examined the graphic representation and converted their solution tonatural language while addressing the mathematical realm (Duval, 2006). Such explanationsare commonly given in Israeli mathematics secondary-school classes, at the early stages ofinvestigations of functions, with good intentions to encourage students’ recognition ofextreme points (where the graph does not continue increasing/decreasing) and students’discrimination between what is mathematically relevant and what is not when addressingextreme points in different representations. At these early stages, the description (e.g., “atthat point the graph keeps increasing, but the slope changes dramatically”) and the labelingof inflection points is merely a superfluous byproduct in the studies of extreme points, butwith time this knowledge becomes a critical attribute of inflection points (Duval, 2006).

Another interpretation of students’ confused ideas may be rooted in their vast mathemat-ical engagement with constructing graphs (converting a verbal representation or a symbolicrepresentation to a graphic representation) and their scarce experience with analyzing givengraphs (converting a graphic representation to a verbal representation). Commonly, whenworking on inverted task, most students systematically fail (Duval, 1999).

The reality-based explanations were explanations embedded in daily context, such as, “ifI drive north on a curved road, the point where I must turn the wheel, but still keep north, isan inflection point”; or “when a plane takes off, it first goes up moderately, and at a certainpoint, at an inflection point, suddenly it changes the slope and keeps going up, yet in asteeper manner”; or “it’s like climbing a mountain that at the bottom has a gentle upwardslope, and at a distinct point becomes a challenging, bold upward slope. That point is aninflection point.” Duval stated that “Cartesian graphs do not work visually for most studentsexcept for giving the naïve holistic information: the line goes up or down, like a mountainroad” (Duval, 1999, p. 16). That is, one source of difficulties might be found in the nature ofvisualization that “consists in grasping directly the whole configuration of relations and indiscriminating what is relevant in it; most frequently, students go no further than to a localapprehension 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 straightforwardvisual representation was actually most problematic for students. So problematic thatstudents’ explanations and reasoning intuitively “crossed the conventional borders of themathematical 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 primarilyencountered such points in realistic settings), and figural concepts (the way they grasped thegraphical presentation of the points) of inflection points in the solutions of advancedmathematics learners (Fischbein, 1987; Tall & Vinner, 1981; Tsamir & Ovodenko, 2004).In daily communication in Hebrew, the realistic explanations that were commonly given toinflection points are consistent with what we commonly label ‘inflection points’ in daily


context. Thus, it is possibly as a result of the primacy effect that this initial model becamerigidly 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 thatparticularly with such students, “one of the goals of teaching mathematics should be changingthe 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 toTasks 1, 2, and 3), and their algorithmic models (as expressed, for instance, in their solutionsto Tasks 4, 5) of inflection points resulted from their past experiences with the concept inmathematics or in realistic occurrences. Students’ solution methods and their justificationsaddressed 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 reactionto Task 2 are labeled inflection points).

Nardi (2008, p. 41) found that university students frequently experienced major tensionsbetween the familiar (concrete, numerical) and the unfamiliar (rigorous, abstract) andbetween the general and the particular, when trying to employ formal, mathematicalreasoning. Our data indicate matching phenomena. The tension between the familiar andunfamiliar revolved around the tension between students’ intuitive, familiar grasp of inflec-tion points either being horizontal (familiar from class) or at peak points on graphs (familiarfrom reality), and from students’ (usually non- or mal) employment of definitions. Thetension between the general and the particular was found in students’ overgeneralization ofthe 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 relatedcounterexamples, with which they were familiar from their mathematics classes (e.g., to refutestatement 1, f ′ (x)=0 as a necessary condition: f (x)=sin(x); to refute statement 2, f ″ (x)=0 as asufficient condition: f (x)=x4). Still, only less than half of the participants correctly answeredthat 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, and42 % presented counter examples to refute the statement 2 (Table 1).

In both cases, a substantial number of students based their correct “false” judgments onincorrect “there is another condition” considerations. In reaction to statement 1, 15 % of thestudents erroneously added the “f ′(x) should not be zero” condition, confusing the logicalterm not necessarily zero with necessarily not zero (see also, Nard, 2008). In reaction tostatement 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 gobeyond the specific content of “inflection”? We would like to address these erroneoussolutions by relating to wide perspectives of logic and proof. Perhaps, students, who areaccustomed to be presented in class with (valid) theorems, or rarely to be informed of thefalsity of a statement, but are never asked to decide themselves true-or-false, might have hadsimilar 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 acounterexample; they were only trying to provide us “validating” justifications to satisfy ourrequest for explanations.


Another reason might be embedded in students’ poor “proving skills” and their lack ofstrategic knowledge of proofs and refutations (i.e., skills of identification of the available rangeof counterexamples or theorems, knowledge of ways to select them appropriately, and to drawon them efficiently when proving). Weber (2001) reported on a study where PhD students hadstrategic knowledge in proving mathematical statements, but undergraduates had difficulties. Itmight 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 istrue for functions which are differentiable at least twice on the range. It was found that whenaddressing 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 “ifyou are in the running shower then you are wet” with the converse statement, so that “if youare 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 ofthe statement, so that they were not open to engage in the logical aspects of the statementsand of the proofs (see also, Selden & Selden, 1995).

Moreover, students “proved” their mistaken “true” judgments to both statements in twoways: (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 twoerroneous proving methods carried a sense of generality, and neither had the form ofempirical proofs. That is, we found no tendencies to base the “proofs” of “the statement istrue” answers, on specific examples. These findings indicate that, unlike widely reported,previous data that pointed to extensive tendencies of students of various ages to providespecific 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 providederroneous, 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 instudents’ 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, andDuval’s frameworks regarding students’ comprehension, intuitions, and images indicate thatthese 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, andin 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 rootedin pseudo-conceptual or pseudo-analytical thought processes, which are “based on the beliefthat 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] thatstatement X will be credited by person Y who is supposed to evaluate it” (ibid., p. 115). Thedesire to get some credit in a certain social setting may guide students to apply fuzzy memoryand superficial generalizations, in theway that theymay 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 forinflection points. When we shared these data with colleagues, their reactions were that these


findings are self-evident. It seems that, while many mathematics educators and high schoolteachers have repeatedly yet randomly encountered such errors, in this study, we provided for thefirst time, to the best of our knowledge, extensive research-based evidence, systematicallycollecting these data from students’ reactions to various representations of tasks and their cleardeclarations about these beliefs. Thus, we regard these findings as valuable research-basedevidence for researchers and for teachers. The two other common errors that we found in thisstudy, i.e., students’ tendencies to regard f ′ (x)≠0 as necessary, or “peak points” on graphs asinflection points were neither reported in the literature nor familiar to our colleagues. Again, thedata 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 precedethe complete acquisition and mastery of a complex concept. In these intermediatestages, some peculiar behaviors are likely to occur. Several cognitive schemes, someeven conflicting with each other, may act in the same person in different situations thatare closely related in time… knowledge of these particular cognitive schemes maymake the teacher more sensitive to students reactions’ and thus improve communica-tion. (Vinner & Dreyfus, 1989, p. 365)

A first step in student-sensitive instruction is an examination of “what do the students that I amgoing to teach know”? One suggestion would be to present students who are about to learn thenotion inflection points with a questionnaire, for instance, like the one we used here to getacquainted with their initial views and common errors. This study provides teachers with ideasregarding errors that should be expected. Since classes vary, and a certain error might be moreevident 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, andpseudo-mathematical ways of reasoning. “It seems that the formation of control mechanisms ina person’s mind might help a lot … this can be done by encouraging the person to reflectivethinking” (Vinner, 1997, p. 127). Conflicts between the concept images and the formal definitionshould be identified and thoroughly discussed, and “if the students are candidates for advancedmathematics then, not only that definitions should be given and discussed, the students should betrained to use them as an ultimate criterion in mathematics tasks” (Vinner, 1991, pp. 80). Takinginto account these recommendations and our findings regarding students’ tendencies to graspf ′ (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?

a1½ �f ðxÞ ¼ sin x; a2½ �f ðxÞ ¼ 1

4x4 � x3 a3½ �f ðxÞ ¼ 12x5


C Reexamination and reflectionLook 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 tovisualize each register, [and] the student can succeed in constructing a graph and beingunable 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 (identifyinflection points on given graphs) and like task 6 identify inflection points of f (x) byexamining the graph of f ′ (x), the latter being more complex and demanding. Duval alsoreferred to the importance of learners’ flexible mathematical performances when roles ofsource register and target register are inverted within a semiotic representation conversiontask. 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 explainthe solutions (conversion from graphic to verbal representation). Then, to present thestudents with a verbal (or symbolic) representation of a related function (e.g., a functionthat has an identical graph), asking the students to investigate the function, to find inflectionpoints, to complete the investigation and draw the graph, and finally to reflect on the entireactivity. Another example for such engagement is task 1, presented in this section.

In the mathematics education community, there is wide agreement that common errorsshould play a significant role in teaching–learning processes, for example, in the design ofpre-teaching, diagnostic tools for identifying students’ difficulties and in the design ofrelated instruction (e.g., Fischbein, 1987; Greeno, Collins & Resnick, 1996). Severalresearchers encourage teachers to challenge students’ mathematical reasoning by askingthem 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 twicedifferentiable 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 acorrect answer for a wrong reason, and sometimes it gives an incorrect answer.Construct examples which exemplify these three situations, and also a family ofexamples 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, thereare additional instructional ideas, and it would be wise to examine what instructionalapproaches are efficient in promoting students’ knowledge and in increasing their awarenessof and abilities to control their pseudo-mathematical, intuitive ideas.

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