text:
Juliane Püschl
email: jpueschl@math.upb.de

Identifying discussion patterns of teaching assistants in mathematical tutorials in Germany

Keywords: Teaching assistants, tutorials, problem oriented instruction, classroom discussion

Mathematical tutorials play an important role in tertiary teaching of mathematics in Germany. However, we do not know a lot about what actually happens in these tutorials. This paper reports from a small study in the context of a PhD project which investigates the work of teaching assistants (TAs). The result of this study is a typology of discussion patterns of tasks in mathematical tutorials. In this study typological analysis, including a hierarchical cluster analysis, was used to identify different ways in how TAs work on mathematical tasks in classroom discussions. The findings suggest that there are five main patterns for the discussion of tasks, differing in focus, length and support of students' learning.

Introduction

In Germany, teaching assistants (TAs) play a vital role in the learning of mathematics at universities. A mathematical course usually consists of three parts: the lecture, homework assignments and small group tutorials. The lecturer delivers the lecture and assigns three to five tasks for the homework assignments. After working on these assignments for one week, the students hand in their solutions to the tasks, TAs correct them and later discuss these tasks in small group tutorials. Usually, the lecturer and TAs meet every week for planning the tutorial session. TAs do not assist the lecturer in his teaching activities, but they are on their own in the tutorials and free to discuss the tasks as they like. TAs in Germany are undergraduate students who are just a few years ahead of the students they teach and they are usually employed for one semester. By leading the tutorials, TAs are close to the students' actual learning and may function as link between students and lecturer, e.g., they shall connect the contents of the lecture to the tasks, they ought to get feedback from the students and report this to the lecturer. TAs are expected to be accurate and to facilitate students' learning by also sharing their own experiences as successful students.

Although TAs play such an important role in the learning of the students, there is hardly any research on German TAs' practices. Internationally, research on TAs has increased in the last 20 years. However, as their role and practices differ in many ways from that of German TAs, it is uncertain in what respect the results of these studies can be transferred to the German context. Therefore, the aim of my PhD project is to understand how German TAs organize their tutorials in order, later, to be able to provide a basis on which we can develop quality criteria for German tutorials and use them to support TAs in weekly meetings and tutorial trainings.

Theoretical background

In the last years, research on mathematical teaching practices has increased and there have been a few studies that compare characteristics and teaching styles of instructors in different levels. [START_REF] Weber | Traditional instruction in advanced mathematics courses: A case study of one professor's lectures and proofs in an introductory real analysis course[END_REF] showed that one lecturer might use different teaching styles, depending on his learning goals and the content he is teaching. Weber could identify three different ways of teaching proofs in a Thematic Working Group 14 Proceedings of CERME10 traditional lecture format: logical-structural, procedural and semantic. This suggests that, at least advanced lecturers, change their teaching styles intentionally.

There have also been studies on comparing teaching styles of different instructors. One example is [START_REF] Pinto | Revisiting University Mathematics Teaching: A Tale of Two Instructors[END_REF] study on two TAs that are even supposed to follow the same lesson plan. Using Schoenfeld's resources, orientation and goals theory [START_REF] Schoenfeld | How we think: A theory of goal-oriented decision making and its educational applications[END_REF], Pinto illustrates how both TAs prepare and hold their tutorials quite differently according to their research background and experience in teaching this particular topic. Also [START_REF] Mali | Characterising university mathematics teaching[END_REF] could accord to the findings of Pinto.

In their study on two tutors in small group tutorials, they found out that teaching seems to be closely linked to research practices of the tutors: the mathematician "uses the graph to make fundamentally mathematical ways of thinking transparent to students", whereas the mathematics educator uses the graph "as an alternative to explain the mathematics" (p. 2193).

Of course, looking at TAs from the German context, they do not have a research background that might influence their teaching. However, their experience from their own tutorials or from teaching tutorials in former semesters as well as their learning goals, beliefs about learning or the mathematical contents might influence the way they discuss tasks. Therefore, the main research question for this paper is: What different "discussion patterns" do TAs use when working on mathematical tasks in tutorials? This includes sub-questions like: Do TAs focus on the problems they identified when correcting the students' work? Do they always discuss the whole task or only parts of it? How much do they support the students in their learning process? For this paper I combine the answers to these questions by trying to find overall patterns for classroom discussions.

Methods and design

The study is being conducted at a German university. Students of mathematics for pre-service teachers are expected to attend lectures and additional small group tutorials of 10 to 30 students. Both, lectures and tutorials are weekly sessions of 90 minutes, the students are supposed to work on tasks out of class which are assessed by the TAs and discussed in tutorials. The videos for this study were generated in tutorial sessions containing 78 task discussions in total (2-3 discussions per tutorial). 32 TAs have taken part in this study, all of them pre-service teachers. The data set includes task discussions in tutorial sessions from different semesters of the years 2010 to 2015 and the discussion covered a range of topics from analysis, arithmetic and geometry. This great variety of tutorials was chosen to get a representative sample of task discussions, thereby reducing other influences (style of lecturer, difficulty of mathematical content, etc.).

The students in the tutorials were all bachelor students studying teaching mathematics in secondary schools. Students were not directly videotaped unless they presented parts of the solution in front of the class. Their questions, comments and other contributions to the task discussions are audible in the videos.

In a previous part of the PhD study, these 78 task discussions were analyzed for different aspects: how TAs start and end the task discussions, what methods they use, how they try to support the students using visualizations, references to the lecture, etc. Using qualitative content analysis, different categories were generated to investigate each aspect in more detail [START_REF] Püschl | An insight into mathematics tutorials[END_REF].

This part of the study focus on the task discussion as a whole. As we want to find out different types of discussions, an empirical typification method, namely the "typological analysis" by Kuckartz (as described in Kluge, 1999), was used for analyzing the data. An important part of typification methods is to identify features which characterize the different discussion patterns. These features for the classification of task discussions are based on some categories from the previous part of the PhD project (see [START_REF] Püschl | An insight into mathematics tutorials[END_REF]. The following four features were considered for the typological analysis: use of didactical elements, completeness of discussion, focus on problems and focus on strategies.

The use of didactical elements is a key feature for the discussion, because it indicates to what extend the TAs support the students in their learning process. In the previous analysis of the PhD study, ten different didactical elements were identified: "use of visualization", "highlighting common mistakes", "reference to lecture, other exercises, school", "giving structural advice", "clarification of expectations", "solving in several ways", "recapitulation of main results", "clarification of student questions", "generating cognitive conflicts", "returning students' questions" and "asking advanced questions". Some were only used in a few discussions (like "returning students' questions") while others could be found frequently (e.g., "use of visualization"). Some didactical elements are rather pedagogical (e.g., "returning students' questions"), others demand more mathematical skills (e.g., like "asking advanced questions"). For the purpose of this paper instances in which a TA used a didactical element were counted, ranging from a minimum of 0 to a maximum of 13 elements per task discussion. The feature completeness of discussion tells, whether the TAs discuss the whole task or only parts of it. Therefore, this is a binary feature, 0 standing for an incomplete discussion. The other two features give insight into the focus of discussion and relate to categories from prior analysis. The focus could be on the mathematical problems the students had when working on the task. The focus on problems feature consists of three categories from the previous part of the study: "giving feedback on the work of students", "highlighting common mistakes" and "telling the students to review specific parts of the task". The focus on strategies feature is quite different: here, the TAs do not discuss mathematical difficulties, but rather pass on strategies that students need to solve a specific type of tasks. It contains the following three categories: "pointing out the task difficulty", "giving structural advice" and "summarizing of task". For both features, the elements in each discussion are counted, ranging from 0 to 4. Based on these four features, agglomerative hierarchical clustering was used to find patterns in the 78 task discussions. Agglomerative clustering starts with all 78 cases, each building one cluster. The algorithm then merges a selected pair of clusters into a single cluster, so that after 77 steps only one big cluster is being left (Hastie, Friedman, & Tibshirani, 2001, p. 472). To equal the relative influence of the four features in the cluster algorithm, they were scaled between 0 and 1. The Squared Euclidean distance was used as a metric to calculate the distance between each cluster. To decide which clusters are joint in each step, the Ward's Method was chosen as linkage criterion. The Ward's Method tries to minimize the total distance from centroids by joining two clusters. This method facilitates the construction of clusters with similar sizes and is frequently used because it has often provided better results than other linkage criterions (Bortz, 2005, p. 573). As the clusters become more heterogeneous in every step of the algorithm, it is often recommended to stop at a number of clusters before the greatest increase of distance (Bacher et al., 2010, p. 241). The statistical analysis was facilitated by the software SPSS (version 23).

The data has been analyzed regarding the four features mentioned above. In addition, other factors have been taken into account as they might influence the results (Kuckartz, 2012, p. 125). Therefore, the discussion patterns identified by the cluster algorithm were analyzed in regard to a variety of other factors. Three factors are presented in this paper: time spent on the discussion, the TAs' individual approaches and their experience in leading tutorials.

Results

The results from the cluster analysis suggest five clusters from the 78 discussions. Using four or less clusters would have resulted in a high increase in distance between the cases in the cluster. Three cases of the 78 discussions were eliminated, because they did not fit one of these clusters for several reasons1 . One case, for example, which showed a great distance to all of the other clusters, was a task discussion in which the TA was really ambitious and wanted to discuss the whole exercise with a lot of student participation, focusing on problems as well as strategies. However, in this way she ran out of time and was not able to discuss even half of the exercise in more than 40 minutes. As there is no similar task discussion in the data and it would have influenced the cluster algorithm too much, this case was eliminated according to these qualitative and quantitative considerations. Table 1 shows that most of the discussions fall into clusters 1, 2 and 5. Only 16 of the 75 discussions have a focus on problems. Cluster 2 and 5 make up more than half of the discussions (55%), both not focusing on either problems or strategies. Cluster 1 is the only cluster with a focus on strategies.

Taking into account the five clusters from cluster analysis five different discussion patterns could be identified in the material:

Heuristic discussion (Cluster 1)

The TA discusses the complete task by focusing on strategies the students have to acquire in order to solve tasks from this specific type. The heuristic discussion is the only pattern with this focus on strategies and used 24% of all task discussions.

Pragmatic discussion (Cluster 2)

The TA discusses the complete task without focusing on strategies or problems and with minimal use of didactical elements to support the students in their learning processes. The pragmatic discussion is the most frequently used discussion pattern (29%).

Student-oriented discussion (Cluster 3)

The TA discusses the complete solution of the task while focusing on the specific problems the students might have had. This pattern is called student-oriented, because the TA satisfies the students' request for a model solution, but also tries to help them to overcome their difficulties. The studentoriented discussion is only used in 13% of the task discussions.

Problem-oriented discussion (Cluster 4)

This type of discussion can be characterized by a focus on problems. The TA highlights the difficulties in the solution process, using many didactical elements to support the students. This pattern is similar to the student-oriented discussion, only differing in the completeness of the discussion. Only 6 task discussions fall into this pattern.

Minimalistic discussion (Cluster 5)

TAs using the minimalistic discussion just discuss parts of the solution without a specific focus. The TAs hardly use didactical elements to support the students. This pattern is quite similar to the pragmatic discussion except for the completeness of the discussion. About 25% of the task discussions fall into this cluster.

Duration of discussions

The average discussion time of one task is about 19 minutes long, ranging from a minimum of 2 to a maximum of 53 minutes. The boxplots in Figure 1 show how long the discussions last in each discussion pattern.

Although the differences in average discussion time between the different patterns is not statistically significant (one-factor ANOVA at a 5% significance level) there are some interesting observations to be made: The pragmatic discussion and the minimalistic discussion are about 10 minutes shorter than the discussions in the other clusters. Especially for the pragmatic discussion, this result is quite surprising as the TAs are going through the complete solution. However, both discussion patterns do not have a specific focus. TAs who discuss problems or strategies probably need some time to concentrate on this focus and need more didactical elements to support the students. Time might be an important factor for TAs to choose a discussion pattern like the pragmatic or minimalistic patters as they often run out of time in the tutorials.

Individual approaches of the TAs

Another factor which might influence the choice of discussion pattern might be the individual approaches of the TA. Depending on their beliefs on learning, some TAs might prefer learning from mistakes and usually choose discussions in the problem-oriented or the student-oriented pattern.

Others rather want to hand on the model solutions and therefore choses the pragmatic discussion pattern. One interesting aspect to investigate might be whether TAs usually choose the same pattern or whether they switch between the patterns when discussing different tasks.

As most TAs in the data only discuss one or two tasks, the data for this analysis is quite small. The following table shows the distribution of discussions from two TAs who discuss seven different tasks: Andrew tends to use complete discussions with no specific focus. Although he discusses seven different tasks, he never focuses on strategies. David's discussions fall in four different patterns. He does not seem to have a preference for any of the different types. This result suggests that some TAs have individual approaches while others might be rather flexible in their choice of discussion pattern.

Both, Andrew and David, are experienced TAs who have led tutorials for several semesters. However, not all of the TAs are as experienced as David and Andrew. Like expert and novice teachers differ in some aspects of their teaching, rather inexperienced TAs might also choose different ways of discussing tasks. Only 11 TAs of the 32 TAs in the data have led tutorials before the semester of this study and are therefore labeled as "experienced" while the other 21 TAs are called "inexperienced". Table 3 suggests that there are some differences between experienced and inexperienced TAs. The experienced TAs, which have lead tutorials in the previous semesters, seem to prefer the heuristic, pragmatic and minimalistic discussions. The inexperienced TAs have a greater focus on problems. They use the student-oriented and problem-oriented patterns more frequently than the experienced TAs. This result is quite surprising as you would rather expect experienced TAs to have a specific focus in the discussion. However, the number of cases is quite small, it would be very interesting to analyze this aspect for a bigger set of data.

Final remarks

This paper presents five different patterns for the discussion of tasks in mathematical tutorials. The "heuristic discussion" focuses on strategy teaching which is very important for learning. This type consumes more teaching time than the other types and is rather used by experienced TAs. As Brophy explains, this kind of instruction is not "only demonstration of and opportunities to apply the skill itself but also explanations of the purpose of the skill (what it does for the learner) and the occasions on which it would be used" (2000, p. 25). Therefore, the students can hopefully gain more than just another solution from this kind of discussion.

In the "student-oriented discussion" and the "problem-oriented discussion" TAs focus on aspects the students struggle with and help them to overcome these difficulties. These types of discussion take some more minutes than the average discussion. Interestingly these patterns are seldom used by experienced TAs. One explanation might be that inexperienced TAs are more aware of the students' problems as they still remember their own problems. It could also be the case that experienced TAs do not believe that these patterns help the students in their learning process. However, this result might only due to the data.

More than half of the discussions have no specific focus and few didactical elements to support students' learning. The "pragmatic discussion" and the "minimalistic discussion" consume the least teaching time, so time pressure in the tutorials might be a reason for the frequent use of these two discussion patterns. However, the TAs might have other motivations for using discussion patterns without focus. Possibly, TAs might pursue learning goals that we are not aware of in this study. This shows that further research on this aspect is needed.

According to the findings of [START_REF] Weber | Traditional instruction in advanced mathematics courses: A case study of one professor's lectures and proofs in an introductory real analysis course[END_REF], some TAs seem quite flexible in using the different discussion patterns. This result gives rise to possibility that we can support TAs in using the appropriate pattern for a specific learning goal. Especially, the heuristic and the problem-oriented patterns could be practiced in tutorial training. Apart from that discussion patterns could be a topic in the weekly meetings, helping the TAs to plan their tutorials more goal-oriented.

Figure 1 :

 1 Figure 1: duration of discussion in minutes each discussion pattern (N=75)

Table 1 : median of four features distributed into five clusters (N=75)

 1

	Cluster		completeness	number of	focus on	focus on
			(1 complete, 0 incomplete)	didactical elements (0-10 elements)	problems -number of elements (0-4	strategies -number of elements (0-4
					elements)	elements)
	1	median	1	4	0	1
		N	18	18	18	18
	2	median	1	2	0	0
		N	22	22	22	22
	3	median	1	6	1	0
		N	10	10	10	10
	4	median	0	5	2	0
		N	6	6	6	6
	5	median	0	2	0	0
		N	19	19	19	19
	overall	mean value	1	3	0	0

Table 2 : distribution of clusters on example of two TAs

 2

			discussion pattern
	TA	heuristic	pragmatic	student-	problem-	minimalistic	overall
				oriented	oriented
	Andrew	1	5	0	0	1	7
	David	2	2	2	0	1	7

Table 3 : distribution of discussion patterns depending on experience of TAs (N=75)

 3

			discussion pattern
	experience of TAs	heuristic pragmatic student-	problem-	minimalistic	overall
				oriented	oriented
	experienced	11	10	3	1	10	35
	inexperienced	7	12	7	5	9	40
	Overall	18	22	10	6	19	75

Proceedings of CERME10

For further explanations on this process of analysis seeHastie, Friedman & Tibshirani (2001, p. 473).Thematic Working Group 14Proceedings of CERME10


domain: math shs

text:
Alon Pinto
email: allalon23@gmail.com

Math teaching as jazz improvisation: Exploring the 'highly principled but not determinate' instructional moves of an expert instructor

Keywords: University math teaching, teaching dilemmas, decision making, problem solving

When engaging students in genuine mathematical problem solving, how can instructors maintain a productive learning environment? In this paper, I examine a series of improvised instructional moves of Alan Schoenfeld, a renowned teacher of mathematical problem solving, and investigate his dilemmas, considerations, and in-the-moment decisions. I use the TRUmath framework to unpack the conflicts that underlie Schoenfeld's dilemmas, and to propose a tacit teaching heuristic that help explains his hard-to-justify moves. I conclude that Schoenfeld's in-the-moment decision making is tacitly oriented towards maintaining certain kinds of balances between his pedagogical principles. On the basis of this analysis, I recommend exploring further the use of TRUmath as a framework for analyzing in-the-moment decision making in the context of conflicting pedagogical principles.

Introduction

I think this kind of teaching is highly principled by not determinate. What I was thinking of is jazz improvisation. It's anything but random; there are moves that the musician will say would or would not be right; but, there may not be a sound justification for any particular in-the-moment move other than 'it just felt right'.

In this quote, Prof. Alan Schoenfeld reflects on a series of in-the-moment teaching decisions he made during a problem solving session. Schoenfeld is an expert teacher of mathematical problem solving (MPS hereafter); he has been studying and teaching MPS for more than three decades now. After so many years, Schoenfeld's instruction seems anything but improvised. For this study, he has reflected on numerous teaching moves that he made during his MPS course, and he was typically able to provide a sound and detailed rationale for his decisions. However, there were certain decisions that Schoenfeld found hard to fully justify, as a key ingredient in their making was a tacit sense of where the class is and how different decisions could work out. In the quote above, Schoenfeld argues that this kind of hard-to-justify decisions makes the instruction of MPS a lot like jazz improvisation, in the sense that both activities are 'highly principled but not determinate'. In this paper we investigate Schoenfeld's jazz-like teaching moves through a case study of three hard-to-justify decisions in one MPS session. In this session, Schoenfeld faced a typical dilemma in MPS-oriented lessons: how should a teacher react when a student comes up with a beautiful and original idea that opens the door to a mathematical exploration that seems worthwhile for some of the students, and a step too far for other students? The aim of this paper is to unpack Schoenfeld's conflicting pedagogical considerations in this case, and to provide insights into his decision making.

The lesson examined in this paper was part of Schoenfeld's MPS course. Schoenfeld's teaching in this course has been studied in several papers. For example, Arcavi, Kessel, Meira and Smith (1998) studied Schoenfeld's teaching in relation to the establishment of classroom norms and MPS heuristics. Schoenfeld's in-the-moment decision making in this course, which is the focus of this paper, has not been studied so far. This MPS course was given to education graduate students and prospective teachers and comprised of paper reading, a small scale research project, and engagement in authentic MPS. In terms of goals and pedagogy, the lesson described in this paper is similar to lessons that Schoenfeld has taught in earlier years to undergraduate mathematics students. Therefore, the dilemmas and instructional moves discussed below are not specific to teacher-education courses, and should be viewed in the context of MPS-oriented instruction at university. There are various approaches for explaining why teachers make the decisions they make as they teach. One approach, which has been gaining much attention in recent years, is to explain teaching decisions in terms of knowledge, goals, and orientations [START_REF] Schoenfeld | How we think: A theory of goal-oriented decision making and its educational applications[END_REF]. This approach has been used at the university level in empirical studies (e.g. Pinto, 2013) and also in professional development programs, as an organizing framework for instructors' self-reflections on their teaching (e.g. [START_REF] Schoenfeld | On understanding and improving the teaching of university Mathematics[END_REF]. However, a notable limitation to the explanatory power of this approach is that instructors' self-reflections are oriented towards what instructors notice in their teaching and have words for. Therefore, there is a need for an organizing framework that would draw attention to various important facets of the work of teaching. One candidate framework is the Teaching for Robust Understanding of Mathematics framework (TRUmath) [START_REF] Schoenfeld | Thoughts on scale[END_REF]. In this paper we analyze Schoenfeld's reflections on his teaching from a TRUmath perspective, and examine the use of TRUmath as an organizing structure for instructors' reflections on their dilemmas and decisions that attends to all the major contributors for productive learning environments.

Setting and methods

This paper examines a lesson taught by Alan Schoenfeld in a "Mathematical Thinking and Problem Solving" course at the Graduate School of Education at UC Berkeley. The lesson took place during the 11 th week of the semester. The class comprised of 21 studentsgraduate students in the school of education and students from teacher preparation programs. The class met once a week for a 3-hour lesson and every lesson included an MPS part where students worked alone or in small groups on a list of problems and then reconvened to share ideas and solutions. The author videotaped the lessons and took notes. After each lesson, Schoenfeld wrote down some reflections on his dilemmas, his instructional moves and decisions, and their impact on the lesson. In addition, the author conducted three 1-hour interviews with Schoenfeld at different stages of the semester that focused on where the class is with respect the learning trajectories for the course.

The analysis in this study is based in part on the TRUmath framework, which seeks to characterize the main contributors for productive learning environments. This framework was derived through a comprehensive literature review by distilling the factors that shape learning in classrooms into a small number of "equivalence classes". These classes are represented through five dimensions: (1) the richness of the mathematics, (2) cognitive demand and opportunities for "productive struggle", (3) equitable access to content for all students, (4) students' opportunities to develop agency, ownership, and positive mathematical identities; and (5) formative assessment. According to the TRUmath framework, these five dimensions are both necessary and sufficient for studying learning environments in the sense that instruction needs to do well along these dimensions in order to produce mathematically proficient students. Figure 1 provides a brief account of each dimension.

In this paper, we explore TRUmath's explanatory power on Schoenfeld's own instruction, and use the five dimensions as an organizing structure for the discussion of his dilemmas and considerations. Schoenfeld is one of the leading developers of the TRUmath framework, and therefore it is particularly suitable for exploring his decision making. We examine a sequence of three hard-tojustify decisions, first from an outside observer perspective based on the videos of the lessons; then from Schoenfeld's inner perspective based on his post-lesson written reflections and the interviews; and finally, from a TRUmath perspective, where the dimensions are used to organize and compare Schoenfeld's various considerations, to unpack the pedagogical conflicts that underlie his dilemmas, and to investigate what kinds of balances he achieved in his in-the-moment decisions.

The Square-ness task

The discussion analyzed in the paper revolves around the Square-ness task, given in the figure below, which was designed by Judah Schwartz in cooperation with members of the Balanced Assessment Group at the Harvard Graduate School of Education. In the lesson, Schoenfeld framed this task as "an introduction to the game mathematicians play", and directed students to start with their intuition as to what it means for a rectangle to be more 'squarish', and then mathematize this intuition by "coming up with a mathematical characterization that would enable anyone to perform some sort of operations on a rectangle […] and obtain a number that would tell, in some sense or other, how close to being a square that rectangle is."

Below you will find a collection of rectangles.

(a) Define a mathematical measure that allows you to tell which rectangle is the "most square" and which rectangle is the "least square".

(b) Define a different measure that achieves the same result.

(c) Is one measure "mathematically superior" to the other? Argue why, and be prepared to defend your choice to the class

Analysis

It's not the note you play that's the wrong note -it's the note you play afterwards that makes it right or wrong. (Miles Davis) Taking a cue from a renowned jazz improviser, I maintain that improvised teaching decisions should not be examined in isolation but rather as part of the flow of instructional moves that teachers make during a lesson. Accordingly, this investigation of Schoenfeld's improvised teaching moves examines three hard-to-justify decisions in the context of Schoenfeld's instruction throughout the MPS session. The analysis is presented as a narrated description of the whole session that comprises of three threads: an outside-observer description of Schoenfeld's moves; a synthesis of Schoenfeld's reflections on his moves, as explicated in the interviews; and a TRUmath perspective on three challenging teaching dilemmas and their hard-to-justify resolutions.

The discussion of the Square-ness task began with a short introduction by Schoenfeld, after which he invited students to present their candidates measures. One of the students, Sophie, approached the board and suggested that the square-ness of a rectangle with side-lengths a and b will be defined by: "The ratio a/b ought to be close to 1". Two students objected to this definition, arguing that it is not well defined and that it should specify that a is the shortest side length. Sophie disagreed at first, claiming that "it does not matter", but was eventually persuaded. She added "where 𝑎 ≤ 𝑏" to the written definition, and walked back to her seat. At this point, Schoenfeld intervened:

Ok, I love it when the class takes over and raises mathematical objections. The question is, if you're characterizing the square-ness of this figure and you're getting a number, shouldn't we get the same number if we happen and bring it down this way instead? (Draws a figure of a rectangle rotated by 90 degrees) It's the same rectangle, so whatever measure you have should bring you the same number. If you have say, a 1 by 3 rectangle, then you get 1 over 3 which is not the same distance from 1 as 3 over 1; so, it begins to be problematic unless you lay it down so that 'a' is the smaller one of the two [side lengths], and then take 'a' over 'b'.

After making this remark, Schoenfeld leaned quietly against the wall and waited for the students to react. In his reflections, Schoenfeld noted that up to this point the discussion took off just as he intended, as the students were engaged in defining, comparing and criticizing measures, and by doing so, expressing their implicit expectations from a measure. Schoenfeld noted that it is quite typical that the first candidate measure is based either on the ratio or on the difference between adjacent side lengths of the rectangle. The measure Sophie suggested has the nice property that it can be defined in a way that makes it invariant under rotations and scaling. The students' debate on whether Sophie's measure was well defined did not address explicitly the properties of the measure. The students seemed more occupied with figuring out the exact routine for computing the measure. Nevertheless, Schoenfeld explained in the interview that this debate provided him with an opportunity to acknowledge, respond to and build on students' ideas, while rephrasing these ideas in a way that fit his goalsto engage students in discussing the desired properties of measures.

Emmy was the first to comment on Sophie's example: "I think that I can probably find a quadrilateral that is not a square, but would have that, hmm… would be a square under that measure of squareness, but it is not a rectangle, is that OK?" Schoenfeld's immediate response was to the entire class: "Do you want to take a vote? Is that OK?" The class seemed divided with some students wondering whether considering quadrilaterals other than rectangles is allowed, while others expressing interest in seeing Emmy's example. Schoenfeld agreed with the students that the problem is stated just for rectangles, but as several students responded in disappointment, he paused to make a quick evaluation and to decide how to proceed.

In his reflections on this moment, Schoenfeld noted that Emmy's example came as a surprise, and too soon with respect to where he felt the class was. He explained that he was more expecting students to propose another candidate measure, which he could compare to Sophie's measure; or to point out that Sophie's measure is invariant under scaling, which would have provided another opportunity to discuss properties of measures. Schoenfeld reflected on his dilemma: on the one hand, the class seemed eager to see Emmy's example; Emmy's comment was well aligned with his own agenda of discussing the desired properties of measures, as it pointed out the fact that Sophie's example might not be generalizable to parallelograms. He also considered this comment as an authentic and beautiful example of 'doing mathematics' and he wanted to acknowledge this; and, coming from a student, this generalization felt natural and organic rather than an artificial teaching move, making it even more appealing. On the other hand, Schoenfeld noted that he was worried that Emmy's example might steer the discussion towards arbitrary quadrilaterals, and he was not sure that the class was ready for this level of abstraction. He wanted more students to participate in the discussion and considered putting Emmy's example on hold so other students could present their candidate measures for rectangles. He also noted that he had examined some of the student work on this problem and that there were a few important insights that he wanted to draw from that work.

In the classroom, Schoenfeld responded almost instantly: Now, before I throw [Emmy] out of class (laughing), let's examine what [she] said. One of the properties one might like for, hmm... any definition is to ask the question: what class of objects this definition applies to? […] So, this measure (points at the Sophie's definition on the board) works for rectangles, but being my psychic self, I think the figure Emmy had in mind was a family of rhombuses (draws two rhombuses on the board), all of which have a measure 1 according to the definition, if we think in terms of side lengths; but they don't look like a square! Note that Schoenfeld's response opened the door for Emmy's example, but that he presented the example himself, as he understood it, rather than letting Emmy present it in her own words. Looking back at his response, Schoenfeld noted that it is hard to justify, claiming that on one hand, it is highly principled in the sense that it is consistent with his goals and orientations, as explicated in his reflections on his dilemma. On the other hand, this decision was not determined by principles, as it was based in part on a tacit sense of where the class was, and how well things could work out.

We now turn to analyze Schoenfeld's hard-to-justify decision from a TRUmath perspective. One option Schoenfeld had was to invite Emmy to present her example in her own words. This option is well aligned with the Formative Assessment and Agency dimensions. Moreover, Schoenfeld considered Emmy's idea to be "a beautiful example of doing mathematics", and since his goals were to discuss properties of measures, he considered this option to also be well aligned with the Mathematics dimension. However, Schoenfeld's reflections suggest that he found this option less appealing from the perspectives of the Access and Cognitive Demand dimensions. He explained that it is essential that students understand and relate to the goals of the exploration. Emmy, who might still be struggling to formulate her idea, could end up leading a discussion that the rest of the students could not engage with productively. Another option Schoenfeld considered was flatly rejecting Emmy's example, or putting it on a back burner. This option would have given Schoenfeld more control over the lesson, which has merits in terms of the Mathematics, Cognitive Demand and Access dimensions. However, Schoenfeld considered this route potentially harmful in terms of Authority, Agency and Identity, and Formative Assessment. Schoenfeld's response represents an alternative to these two options. He acknowledged and built on Emmy's idea (Formative Assessment), lowering the risk of being perceived as rejecting her thinking (Authority, Agency and Identity). However, he did so by proposing two visual examples of rhombuses, making the discussion more concrete and accessible (Cognitive Demand and Access). Moreover, Schoenfeld provided a crisp outlining of the topic of the discussion: "what class of objects this definition applies to" (Access), orienting the discussion towards the properties of measures (Mathematics). To summarize, Schoenfeld considered the first two options to be potentially beneficial as well as potentially harmful; his reaction chose a middle ground that he still considered beneficial, and safe.

The lesson continued with Sophie, Emmy and a few other students discussing how to modify Sophie's measure to make it 'more square'. This discussion led to a new candidate measure for square-ness: the product of the ratio between adjacent side lengths and the ratio between adjacent angles. However, Emmy criticized this measure, claiming that while this measure has the nice property that squares are separated from other shapes, she can no longer see what kind of ordering this measure induces on parallelograms, and whether this ordering has anything to do with her initial intuition as to what square-ness should mean. Several students endorsed this criticism, and the class abandoned this measure. One student suggested that it might not be possible to find a measure that works for both rectangles and rhombuses, and the whole classroom discussion started to break up into several concurrent discussions. At this point, Schoenfeld intervened: I'll point out that what we're doing right now is exactly the business mathematicians are engaged in. […] We start with rectangles and see candidate measures for rectangles; then the question is, what about parallelograms? Trapezoids? Arbitrary quadrilaterals? Is it possible to find a measure that could characterize square-ness for all of those? We only got one definition of square-ness of rectangles so far, and I want to see a few more. It is possible that if we are just looking at rectangles any of the candidate measures will do, although some might be easier to calculate, some might correspond more to your intuition in terms of how square something is. And then, as we move on, only some of those definitions work for more objects. That's the game mathematicians play. So, we have two directions to go. We have this definition (wipes the board clean and writes 'a/b closest to 1 where 𝑎 ≤ 𝑏'). We can ask, are there any other characterizations, or reasons to like them more or less; and we can ask do they generalize and how much, which can also get us to a discussion about just what properties of definitions in general do we want, and what properties do we want in this particular case. The floor is open.

Reflecting on this intervention, Schoenfeld noted that this was a point where he sensed the class was indeed not ready for the exploration they initiated, as he anticipated might happen; in his message to the students, he was trying to steer the discussion back to rectangles, while making sure he is still giving due credit to the exploration the students were engaged with, framing it as the 'game mathematicians play'. Schoenfeld considered this decision as essentially based on a tacit evaluation of where the class is. In the interview, he used TRUmath terms to make this evaluation somewhat more explicit: He explained that he was reading from the students' facial expressions that some students were getting disconnected, signaling Access was becoming an issue; he also noted that the fact students starting to question whether the problem could be solved at all was for him a signal that the Cognitive Demand of the task might be too high. Schoenfeld concluded that in his intervention he was implicitly trying to attend to the Access and Cognitive Demand dimensions.

At this point, Emmy suggested: "I have an idea, but I don't know how to turn it into a measure […] I have a measure that would split out squares, but I don't know how to make it order everything else. Should I share it? (Schoenfeld nods) OK, my theory is that if you have a given a perimeter for a quadrilateral, the square will have the maximum area. So, I want something that takes perimeters and determine whether or not, hmm… determines whether or not that's the maximum area for that circumstance and then order everything else according to how not maximum it is, or something."

Emmy's suggestion led to a rapid exchange between her and Sophie, while the rest of the students remained quiet. In the interview, Schoenfeld referred to this moment as another challenging dilemma that led to a hard-to-justify decision. While he considered Emmy's comment to be mathematically inspiring, it also impeded his attempts to lead the discussion back to rectangles. Schoenfeld explained that he guessed Emmy's idea is intuitively based on the isoperimetric theorem, and he estimated that forming a measure for arbitrary quadrilaterals on the basis of this intuition might prove too difficult for most students, potentially reducing their confidence and sense of efficacy even further. Thus, even though following up on Emmy's comment was very appealing from the Mathematics and Formative Assessment perspectives, this option seemed very risky from the perspectives of Access, Cognitive Demand and Identity. However, Schoenfeld found that while his principles directed him to object to exploring Emmy's idea, his sense of the class suggested otherwise: he sensed that the students were quiet but not passive, that they were actively listening to Emmy and Sophie. Consequently, Schoenfeld explained he decided to try and provide the class with just enough scaffolding to enable more students to engage productively in the new exploration: Ok, let's take what we do know and see if we can turn this to a measure. Hmmm… you may have heard […] of this thing known as the isoperimetric theorem … the general theorem is that if you take any figure whatsoever for a fixed perimeter, the circle is the figure with the largest area. If you limit yourself to quadrilaterals, to rectangles, it turns out that for any given perimeter the square is the figure with the largest possible area. So, the question is whether we can turn that into a measure we can use, and then think about abstracting this into some of these other figures.

In TRUmath terms, Schoenfeld's decision can be expressed as an attempt to amend the level of Cognitive Demand so to increase Access. The intervention paid off. Four more students joined Sophie and Emmy and participated in the exploration. It took just a few minutes of discussion for Sophie to come up with a measure that works: "The perimeter over four, squared, over the area of the shape". The class enthusiastically picked up on this suggestion, and eventually endorsed it.

Discussion

In this paper, we examined a sequence of three in-the-moment decisions. Schoenfeld's first decision was to open the door to Emmy's original idea, but present it in his own words; when the exploration of Emmy's idea seemed too challenging for the class, Schoenfeld's second decision was to try and steer the discussion back to the original problem; and finally, a quick evaluation of where the class was led Schoenfeld to reverse his second decision and allow an even more challenging exploration. Schoenfeld considered his decision making to be highly principled in the sense that his decisions were well aligned with his explicit orientations and goals; however, in his reflections, he also observed that some of these decisions were hard to fully justify since they were strongly influenced by a tacit sense of where the class is and how things could work out. This sense of the class is a resource Schoenfeld developed over years of teaching the course; his reflections suggest that this resource has a crucial role in his decision making when faced with challenging dilemmas: it helps resolve pedagogical conflicts that rise from tensions between competing goals and orientations. The TRUmath framework proved to be useful for unpacking these tensions by providing an organizing structure for the different considerations and the conflicts they present. For example, in the context of Emmy's original idea, when examining three alternative options, we found that Schoenfeld considered two of the options to very well aligned with some of the dimensions, but also potentially harmful from the perspective of the other dimensions. The TRUmath analysis suggested that Schoenfeld chose a path that he considered more moderate across all five dimensions in terms of potential gains and risks. This analysis led Schoenfeld to suggest a teaching heuristic that may have tacitly guided him: keep the lesson productive from the perspective of each dimension, and avoid the temptation to excel in just one or two dimensions at the expense of the other dimensions.

This paper illustrates the theoretical potential of TRUmath as a framework for explaining decisions made in light of conflicting goals and orientations, and the practical potential of TRUmath as an organizing structure for teacher reflection that highlights the gains and risks entailed in different instructional moves. As Schoenfeld is both the subject of this study and one of the developers of TRUmath, further research is required to assess TRUmath's explanatory power for other instructors.

Figure 1 -

 1 Figure 1 -The five dimensions of the TRUmath framework (Schoenfeld, 2015)

Proceedings of CERME10


text:
Regina Ovodenko
email: reginao@cet.ac.il

Pessia Tsamir
email: pessia@post.tau.ac.il

To be or not to be an inflection point

Keywords: Inflection point, formal knowledge, algorithmic knowledge, intuitive knowledge

This paper describes students' grasp of inflection points. The participants were asked to define inflection points, to judge the validity of related statements, and to find inflection points by investigating (1) an algebraic representation of a function, and (2) the graph of the derivative. We found that participants provided their own "filtering conditions" to determine or deny the existence of an inflection point In order to analyze participants' conceptions of inflection points, we used the lenses of Fischbein's theoretical framework.

Introduction

Functions receive considerable attention in secondary school, commonly in algebra and calculus lessons. Inflection point is one of the function-related notions addressed in high school and in further mathematics studies. In preliminary studies, we found some indications of common erroneous conceptions of the notion (e.g. [START_REF] Ovodenko | Possible causes of failure when handling the notion of inflection point[END_REF][START_REF] Tsamir | Prospective teachers' images and definitions: The case of inflection points[END_REF][START_REF] Tsamir | University students' grasp of inflection points[END_REF]. These findings encouraged us to expand our research regarding the grasp of the notion of inflection point, and regarding possible sources of related common errors, while addressing a larger and diverse population who was given richer types of tasks (elaborated upon in the methodology section). In this paper, we present part of the findings from the extended study [START_REF] Ovodenko | Students' Images and Definitions of Extrema and Inflection Points[END_REF]. The research tools were designed and the findings analyzed with reference to a number of theoretical frameworks, including Fischbein's theory of algorithmic, formal, and intuitive components of mathematical knowledge [START_REF] Fischbein | Intuition in Science and Mathematics. An Education Approach[END_REF](Fischbein, , 1993a) ) and his theory of figural concepts [START_REF] Fischbein | The theory of figural concepts[END_REF]. Specifically, our research questions are: In the students' opinion, (1) When is a point an inflection point? (2) When is a point a non-inflection point?

What does research tell us about students' conceptions of inflection points?

Literature gives some indications of students' difficulties when using the notion of inflection point. Some researchers (e.g. [START_REF] Carlson | A cross-sectional investigation of the development of the function concept[END_REF] have reported that students tend to use fragments of phrases taken from earlier-learned theorems, such as "if the second derivative equals zero [then] it is an inflection point" when solving problems in the context of dynamic real-world situations.

Other researchers have reported that early experiences with the tangent to a circle contribute to the creation of the intuitive grasp of the tangent as a line that touches the graph only at one point and does not cross it (e.g. [START_REF] Vinner | Conflicts between definitions and intuitionsthe case of the tangent[END_REF]. This intuition was evoked when students were asked to identify and draw a tangent line to a curve's points that included non-differentiable and differentiable inflection points (e.g. [START_REF] Biza | First year mathematics undergraduates' settled images of tangent line[END_REF].

In a previous study, we examined students' conceptions of inflection points, in which we came across a novel tendency to regard a "peak point" as an inflection point (e.g. [START_REF] Tsamir | Prospective teachers' images and definitions: The case of inflection points[END_REF]. We found tendencies to regard f '(x) = 0 as a necessity for the existence of an inflection point [START_REF] Ovodenko | Possible causes of failure when handling the notion of inflection point[END_REF], as well as tendencies to view f '(x) ≠ 0 as a necessary condition and Thematic Working Group 14 Proceedings of CERME10 f ′′(x) = 0 as a sufficient condition for an inflection point [START_REF] Tsamir | University students' grasp of inflection points[END_REF]. Consequently, we designed a large study to examine students' conceptions of the inflection point when solving a rich variety of problems. Here we report on part of the findings [START_REF] Ovodenko | Students' Images and Definitions of Extrema and Inflection Points[END_REF].

Methodology

The research population included 223 participants from different educational levels of mathematics: high school students studying mathematics at the intermediate level, high school students studying mathematics at the advanced level, university students and university graduates (the latter majoring in mathematics-rich subjects, such as mathematics, computer science, and electronic engineering). All participants had studied the notion of inflection point during their calculus lessons.

According to the Israeli mathematics curriculum for secondary schools, an inflection point is defined as a point on a curve at which the curve changes from being concave up to concave down or vice versa, usually relating to functions that are at least twice differentiable in a small neighborhood of the point. It is important to note that the first encounter with inflection points occurs before the term is defined. It happens when students start investigating functions: they solve f '(x) = 0 to find possible x-s of extreme points and accidentally encounter cases where f '(x) = 0, but there is no extreme point because the function is monotonic in the interval that includes this point. In such casesf '(x) = 0 but the point is non-extremestudents are first guided to label these points as inflection points for purposes of communication and to distinguish them from extreme points. Afterwards, useful theorems are introduced, e.g., a necessary condition for x0 to be an inflection point is f ''(x0) = 0; a sufficient condition may be: (1) f ′′(x0 + ), and f ′′(x0 -) have opposite signs in the neighborhood of x0; or, (2) f ′′′(x0) exists and f ′′′(x0) ≠ 0.

In order to widen the scope of the gathered data, two types of tasks were designed: Produce-a-Solution (production) tasks (i.e., solve mathematical problems), and Evaluate-a-Solution (identification) tasks (i.e., examine the correctness of given solutions). Tasks in each of the two types were presented in verbal, graphic, and algebraic representations (see Figure 1). This systematic structure of tasks provided insight to the participants' ideas and reasoning. It was developed and empirically validated (during preliminary pilot studies) as a tool that allows students to explain related formal, algorithmic and intuitive components of their mathematical knowledge [START_REF] Fischbein | Intuition in Science and Mathematics. An Education Approach[END_REF](Fischbein, , 1993a)). The contribution of these structured tools may go beyond the broad exploration of students' conception of inflection points; such structured tools could be useful to reveal students' conceptions of additional mathematical notions. That is, the research tools offered in this study may serve as a model when designing research tools aimed at investigating students' conceptions of other mathematical notions.

In the following, we present four tasks (Produce-a-Solution) from the questionnaire. Task 3: Find (if possible) the inflection points of the functions:

(1)

f (x) = x 4 + 2x 3 -1; (2) f (x) = x 4 + 32x; (3) f (x) = | x 3 -1|; (4) 3 3 ,0 () ,0         xx fx xx (5) 2 2 ,0 () ,0         xx fx xx
Task 4: Find (if possible) the inflection points of f(x), g(x), t(x), based on the graphs of f '(x), g '(x), t '(x)see Figure 2.

Task 1 and Task 2 are in a verbal representation, addressing participants' formal knowledge, Task 3: in an algebraic representation, addressing what we expected to be algorithmic knowledge; and Task 4 in a graphic representation, addressing participants' figural conceptions. Intuitive knowledge may be expressed in all four tasks.

Before giving our high school participants the tasks, we asked their teachers whether these tasks would be familiar to them. We learnt that Task 3 was expected to be most familiarstudents usually investigated algebraic expressions of functions. They seldom analyzed graphs of the derivative, as required in Task 4; and were rarely asked to determine and prove the validity of a statement, as required in Task 2. Students' modest experience with such tasks and the impossibility of applying routine algorithms in the related solutions led us to assume that students' knowledge might be challenged.

Based on the analysis of their solutions, 20 participants were invited to individual, semi-structured, follow-up interviews, where they were asked by the researchers to elaborate on their written solutions. Interviewees were chosen according to their solutions in the questionnaires, focusing on interesting, correct, and incorrect ideas, while aiming to understand their reasoning. Interviewees were asked, among other things, to explain their solutions and to analyze solutions proposed by other participants. The interviews took 30-45 minutes and were audiotaped and transcribed.

Results

In this section, we answer the research questions: In the students' opinion (1) When is a point an inflection point? (2) When is a point a non-inflection point? We discuss each of the ideas as presented in relevant tasks. More specifically, we present each of the conceptions found as expressed in the four tasks, in descending frequency of the phenomenon.

When is the point an inflection point?

Our data indicates six sets of conditions that in the participants' opinions guarantee the presence of an inflection point.

Passage point from convex to concave or vice versa ⇒ Inflection point -This conception was mainly expressed (55%) in Task 1 (Define): "Inflection point is a point where the graph shifts from concave to convex (or vice versa)". However, no reference was made to characteristics of the function (domain, continuity or differentiability).

f ′′(x0) = 0 and a passage from convex to concave ⇒ Inflection point -This conception prevented participants from finding the non-derivative inflection point of the functions, based on the graphs of derivatives in Task 4 (24%). In addition, it was also expressed in solutions to the algebraic representation of function f (x) = |x 3 -1| (16%) in Task 3.

f ′′(x0) = 0 ⇒ Inflection point -This conception was evident in the participants' solutions to three tasks. In Task 3, f ′′(x) = 0 considerations were erroneously used as sufficient for inflection points of polynomial functions and even of piecewise functions. For example, participants correctly found A(0, -1) and B(-1, -2) as inflection points of f (x) = x 4 + 2x 3 -1 by examining only f ′′(x) = 0 (30%).

Similarly, the point (0, 0) was erroneously claimed to be an inflection point of 3 3 ,0 () ,0

        xx fx xx
by examining only f ′′(x) = 0 (15%). About 30% incorrectly claimed that Statement 2 is correct (Task 2). Some of them added: "That is the definition: P is an inflection point if and only if f ′′(xP) = 0". Others provided algorithmic considerations, mentioning solutions to investigate-the-function and stopping after solving f ′′(x0) = 0, e.g.: "We find inflection points when looking for extreme points. If f ′′(x0) = 0, then the point is not minima or maxima, and that is why it is an inflection point". In their reactions to Statement 3 (Task 2) participants (10%) incorrectly answered that the statement is valid and explained that "f ′′(x0) = 0 is a sufficient condition for the existence of an inflection point".

f '(x0) = 0 and f ′′(x0) = 0 ⇒ Inflection point -This conception was mainly expressed in Task 2; about half of the participants incorrectly claimed that Statement 3 was valid. Most of them provided the methods that they used to solve investigate-the-function tasks. They wrote: "[We] always find inflection points when looking for extreme points, thus starting this search with f ′(x) = 0", then: "An inflection point is a point where f ′′(x) = 0". In addition, in Task 4 (Investigate the graphs of derivative), about 20% incorrectly found an inflection point only at x = 6, exhibiting an erroneous assumption: "We see that t ′(6) = 0, that is, the slope of the tangent of y = t ′(x) at x = 6 is zero, so, t ′′(6) = 0, therefore at x = 6 there is an inflection point."

f ′′(x0) = 0 and monotonicity in the neighborhood of x0 ⇒ Inflection point -Participants (42%) incorrectly claimed that Statement 4 was valid (Task 2). They explained: "Those are sufficient conditions for an inflection point", or gave a supporting example, like f (x) = x 3 + 5. In their interviews, several of the latter explained their solution in terms of: "If the second derivative is zero and the function continues to increase when increasing and to decrease when decreasing, there is a change of convexity-concavity". The combination of these two conditions is likely to determine an inflection point. However, this answer can be refuted by a counter example, like f (x) = x 4 + 32x (this function appeared in Task 3, but it was not used to refute this statement), f ′′(0) = 0 and the function is increasing monotonically in the neighborhood of x = 0 but (0, 0) is not an inflection point.

Evidence about this set of insufficient conditions that seem to be "allegedly certain" for ensuring an inflection point, to the best of our knowledge, was found for the first time in this research.

f ′(x0) = 0 and monotonicity in the neighborhood of x0 ⇒ Inflection point -Participants (16%) wrote, for instance: "A point where f ′(x) = 0 and the graph keeps increasing (or decreasing) before the point and after it is an inflection point" (Task 1). In reaction to Statement 1 (Task 2), participants correctly answered that the statement is false, but their explanation was: "f ′(x0) = 0 is necessary but not a sufficient condition for an inflection point. If in addition to f ′(x0) = 0 the function increases (or decreases) before and after the point, only then is the point an inflection point" (10%). It should be noted that a combination of these conditions define a particular type of notiona horizontal inflection point. This grasp of inflection point probably ignores non-horizontal inflection points (Task 3), like the inflection point of the function f (x) = x 4 + 2x 3 -1 (10%).

When is the point a non-inflection point?

We found three conditions that deny the existence of an inflection point:

No differentiability ⇒ No inflection point -In reactions to Task 3, investigate the function f (x) = |x 3 -1|, most participants (63%) found an inflection point only at x = 0, providing algorithmic considerations of solutions to investigate-the-function, such as: "f ′(0) = 0, f ′′(0) = 0, before x = 0, f ′′(x) is positive, so f (x) is convex and after x = 0, f ′′(x) is negative, so f (x) is concave, thus x = 0 is an inflection point". Some of them added a correct graph to their investigation and, with relation to x = 1, wrote that "although at this point the function changes from concave to convex, it is a noninflection point, because the function is not differentiable at x = 1". In their interviews, several participants explained: "The function is non differentiable at x = 1 and therefore there is no inflection point", or: "No differentiability, no inflection point." Following this, a quarter of the participants defined "inflection point" (Task 1) as requiring differentiability. For example: "A point where f ′′(x) = 0 and the function turns from concave to convex or vice versa", or: "The slope of the tangent of the function at this point is zero, and the function is either increasing on both sides of the point, or decreasing on both sides of the point."

No second derivative ⇒ No inflection point -This conception, that is consistent with the condition f ′′(x0) = 0, is used as a filter to reactions to Task 3, Investigate the function 2 2 ,0 () ,0

        xx fx xx ,
included explanations (23%) such as: "When there is no second derivativethere is no inflection point. All inflection points must satisfy the condition f ′′(x0) = 0. Otherwise there is no infection point". It was also found in reactions to Task 4 (15%), Investigate the graphs of f ′(x), that only the point x = 6 that satisfies the condition f ′′(x0) = 0 was identified as an inflection point. The other point (x = 10), where the second derivative is not defined, was ignored.

f ''(x0) = 0 and f '''(x0) = 0 ⇒ No inflection pointexpressed in the investigation of the function f (x) = x 4 + 32x (7%). Note that this function really has no inflection points. Yet, this correct judgment was based on a wrong consideration. A counter-example is f (x) = x 5 has an inflection point, yet both f ′′(0) = 0 and f ′′′(0) = 0.

Commonly studies report of students' conception of mathematical notions by reporting on criteria that lead to regard the notion as defined. Here we show a new angle of criteria that regards the notion as undefined.

Discussion

We discuss the findings by using Fischbein's (1993a[START_REF] Fischbein | The theory of figural concepts[END_REF] theoretical framework for analyzing students' errors and for examining possible related sources.

What are possible sources of students' mathematical errors?

Fischbein studied broad aspects of students' mathematical reasoning, claiming that an analysis of students' performance has to take into account three basic aspects: algorithmic, formal and intuitive [START_REF] Fischbein | Intuition in Science and Mathematics. An Education Approach[END_REF](Fischbein, , 1993a)). The algorithmic aspect includes knowledge of (a) "how" to solve a problem, and (b) "why" a certain sequence of steps is correct. The formal aspect includes knowledge of axioms, definitions, theorems, proofs and knowledge of how the mathematical realm works. The intuitive aspect of mathematical knowledge is an immediate and self-evident, though not necessary correct, knowledge, accepted with certainty. Fischbein's three components of mathematical knowledge and their interrelations play a vital role in students' mathematical performances. However, "sometimes, the intuitive background manipulates and hinders the formal interpretation or the use of algorithmic procedures", causing inconsistencies in students' solutions (Fischbein, 1993a, p. 14). Fischbein further addressed the impact of drawings (e.g., in geometry) on learners' mathematical reasoning by explaining that the figural structure may dominate one's reasoning instead of being controlled by the corresponding formal constraints [START_REF] Fischbein | The theory of figural concepts[END_REF].

What are the possible sources for students' errors with the concept of inflection points?

We found tendencies to determine or deny the existence of an inflection point under certain conditions. It is important to note that during the study we did not ask directly: Under which conditions, does or does not one get an inflection point? Participants provided "filtering conditions" by their own initiative. So, if one of the following sets is true: An initial evaluation of the reasons underlying erroneous conceptions suggested two main causes: algorithmic experience with investigations of functions, and the impact of the drawing. Students tended to explain that: "This is how I find an inflection point when I investigate a function", or, [in relation to Task 4] "According to the graphs, each function has one inflection point at x = 7 where the graphs shift from concave down to concave up". Thus, it seems that the answers may intuitively evolve from the participants' mathematical, algorithmic experiences (Fischbein, 1993a) and from their figural concept of inflection point [START_REF] Fischbein | The theory of figural concepts[END_REF].

Four of the six sets of conditions that participants presented for "being an inflection point" do not necessarily lead to inflection points (sets 1, 3-5); the other two sets determine inflection points only for a limited family of functions (sets 2, 6). For example, in set 1, the participants provided intuitive definitions, without reference to the type of the functions (e.g., continuous or differentiable). In set 4, participants exhibited slope-zero figural concepts [START_REF] Fischbein | The theory of figural concepts[END_REF] in their reactions to Task 4, when they incorrectly found an inflection point "where slope of the tangent is zero...", or, "where the first derivative and second derivative cross the x-axis". In set 5, the necessary condition f ′′(x) = 0 was presented as a critical step in the algorithmic offering (Fischbein, 1993a), but in combination with the condition of monotonicity, that at first sight seems "sufficient" for an inflection point, surprisingly this does not necessarily lead to an inflection point (as presented in the results section).In set 6, the unnecessary condition f ′(x) = 0 was possibly used as a result of the "primacy effect". That is, it might be the case because these are usually the first inflection points addressed in calculus lessons [START_REF] Fischbein | Intuition in Science and Mathematics. An Education Approach[END_REF]; but it was presented with the condition of monotonicity, and thus defined a particular type of notiona horizontal inflection point.

The three sets of "denying conditions" сan shed some additional light on students' conception of inflection point from two perspectives: (1) types of functions that are usually investigated, and, (2) logical constraints of their knowledge. Requesting the necessity of differentiability (set 7) can be related to functions that are usually investigated or presented graphically in textbooksmost of these are differentiable at the inflection point. Thus, an intuitive image of a "smooth inflection point" was created. Here, as in many other cases, students recognize the concept "by experience and usage in appropriate contexts" (Tall & Vinner, 1981, p. 151; their emphasis). The necessity of twice differentiability (set 8) might be rooted in intuitive ideas that interfere with students' formal knowledge [START_REF] Fischbein | Intuition in Science and Mathematics. An Education Approach[END_REF]1993a). That is, from the theorem, "If f (x) is twice differentiable in some neighborhood of x0, and if x0 is an inflection point, then f ′′(x0) = 0" students erroneously conclude that "if no second derivative then there is no inflection point".

Here, this answer can be refuted by a counter example, like f (x) = x 5/3 . In set 9, the inadequate declaration: 'f ′′(x0) = 0 and f ′′′(x0) = 0 ⇒ No inflection point', might be rooted in intuitive ideas that interfere with students' algorithmic knowledge [START_REF] Fischbein | Intuition in Science and Mathematics. An Education Approach[END_REF]. That is, from the theorem "f ′′(x0) = 0 and f ′′′(x0)  0 ⇒ Inflection point" students erroneously create the rule "if ... then..., if not … then not…".

This study considerably enriches the existing body of knowledge regarding high school students', university students', and university graduates' conceptions of inflection points. Only a small number of studies have dealt with students' conceptions of inflection points directly (e.g. [START_REF] Rivel | Students' identification and characterization of inflection points of a function[END_REF][START_REF] Tsamir | Prospective teachers' images and definitions: The case of inflection points[END_REF] and indirectly (e.g. [START_REF] Biza | First year mathematics undergraduates' settled images of tangent line[END_REF][START_REF] Vinner | Conflicts between definitions and intuitionsthe case of the tangent[END_REF]. Reported studies usually addressed a limited population and dealt only with specific conceptions. The current research offers a broad collection of related correct and incorrect conceptions found among participants with suitable mathematical backgrounds (as specified in the methodology section), with reference to the type and the representation of the given tasks.

Figure 1 :

 1 Figure 1: Structure of research questionnaire

Task 1 :

 1 Define: What is an inflection point? Task 2: True or false? Prove: Statement 1: f: R → R is a differentiable function. If f '(x0) = 0, then P(x0, f (x0)) is an inflection point. Statement 2: f: R → R is a continuous, (at least twice) differentiable function. If f ′′(x0) = 0, then P(x0, f (x0)) is an inflection point. Statement 3: f: R → R is a continuous, (at least twice) differentiable function. If f ′(x0) = 0 and f ′′(x0) = 0, then P(x0, f (x0)) is an inflection point. Statement 4: f: R → R is a continuous, (at least twice) differentiable function. If f ′′(x0) = 0 and the function is monotonically increasing (decreasing) in the neighborhood of x0, then P(x0, f (x0)) is an inflection point.

Figure 2 :

 2 Figure 2: Graphs in Task 4

  (1) convex-concave; (2) f ′′(x0) = 0 and convex-concave; (3) f ′′(x0) = 0; (4) f ′(x0) = 0 and f ′′(x0) = 0; (5) f ′′(x0) = 0 and monotonicity in the neighborhood of x0; (6) f ′(x0) = 0 and monotonicity in the neighborhood of x0; on the other hand, if there is (7) No differentiability; (8) No second derivative; (9) f ′′(x0) = 0 and f ′′′(x0) = 0then there is no inflection point. During the interviews, students reinforced these views.

Proceedings of CERME10