Nele Stubbemann
email: nele.stubbemann@uni-bremen.de

Christine Knipping
email: knipping@math.uni-bremen.de

Metacognitive activities of pre-service teachers in proving processes

Keywords: Metacognition, monitoring, proving processes, pre-service teachers

Proving processes are often complex and difficult. Problems during the phase of proof finding are not unheard of. The ability to find out which knowledge and strategies we need to prove a given statement and how to use knowledge and strategies in proving is called metacognition. A study with pairs of pre-service teachers shows that content knowledge is an important factor whether or not metacognitive activities like monitoring can help handling problems in the proving process.

Introduction

To be able to successfully prove a geometrical statement, one needs more than just subject-specific knowledge and methodological knowledge of proving procedures [START_REF] Heinze | Reasoning and proof: Methodological knowledge as a component of proof competence[END_REF]. It is important to know if what one is doing makes sense, is promising and productive. This ability to reflect on one's own knowledge is called metacognition, which is important in general life as well as in mathematics and allows one to find out what knowledge and strategies are needed to solve a problem and how to use knowledge and strategies in proving.

In mathematics education a lot of research exists about metacognition in problem-solving (e.g. [START_REF] Kuzle | Patterns of metacognitive behavior during mathematics problem-solving in a dynamic geometry environment[END_REF][START_REF] Schoenfeld | What's all the fuss about metacognition[END_REF][START_REF] Schoenfeld | Learning to Think Mathematically: Problem Solving, Metacognition, and Sense Making in Mathematics[END_REF]. But there is little research about the significance of metacognitive activities for proving processes, in particular there is nearly no research on preservice primary school teachers' metacognitive activities, even though they will later in their classrooms be responsible to establish the basis for argumentation. Our research project addresses the need of future teachers to be able to understand and perform proving processes, as well as to perceive their argumentation basis. Metacognitive competences can in our view be useful and fruitful, if taken care of in a sensible way.

The metacognitive activity of monitoring is the focus of this paper. How monitoring is in play during the proving process of students, particularly in what we call loops, elongations, circular reasoning and wanderings is of interest to us. We look specifically at what role content knowledge plays in this and what impact content knowledge has on monitoring.

Theoretical background

The colloquial meaning of metacognition is approximately "reflections on cognition or thinking about your own thinking" (Schoenfeld, 1987, p. 189). [START_REF] Schraw | Promoting general metacognitive awareness[END_REF] divides metacognition into two big fields. Knowledge of cognition is the knowledge about one's own cognition or about cognition in general. The regulation of cognition is an accumulation of activities that help to control one's own learning. It includes monitoring, which is the activity of controlling and evaluating one's understanding and one's own performance during a process, in our case the proving process. According to [START_REF] Heinze | Reasoning and proof: Methodological knowledge as a component of proof competence[END_REF] the ability to prove a statement includes metacognition. [START_REF] Wittmann | Beweisen und Argumentieren [Proving and Argumentation[END_REF] focusses on two things that are important for proving statements, factual knowledge and metaknowledge about proofs. According to Wittmann students often do not have sufficient metaknowledge about proof, which includes knowledge of the functions of proof, characteristics of proving as well as knowledge of how to prove.

The difference between proving as a process and proof as a product is important as well. [START_REF] Wittmann | Beweisen und Argumentieren [Proving and Argumentation[END_REF] sees the process of proving as the activity of finding a proof. The process is complex, needs creativity and problem solving competencies. It can go in leaps and bounds and includes sudden success as well as phases where one does not know what to do. The proof product on the other hand is the final product of the proving process. It is succinctly verbalised and neatly written down. Looking at the written proof one cannot see the efforts of the proving process.

Empirical methods

The base of this research project are case studies of four pairs of pre-service primary school teachers. These students participated in a lecture course on elementary geometry in the third semester of their Master Studies. Participation in the study was voluntary and took place in the spring of 2018. Video recorded interviews are the basis of the case studies. The students worked in pairs on proof tasks so that discussions between them provided insight into their metacognitive activities. In the first part of the semi-structured interview, questions about their own proving experiences and proving in general were asked. Then the students were given two statements from plane geometry, one after the other, which could be proven with knowledge from the lecture. Afterwards the process of finding the proof was discussed with the students.

The transcripts of the interviews were looked through for obstacles that can occur in the proving process. In a second step these obstacles were scanned for monitoring activities, which were then coded with a system for categorizing metacognitive abilities developed by Cohors-Fresenborg and Kaune (2007a), that divides monitoring into eight subcategories, e.g. the controlling of calculations and the controlling of argumentation. This allowed us a differentiated view on monitoring activities. After coding the data, the effect of monitoring in the identified phases with obstacles were analysed. Particularly the influence of content knowledge was focused on in order to clarify the impact of the assumed relationship between monitoring and content knowledge for the proving processes and specifically their successful performance.

Prototypes for obstacles in proving processes

Proving processes are rarely linear. Having difficulties, getting stuck or trying something that is not constructive is normal. After looking through the data we can distinguish four specific obstacles that the students encountered in their proving processes: loops, elongations, circular reasoning and wanderings. In the following we are going to characterise these obstacles with data from all four groups. With these examples we explain to what extent monitoring influences these obstacles and show the important role of content knowledge.

Loops

Loops are one type of obstacle in proving processes. In a discussion one might have a kind of side discussion that is still part of one's proof idea and attempt, but that is not really constructive for the proof. To resolve a loop situation one must rein in this discussion and pursue the original proof idea instead.

A typical example for a loop is the following discussion of Pia and Charlotte (Group 3). The two students attempt to prove the second statement: "A parallelogram is a rectangle if and only if its diagonals have the same length." During their discussion they draw a rectangle and a parallelogram and argue whether or not they should use the drawing of the parallelogram to prove the statement or if they should use the drawing of the rectangle instead. While Pia is arguing that in a parallelogram the diagonals cannot have the same length, Charlotte suddenly has a new idea; here a loop starts. Charlotte wants to change the statement they have to prove to the following: "A parallelogram is a square if and only if its diagonals have the same length". She describes her thinking.

At that moment Pia uses the definitions of a square and a parallelogram and the connections and differences between the two (every square is a parallelogram, but not every parallelogram is a square). She intends to make clear to Charlotte why her idea, to substitute the rectangle in the statement with a square, is not helpful for the proof of the original statement. Pia's behaviour is a reaction to a monitoring activity, which is the examination of her partner Charlotte's understanding of mathematical objects and their role for their argumentation. According to the system of Cohors-Fresenborg and Kaune (2007b) this can be categorized as the monitoring activity M4c: "Controlling of Argumentation -uncovering mistakes in the argumentation" or M2b: "Controlling of an assessment of (assumed) mistakes -uncovering one's own false ideas or those of others". Charlotte then applies this knowledge to squares and rectangles and understands why her idea will not work out as a proof. Both students are back at the "starting point" (arguing about which drawing to use for their argumentation), the loop is closed. Their side discussion ends, they do not lose themselves in it. They are back on to their original path.

In this example of a loop, Pia's monitoring activities match the "problem" Charlotte has (Why can we not use a square in our argumentation?) and helps them to end their side discussion. They go back to their original proof idea. Their monitoring was quite precise regarding the comprehension of the geometrical objects square, rectangle and parallelogram. Pia showed that she possessed the content knowledge necessary to make use of the monitoring activity and improve their work.

Elongations

An elongation in a proving process occurs when one does not pursue the shortest possible way, but instead substitutes several steps for one step.

Elongation simply means to lengthen an action or a process. Often these are hard to determine in data, as a shortest possible way needs to be defined first, in order to be able to find variations of it.

To define this, one has to consider (theoretically) available knowledge which may differ between people and in time. The shortest way of a grade 5 student may differ from that of a university professor; the shortest way in a first proof of a new theorem may be an elongation later on, when shorter ways have The first statement our students had to prove was: "Given is an angle with arms g and h. If you draw a line k parallel to arm g that crosses arm h, then h and k together with the angle bisector form an isosceles triangle". The shortest way to prove the statement that one might follow is using the Alternate Interior Angles Theorem with the angle to show that the base angles have the same size, then the Isosceles Triangle Theorem can be used. Nina and Maja (Group 1), as well as Daria and Leonie (Group 4), proved the statement using the Corresponding Angles Postulate with the angle (see Figure 1). Then supplementary angles and the Triangle Postulate are used to be able to show that the base angles are equal. This is an example of an elongation, as the students had the same basic idea, but for the elongated proof more steps are used than needed.

Neither in the proving process of Group 1 nor in the process of Group 4 a monitoring activity could be found that triggered an elongation. The cause of the observed elongation seems instead to be the angle the students focussed on ( instead of the angle ) and therefore more steps were needed to show that the base angles of the triangle are equal. No indication was found why a group decided to focus on instead of . Sufficient content knowledge seems to be the reason that the groups were still able to prove the statement. The students applied all necessary theorems and did not get to a point where they did not know how to proceed.

Circular reasoning

While involved in the proving process circular reasoning is hard to see for the students engaged in proving. Circular reasoning makes it impossible to produce a correct proof, because one falsely assumes what is meant to be proven by the proof.

Dennis and Julius (Group 2) showed circular reasoning in their 'proof' for the first statement. After the exploration and discussion of the statement, Dennis and Julius began to write down their 'proof'. In the first part of their text the students attempted to show that two of the angles in the triangle had the same size. Julius had noted that the angle bisector divided the angle α into two angles of the same size, labelled as base angles of the triangle in his drawing (see the two " " written in red, Figure 2), instead of labelling them as two halves of (two equal parts of an angle bisector). His argumentation was questioned by Dennis: "Then the question is, whether we can assume that they [Dennis is pointing more at the base angles than at the sides h and k] have the same length".

His question shows a monitoring activity, as Dennis is checking what they are doing while they are still in the proving process. Both students have just started writing down the proof. With his question Dennis controls the effectiveness of their approach and if they are still on target. According to the system of Cohors-Fresenborg and Kaune (2007b) this is the monitoring activity M4a: "Controlling of Argumentation -local examination". As a reaction to the monitoring question Julius writes a second explanation about why the base angles have the same size. As a reason he mentions the equality of the two sides of the triangle. This explanation is unfortunately circular as he assumes that the length of the two sides of the triangle are equal, which still needs to be proven.

In circular reasoning monitoring affects the proving process differently than in the loop. Dennis and Julius end up with a circular argument in their proof of the first statement because of a monitoring activity. While in the loop the monitoring brings the students back to a reasonable way. Dennis' monitoring activity is also rather imprecise, more like a bad feeling instead of a wellfounded suspicion, and the reason for the circular reasoning. Both students lack the necessary content knowledge and cannot not use monitoring activities to their advantage. Their monitoring activity did not improve the quality of their proof, but instead guided the students into circular reasoning, which also demonstrates their lack of knowledge about logic and the structure of a proof.

Wanderings

The abandoning of a good proof idea in favour of ideas that are not constructive is what we call wandering. An example of wandering is shown by Pia and Charlotte (Group 3). During their discussions Pia and Charlotte have a productive idea to prove the first statement. They use the Alternate Interior Angles Theorem to show that the base angles of the triangle are equal. For the next step in their proof they need to show that a triangle with two equal base angles has two sides of the same length. For this they have the idea to use the Isosceles Triangle Theorem. But Pia is not sure, if the theorem can be used in both directions and asks: "Can you say that?", which is actually a monitoring question. According to the system of Cohors-Fresenborg and Kaune (2007b) this would be the monitoring activity M5a: "Determination of position: naming of Deficiency of Understanding -defined step". This labelling of a deficiency in her understanding is an important monitoring activity as Pia can only follow and support their argumentation if she understands the concepts and the connections between them.

In this case the monitoring question is the starting point for the wandering. Because of Pia's monitoring question her partner Charlotte is now insecure in her belief and understanding of the Isosceles Triangle Theorem. Even though Pia affirms that she believes that they are allowed to use it, Charlotte abandons this proof idea in favour of another idea. Their new idea does not work out; their written proof is more a construction protocol with a proof idea but not a proof. For this statement, both first have a good idea how to prove it, but then they discard their idea and start with new ideas which in the end do not work out. They end in a wandering.

Like in the loop, and in contrast to the circular reasoning, Pia's monitoring here is quite precise. She questions the use of one specific theorem, the Isosceles Triangle Theorem, which is important for their proof. Unfortunately, both students lack the necessary content knowledge, in this case the content of the theorem, to make use of their monitoring.

Preliminary results

One can raise the question whether or not metacognitive activities, especially monitoring activities, can help students to prevent problems in their proving processes. In theory, monitoring one's work and progress should help students to prevent or find problems and mistakes.

Our prototypes can be systematized in a fourfold table along the dimension "monitoring" and "content knowledge" (see Table 1). Loops are in the field of monitoring and content knowledge, elongations in the field of content knowledge without monitoring. Circular reasoning and wanderings are in the area of monitoring, but with no content knowledge. In the area without monitoring or content knowledge we suspect another prototype, dry spells. A dry spell is a part of one's proving process where one hits a barrier and does not know how to go on for quite some time. Not actively working at the proof anymore can be a sign for a dry spell, as well as remaking drawings, calculating something or recapping what one already knows in the hope of getting a new idea. Looking at the success that is (or not) shown in the proving process, it is possible to group the observed obstacles (loops, elongations, circular reasoning and wanderings). Both loops and elongations are parts of a proving process where one still reaches the proof in the end, but not in the shortest or easiest way as one's ideas would have allowed. These two are detours in the proving process. Circular reasoning and wanderings are parts of one's proving process where one follows an idea that does not end in a proof. These are drifts in the proving process. Dry spells can be seen as a third group where the proving process pauses, but this still remains to be looked at in our data. Whether or not one follows detours or drifts in a proving process seems to depend on one's monitoring activities. Looking at the monitoring activities described above, one can see that monitoring has many different facets, for example controlling an argument, naming problems in the comprehension of arguments and their connection or uncovering deficiencies in understanding. But our examples show that the mere existence of monitoring activities is not enough to indicate the success of a proving process. There seem to be differences in the way that monitoring activities are expressed. Monitoring formulated in a general way like "Is that possible? Is that right?" like the example of a circular reasoning by Dennis and Julius (Group 2) seem to be about a gut feeling that something is not quite right, without knowing exactly what. This type of monitoring activities we shall call lower-level monitoring. It is not specific enough to help pinpoint a concrete mistake and correct it. Dennis' question "Then the question is, whether we can assume that they have the same length" is one example for this. What we call higher-level monitoring is a metacognitive activity that is more precise and aims at specific parts of an explanation or argumentation. For example, if one can use a specific theorem the way it is used in the argumentation or if the comprehension of a specific geometric object is correct. Pia and Charlotte (Group 3) showed higher-level monitoring in both cases, in the loop and in the wandering. In one case Pia questioned the use of a specific theorem, the Isosceles Triangle Theorem, at a specific point of their argumentation. In the other case Pia monitored and corrected Charlottes' understanding of the definition of and the connection between squares, rectangles and parallelograms. These interventions were precise and specific and therefore higher-level monitoring.

The level of monitoring does not seem to be the only factor influencing the quality of a proof as one reason for ending up in drifts can be monitoring activities. In cases where the monitoring could not be used profitably, monitoring activities seem to have a negative effect on the proving process.

Looking at the two examples of drifts, in one case, Dennis and Julius (Group 2) show lower-level monitoring, the other students, Pia and Charlotte (Group 3) demonstrate higher-level monitoring. Nevertheless, both groups ended up in a drift. Content knowledge is the other factor that influences the success of the proving process. The circular reasoning in Dennis and Julius' proof of the first statement happens because they do not know how to prove the equality of the base angle in the triangle (and lack the logical and methodological awareness that you cannot use what you want to prove). Pia and Charlotte end up in a drift after Pia's monitoring activity, because both cannot remember all of the content of the Isosceles Triangle Theorem, which led them to abandoning their proof idea. Looking at detours, monitoring activities can also show a positive effect, for example ending detours by closing loops. Here the monitoring and the content knowledge used afterwards ended the side discussion and brought the students back to their original proof idea, thereby helping them not to end in a drift. Thinking about these examples, content knowledge seems to be what enables one to find a solution to the obstacles that monitoring activities reveal. One can only benefit from monitoring activities if one has the content knowledge necessary to "fix the problems", for example to discern the problematic parts, to recognise what is wrong or missing or to see possible corrections. Without content knowledge one is less likely to find and correct mistakes, misunderstandings or errors through monitoring.

Monitoring is important in mathematics and especially in proving. It allows to control and evaluate one's own performance and understanding during the process. In the proving process three different categories of obstacles can appear: detours, drifts and dry spells. The interviews with the pre-service primary school teachers show that monitoring activities have a relevant influence. But not every monitoring activity has the same influence on proving processes. One can differentiate between monitoring of a lower and a higher level. Whether monitoring activities support or inhibit the success of the proving process is also highly dependent on one's content knowledge.

Figure 1 :

 1 Figure 1: Drawing for the first statement (Group 4)

Figure

  Figure 2: Drawing for the first statement with "red angles" (Group 2)

Table 1 : Overview of the obstacles and the relation to monitoring and content knowledge

 1

	Content knowledge
	yes	no


Fiorenza Turiano
email: fiorenza.turiano@gmail.com

Paolo Boero
email: boero@dima.unige.it

Beyond direct proof in the approach to the culture of theorems: a case study on 10-th grade students' difficulties and potential

Keywords: Culture of theorems, euclidean geometry, indirect proofs, figural concepts, metaknowledge of proof

In this paper, we aim to contribute to the discussion on the students' cognitive processes inherent in proof by contradiction in geometry, particularly as concerns the role played by images. We will present an episode from a one-year experimental pathway in Euclidean geometry designed to develop the culture of theorems in 10 th grade: in crucial moments of proof by contradiction, we observed some students' gestures and reasoning aimed to restore the harmony between visual and conceptual aspects, which had been broken by conflicts due to specific features of that proof in geometry. The students' need of keeping such harmony poses didactical questions, which, if conveniently taken in charge by the teacher, might contribute to design smoother pathways to the indirect proofs.

Introduction

The development of students' competencies concerning theorems is one of the most important goals in the high school. Its relevance is stressed by the Italian national guidelines for curricula in the high school (MIUR, 2010), according to the current definitions of competency (European Community, 2018), in particular, its requirements of autonomy and awareness. In this perspective, developing Culture of Theorems (CoT;[START_REF] Bartolini Bussi | Approaching and developing the culture of geometry theorems in school: A theoretical framework[END_REF] means to acquire, not only knowledge, but also meta-knowledge about theorems: the kind of statements (declarative or hypothetical); the role of hypothesis and thesis; the different methods of proving; the role of previously validated statements and axioms; the distinction between the construction of a conjecture (based on different processes: induction, abduction, analogy, etc.) and the construction of a proof (within a theory). Therefore, autonomy and awareness are two co-present basic requirements in the activities inherent in CoT: exploring to get conjectures; tackling construction problems and their theoretical validation; constructing proofs. We chose the Euclidean geometry as a domain to set up a learning environment suitable for the development of the CoT in two 10 th grade scientific oriented classes. In Italian school, the passage from intuitive to deductive geometry takes place in the 9 th grade: the students must be educated to justify geometrical facts and to prove theorems. These goals require an evolution in the students' ways of looking at geometrical figures and their awareness of links between a proof of the statement and the theory within which the proof makes sense. Thus, we chose the construct of figural concepts [START_REF] Fischbein | The theory of figural concepts[END_REF] and the definition of mathematical theorem (Mariotti, 2001) to frame our experimental pathway. In agreement with the literature, in our pathway the indirect proofs (by contraposition and by contradiction) posed specific cognitive and didactical problems that had been absent in situation of direct proof. We tried to deal with them by using the model for indirect proof elaborated by [START_REF] Antonini | Indirect proof: what is specific to this way of proving[END_REF]. In particular, in this paper we consider a classroom episode concerning proof by contradiction, in which the analysis of students' behaviors, performed according to that model, allowed us to identify specific instances of the obstacles deriving from the drawings that should support the reasoning in its different phases, and some spontaneous students' ways to escape them. The discussion will concern the choice of suitable tasks for smoother approaches to proof by contradiction, and some aspects of the potential and limitations of Euclidean Geometry, as the privileged field to approach the culture of theorems in high school and, in particular, the issues related to indirect proofs.

Theoretical background

The Theory of Figural Concepts (TFC; [START_REF] Fischbein | The theory of figural concepts[END_REF] highlights that geometrical figures possess simultaneously conceptual qualities (controlled by constrains within the realm of an axiomatic system) and images features (based on the perceptive-sensorial experience). The interplay between image component of a drawing and theoretical knowledge may (and should) implement a harmonious fusion between concept and image: the theoretical knowledge transforms the image into a geometrical figure. Fischbein calls Figural Concept this complete harmonious fusion. TFC is a suitable construct for our study for at least three reasons: to investigate how and if the students manage to deal with the relationships between different components in the geometrical figures; to interpret the difficulties deriving from a missing or incomplete fusion; to explain different students' behaviors concerning proof by contradiction, when they deal with it in two different domains: elementary number theory (e.g. when proving the irrationality of ), Euclidean geometry.

Mariotti ( 2001) defines a theorem as the triad (S-P-T) statement-proof-theory: a statement, its proof, and a theory (as a system of shared principles and deduction rules) within which this proof makes sense. [START_REF] Antonini | Indirect proof: what is specific to this way of proving[END_REF] refine this triad in order to better describe and explain the process involved in an indirect proof, due to its logical structure. According to them, proving by indirect method requires two phases: 1) a shift from a statement S (called principal) toward a statement S* (called secondary, which is proved by a direct method) obtained assuming as hypothesis the negation of the thesis of the principal one (in proof by contradiction) or of its thesis (in proof by contraposition), and as thesis, respectively, a contradiction or the negation of the hypothesis of the principal one; 2) the validity of the implication S*S depends on the logical theory, that is external to the theory in which the principal and secondary statements are formulated.

The two authors call meta-statement S*S, meta-proof the proof of S*S, meta-theory the logical theory into which the meta-proof makes sense. Then, any theorem with indirect proof consists of a couple of subtheorems belonging to two different levels: the level of the mathematical theory and the level of the logical theory. Unfortunately, usually teaching practice takes for granted the metatheorem.

Method

Euclidean geometry as a suitable domain for a teaching-learning pathway to the CoT

We chose Euclid's plane geometry to make 10 th grade students approach the CoT for at least three reasons: 1) in Italy, since the end of the XIX century, Euclidean geometry has been considered to be the most suitable domain to allow high school students to tackle theorems, proofs, proving; 2) Euclidean geometry offers the possibility to deal with different kinds of proof; 3) the validation task of a geometric constructions provides the teacher with the opportunity of making the students aware of the nature and relevance of theoretical thinking in mathematics.

The structure and tasks of the pathway

In the schoolyear 2016-17 our project was designed and implemented for the first time in a 10 th class scientific-oriented. It took about 46 hours. In the same school, in the next year, the experiment involved two 10 th classes (named 2A and 2D, respectively of 31 and 20 students). It was 7 months long, 2 school-hours (100 minutes) per week, summing up to about 56 hours. All the lessons were held by the same teacher (the first author), in the same day: first in 2A, then in 2D; a researcher (the second author) played the role of participant observer. The pathway was divided into 3 modules: 1) from the construction of a tangent circle to the two sides of a given angle, to the construction of the tangent circle inside a given triangle; 2) from conjecturing about sufficient conditions that create different kinds of triangles inscribed in a circle, to the proof of the relation between inscribed angle and central angle that subtends the same arc; 3) given a circle, from exploring and proving the relationships between its tangent and intersecting straight lines, to solving some problems concerning inscribed and circumscribed quadrilaterals. The classroom activities alternated individual tasks (presented on worksheets to be filled by the studentsenough time was allocated to them, in order to develop specific linguistic skills) and collective discussions orchestrated by the teacher. They were addressed to identify gaps, mistakes, linguistic issues and elements of metaknowledge of proof; and to reflect on different strategies of proof and on the organization of the proof text according to its specific structural constraints. In some cases, the discussion concerned also the comparison and the analysis of a few students' individual productions, selected and made anonymous by the teacher. The individual activities concerned different kinds of tasks: construction problems and related theoretical justifications; proving of produced conjectures (after having been shared and refined under the guide of the teacher in collective discussions); analysing and refining some mates' productions; cloze activities on proof texts; identifying salient meta-mathematical aspects of produced texts. At the end of each module, the summative assessment was made up of two parts: 1) a homework including: a) the written revision of all the personal worksheets, with careful identification of gaps or mistakes, comments on their specific causes of difficulties, with correction or reconstruction; b) a written report on her own experience: if she met difficulties during the work of the module, and when and how (and if) she overcame them; 2) a classroom work: the students were asked to correct and refine 3 worksheets, chosen by the teacher: one produced by a schoolmate, two produced by herself.

The collected data consist of all the individual productions (including home-works and selfreflective reports) and audio-recordings and field notes taken by the teacher and the participant observer.

The student's specific background on proof

The pathway followed a curricular 9 th grade geometric course (triangles and quadrilaterals), approaching the basic notions of CoT, included construction problems and theorems with direct proofs. The teacher estimated that more than 65% of the students were at ease with this kind of proof.

The episode: a construction problem for the approach to proof by contradiction

From the first module of 2017/18 pathway we focus on an episode, which concerns the validation of the construction of a circle tangent to the sides of a given angle; it consists of 5 steps carried out in parallel in the two classes. The related Euclid's proof by contradiction consists of a few steps, and most students already knew all the theorems and definitions used in this proof, including Euclid' definition of tangent to a circle (only one point of intersection). They were also familiar with the construction of the bisector of a given angle. Therefore, in our a priori analysis, we judged that proof suitable for beginners' first approach to proof by contradiction in geometry, and we predicted that at least some students would have been able to perform such mathematical reasoning. Here, we report only that part of Euclid's proof which proves by contradiction that if we assume EA perpendicular to diameter BA (see Figure 1) then it meets the circle in only one point. We didn't show it to the students.

Euclid' Statement (proposition 16 in Book 3): The straight line drawn at right angles to the diameter of a circle from its extremity will fall outside the circle. (Heath's translation) (Proof) For suppose it does not, but, if possible, let it fall within as CA, and let DC be joined. Since DA is equal to DC, the angle DAC is also equal to the angle ACD (prop. I.5). But the angle DAC is a right angle; therefore, the angle ACD is also right: thus, in the triangle ACD, the two angles DAC, ACD are equal to two right angles; which is impossible (prop. I-17). Therefore, the straight line drawn from the point A at right angles to BA will not fall within the circle. Figure 1 First step: construction of the tangent circle to both sides of a given angle Lesson 1 (50'). We didn't propose any method of construction and let the students propose it. We collected their worksheets.

Lesson 2 (100'). By collective discussion, we analyzed all their methods (be they right or wrong) and, among them, we chose the following one, because it was aligned with the classroom background:

1) Take a point C on the bisector of the angle aVb.

2) From C draw the perpendicular line to side a and name T 1 the intersection point. 3) From C draw the perpendicular line to side b and name T 2 the intersection point. 4) Draw the circle of center C and radius CT 1 : it is tangent to both the sides of the angle given. Figure 2 Second step: theoretical justification of the construction Lesson 3 (100'). Here is the text of the proof task; two hints are added: By using definitions and theorems that you know, explain why (i.e. prove that) the obtained circle is tangent to both sides of the angle.

Hints: 1) In the 3 rd step of the construction we took for granted that also T 2 belongs to the constructed circle. Now you must prove that CT 1 =CT 2 . 2) Given the definition of tangent line, in order to prove that each side of the angle is tangent to the circle, you must prove that each side of the angle has only one point in common with the circle: the side a has, in common with the circle, only T 1 , the side b has, in common with the circle, only T 2 . Try to find a reasoning by contradiction about only one side; for the second one you can re-use it.

Lesson 3) in the 2A class: we observed that almost all students proved that CT 1 =CT 2 , but no one was able to continue. The difficulty was to find the starting point of proving by contradiction, so, we decided to offer it: "Let us suppose that the circle has two points, T 1 and T 1 * , in common with side a. What consequence can you derive from this new figure?". While we were waiting for some contributions, we observed gestures simulating reciprocal position of triangle and line, and listened fragments of speech left open. But, the situation reached a stalemate. Thus, the teacher drew the Figure 2 on the blackboard, added the segment CT 1 * (like in figure 3see later), and said: "Let's look at the angles: in T 1 and in T 1 * . Which kind of angles do you think they are?". Her goal was to hint an isosceles triangle, due to two equal angles (opposite to equal sides). She suggested the students to think about the kind of the resulting triangle, based on the reasoning, in spite of how it looked. But most students saw a scalene one. They looked very uncomfortable, because of the conflict (cf. TFC) between the drawing and the reasoning guided by the teacher. She observed that Fed was slowly moving his index and medium fingers, a scissors-like movement; she interpreted it as a representation of CT 1 * that was collapsing on CT 1 . She asked him the meaning of his gesture.

He said: "I see an isosceles triangle, then it has two right angles, (…), I must refuse it; (…)?". Thus, the teacher mirrored that gesture and asked: "In which case, could the angle in C be the zero angle? Look at your fingers moving". Fed: "If the two sides collapse in one… and T 1 * in T 1. ". Fed smiled.

This interpretation spread quickly in the classroom and revitalized the atmosphere: almost all the students restarted to work. But, the lesson-time ended: we had to stop and collect their worksheets.

Lesson 3) in 2D: we decided not to distribute the hints sheet, because it had interfered in an ineffective way with the students' reasoning, and to guide in interaction with students the construction of the proof by contradiction in a smooth way, drawing the side CT 1 * . We proposed to consider the triangle CT 1 T 1 * as isosceles. Suddenly, Bea said: "CT 1 * can't be a radius of the circle.

It's the hypotenuse!". By collective interaction, the teacher replied the same gesture made in 2A: several students saw in it the zero angle and carried on their work. Meanwhile, Fra showed his worksheet to the teacher, where he had written: "if there were two points of intersection, then CT 1 would have not been perpendicular to side a (the negation of the hypothesis)". The teacher told him that his reasoning was the incipit of another kind of proof and encouraged him to complete it. However, the teacher did not share it with the class, in order to not interfere with another way of reasoning, especially because, in that moment, her goal was the proof by contradiction. The discussion was oriented in that direction.

Analysis of the texts: only 6 students out of 51 produced a rather complete proof by contradiction.

Students' difficulties to deal with the impossible figure at the beginning of the proving process were well represented by three students who, when they had to draw the CT 1 T 1 * triangle, wrote that they choose T 1 * "very near" to T 1 . Concerning the subsequent phase of proving, in some cases, students' words resonated with the gesture of closing the two sides of the angle: "The triangle must become the radius CT 1 "; "The angle goes to 0 and the triangle does not exist anymore". Some students wrote that the triangle was impossible, because the width of one angle was 0; other students transformed the "impossible" triangle by making it collapse on the segment CT 1 , because "the measure of the angle T 1 CT 1 * must be 0".

Third step: reviewing some proofs produced by mates.

Lesson 4) (100'). In both classes, students were provided with the photocopies of proofs produced by some of them: 2 for the first part of the task, and 5 proofs for the part that concerned proof by contradiction. They were asked to review, correct and improve their mates' proof texts. Most students worked in a rather exhaustive way on the first group of productions (direct proofs), while they had difficulties with the other five productions. A collective discussion followed, still oriented in the direction of putting proving by contradiction into evidence.

Fourth step: individual written refining the proof

Lesson 5) (50'). Task: "Taking into account the discussion, write down in an accurate and complete way the theoretical justification that the circle is tangent to both sides of the angle."

First part:

The circle of center C and radius CT 1 meets side b in a point T 2

Second part:

The circle of center C and radius CT 1 is tangent to the side a of the angle. Lesson 6) (50'). We provided students with a written theoretical justification of the construction; all the steps were reported, but their justifications were lacking. The conclusion after the emergence of the contradiction was already complete. The drawings of Figure 3 were reported on the worksheet. The students were requested to complete the text by writing the justifications of the statements, which they considered lacking. In this case, most productions were exhaustive and satisfactory.

Interpreting students' difficulties

According to [START_REF] Antonini | Indirect proof: what is specific to this way of proving[END_REF] model of indirect proofs, the students were expected to meet difficulties due to the transition from a direct proof initiated by falsifying the thesis of the theorem, to the management at the meta-level of the resulting contradiction. But another, strong difficulty intervened at the beginning of the reasoning: the existence of an impossible figure T 1 * , which contrasted with perceptual evidence (hence the choice of two "very near" points T 1 and T 1 *

). It is like if the choice of very near points might make the existence of a second meeting point between line a and the circle more reasonable! In this case, a complementary interpretation of students' difficulties might be provided by TFC, particularly when Fischbein (1993, p. 148) says: "There is certainly a conflict here generated by the fact that the two systems, the figural and the conceptual, did not yet blend in genuine figural concepts […]. The figural effect is too strong and it seems to cancel the conceptual constraints". Accordingly, in our case, we might interpret students' difficulties in accepting the impossible figure by considering the conceptual assumption of the existence of two points of intersection as the starting point of the direct reasoning that brings to the contradiction, and the conflict with its figural representation. Indeed, in the case of the proof of irrationality of , most of the same students had easily accepted to start their reasoning from the assumption that =m/n, with m and n coprime integers. The direct proof could be developed up to the conclusion that m and n should have had 2 as a common factor. Then, the transition to the logic meta-level inherent in the management of the contradiction made some students hesitant just for a while, without preventing them from reaching a conclusion. On the contrary, in our geometric case the transition to the logic meta-level was problematic: another "impossible figure" worked again as an obstacle, and some students felt the need of collapsing that figure to eliminate the source of the contradiction, instead of accepting the contradiction and coming to the conclusion of refuting the negation of the thesis.

Discussion

The 2017/18 teaching experiment allowed us to better understand some reasons for the difficulties met by students approaching proof by contradiction through a typical theorem of Euclidean geometry. The difficulties in two different phases of proving depended on the figures of the (necessary) visual support to develop the proof (impossible figures!), intertwined (in the second phase) with the difficulties inherent in the management of the contradiction at the meta-level. In order to deal in the future with the approach to proof by contradiction in Euclidean geometry we are now considering three possibilities: First, to lessen the difficulty inherent in the impossible figure by changing the geometry theorem at stake. [START_REF] Tall | Cognitive development of proof[END_REF] suggest a proving situation related to the proof that the angular bisector of any angle in any triangle intersects the axis of the opposite side in a point that cannot be internal to the triangle. The proof is based on the assumption of an internal point of intersection, and on the consequent reasoning that brings to the conclusion that all the triangles are isosceles.

Second, to approach indirect proofs in geometry through a different treatment of the same situation considered in this paper: the starting point would be to prove that a secant line meeting the circle in T 1 and T 1 * cannot be perpendicular to the radius CT 1 (like the student Fra. did). By this way students would learn to move to the meta-level of logic considerations inherent in indirect proofs without the obstacle of impossible figures; then they might move to a proof by contradiction for another theorem.

Third, to take into account some students' tendency to reason dynamically 1 , in order to escape the contradiction related to the impossible figures, and to show how it may result in a valid proof; and then to move either to a proof by contraposition, or to a proof by contradiction as another way of proving the same theorem by reasoning on the impossible figure.

These possibilities are not mutually exclusive. More generally, we may ask ourselves if the domain of Euclidean geometry is suitable for students' approach to indirect proofs. Indeed, in the literature many examples of such approach concern other domains, and even the first encounter of our students happened in the arithmetic field, without any relevant difficulty. For us, two reasons still suggest to privilege Euclidean geometry for a full-fledged approach to indirect proof (in particular, to proof by contradiction). First, the nature of proof by contradiction: the obstacle of the "impossible figure", if conveniently dealt with thanks to a suitable preparation, might put into better evidence the nature of the proof by contradiction and the necessity to move to the meta-level [START_REF] Antonini | Indirect proof: what is specific to this way of proving[END_REF]. Second, during the process of proving by contradiction, impossible figures do not intervene only in Euclidean geometry, but also in other fields of mathematics (e.g. in Calculus) when students represent the negation of the thesis. Thus, it would be good that students meet impossible figures and deal with them in high school. Moreover, we may add that proof by contradiction entered Western mathematics through Euclid's elements, while it was absent in other important historical developments of the discipline (like in the case of Chinese mathematics; see [START_REF] Siu | Proof in the Western and Eastern Traditions: Implications for Mathematics Education[END_REF]. Thus, it is a cultural product that needs a suitable teacher's mediation. The field of Euclidean geometry seems to be suitable for it, provided that a smooth approach to indirect proofs is planned.

Figure 3 :

 3 Figure 3: the two parts of the statement This time, more than one third of students produced more or less exhaustive proofs by contradiction, but in some cases the texts looked like mere transcripts of what the teacher had put into evidence.


Kitty Yan

Gila Hanna
email: gila.hanna@utoronto.ca

Identifying key ideas in proof: the case of the irrationality of √k

Keywords: Key idea, proof reconstruction, undergraduate mathematics

The mathematics education literature reveals an ongoing interest in fostering students' ability to construct and reconstruct proofs. One promising tool is the concept of "key idea". This study examined the relationships between how well students identified and formulated key ideas of the proof the irrationality of √k for non-square k and their subsequent ability to reconstruct the proof. The findings show that students' proof reconstructions were closely aligned with key ideas that they identified. Very few students were able to use precise language and point to an idea that helped them both understand the proof and reconstruct it. The findings suggest that mathematics educators, in their desire to see students enhance their understanding of proof and proving by the use of key ideas, need to create opportunities for key idea identification and proof reconstruction with attention to students' differing approaches to reconstructing a proof.

Introduction

Since [START_REF] Hanna | Some pedagogical aspects of proof[END_REF] highlighted the distinction between proofs that prove and proofs that explain, it has been widely accepted that the purpose of using proof in the classroom is not merely to convince students that a mathematical statement is true, but also, and more importantly, to provide them with mathematical insights (e.g., [START_REF] Hanna | Challenges to the Importance of Proof[END_REF][START_REF] Mariotti | Argumentation and proof[END_REF][START_REF] Mejia-Ramos | An assessment model for proof comprehension in undergraduate mathematics[END_REF][START_REF] Stylianides | Research on the teaching and learning of proof: Taking stock and moving forward[END_REF][START_REF] Thurston | On proof and progress in mathematics[END_REF]. Closely associated with the notion of "proofs that explain" is that of a key idea in a proof. For some scholars this term refers to a central mathematical idea, method, or strategy used in a proof, whereas for others it connotes the outline, overview or architecture of a proof. Despite these different interpretations of key idea, this notion has had pedagogical implications that are consistently aligned with the promotion of students' ability to undertake proof construction and reconstruction on the basis of richer understanding [START_REF] Detlefsen | Proof: Its nature and significance[END_REF][START_REF] Duval | Cognitive functioning and the understanding of mathematical processes of proof[END_REF][START_REF] Gowers | Mathematics, memory, and mental arithmetic[END_REF][START_REF] Hanna | Key ideas and memorability in proof[END_REF][START_REF] Koichu | Decoding a proof of Fermat's Little Theorem via script writing[END_REF][START_REF] Raman | Key ideas: What are they and how can they help us understand how people view proof[END_REF][START_REF] Robinson | Proof = guarantee + explanation[END_REF].

Despite the increasing interest among mathematics educators in fostering the ability of students to construct and reconstruct proofs, and in proof reconstructability in particular, there are limited studies that investigated how university mathematics students go about identifying the key idea(s) of a proof, consciously or unconsciously, or how key idea(s) are actually used to help students grasp the overall structure of a proof or the mathematical insights embedded in it. Of those empirical studies that have investigated students' comprehension of proof (e.g., [START_REF] Weber | Why and how mathematicians read proofs: An exploratory study[END_REF], very few have probed the identification of key ideas as a factor that promotes the ability of undergraduate university mathematics students to construct and reconstruct proofs. This paper examines the relationships between the students' ability to identify and formulate key ideas of a proof and their subsequent ability to reconstruct the proof, with specific reference to the use of key ideas.

Theoretical underpinnings

Practicing mathematicians have paid considerable attention to the concepts of understanding, remembering, and reconstructing proofs [START_REF] Byers | How mathematicians think: Using ambiguity, contradiction, and paradox to create mathematics[END_REF][START_REF] Manin | Truth, rigour, and common sense[END_REF][START_REF] Thurston | On proof and progress in mathematics[END_REF]. [START_REF] Gowers | Mathematics, memory, and mental arithmetic[END_REF], a Fields medalist, suggests that focusing on the main idea(s) of a proof would enhance its memorability and reconstructability.

Key ideas

The notion of key idea in proof construction and reconstruction stems from both mathematical and pedagogical considerations. [START_REF] Lai | Factors mathematicians profess to consider when presenting pedagogical proofs[END_REF] reveal that mathematicians value and emphasize the presentation of the main ideas of a proof over that of a completely rigorous proof. Similarly, mathematics educators have discovered that to help students understand and remember proofs it is more beneficial to put an emphasis on the main ideas contained in the proofs than to teach the students how to build valid sequences of logical steps [START_REF] Durand-Guerrier | Argumentation and proof in the mathematics classroom[END_REF][START_REF] Hanna | Key ideas and memorability in proof[END_REF][START_REF] Hemmi | Students' encounter with proof: The condition of transparency[END_REF][START_REF] Knipping | A method for revealing structures of argumentation in classroom proving processes[END_REF][START_REF] Malek | The effect of using transparent pseudo-proofs in linear algebra[END_REF]. Furthermore, [START_REF] Duval | Cognitive functioning and the understanding of mathematical processes of proof[END_REF] suggests using main ideas to overcome students' mental blocks in proof construction. Such ideas described above are "key ideas" which carry the flow of information in mathematical proof [START_REF] Detlefsen | Proof: Its nature and significance[END_REF] and capture the gist of a proof [START_REF] Robinson | Proof = guarantee + explanation[END_REF].

The term key ideas may mean different things to different people. Some scholars refer by that term to the most important mathematical ideas, methods, or strategies used in a proof, whereas others have in mind an outline, overview or architecture of a proof. The former maintain that a proof can foster understanding more successfully if it is constructed on the basis of well understood and internalized key mathematical ideas [START_REF] Gowers | Mathematics, memory, and mental arithmetic[END_REF][START_REF] Hanna | Key ideas and memorability in proof[END_REF][START_REF] Mason | Thinking mathematically[END_REF]. The latter, while also focusing on the central constructive idea, propose a general approach in which a proof task is broken into chunks to highlight its overall structure [START_REF] Leron | Heuristic presentations: The role of structuring[END_REF][START_REF] Mejia-Ramos | An assessment model for proof comprehension in undergraduate mathematics[END_REF][START_REF] Robinson | Proof = guarantee + explanation[END_REF][START_REF] Selden | A theoretical perspective for proof construction[END_REF]. [START_REF] Robinson | Proof = guarantee + explanation[END_REF] suggests that it is essential to ignore low-level details while highlighting the overall structure of the proof. [START_REF] Selden | A theoretical perspective for proof construction[END_REF] suggest the construction of a proof framework or outline specifically to reduce the burden on the working memory of the mathematician or student. [START_REF] Raman | Key ideas: What are they and how can they help us understand how people view proof[END_REF] understands the term key idea differently. For her, it refers to a heuristic idea that one can map to a formal proof with an appropriate degree of rigor.

In this study, the term key idea in a proof was presented to the students as an idea, such as mathematical fact or concept, a specific proving approach or technique that one actually needs to remember in order to know how to go about re-constructing a proof. One might ask "why reconstruction?" At the undergrad level, students often apply a previously learned proof to reconstruct a proof of that type on a new problem. In this sense, proof reconstruction is an outcome measure of the power of key idea.

Here are examples of key ideas. 1) To prove that √2 is irrational, a key idea might be: to start with a minimal fraction p/q and look for a smaller such fraction. 2) To prove that the sum of the first n positive integers, S(n), is equal to n(n+l)/2, a key idea might be: to notice that each pair (1 + n), (2 + (n-1)), (3 + (n-2)) ... has the same sum; another key idea might be: to represent S(n) in a triangular form of dots. 3. To prove Viviani's theorem, a key idea might be: to use the fact that the area of a triangle is half its base times its height.

Method

Both qualitative and quantitative methods were used in the larger exploratory study on the key ideas of proofs. This paper reports the results based mainly on the qualitative data collected from 17 first year undergraduate mathematics students, who attended a transition-to-proof class in the honours program of mathematics or mathematics education at an urban university in central Canada. The age range of the participants was from 19 to 38.

Data Collection and Analysis

The purpose of the quantitative survey' was to gather data on the use of key idea(s) in proof construction and reconstruction. For example, students' agreement to some statements about key ideas were as follows: 1) When I read a proof, I try to identify its key ideas (76%); 2) After constructing a proof, I tend to identify the key ideas used in it (59%); 3) When I reconstruct a proof, I am less likely to get lost if I already know the key idea used in the proof (53%). Student work was the major source for investigating how students identify key ideas in a proof and how they use the key ideas to reconstruct it. It included worksheets, individual assignments, group work, mid-term tests, final exams, as well as snapshots of students' proving products on a board or on paper. Five complete proofs were used for students to identify and formulate key ideas. The paper focuses on the proof of irrationality of √k for non-square k for investigation and discussion.

The proof of the irrationality of √k for non-square k (see Figure 1) was selected from the book Charming Proofs: A Journey into Elegant Mathematics [START_REF] Alsina | Charming proofs: A journey into elegant mathematics[END_REF]. This proof was given to the students who attended "Problem, Conjectures and Proofs" course as an in-class exercise which consisted of three components: 1) reading the proof, 2) responding to seven prompt questions such as, "What is the method of the proof?"; "Explain how to use to prove that and What is the relevance of these inequalities?"; and "What is the idea of the proof? Limit yourself to three or four sentences", and 3) reconstructing the proof. The instructor gave the students approximately 40 minutes to read the proof and to respond to the prompt questions designed by the instructor. After the students handed in the worksheets, they had approximately 30 minutes to reconstruct the proof. For all the three components of the exercise, the students were asked to work individually.

Fifteen responses were collected. At the end of the course, follow-up question sheets designed to gather more information about students' perceptions of the notion of key idea were also collected. Email discussions were also conducted with three participants.

Findings

The findings are based on the work of 15 students for whom there is complete data. The students' work on this proof is organized by the relationship between how well the students identified and formulated the key ideas of the proof and how well they subsequently used them to reconstruct the proof (see Table 1). Each of four qualified judges (two mathematics educators and two mathematicians) assessed the students' responses independently and then the four settled on a grouping of the responses into three categories: 1) clear key ideas (well-formulated mathematical ideas useful in the proof), 2) unclear key ideas (imprecisely formulated mathematical ideas with no clear way to use them in the proof), and 3) unhelpful key ideas (well-formulated mathematical ideas irrelevant to the proof). The main criteria used to categorize how well the students identified and formulated the key ideas of the proof were relevance and clarity in stating where the contradiction comes from in the proof.

Proof Reconstruction

Identification Nine out of 15 students identified the key ideas of the proof inappropriately and were unsuccessful in reconstructing the proof. These key ideas, irrelevant to the proof presented, fall into three categories (see of 9 students improperly considered that the key idea of the proof involved the properties of straight lines or linear functions, while 3 students claimed incorrectly that the Well Ordering Property was used in the proof. The third type "mechanical procedures" shows that the student simply listed the procedures to be followed to complete the proof without addressing the mathematical ideas used.

Student Kathy displayed a clearly identified key idea of the proof followed by a successful reconstruction. As shown in Figure 2, she offered a well-articulated key idea of the proof by breaking down the key idea into three related elements: 1) an assumption being made -"assume √ can be presented as a rational number in lowest terms", 2) the consequence of the assumption -"then point (n, m) should be the closet point to the origin on the line for x and y integers", and3) what contradicts the assumption -"point (m-pn, kn-pm) is also on the line and that (m-pn) and (km-pm) are both integers less than n and m respectively, so there is a contradiction". The logical progression of the three elements was clearly stated and well presented.

In Kathy's proof reconstruction (see Figure 3), the same structure and wording of the key ideas appeared. In fact, the reconstruction was an expanded version of the key ideas with adequate details of reasoning and verifying. Notice that, from the instructor's perspective, the proof could have stopped when Kathy reached the point where "the point (m-pn, kn-pm) has integer coordinates and is closer to the origin than (n, m)". However, Kathy continued on to explain that " is linear argue that now a new point was found such that √ = in lower terms than . This intention and action to be more explicit and transparent than the original proof shows that the student reconstructed a proof that she understood, and needed to communicate with herself and the instructor about what she considered critical to reach the contradiction. I have to break the proof into related steps so that I can understand it when I am reading it, and communicating that is just the process of putting the ideas of those steps back into words. When doing this, I choose phrasing that would help me to understand the proof better. So I think that I did well on those questions because they are similar to my own process of understanding the proofs presented.

There was one case of an incorrect key idea followed by a successful reconstruction. A student stated that, "The main idea of this proof is we won't have a line when the slope of it is irrational. The graph will be a curve." However, his reconstruction of the proof was complete and clear, despite the mis-identified main idea. To probe what was going on in the student's mind, attempts were made to discuss the work with him. Unfortunately, the student was not interested in such discussions.

Discussion

The findings indicate that compared to the original proof, students' proof reconstructions varied in some ways. These variations deserve some pedagogical attention given the fact that a deep understanding of a proof often leads to a personal interpretation of the proof. In Kathy's case, she deliberately added some details that did not appear in the original proof. The "unnecessary verification" served her as a necessary self-explanation to make a strong argument. Similarly, another student interpreted "a closer point to the origin" presented in the original proof as the integral coordinates "below m and n" so that he could link the point (m-pn, kn-pm) back to the straight line.

Three out of 15 participants identified the key ideas and reconstructed of the proof with limited success, while nine students did not manage to state a key idea clearly. Consequently, these nine students' reconstructions of the proof tended to lack clarity as well. What seems to have thwarted their ability to reconstruct the proof was their lack of success in identifying clearly a key idea of the proof with the intention of re-using it in future reconstruction. The inadequate level of clarity in these students' work might be due to a lack of full or partial understanding of the proof, or simply due to a struggle with being precise in expressing their ideas.

The findings are somewhat in line with [START_REF] Raman | Is that a proof?": Using video to teach and learn how to prove at the university level[END_REF] in that a proof should not be thought as having only one particular key idea: "We refer to "a" key idea rather than "the" key idea, because it appears that some proofs have more than one key idea" (p. 2-156). The findings do support Gowers' ( 2007) view that a key idea provides not only a clue how to remember a proof but also to write it up. In fact, the findings bring evidence that the students' proof reconstructions were closely aligned with the key ideas they themselves identifiedvaried in type as those key ideas may have been. It is therefore important for mathematical educators to be aware of these variations so as to understand students' work better. In addition, instructors could be attentive to students' differing approaches to reconstructing a proof and aim at creating more opportunities for students to identify key ideas and engage in proof reconstruction.

Figure 1 :

 1 Figure 1: The proof shown to the students

Figure 2 :

 2 Figure 2: Kathy's key idea of the proof of the irrationality of √ and passes through the origin then and is in lower terms ", which did not appear in the original proof presented to the students. It seemed that in order to come to the conclusion, Kathy needed to see the integer coordinates of the two points in the form = to

Figure 3 :

 3 Figure 3: Kathy's reconstruction of the proof of the irrationality of When asked to reflect on what led to the success in understanding and communicating the key ideas of the proof, Kathy stated that:

Table 1 : Relationships between key idea identification and proof reconstruction

 1

		Clear	of Key Idea Unclear	Unhelpful	Total
	Successful	2		1	3
	Unclear		3		3
	Problematic			9	9
	Total	2	3	10	15

Table 2

 2

		). The first two types reflect students' misunderstandings of the proof: 5 out
	Type	What key ideas focused on	Number of Students
	1	Properties of straight lines or linear function	5
	2	Well ordering property	3
	3	Mechanical procedures	1
	Total		9

Table 2 : Types of unhelpful key ideas

 2