domain: math shs shs.edu

text:
Katharina Kirsten
email: k.kirsten@uni-muenster.de

Identifying phases and activities in the proving process of first-year undergraduates

Keywords: Proof, process-oriented perspective, proving cycle, undergraduate

. In the presented study, students' approaches to proving are analysed from a process-oriented perspective. The primary aim is to empirically confirm a process-oriented model for the proving competence of undergraduates. This model may be used to analyse proving processes at an individual level and, therefore, to gain more detailed information about proving process and its phases in general.

Theoretical framework

The study is based on a theoretical model of the proving process that describes different phases and activities in proof construction and which is mainly following the considerations of [START_REF] Boero | Argumentation and mathematical proof. A complex, productive, unavoidable relationship in mathematics and mathematics education[END_REF]. [START_REF] Boero | Argumentation and mathematical proof. A complex, productive, unavoidable relationship in mathematics and mathematics education[END_REF] describes a proving process that starts with conjecturing and exploration activities and ends up in selecting arguments and linking them to a deductive chain. However, proving tasks at university level often consist of a statement estimated to be true, especially during the first year of studies. The construction of a proof is in this case rather aimed at justification instead of conjecturing and problem exploration. According to this, we suggest the following variation of Boero's model. 1. The model focuses on proving activities concerned with the justification of a given statement. In particular conjecturing activities are excluded. However, reducing the process described above does not mean to exclude all exploration activities, but locating them at the beginning of the proof construction. Leaning on [START_REF] Reusser | Erwerb mathematischer Kompetenzen[END_REF] approach of a situation model, a mental representation of the given statement is estimated to be developed by exploring the proving task. This representation could affect the proving process in a meaningful way.

2. The model includes validating activities at the end of the process. In this phase, which is already implicitly considered by [START_REF] Boero | Argumentation and mathematical proof. A complex, productive, unavoidable relationship in mathematics and mathematics education[END_REF], the final proof is reviewed regarding content, structure and linguistics. Besides, further (shorter or more elegant) proofs can be considered.

3. The underlying structure of the model is a cycle. This kind of structure provides the assumption that proving processes are not supposed to be linear. In fact, the proving process is shaped by interruptions, revisions and turns.

Research question and method

The modifications lead to the following research question: Is the proving cycle an appropriate tool for analysing proving processes? That means, is it possible to reconstruct the different phases and activities stated in the proving cycle empirically? Is there in particular evidence for the existence of an exploration phase? In accordance with the research question, a qualitative study has been designed with the purpose to provide evidence of the proving cycle and to gain more detailed information about the different phases. Therefore, first year undergraduates and first year pre-service mathematics teachers (grammar school) are asked to work on proving tasks in the field of real analysis. To encourage the participants to talk about their ideas and approaches, the working processes are organized in pairs. The proving process of each pair is videotaped, transcribed and finally encoded according to Mayring's (2007) structuring content analysis. The coding is based on a system of categories, which consists of the five theoretical stated phases in the proving cycle.

Results

The analysis of data from six cases shows that the system of categories seems to be well suited to describe proving approaches of undergraduates. Each of the suggested phases could be empirically confirmed in nearly all cases. However, formulating a precise and clear proof is an activity, which is sometimes omitted. An exploration phase could be reconstructed in all cases, although it varies in quality and quantity. The structure of the analysed proving processes is linear insofar as many phases could be reconstructed in the suggested order and turns mainly concern consecutive phases. Only in those cases, where the identified key ideas turn out to be inadequate, the proving process starts with repeated exploration cyclically. As there is no need for further categories the proving cycle can be used as a tool for analysing proving processes. Additionally, this tool can serve as a basis for deeper inductive investigations of the proving competence from a process-oriented perspective.

Figure 1 :

 1 Figure 1: Proving cycle This model differs from the existing model in three ways:

Proceedings of CERME10


text:
Xiaoheng Yan
email: xh.yan@mail.utoronto.ca

John Mason
email: john.mason@open.ac.uk

Gila Hanna
email: gila.hanna@utoronto.ca

An exploratory teaching style in promoting the learning of proof

Keywords: Explanation, exploration, proof, proof construction

Our overall concern is with helping students learn to construct and re-construct proofs. Here we investigate an exploratory style which invites learners to think for themselves, with the instructor circulating among them while listening, probing, and suggesting. The objectives of this investigation are, to understand how the actions of teachers can contribute to the development of their students' thinking, and to provide explicit pedagogic strategies that teachers can use to promote their students' appreciation and understanding of mathematical proof.

Theoretical underpinnings

Our paper adopts the notion of acts of teaching [START_REF] Mason | Teaching as disciplined enquiry[END_REF][START_REF] Mason | Teachers' roles[END_REF]: something that initiates and directs; something which is acted upon; and something which mediates between these, enabling the action to take place. We proceed from this stance to analyse an exploratory style as it was observed in an advanced undergraduate geometry class, with the aim of identifying what specific contributions this style brings to the learning process.

When teachers introduce a proof task, they are likely to have a complex set of expectations of what learners will get from engaging with the task. They are (we hope) aware of, or have access to connections with pervasive mathematical themes, with other contexts in which similar ideas arise, and with the specific powers that learners have. Teachers will have views on how these powers might be developed through working on the task, and on opportunities to interact with learners during which both mathematical thinking and appreciation and comprehension of some particular mathematical topic will be deepened and enriched. Tasks can vary from following an exposition, through exploring relationships, exercising new-found procedures, and making use of newly encountered technical terms as part of their personal and collective developing narrative.

Following the Systematics of Bennett (1956Bennett ( -1966;;[START_REF] Bennett | Elementary systematics: A tool for understanding wholes[END_REF], we distinguish six modes of interaction which arise from the teacher, the learner and the content playing the roles of the three impulses comprising any action: initiating, responding and mediating, and conveniently labelled by six 'exs': expounding, explaining, exploring, examining, expressing and exercising [START_REF] Mason | Teachers' roles[END_REF]. In expounding, the initiative is with the teacher who uses the presence of learners (actual or virtual) as the mediator to make contact with the mathematical content in a significant way. The teacher draws the learners into the teacher's world and ways of perceiving and acting. By contrast to expounding, and also in contrast to its every-day sense, explaining in this framework involves the teacher using the content as a mediator in order to make contact with the student, through listening, watching and probing. The teacher tries to enter the world of the learner. As soon as the teacher thinks they 'know' where the learner's difficulty lies, the action usually reverts to exposition. Exploring involves students taking initiative, mediated by the teacher who may suggest a starting point for exploration, and may Thematic Working Group 01 Proceedings of CERME10 make suggestions based on what students are saying and doing. Examining in this framework involves the student seeking to validate their own criteria against those of the teacher'. Expressing is what the student does when they feel the urge to articulate insights or make conjectures. This can occur for example in response to a teacher asking probing questions. Exercising is what a student does when they feel the desire to practice in order to gain fluency. Our interest here is on the interactions labelled as exploring, expressing, and explaining in the technical sense used in this framework.

Our investigation builds upon previous research. [START_REF] Grenier | Research situations to learn logic and various types of mathematics reasoning and proofs[END_REF] has shown that an experimental teaching approach focusing on various "research situations" can succeed in helping students master mathematical reasoning and proving. [START_REF] Selden | Proof and problem solving at university level[END_REF] considered a division of proofs into a formal-rhetorical part and a problem-centered part. In their view, the formal-rhetorical part of a proof depends only on unpacking and using the logical structure of a theorem, while the problemcentered part depends on exploration and understanding, which is essential to the learning of proof.

Several researchers have investigated the use of explorations in various learning contexts. In the use of dynamic geometry, for example, [START_REF] Mariotti | Justifying and proving in the Cabri environment[END_REF], and others have shown the value of teaching proof through exploration. In the context of argumentation and proof and of the axiomatic organization of mathematics, several other researchers have examined students' use of empirical explorations [START_REF] Durand-Guerrier | Argumentation and proof in the mathematics classroom[END_REF]Hanna, 2010;[START_REF] Hemmi | Three styles characterising mathematicians' pedagogical perspectives on proof[END_REF][START_REF] Jahnke | Understanding what a proof is: A classroom-based approach[END_REF][START_REF] Reid | Proof in mathematics education: Research, learning and teaching[END_REF][START_REF] Stylianides | Facilitating the transition from empirical arguments to proof[END_REF]. Additionally, [START_REF] Garuti | Cognitive unity of theorems and difficulty of proof[END_REF] and [START_REF] Pedemonte | How can the relationship between argumentation and proof be analysed?[END_REF] highlighted what they termed cognitive unity, the continuity between the process of conjecturing achieved through exploration and the production of an acceptable proof.

The exploratory style in this study

We investigate a particular exploratory style in which university students already familiar with linear algebra are taking an advanced geometry course in which they are exposed to proofs by means of explorations that invite them to think for themselves, with the instructor circulating among the students, listening to their discussions, asking probing questions intended to help the students think more deeply about the issue at hand, and at times offering suggestions.

Participants and classroom setting

The 24 participants were undergraduate mathematics students in a mixed-year class (2 nd to 4 th year) at a large urban university in central Canada. The advanced geometry course covered plane geometry, spherical geometry, and briefly, some hyperbolic geometry. It addressed the critical role of transformations (symmetries and isometries) in all of these geometries, the use of dynamic geometry software (The Geometer's Sketchpad), and proof. The instructor of the course was a geometer. The class met once a week for 3-hour sessions, 20 classroom meetings in total. The course was intended to keep equal proportion of instruction time and exploration time to facilitate the exploratory style of teaching and learning. Three classroom sequences were assigned to the investigations on conic sections. This paper focuses on one of the sessionsthe exploration of ellipse.

To facilitate investigations and communication, the classroom setting of the geometry course was unconventional: 7 large round tables with chairs around each filled up the classroom, with 5 large blackboards mounted on three walls. In addition, manipulatives and visual aids associated with geometry were kept in the closet at the back of the classroom with free access for the students.

Data collection and analysis

The exploratory style of learning proofs was documented through (1) classroom observations in the form of audio recordings and field notes; (2) follow-up questions for students; (3) students' course reflections on the explorations; and (4) the researcher's research journal. In particular, the follow-up questions consist of 4 open-ended questions about students' explorations in geometry throughout the course. 17 out of 24 students completed the follow-up question sheets. All students submitted the course reflections.

The data drawn through classroom observations was organized and analyzed by the framework of 6 'exs'. The data from students' written reflections was analyzed using NVivo 10 software to explore themes and patterns of responses. The unit of analysis was a statement. Each participant's work was divided into statements and grouped in categories.

Investigations of conic sections: Findings and analysis

This paper focuses on how proving was promoted through the initiative taken by the instructor and the students, while the content connects the instructor and the students. It does not measure the students' achievement because it is concerned with perceptions of their own understanding of proof.

Initiatives of the instructor -Expounding and Explaining

The class made use of a hands-on investigation involving flashlights to explore conic sections. The instructor held a flashlight aimed at a wall at different angles. "The flashlight bulb and reflector make a "cone" of light. The wall cuts the cone with a plane, making a conic section. So moving the light changes which section we have," the instructor explained. As the beam was forming a circle, ellipse or parabola, he asked students to identify the particular shape made on the wall and to pay attention to the critical points where there was a change from one conic section to another, as the angle changed. Then the instructor raised a question about hyperbolae with a suggestion, "Now, what features will confirm the shadow is a hyperbola? You may look for asymptotes -lines which the light approaches as it goes up the wall."

Initiatives of the students -Exploring and Examining

After the demonstration, students worked in groups to create all four conic sections by using the light source of their smartphones or the flashlights provided. Students quickly discovered that when the beam was perpendicular to the wall, it gave a circle; when it was tilted a bit, it gave an ellipse; when it was tilted more, with the ellipse vanishing, a parabola emerged. Group discussions mainly focused on the creation of a hyperbola and the difference between hyperbolae and parabolae.

S8:

How do you know it is a parabola or hyperbola? S22: It depends on the angle you hold it at.

S8:

Right, but how does an angle tell us whether it is a parabola or hyperbola? S12: Well… if you look at this graph I found online, the parabola's axis is parallel to the cone's side. If it were not parallel, it would become a hyperbola.

S8: I see. So how is this related to what we are doing? The wall is the cutting plane and the light source is the cone. When the wall is not parallel to the borderline of the beam, it is a hyperbola. I cannot make them parallel precisely, but I get it.

This group discussion shows that in the course of their exploration, students did not limit their exploration to the flashlight demonstration but went on to research the problem by retrieving information online so as to better understand the features that confirm that the shadow was a hyperbola, the difference between hyperbolae and parabolae, and also to explain it to their peers.

Following the flashlight investigation, a series of paper folding activities was carried out. Taking the ellipse as an example, each student was given a clean sheet of paper with a circle and a point P inside the circle but away from the center (Figure 1a). Students were first asked to pick a point on the circle, say G, and fold the paper until P was lying directly on top of G (Figure 1b), and then to make a neat crease. Then students were asked to repeat the fold and crease action for a few dozen relatively evenly spaced points on the circle and to observe what shape emerged (Figure 1c). Some students struggled to work out which point was being folded and to wheremistakenly folding a chosen point onto some other point on the circle, while others struggled to create a precise fold, due to the nontransparent nature of paper sheets. Although students worked on the folding at their own pace, within groups, they were talking to and helping each other as they proceeded, which allowed the ones who were struggling to listen, watch and move forward. With a number of creases created, conjectures were emerging in groups. For instance:

S6:

I know that it is not going to be a circle. It is not circular. It would only be a circle if you can fold it onto the middle point. If you can fold it onto other point, it will be off-site the shape.

S17: I think it is going to be a parabola and P is going to be the focal point of the parabola. It makes sense.

Instructor: You need more folds. You can select a few more points on this side of circle (pointing at the sheet that S7 was holding).

S7:

Oh, wait. It is an ellipse! I have a lot of lines. You can envision other points are going to be there. It is very clear it is an ellipse. You can really see it! S17: (looking at S7's paper) Yeah, it is an ellipse… P looks like one of the focal point.

S6:

Where is the other focal point?

Here we see that the students did not always do more than offer a conjecture about the shape of the conic created by the creases. The instructor felt it necessary to intervene and re-direct the students' attention to the core idea (the content) of the session.

Initiatives of the content -Exercising and Expressing

With more questions raised, the instructor asked each group to focus on the following questions:

(1) Now you have this ellipse, you know how to paper fold it. How do you prove it is an ellipse?

(2) If you pick one of the folds, how does this fold help us prove it is an ellipse?

(3) What can you say about this ellipse and the circle? How and why are they related?

S17:

If you have this fold, and you have this distance from point P, then this distance (PE+EC) is going to equal to that distance (GC) because this is a reflection.

S6:

Yes, but what does that have to do with the ellipse?

Instructor: Note: GC is the radius. These two distances (PE and EC)…

S6:

No, I didn't get it.

Though the students had all the information they needed, they still had difficulty reaching the final step of the proof. By posing a prompt and question, the instructor tried to direct students' attention to the sum of the two distances (PE+EC) and the fact that it is equal to the radius. Then, a GSP graph, similar to the folding sheet in Figure 1 above, was shown to have students focus on the relationship between the two (see Figure 2a).

S17:

OK. We are looking at P to E and E to C. The distance is equal to GC, which is the radius of the circle. Oh… That makes an ellipse because this distance (PE) plus this distance (EC) is fixed, the radius of the circle.

S6:

And the ellipse must have something to do with the center as a focal point. If P is moving around according to the center point, the ellipse will just move as you move the P. So the center is another focal point. You can take any circle and a point off the center of the circle, and it will always be the case. A Geometer's Sketchpad (GSP) animation was shown at the very end of the class. Figures 2(b) and 2(c) were two snapshots of the GSP animation indicating how an ellipse was constructed. This process taught them that, "A hypothesis is evaluated by deductively drawing consequences and by investigating whether these consequences agree with experience or should be accepted for other reasons" (Jahnke & Wambach, 2013, p. 469).

In a subsequent class, students were shown a demonstration in which sand was poured onto a circular disk with an off-center hole in it corresponding to the point in Figure 1a. A number of the students voluntarily poured the sand through a plastic strainer. As more sand poured onto the board, more a "ridge" clearly emerged. Looking down on it from above, it appeared to be an ellipse.

Students' reflections

Paper folding and sand pouring

More than half of the students admitted that they found paper folding complicated. While struggling with the first step, they missed the instructions for the second step. Following what was being said and doing folds correctly was an obvious challenge. Despite the confusion and the errors they made in folding, students appreciated the geometry embedded in the experiments. As S3 put it: "The reason for why the specific folds result in the shape never bothered me until I had this experience. I felt that I had a teacher explained the mathematical relevance." S4 observed that seeing the ellipse emerge during the sand pouring, was completely unexpected: "You can actually see an ellipse due to gravity pulling the excess parts down, forming a 'hill'." When asked about the definitions of conic sections, S14 stated that, "it is much easier, at least for me, to recall a process or property that I have physically manipulated or seen carried out visually than to recall a written definition."

The GSP demonstrations

The students were asked whether the GSP graph and demonstration directly or indirectly helped them with the proof of ellipse. The majority of the students claimed that the animation directly helped with the proof. The responses that support this claim can be categorized as follows:

 Have attention focused. "We had all the information but we couldn't prove it until we saw this picture (Figure 2a) which has all the lines with the colors" (S6).  Accuracy. "The animation is more visual and accurate than the paper folding" (S7); "The animation showed an infinite number of straight lines without making mistakes" (S12).  Legitimate process. "The animation presented a 3D visual to experiment with" (S6); and "It provided an extended version of the folds we did in class and allowed me to continue my experience in a less time consuming and more efficient way (S17).  Exposure of the final product. "It showed the final product of paper folds when you fold 100 or 1000 times" (S4).

However, 2 students claimed that the GSP indirectly helped with the proof. One student explained that the animation "helped more with my understanding as opposed to helping me write down the formal proof" (S16), whereas the other student believed that the role of the GSP was to show how the ellipse was constructed and to show how the proof connects to the demonstration (S5). Compared to the hands-on investigations, S7 and S20 claimed that the GSP animation presented in class did help with their understanding, but they had difficulty interacting with it. This is so because they did not know how to use the geometry software. As S11 said, "If I knew how to use the software, I would definitely use it more".

Discussion

Adopting an exploratory style of teaching is metaphorically a bit like heading off into unknown territory; perhaps a city or forest not previously visited, and coming across blaze marks. On the surface, the 'exploratory style' of teaching involved initial stimuli provided by the instructor, then the instructor circulating listening, probing and suggesting. Questions raised by group members were fundamental and critical for the trajectory of the sessions. Beneath the surface lie the subtleties in how much time students were given to think for themselves, to discuss with each other, to try to resolve questions that arose, and to seek assistance from the instructor. Three different contexts in which the same shapes emerge can be seen as a form of variation [START_REF] Marton | Necessary conditions for learning[END_REF] with both conceptual and procedural aspects. The instructor's commitment to experiential style of engaging students with the mathematical content provided opportunity for different forms of interaction: students were stimulated to explore, to express, to seek explanation when they felt they needed it, and even to exercise their developing ideas. This prepared them to be able to make sense of what little exposition was provided during the sessions. Being fully engaged, with their hands, their own thoughts, and discussing with their peers, enabled them to produce a proof in which they had a high degree of confidence. Because the conjectures came mostly from them, they had an interest in proving, and a desire to find a proof. The physical and virtual phenomena directed student attention to the dynamic changes and alterations of the objects that they were creating, particularly when the instructor noticed students' struggles during the exploration. Student attention was directed by their peers through formulating conjectures, raising questions, and communicating their thoughts. These allowed learners to be immersed in an environment which engaged them to make conjectures, to try to express their vague thoughts, to modify their own conjectures and to challenge the conjectures of their peers, which is in line with the observations of [START_REF] Grenier | Research situations to learn logic and various types of mathematics reasoning and proofs[END_REF] and [START_REF] Hemmi | Three styles characterising mathematicians' pedagogical perspectives on proof[END_REF].

Pedagogical implications

One of the features that distinguish mathematics from other disciplines is that mathematical conjectures ultimately require proof. One importance of exposure to mathematical reasoning and proof is that it provides learners with an opportunity to "know that they know", not because someone has asserted something but because they can justify it on the basis of previously agreed properties. In the case of the paper folding and sand pouring, it is a means to provoke students' curiosity of why it works and invite them to discover a geometric proof of ellipse on their own. The exposure to the necessity of 'why' could have great impact on promoting students' learning of proof when the activity is carefully designed and chosen.

The Geometer's Sketchpad was used in the classroom throughout the course. Introducing and using geometry software in the classroom at a regular basis can gradually change the way that students approach geometry. However, as a teacher, exploiting manipulatives and geometry software effectively requires familiarity with the materials. Each context has its own trajectory, in terms of time required to make sense of the actions and to interpret the effects. Perhaps the most important pedagogic implication is the need to stimulate students to make connections, to develop their own personal narrative concerning the connections between different manifestations of the same mathematical object.

For the student teachers in the class, we believe that the exploratory teaching style allowed them to grow as a student and as a teacher. At first they looked to be told what they must do and how they will be assessed. By engaging them in mathematical activity, they had a chance to experience themselves as mathematician-learners, and to exercise and develop their own powers to imagine and express, specialize and generalize, conjecture and convince [START_REF] Mason | Teachers' roles[END_REF]. As S3 put it, "My focus throughout the course remained on learning rather than passing, as it should be."

Thinking in terms of modes of interaction has enabled us to add a little bit of detail to the notion of an 'exploratory style' of teaching. However, in order to provide specific advice for teachers, it will be necessary to discern even finer details, which deserves further studies.

  sheet of paper for folding to create an ellipse

  graph (a) and animation of the formation of an ellipse (b and c)

Proceedings of CERME10

Acknowledgment: Preparation of this paper was supported in part by the Social Sciences and Humanities Research Council of Canada.


domain: math shs

text:
John Selden
email: jselden@nmsu.edu

Annie Selden
email: aselden@nmsu.edu

An expanded theoretical perspective for proof construction and its teaching

Keywords: Proof construction, behavioral schemas, automaticity, consciousness, feelings

This paper presents a theoretical perspective for understanding and teaching university students' proof construction. It includes features of proof texts with which students may be unfamiliar. It considers psychological aspects of proving such as behavioral schemas, automaticity, working memory, consciousness, cognitive feelings, and local memory. We discuss proving actions, such as the construction of proof frameworks that could be automated, thereby reducing the burden on working memory and enabling university students to devote more resources to the truly hard parts of proofs.

Introduction

We report an expanded theoretical perspective to better notice, understand, and alleviate difficulties of university students' proof construction including: features of proof texts, types of proofs, situation-action links, behavioral schemas, automaticity, non-emotional cognitive feelings, and local memory. Most difficulties were observed and documented during a 10-year teaching experiment-a proof construction course. Our explanations call on the psychological as well as the education literatures. Local memory (Section 4.10) arose from observing our own proving experiences.

Features of proof texts

The proving process involves many more actions (both physical and mental) than appear in the final proof text (e.g., [START_REF] Mamona-Downs | Proof status from a perspective of articulation[END_REF][START_REF] Selden | A perspective for university students' proof construction[END_REF]. Indeed, researchers have distinguished argumentation from proof, noting that an informal line of reasoning may "justify" a theorem to the prover's satisfaction, but this often differs from the corresponding final proof text written to the standards of the mathematical community (e.g., [START_REF] Pedemonte | How can the relationship between argumentation and proof be analyzed?[END_REF].

The genre of proofs

Students sometimes find the manner in which proofs are written perplexing, as it is often at variance with other genres of writing. We have identified some significant features that generally occur in proof texts: (1) Proofs are not reports of the proving process. (2) Proofs contain little redundancy.

(3) Symbols are (generally) introduced in one-to-one correspondence with mathematical objects. (4) Proofs contain only minimal explanations of inferences, that is, warrants are often left implicit. (5) Proofs contain only very short overviews or advance organizers. (6) Entire definitions, available outside the proof, are not quoted in proofs. (7) Proofs are "logically concrete" in the sense that quantifiers, especially universal quantifiers, are avoided where possible. (Selden & Selden, 2013a).

Structures of proofs

A proof can be divided into a formal-rhetorical part and a problem-centered part. The formalrhetorical part is the part that depends only on unpacking the logical structure of the statement of Thematic Working Group 01 Proceedings of CERME10 the theorem, associated definitions, and earlier results. In general, this part does not depend on a deep understanding of the concepts or genuine problem solving in the sense of Schoenfeld (1985, p. 74). We call the remaining part of a proof the problem-centered part. It does depend on problem solving, intuition, heuristics, and understanding the concepts involved [START_REF] Selden | The role of procedural knowledge in mathematical reasoning[END_REF].

Proof frameworks

A feature that can help write the formal-rhetorical part of a proof is what we call a proof framework, of which there are several kinds, and in most cases, both a first-and a second-level framework. For example, given a theorem of the form "For all real numbers x, if P(x) then Q(x)", a first-level proof framework would be "Let x be a real number. Suppose P(x). … Therefore, Q(x)," with the remainder of the proof ultimately replacing the ellipsis. A second-level framework can often be obtained by "unpacking" the meaning of Q(x) and putting the second-level framework between the lines already written for the first-level framework. Thus, the proof would "grow" from both ends toward the middle, instead of being written from the top down.

Operable interpretations

Another feature that can help write the formal-rhetorical part of a proof is converting definitions and previously proved results into operable interpretations. These interpretations are similar to [START_REF] Bills | Operable definitions in advanced mathematics: The case of the least upper bound[END_REF] idea of operable definitions. For example, given a function f: X →Y and A ⊆ Y, one defines f -1 (A) = {x ∊ X | f(x) ∊ A}. An operable interpretation would say, "If you have b ∊ f -1 (A), then you can write f(b) ∊ A and vice versa." One might think translation into an operable form would be unnecessary or easy especially because the symbols in {x ∊ X | f(x) ∊A} can be translated into words in a one-to-one way. But for some students this requires practice.

Dimensions of potential proof construction difficulty

The need for previous results-proof types: 0, 1, 2, 3

We have classified theorems of increasing difficulty to refine our inquiry-based "proof course" notes [START_REF] Selden | Proof and problem solving at university level. The Mathematics Enthusiast[END_REF]. Type 0 often follows immediately from definitions. Type 1 may need a result in the notes. Type 2 needs a lemma, not in the notes, but relatively easily to discern, formulate, and prove. For Type 3, at least one of discern, formulate and prove should be difficult. A sample Type 3 theorem is: A commutative semigroup S with no proper ideals is a group, when provided only the definitions of semigroup and ideal. One needs to observe that, for a∊S, aS is an ideal, so aS=S. This implies equations of the form ax=b are solvable for any b∊S. Using some clever instantiations of this equation, one can obtain an identity and inverses, and conclude S is a group.

The need for unguided exploration

In constructing some proofs, one may reach a point where there is no "natural" way forward. In what we call unguided exploration, one may need to find, or define, an object and prove something about it, with no idea of its usefulness, that is, one may need to "explore" the situation. For example, in proving the above semigroup theorem, this can happen three times. First, one notes aS is an ideal and thus aS=S. Then one sees equations of the form ax=b are solvable for any b∊S. Such exploration may require self-efficacy [START_REF] Bandura | Self-efficacy[END_REF], which can be encouraged by arranging early student successes.

The need to unpack the logical structure of a theorem statement

An informal statement is one that departs from the usual use of predicate and propositional calculus or fails to specify variables. For example, Differentiable functions are continuous, is informal because a universal quantifier and a variable are omitted, and because it departs from the usual "ifthen" form of the conditional. Such statements are commonplace in everyday mathematics. They are not ambiguous or ill-formed because widely understood, but rarely articulated, conventions permit their precise interpretation by mathematicians and less reliably by students. An informally stated theorem can be memorable and easily brought to mind, but it may be difficult to unpack and prove [START_REF] Selden | Unpacking the logic of mathematical statements[END_REF].

Psychological aspects of the proving process

We view proof construction as a sequence of actions that can be physical (e.g., writing a line of the proof or drawing a sketch) or mental (e.g., changing one's focus from the hypothesis to the conclusion or trying to recall a theorem). The sequence of actions that eventually leads to a proof is usually considerably longer than the final proof text and is often not constructed from the top down.

Situations and actions

When considering proving, we use the term, action, broadly as a response to a situation in a partly completed proof. We include not only physical actions, but also mental actions. The latter can include trying to recall something or bringing up a feeling, such as a feeling of caution or of selfefficacy [START_REF] Selden | The roles of behavioral schemas, persistence, and self-efficacy in proof construction[END_REF]. In addition, we include "meta-actions" meant to alter one's own thinking, such as changing focus to another part of a developing proof construction.

Situation-action links, automaticity, and behavioral schemas

If, during several proof constructions in the past, similar situations have corresponded to similar reasoning leading to similar actions, then, just as in classical associative learning [START_REF] Machamer | Learning, neuroscience, and the return to behaviorism[END_REF], a link may be learned between them, so that another similar situation evokes the corresponding action in future proof constructions without the need for the earlier intermediate reasoning. Using such situation-action links strengthens them, and after sufficient experience/practice, they can become overlearned and automated. We call automated situation-action links behavioral schemas.

Features of automaticity

In general, it is known that a person executing an automated action tends to: (1) be unaware of any needed mental process; (2) be unaware of intentionally initiating the action; (3) execute the action while putting little load on working memory; and (4) find it difficult to stop or alter the action [START_REF] Bargh | The four horsemen of automaticity: Awareness, intention, efficiency and control in social cognition[END_REF]. However, not necessarily all four of these tendencies occur in every situation.

Forming behavioral schemas converts S2 cognition, which is slow, conscious, effortful, evolutionarily recent, and calls on considerable working memory, into S1 cognition, which is fast, unconscious, automatic, effortless, evolutionarily ancient, and places little burden on working memory [START_REF] Stanovich | Individual differences in reasoning: Implications for the rationality debate?[END_REF]. This conversion into S1 cognition conserves working memory resources.

Behavioral schemas as a kind of knowledge

We view behavioral schemas as belonging to a person's knowledge base. They can be considered as partly conceptual knowledge (recognizing and interpreting the situation) and partly procedural knowledge (doing the action), and as related to [START_REF] Mason | Beyond mere knowledge of mathematics: The importance of knowing-to-act in the moment[END_REF] idea of "knowing-to-act in the moment". We suggest that, in using a situation-action link, or a behavioral schema, almost always both the situation and the action (or its result) will be at least partly conscious.

Here is an example of a behavioral schema that can conserve resources. One might be starting to prove a statement having a conclusion of the form p or q. This would be the situation. If one had encountered this situation a number of times before, one might readily write into the proof "Assume not p" and prove q or vice versa. While this action can be warranted by logic (if not p then q, is logically to, p or q), there would no longer be a need to bring the warrant to mind.

The genesis and enactment of behavioral schemas

The action produced by the enactment of a behavioral schema might be simple. It might also be compound, such as a procedure consisting of several smaller actions, each produced by the enactment of its own behavioral schema that was "triggered" by the action of the preceding schema in the procedure. We have developed a six-point theoretical sketch of the genesis and enactment of behavioral schemas (Selden, McKee, & Selden, 2010, pp. 205-206). Very briefly, here are the six points: 1) Within very broad contextual considerations, behavioral schemas are immediately available. 2) Simple behavioral schemas operate outside of consciousness. One is not aware of doing anything immediately prior to the resulting actionone just does it. 3) Behavioral schemas tend to produce immediate action, which may lead to subsequent action. One becomes conscious of the action resulting from a behavioral schema as it occurs or immediately after it occurs. 4) Behavioral schemas were once actions arising from situations through warrants that no longer need to be brought to mind. Behavioral schemas cannot be "chained together" and act outside of consciousness, as if they were one schema. 5) An action due to a behavioral schema depends on conscious input, at least in large part. In general, a stimulus need not become conscious to influence a person's actions, but such influence is normally not precise enough for doing mathematics. 6) Behavioral schemas are acquired (learned) through (possibly tacit) practice. That is, to acquire a beneficial schema a person should actually carry out the appropriate action correctly a number of timesnot just understand its appropriateness. Changing a detrimental behavioral schema requires similar, perhaps longer, practice.

Implicit learning of behavioral schemas

It appears that the process of learning a behavioral schema can be implicit, although the situation and the action are in part conscious. That is, a person can acquire a behavioral schema without being aware that it is happening. Indeed, such unintentional, or implicit, learning happens frequently and has been studied by psychologists and neuroscientists (e.g., [START_REF] Cleeremans | Mechanisms of implicit learning: Connectionist models of sequence processing[END_REF]. In the case of proof construction, we suggest that with the experience of proving a considerable number of theorems in which similar situations occur, an individual might implicitly acquire a number of relevant beneficial behavioral schemas. As a result, he or she might simply not have to think quite so deeply as before about certain portions of the proving process, and might, as a consequence of having more working memory available, take fewer "wrong turns".

Bandura's ideas "ring true" with our past experiences as mathematicians teaching courses by the Moore Method [START_REF] Mahavier | What is the Moore Method?[END_REF]. Such courses are taught from a brief set of notes consisting of definitions, requests for examples, and statements of major results, together with lesser results needed to prove them, but no proofs. The students provide the proofs and present them in class.

The development and uses of local memory

Some may think that proof construction consists mainly of conscious thought (i.e., as communication with oneself or others using speech, vision, etc., or their inner versions, as suggested by [START_REF] Sfard | Thinking as communicating: Human development, the growth of discourses, and mathematizing[END_REF]. However, we take a somewhat different view. In constructing a proof of some complexity, often much more relevant information can be activated than can be held in one's short-term working memory (ST-WM). When such information is lost from consciousness, it may not return to its original state, but rather to a state of partial activation. Nonetheless, conscious thought can sometimes influence the activation of related information in long-term memory (LTM), that is, help bring something to mind. [START_REF] Ericsson | Long-term working memory[END_REF] stated that "reliance on acquired memory skills will enable individuals [experts] to use LTM as an efficient extension of ST-WM in particular domains and activities after sufficient practice and training." We speculate that mathematicians can do this when conducting their own research. We have observed of ourselves, when attempting an intricate complex proof, that a considerable amount of information is generated, but cannot all be kept in mind; however, it is easily recalled. We refer to such partially activated information as local memory --it is available as long as we are seriously engaged with the proof. It seems analogous to [START_REF] Ericsson | Long-term working memory[END_REF] idea of long term working memory (LT-WM).

Teaching and future research considerations

We believe this perspective on proving, using situation-action links and behavioral schemas, together with information from psychology, is mostly new to the field. Thus, it is likely to lead to additional insights and teaching interventions, which brings up the question of priorities. Which proving actions of the kinds discussed above are most useful for mid-level university mathematics students to automate when they are learning how to construct proofs? Since such students are often asked to prove relatively easy theorems-ones that follow directly from definitions and recently proved theorems-it would seem that noting the kinds of structures that occur most often might be a place to start. Indeed, since every proof can be constructed using a proof framework, we consider constructing proof frameworks as a reasonable place to start. Furthermore, we have observed that some students do not write a second-level proof framework, perhaps because they have difficulty unpacking the meaning of the conclusion. This may be because a relevant definition needs to be converted into an operable interpretation in order to construct the second-level proof framework. Thus, helping students interpret formal mathematical definitions so that these become operable might be another place to start, even though students should eventually learn to make such operable interpretations themselves.

Finally, this theoretical perspective is likely to allow one to see parts of the teaching of proof construction in unusual ways and lead to new questions. For example, unguided exploration can be helpful for some proofs, but a student could easily feel the time required for exploration might reduce (timed) test grades. A feeling of self-efficacy might overcome that, but how are feelings "taught"? Early successes with proofs can help, but arranging for these might require detailed planning of the course before it starts. Such planning could perhaps be aided by following a textbook, but most advanced mathematics textbooks prove the most important and useful theorems themselves, thereby taking away from students the opportunity to experience the proving of even parts of such theorems.

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Detrimental behavioral schema

Many teachers can recall having a student write √(a 2 + b 2 ) = a + b, giving a counterexample, and then having the student make the same error somewhat later, perhaps in a different context. Rather than being a misconception (i.e., believing something that is false), this may well be the result of an implicitly learned detrimental behavioral schema. If so, the student would not have been thinking very deeply about this calculation when writing it. Furthermore, having previously understood the counterexample would also have little effect in the moment. It seems that to weaken/remove this particular detrimental schema, the triggering situation of the form √(a 2 + b 2 ) should occur a number of times when the student can be prevented from automatically writing "= a + b" in response.

Feelings and proof construction

The word "feeling" is used in a variety of ways in the literature so we first indicate how we use it. Often feelings and emotions are used more or less interchangeably, perhaps because both appear to be conscious reports of unconscious mental states, and each can, but need not, engender the other. We will follow [START_REF] Damasio | Looking for Spinoza: Joy, sorrow, and the feeling brain[END_REF] in separating feelings from emotions because emotions are expressed by observable physical characteristics, such as temperature, facial expression, blood pressure, pulse rate, perspiration, and so forth, while feelings are not.

Feelings, such as a feeling of knowing, can play a considerable role in proof construction [START_REF] Selden | Affect, behavioural schemas and the proving process[END_REF]. For example, one might experience a feeling of knowing that one has seen a theorem useful for constructing a proof, but not be able to bring it to mind at the moment. Such feelings of knowing can guide cognitive actions because they can influence whether one continues a search or aborts it (Clore, 1992, p. 151). We call such feelings non-emotional cognitive feelings.

For the nature of feelings, we will follow [START_REF] Mangan | Sensation's ghost: The non-sensory 'fringe' of consciousness[END_REF], who has drawn somewhat on William James (1890). Feelings seem to be summative in nature and to pervade one's whole field of consciousness at any particular moment. Non-emotional cognitive feelings, different from a feeling of knowing, are: a feeling of familiarity and a feeling of rightness. Rightness is "the core feeling of positive evaluation, of coherence, of meaningfulness, of knowledge". [START_REF] Mangan | Sensation's ghost: The non-sensory 'fringe' of consciousness[END_REF]. About such feelings, [START_REF] Mangan | Sensation's ghost: The non-sensory 'fringe' of consciousness[END_REF] has written that "people are often unable to identify the precise phenomenological basis for their judgments, even though they can make these judgments with consistency and, often, with conviction." Finally, we conjecture that feelings may eventually be found to play a larger role in proof construction than indicated above, because they provide a direct link between the conscious mind and the structures and possible actions of the unconscious mind.

The roles of affect and self-efficacy

In order to prove harder theorems--ones with a substantial problem-centered part--students need to persist in their efforts, and such persistence is facilitated by a sense of self-efficacy. According to [START_REF] Bandura | Self-efficacy in changing societies[END_REF], self-efficacy is "a person's belief in his or her ability to succeed in a particular situation". Of developing a sense of self-efficacy, [START_REF] Bandura | Self-efficacy[END_REF] stated that "The most effective way of developing a strong sense of self-efficacy is through mastery experiences," that performing a task successfully strengthens one's sense of self-efficacy. Also, according to Bandura, "Seeing people similar to oneself succeed by sustained effort raises observers' beliefs that they too possess the capabilities to master comparable activities to succeed."


text:
Kerstin Hein
email: kerstin.hein@math.tu-dortmund.de

Susanne Prediger
email: prediger@math.uni-dortmund.de

Fostering and investigating students' pathways to formal reasoning: A design research project on structural scaffolding for 9 th graders

Keywords: Formal proof, logical structure, verbal representation, structural scaffolding

Major obstacles for students learning formal reasoning are the lack of transparency of the logical structure of formal deductions, their theoretical status, and their verbal representation. For fostering students' learning of formal reasoning, making explicit the logical structures and unpacking their verbal representations is therefore necessary. In the design research project presented, a teachinglearning arrangement of angle theorems was designed in which given if-then-statements were to be connected with formal deductions based on the design principle of structural scaffolding. A case study of a pair of 9 th graders investigated students' pathways towards becoming aware of and using the logical structures and exemplifies the functioning of structural scaffolding.

Introduction

Formal reasoning, the logical deduction of new theorems from other theorems, has been shown to be a huge challenge for many students at both secondary and tertiary level. Empirical research studies have identified different reasons for these difficulties [START_REF] Harel | Students' proof schemes: Results from exploratory studies[END_REF]. The design research project presented here focuses on one major obstacle, namely understanding the logical structure of deductions and deductive theory development, which is rarely explicit in mathematics classrooms [START_REF] Durand-Guerrier | Examining the role of logic in teaching proof[END_REF]. For this obstacle to be overcome, researchers have suggested that it is important to make the logical structures of deductions and their verbal representations explicit [START_REF] Durand-Guerrier | Examining the role of logic in teaching proof[END_REF].

This design research study follows this general suggestion. It draws upon the design principle of structural scaffolding (following general ideas of scaffolding, cf. [START_REF] Lajoie | Extending the scaffolding metaphor[END_REF]. It pursues two main research questions: (1) How can a teaching and learning arrangement be developed to make logical structures of deductive reasoning explicit? (2) Which typical pathways towards formal reasoning can be initiated by such a teaching-learning arrangement, and which obstacles appear along the pathways? The first two sections present the theoretical background and the methodological framework. The design outcome (a teaching-learning arrangement based on structural scaffolding) and a case study of two students in 9 th grade on their pathway is presented afterwards.

Theoretical background: Approaching logical structures by structural scaffolding

Formal reasoning is crucial in mathematics, not only for convincing one self and others of the truth of theorems or for explaining connections, but mainly for building (at least locally) deductive theories [START_REF] De Villiers | The role and function of proof in mathematics[END_REF]. Even if, for example, students immediately are convinced of the truth of angle theorems, deducing them from each other encourages students to organize them in a logical and deductive sequence and give insights in mathematical evidence instead of empirical (cf. Fig. 1).

Thematic Working Group 01

Proceedings of CERME10 Missing learning opportunities for formal deductions. In school mathematics, in contrast, most reasoning activities do not refer to formal deductive reasoning but to semantical reasoning where the epistemic value is prioritized over the validity of a statement [START_REF] Duval | Structure du raisonnement deductif et apprentissage de la demonstration[END_REF]. Formal deductions are presented mostly in a ready-made form [START_REF] Harel | Students' proof schemes: Results from exploratory studies[END_REF]. This does not allow students access to awareness of how to compose an argumentation using logical structure. Although composing the deductions is only the last step of proving [START_REF] Boero | Argumentation and mathematical proof: A complex, productive, unavoidable relationship in mathematics and mathematics education[END_REF], it is still necessary to offer learning opportunities for this last step, e.g. by linearly ordering all elements and formulating their logical relations in written forms [START_REF] Russek | Writing to learn mathematics[END_REF].

Logical structure of formal deduction and everyday argumentations. Everyday argumentations have often been described by the argumentative structure of data, warrant, and claim [START_REF] Toulmin | The uses of arguments[END_REF]. While in everyday argumentation the warrant may be omitted and only made explicit when an opponent raises doubts [START_REF] Rapanta | What is meant by argumentative competence? An integrative review of methods of analysis and assessment in education[END_REF], it is crucial in formal reasoning and has a theoretical rather than semantic status [START_REF] Duval | Structure du raisonnement deductif et apprentissage de la demonstration[END_REF]. That means that the existence of mathematical theorems and their statements, which are not characteristics of reality, are of relevance. Thus, the range of possible warrants must be made explicit in mathematics [START_REF] Douek | Some remarks about argumentation and mathematical proof and their educational implications[END_REF]. Also, the status of preconditions in if-then-statements differ: in most everyday argumentations, if-thenstatements are only formulated when the conditions are satisfied (Nunes, Schliemann, & Carraher, 1993, p. 130ff). In mathematics, in contrast, if-then-statements are hypothetical, so the validity of the preconditions always have to be checked before applying an if-then-statement as an argument.

Making explicit logical structures for students. Due to these differences, many researchers have suggested explicating the logical structures of formal reasoning in the learning process [START_REF] Durand-Guerrier | Examining the role of logic in teaching proof[END_REF]. [START_REF] Cho | The effects of argumentation scaffolds on argumentation and problem solving[END_REF] used Toulmin's (1958) argumentation scheme for this purpose for college students in non-mathematical contexts, and we will extend this approach for 9 th graders in a geometrical context by including the check of preconditions.

Structural scaffolding as a design principle. Explicating alone is not enough. For students to become acquainted with the logical structure, and to produce it in their own deductions, this study draws on the design principle of scaffolding. Scaffolding is characterized as enabling learners to realize supported activities before they can conduct them independently [START_REF] Wood | The role of tutoring in problem solving[END_REF]. Initially only applied to one-to-one interaction for language learning, the idea of scaffolding has increasingly been elaborated into a design principle for materials and computer tools, whole-class contexts, for open geometrical proofs [START_REF] Miyazaki | Students' understanding of the structure of deductive proof[END_REF], and other learning contents (e.g. [START_REF] Lajoie | Extending the scaffolding metaphor[END_REF].

Methodology of the design research study

Design research as methodological framework. Because the study has the dual aim of designing a teaching-learning arrangement (here: on the logical structure of formal reasoning with the design principle of scaffolding) and developing an empirically grounded local theory of students' learning pathways, we chose the methodological framework of design research with a focus on learning processes [START_REF] Gravemeijer | Design research from the learning design perspective[END_REF]. The concrete model of Topic-Specific Didactical Design Research (cf. [START_REF] Prediger | Topic-specific design research with a focus on learning processes: The case of understanding algebraic equivalence[END_REF] relies on the iterative and intertwined interplay of four working areas: (a) specifying and structuring learning contents; (b) developing the design of the teaching-learning arrangement, (c) conducting and analyzing design experiments; and (d) (further) developing local theories on teaching and learning processes.

Design experiments for data collection. Design experiments are the methodological core of design research studies [START_REF] Gravemeijer | Design research from the learning design perspective[END_REF]. For this project, 3 design experiment cycles were conducted with 20 ninth and tenth graders (age 14-16 years) in total. The case study reported here stems from Cycle 3 in which design experiments in laboratory settings were conducted and videotaped with 5 pairs of students, comprising two sessions of 60 minutes each (in total about 600 minutes video material). The students were familiar with the geometrical topic of angle theorems. The empirical part of this paper focuses on the case study of two female students, Katja and Emilia, from grade 9 and the first author as design experiment leader (in the following called tutor).

Methods for qualitative data analysis. The transcript of the video was analyzed with respect to students' development of explicating elements of the logical structure (using the analytic scheme of data, warrant, claim, cf. [START_REF] Krummheuer | The ethnography of argumentation[END_REF] and to how students articulate relations between these elements (linguistic analysis, not presented here). This makes it possible to investigate the functioning of the scaffolding tool and typical pathways and obstacles.

Design Outcome: Teaching-learning arrangement with structural scaffolding

Mathematical topic. Within the iterative design experiment cycles, a teaching-learning arrangement was developed for the mathematical topic of angle sets. This topic was chosen because the if-thenstatements and the set of possible warrants are well limited in this field and locally organized (cf. Fig. 1).

Structural scaffolding. For structural scaffolding, we use materialized argumentation structure forms on paper as depicted in Fig. 2. In addition to [START_REF] Toulmin | The uses of arguments[END_REF] argumentation structure, the materialized structure also makes explicit why the preconditions of the if-then-statements (named arguments) are satisfied. Every theorem that is already proven is offered as warrant for the next step of formal reasoning. Working with this materialized structural scaffold in each step allows the students to make explicit their often implicit ideas. In the following, the boxes (from above to below) are named data box, condition check box, argument box, and conclusion box.

Learning trajectory for introducing the structural scaffold. The intended learning trajectory starts by activating students' previous knowledge on angle sets in cases of determining angles for concrete constellations ("Find  if °…"). When first asked to prove the general vertically opposite angle theorem, students' initial argumentative resources often include the critical feature, but are limited mostly by their semantic nature ("because supplementary angle"). Starting from these initial reasoning resources, the tutor introduces the structural scaffold by explaining the new practice of formal reasoning as making explicit all aspects implicitly contained in the students' brief argumentation. The condition check box for checking if the precondition of the if-then-statement is satisfied had to be introduced after the first design experiment cycle in order to clarify the theoretical and hypothetical status of if-then-statements in mathematics. The structural scaffold serves different roles along the learning trajectory, (I) as a visualizer for the extended structure; (II) as a working tool for the students to check the completeness of their explicit reasoning; and (III) as a framework for writing down the proof. In our design experiment, after 120 minutes, the students write proofs with deductive chains of reasoning, even though they do not yet find deductive chains for more complex proofs on their own.

Empirical insights into Katja's and Emilia's pathways to formal reasoning

Katja and Emilia start their learning pathway in the way described above. Figure 2 shows the product of the phase of jointly introducing the structural scaffold ending with Sequence 1.

Sequence 1: Reasoning determined by empiricism instead of validity of statements

When asked to prove that α and β are equal, Emilia and Katja offer a typical initial, semantic three-word answer "vertically opposite angles" (unprinted Turn 339), assuming that classifying the type of relation between the two angles is enough. Becoming aware that they are supposed to prove the vertically opposite angle theorem by using arguments like the argument of supplementary angles (cf. Yes. 366 Katja:

Yes. 367 Emilia: And that, uh. 368 Katja: γ plus β 369 Emilia: Yes, so actually this can be -Yes, precisely -But we have no concrete numbers [points to the conclusion box previous task]and then we can go on -so, I don't know, whether we can do this in such small steps, because we have no numbers at all, but then we could say, α plus γ equals 120, umm, 180 degrees. And β plus δ equals 120, umm, 180 degree 370 Tutor:

Mmm. 371 Katja: And… 372 Emilia: And accordingly 373 Katja: γ plus β -plus δ and then 374 Emilia: yes, okay, but actually, actually we need only one, don't we? Then it is just unnecessary, this angle. [points to the angle δ] -So I would say… … we have tothis with [6 sec break] yes, α plus γ 180 degree, then 180 degrees minus β 376 Emilia: No, so I would easily write 377 Katja:

["unintelligible"] 378 Emilia: α plus γ 180 degree and β plus γ 180 degree. 379 Tutor:

Yes. 380 Emilia: And then, if, a system of equations could be created.

When asked to explain in more detail, the students offer details of steps of their calculation ("α plus γ equals 120, umm, 180 degrees", Turn 369), but do not explicate the warrants for these relations (here the argument of supplementary angles). In this way, they find out that they do not need the angle Interestingly, they formulate steps of action or calculation instead of general relations, and consequently, these steps are combined temporarily ("and then" in Turn 373) instead of logically.

Sequence 2: Filling the argumentation structure form without verbalizing the connections

When filling in the materialized argumentation structure form (Fig. 2), the students discuss whether they need δ and organize their process: 403 Emilia: Well theneh, I would say -I know, I think, that here [points to the argument box], we first write that the supplementary-argument is our argument. Then we think which has to be there [hints to the condition-check box] 404 Katja:

[writes "supplementary angle" in the argument box, 21 sec break] Yes, that here 405 Emilia: Ah, I wanted to write that 406 Katja:

… that we γ here 407 Emilia: Yes, that α and β have the same supplementary angle. 408 Katja:

[3 sec break] Where? 409 Emilia: Here [hints a finger at the condition check box] 410 Katja:

[

writes in the condition check box: α and β have the same supplementary angle γ] […]

[Discussion with the tutor, if the second angle δ is necessary] 417 Emilia: Yes, okay. -Umm, then I would now write here, umm, -α plus γ equals 120 degrees and β plus γ equal 120, umm, 180 degrees. Why do I always say 120? Yes, 418 Katja:

[writes both equations in the conclusion box, cf. Fig . 2]

The students succeed in filling in the argumentation structure form mostly without help from the tutor.

In particular, they correctly identify all elements of the logical structure, first choosing the argument and then checking whether its precondition is satisfied (Turn 403). After filling in the form, they condense the proven theorem as a new argument to be used for further proofs (in non-printed Turns 452-471): "Argument of vertically opposite angles: If two lines cross each other, then the opposite angles are equal. (They are called vertically opposite angles.)". This illustrates how the scaffold supports them to produce a complete argumentation and to understand the logical structure. However, it is remarkable that they still do not use any logical connectors to relate the different elements to each other. The language is rather deictic ("here", "there" in Turns 403, 404, 409, 417), but the logical relation between the elements is not verbalized by the students. To give an expert model of how the connections could be expressed, the tutor finally intervenes as follows:

474 Tutor:

[…] Also this condition of point of intersection was considered, so that we have two times two supplementary angles. Here as condition, and because we have supplementary angles, we could use the supplementary-argument that says that a pair of supplementary angles add up to 180 degrees. Therefore, it can be used for our supplementary angles and umm, here two times two were regarded, this means we have two times this equation with, umm, our angles. […]

Sequence 3: Mastering formal reasoning

After determining a specific alternate interior angle, the next task for Emilia and Katja is to prove the general alternate interior angle theorem (Fig. 4). For constructing their formal argumentation structure, the students are given the equality argument (If δ = μ and μ = π, then δ = π. (transitivity)) and the corresponding angle argument (which can only be derived from the parallel axiom and is hence left unproved for the students, cf. Fig. 1). Again, the students successfully construct a complete argumentation structure supported by the structural scaffold of the form. Based on an enriched sketch, they deduce the theorem in three steps (cf. The written text produced by Katja for this last step shows what she has learned (cf. Fig 6). Katja's text provides at least situational evidence that she has grasped the logical structure of formal reasoning and can express some of the logical connections. In contrast to the beginning of the students' learning pathway, she makes explicit the warrant ("the equality argument says that") and the conditions of its application ("Now, we know that  and β have the same measure and α and β."). For expressing the logical connections, she adopts elements of a language offered by the tutor in Turn 747 ("the supplementary-argument that says"). She also expresses the deduction from the argument to the conclusion: "from this we can conclude". However, the order of aspects is still the order of discovery, not yet the strict order of formal reasoning as the conditions are again guaranteed after using the argument. In total, these three sequences from the students' learning process provide insights into the students' pathway from their everyday argumentative resources towards formal reasoning, their induction into mathematical proof as a cultural practice. The structural scaffold strongly supports comprehending of the logical structure, the designated Function (I). The students also capture the norm that the practice of formal reasoning is characterized by making explicit every element in the logical structure (every box must be filled, Function II). However, the scaffold alone does not sufficiently support the process of talking about the logical structures, as visible in Sequence 2. Hence, the structural scaffold had to be complemented by language scaffolds (in this case oral expert modelling offered by the tutor). The written product from Sequence 3, finally, shows that the students can adopt the language scaffolds for communicating about the formal deductions.

Discussion and outlook

The case study of Katja and Emilia gives a first indication for the potential efficacy of the structural scaffolding. Other pairs of the 10 students in Design Experiment Cycle 3 also succeeded in mastering formal reasoning, supported by the scaffold. Filling the boxes serves as prompts for identifying every single aspect of the logical structure (data, warrant, and conclusion) and the satisfaction of preconditions of argumentations. The specific strength of the materialized structure form is that it not only makes the logical structure visible, but also permits students to complete the form in non-linear order. Based on this structural scaffold, the students' written texts are mostly produced in linear, deductive order. As with any provided format, it can be done non-generatively, passively, locally filling each box but not attending to what role the boxes play in formatting the reasoning. However, not only Katja and Emilia but also other students we observed benefited from the scaffolding as they learnt to distinguish between preconditions, if-then-statements and conclusions. The scaffold supported the students to express the relations between the elements of the logical structure verbally and to reflect amongst other things about the generality of the statements or which characteristics of the sketches are important. These features are crucial to increase awareness of formal reasoning.

Of course, the study still has methodological limitations which have to be overcome in later cycles or future research. Limitations concern the sample size (2 students presented, 10 in total) which is not yet representative. So far, the teaching-learning arrangement is focused on one specific topic, the angle theorems, which need to be extended to other topics in future research in order to gain evidence of the overall claim of efficacy. The most important limitation in view for the next cycle of the presented project concerns the language: we intend to identify the phrases and syntactic structures which appear to be necessary for students to realize the need to articulate the logical connections "Now we know, that γ and β have the same measure and α and β. The equality-argument says, that if the first angle has the same measure as a second and this has the same measure as a third, that the first and the third have the same measure. Here the first is γ, the second β and the third α. From this we can conclude that γ has the same measure as α."

Figure 1 :

 1 Figure 1: Locally deductive theory of angle theorems

Figure 2 :

 2 Figure 2: Materialized argumentation structure forms as structural scaffolds

  Fig 2) and the calculating argument ("If there are angle measures, then it is possible to calculate with them like numbers."), they start by naming the angles γ and δ (Figure 3). Then they discuss the necessary conditions and conclusions. 362 Emilia: […] And now we could say actually that α plus 𝛾 results in 180 degrees. 363 Tutor: Mmm. 364 Emilia: Also like here [points to conclusion box of the last task with α + 120° = 180°] 365 Tutor:

Figure

  Figure 3: Marking angles as first step

  Fig. 5): In Step 1, the use the vertically opposite angle for deriving that  =  In Step 2, they use the corresponding angle argument for deriving  = . For deriving that  = , they use the equality argument and produce the last chain of reasoning in Step 3.

Figure 4 :

 4 Figure 4: Alternate interior angle theorem

Figure

  Figure 5: Katja's and Emilia's three steps of formal proof for the alternate angle theorem

Proceedings of CERME10

between the elements in the argumentation. This will provide support for the students on the linguistic level as well as on the logical-structural level.