Erik Hanke
email: erik.hanke@uni-bremen.de

"A function is continuous if and only if you can draw its graph without lifting the pen from the paper" -Concept usage in proofs by students in a topology course

Keywords: Teaching and learning of specific topics in university mathematics, teaching and learning of analysis and calculus, topology, continuity, epistemic actions

 contextual layers of objects. A "propositional" layer to describe relationships between objects used in proofs is introduced and used to perform case studies of students' solutions.

INTRODUCTION

The concept image of a continuous real function of one real variable as one whose graph is "in one piece" or "can be drawn without lifting the pen from the paper" (provided that the function is defined on an interval) is held by many students in school or university [START_REF] Tall | Concept image and concept definition in mathematics with particular reference to limits and continuity[END_REF][START_REF] Hanke | Students' view of continuity: An empirical analysis of mental images and their usage[END_REF].

This piece of research is intended to expand the viewpoint from students' concept usage of continuity from first-year analysis to higher courses and sensitise for some students' thinking processes. To a large extent, this paper is of philosophical nature and suggests a refinement of [START_REF] Schäfer | Towards a refinement of epistemic actions: A case study of processes of recognition on spirolaterals[END_REF] approach to describe epistemic actions with the RBC-model, namely by introducing a new specific layer called propositional layer. This is relevant for the specifically mathematical procedure of deduction from theorems about relationships between objects. This refinement is then applied to students' (partial) proofs of the fact that a real function defined on ℝ is continuous if and only if it has a path-connected graph. Thus, this study begins to fill a gap in the literature on this highly recurring concept image and the difficulties in finding a rigorous proof which requires some topological knowledge.

The general research question before starting this investigation was which mental images students use in a proof to a prevalently, vividly acceptable theorem. Here, the aim of this article is to display students' proofs and moot a way to dissect these according to the levels of concreteness of the objects the students used.

The task

The exercise in question was (translation E.H.; see Ross (2013, p. 182)):

"In school, one often says 'A function is continuous if you can draw its graph without lifting the pen.' Prove the following exact version of this proposition: A function 𝑓: ℝ → ℝ is continuous if and only if its graph Γ 𝑓 = {(𝑥, 𝑓(𝑥)): 𝑥 ∈ ℝ} ⊂ ℝ 2 is path-connected."

The intuition of "without lifting the pen" has to be translated into a valid statement carefully. Path-connectedness really is required instead of connectedness, and the theorem is no longer valid for functions from an arbitrary path-connected space to the real numbers. It is tacitly assumed in this task that the topologies for ℝ and ℝ 2 are Euclidean and Γ 𝑓 inherits the induced topology. Note that the graph Γ 𝑓 is the image of the function id × 𝑓: ℝ → ℝ 2 , 𝑥 ↦ (𝑥, 𝑓(𝑥)). Thus, the forward implication in the exercise follows from the facts that functions into products of topological spaces are continuous (with respect to the product topology) if their components are continuous, and continuous images of path-connected sets are path-connected. For the reverse direction, continuity of 𝑓 at 𝑝 ∈ ℝ can be proven by contraposition via the 𝜀-𝛿definition using the existence of a path of the form 𝛾 = (𝜑, 𝑓 ∘ 𝜑) between (𝑢, 𝑓(𝑢)) and (𝑣, 𝑓(𝑣)) in Γ 𝑓 for some 𝑢 < 𝑝 < 𝑣.

Since this paper has theoretical aims next to the empirical investigation and due to page restrictions, I omit an à-priori-analysis of different possibilities of proving this theorem and prerequisites.

FRAMEWORK: THEORY AND METHODOLOGY

Concept usage, object layers and the model of nested epistemic actions

The concept image of a learner for a mathematical object, class of objects or procedures is "the total cognitive structure that is associated with the concept, which includes all the mental pictures and associated properties and processes" (Tall & Vinner, 1981, p. 152). In contrast, (personal) concept definitions are students' attempts to specify a concept. [START_REF] Moore | Making the transition to formal proof[END_REF] claims that besides "mathematical language and notation" and "getting started" one of the major difficulties in proving is "concept understanding" (mix of concept images, concept definition and concept usage) (p. 249): This means, many students who fail in a proof "lack intuitive understanding of concepts", "cannot use concept images to write a proof", "cannot state the definitions [properly, E.H.]" or "do not know how to structure a proof from a definition" (Moore, 1994, p. 253). The term concept usage "refers to the ways one operates with the concept in generating or using examples or in doing proofs" (Moore, 1994, p. 252).

The Abstraction in Context Methodology (AiC) [START_REF] Dreyfus | Introduction to abstraction in context (AiC)[END_REF] offers a theory about learning, originating in the need for a new mathematical construct, its construction of knowledge and its consolidation taking into account a specific form of context. In this setting, construction of knowledge means performing nested epistemic actions (RBC-model): Recognising, building-with and constructing. Recognising is "seeing the relevance of a specific previous knowledge construct to the problem at hand", building-with "comprises the combination of recognized constructs, in order to achieve a localized goal such as the actualization of a strategy, a justification, or the solution of a problem" and construction "consists of assembling and integrating previous constructs by vertical mathematization to produce a new construct" (Dreyfus & Kidron, 2014, p. 89). The actions are nested in the sense that building-with something requires its recognition and construction requires building-with. Hence, the word construction is meant globally within the AiC methodology and locally in the RBC-model. It is noteworthy that flexibility and availability of a construct does not stem from construction itself but consolidation.

Based on [START_REF] Oerter | Interaktion als Individuum-Umwelt-Bezug[END_REF] theory of activity, [START_REF] Schäfer | Towards a refinement of epistemic actions: A case study of processes of recognition on spirolaterals[END_REF] differentiated between three layers which help to describe recognition processes of objects more precisely. Next to objectification [orig. Vergegenständlichung] which is the creation of objects, objectual concern [orig. Gegenstandsbezug] towards previously constructed objects can be classified on three layers [START_REF] Oerter | Interaktion als Individuum-Umwelt-Bezug[END_REF]: On the singular layer objects are not distinguishable from the action of an individual itself (e. g. recognising numbers in a table); for the actor, the objects are not yet seen as objects and do not need to have names. On the contextual layer objects are characterised by their usagenot anymore restricted to an individual but shared within a communityand the usage is performed within a specific contentual context and similarity of situations; the objects gain persistence beyond singular action. Finally, on the formal layer objects are disengaged from specific actions or context [START_REF] Oerter | Interaktion als Individuum-Umwelt-Bezug[END_REF][START_REF] Schäfer | Towards a refinement of epistemic actions: A case study of processes of recognition on spirolaterals[END_REF]. The objectual concern of previously constructed objects (of a learning process) is reflected in the way someone can use this object. In this article, this activity theory oriented standpoint is specified to mathematics practice in students' attempts to prove a topological fact.

The approach to consider object layers and actions on them seems related to [START_REF] Sfard | On the dual nature of mathematical conceptions: Reflections on processes and objects as different sides of the same coin[END_REF] idea of the duality of operational and structural conceptions (views of an individual of a concept) within the process of concept formation; concept here means a "mathematical idea […] in its 'official' form" (p. 5). Structural conception considers concepts as "abstract objects" and its dual form, the operational conception, is about "processes, algorithms and actions", not the notion as an object itself (Sfard, 1991, p. 4;emph. orig.). Both of these views of conception can be seen in mathematical practice: In the action of formal recognition as well as propositional recognition and building-with (see below) objects are conceptualised as structural and through operational conceptions a conclusion is achieved. Propositions themselves are structural, and their scope is reflected in their use, allowing a deduction or justification (not necessarily mathematically correct though).

Founded in the Anthropological Theory of the Didactic, Hausberger (2018) described "structuralist praxeologies" which characterise mathematical justification practices that are oriented towards replacing a statement about the particular with one about the general: "Structuralist thinking is characterized by reasoning in terms of classes of objects, relationships between these classes and (structural) stability of properties under operations on structures " (pp. 82f.; emph. orig.). What I describe here as propositional building-with action would fall most likely under his levels 2 and 3 of "structuralist dimension" of a proof (Hausberger, 2018, p. 81;emph. orig.) which describe the application of theorems to a task at hand, therefore reasoning on structuralist rather concrete object level. For example, the task of showing that an object O (e. g. ℤ) possesses property A (e. g. unique factorisation domain) can be changed to the task of showing that O belongs to a class of objects C (e. g. Euclidean rings) in order to apply a theorem which states that each member of C possesses A (e. g. every Euclidean ring is a unique factorisation domain). The identification of O's membership to C resembles in the proofs in this paper to recognition actions (of different layers) that O possesses property C. This procedure is "illuminating as to the 'root causes' behind the result" (Hausberger, 2018, p. 83).

Mathematical notions such as "continuous real function" are on the formal layer for experienced students and mathematicians, and instances thereof can be recognised as having the general properties of elements of the class they belong to. However, mathematical theorems can also be seen as objects on the formal layer. If they come into use, e. g. by specialisation to a concrete situation in an exercise, they become an object one builds-with. In fact, since the usage of objects is of particular interest here, I claim that a new object layer should be included, the propositional layer: On the propositional layer we find theorems as objects that describe properties of objects on the formal layer. Thus, it does not merely contain abstract mathematical objects but relationships between objects as own objects. These relationships can then be applied in a propositional building-with action towards objects on the formal or the contextual layer. [1] In practice, the decision for which object layer occurs at which place can be guided by the instantiation of objects ("let 𝑓 be given by 𝑓(𝑥) = 𝑥 2 + 1"), which mostly suggests the singular or contextual layer, or their declaration ("let 𝑓 be a function such that …"), which highlights an object rather on formal layer. In proofs, the identification of propositional recognition and building-with actions may be assisted by the writer, e. g. referencing the lecture, the number of a theorem etc. However, somebody's "personal mathematical toolkit" may determine which layer really is in use. The analyses below reflect an interpretation of the written product, not the way of finding the proof.

Examples for the identification of the object layers

If one wants to show that the unit circle 𝑆 1 is compact, one can directly use the definition of compactness by taking any open cover of the circle, assuming there was no finite subcover and, using concrete, contextual properties of 𝑆 1 , evoking a contradiction. This way, propositional objects are not necessarily involved (except for the definition of course, and depending on the particular argumentation). On the other hand, identifying 𝑆 1 as continuous image of [0,1] → ℝ 2 , 𝑡 ↦ (cos(2𝜋𝑡) , sin(2𝜋𝑡)), is a contextual recognition of 𝑆 1 to the context of the map, and together with the propositional recognition of the fact that continuous images of compact spaces are compact the propositional building-with action of this fact to the situation at hand yields the compactness of 𝑆 1 . Structurally, these two proofs are completely different.

Next, consider the above proof that the graph of a continuous function is pathconnected. First, the graph is recognised as the image of a certain map id × 𝑓 (contextual layer), and this map is recognised as a product of continuous maps (contextual layer). The theorem that products of continuous maps are continuous is recognised on the propositional layer and used to deduce the continuity of the map written down in the proof (building-with on the propositional layer to further recognise the continuity of id × 𝑓 on formal layer: The concrete map is no longer important, simply its continuity). Finally, the theorem that continuous images of path-connected spaces are path-connected (recognition on the propositional layer) is used in a buildingwith action on the propositional layer yielding that the graph of 𝑓 is path-connected by recognising that the theorem is applicable to the situation at hand. Even though this proof is very short, many recognition and building-with actions had to be completed, brought into order and were compressed in only two sentences.

The "if"-direction of the given task can be proved in the formal/propositional manner as described and is a special case of the theorem "If 𝐹: 𝑋 → 𝑌 is any function between a path-connected topological space 𝑋 and any topological space 𝑌, then the graph of 𝐹 is path-connected if 𝐹 is continuous." The main ingredient is the fact that for any pathconnected space 𝑈 and any function 𝑔: 𝑈 → 𝑉 between topological spaces, continuity of 𝑔 implies the path-connectedness of 𝑔(𝑈). Since the reverse directions of these two statements are not true [2], the reverse direction of the students' task cannot be proved (completely) in the formal/propositional manner considering solely (path-connected) topological spaces, continuous maps and their properties as above. Nevertheless, it is surely possible to argue with propositional objects, e. g. using the intermediate value theorem or the intermediate value property of functions (Ross, 2013, p. 182f.).

DATA COLLECTION

The data collection for this study took place during the spring semester 2017 at the University of Bremen within the topology class for Bachelor students in pure mathematics. I was not involved in this class but was informed by the lecturer about the contents. During the third week of the semester the students had to solve (besides others) the task presented above. Construction is not directly observable in the analyses below because all notions involved are not new to the students. The context of the contextual layer is understood very locally, depending on the objects already available in the particular solution, for instance those that have been declared before.

RESULTS

The following cases are supposed to illustrate the work with the above framework (for groups 2 and 3 only one implication is shown). Overall, the solutions were very different regarding the approaches used. More details cannot be included here. All of the following transcripts were translated from German respecting (unusual) syntax.

Notational and language errors are mostly ignored. Abbreviations of German words were often not abbreviated. Small notational errors like forgetting a closing bracket were corrected. An open ball of radius 𝜔 centred at 𝑎 is denoted by 𝑈 𝜔 (𝑎).

Case study: Group 1

The following is a transcript of the solution of group 1 with a sketch redrawn by myself. As a first step, the students declare a positive 𝜀 (line 1) which indicates that they would like to test the 𝜀-𝛿-definition of continuity. It looks like a recognition on the formal layer, but it is not stated explicitly at which point they want to check continuity; most probably it is "𝑥". Afterwards, the students assume that there exists a function (most likely a path, even though not stated) [0,1] → Γ 𝑓 connecting the points (𝑥 -𝛿 1 , 𝑥 -𝜀) 𝑇 and (𝑥 + 𝛿 2 , 𝑥 + 𝜀) 𝑇 . It is not clear why these points should lie on the graph of 𝑓 but this should be the case since the path lies in Γ 𝑓 by their assumption (recognition on contextual layer) (lines 2-3). Interpreting the students' sketch of the graph, which they do not refer to, I hypothesise that they actually mean the points (𝑥 -𝛿 1 , 𝑓(𝑥) -𝜀) 𝑇 and (𝑥 + 𝛿 2 , 𝑓(𝑥) + 𝜀) 𝑇 , and 𝛿 1 and 𝛿 2 are chosen such that 𝑥 -𝛿 1 and 𝑥 + 𝛿 2 are preimages of the corresponding second components of the points on the graph under 𝑓. Thus, the path is recognised on contextual layer using false assumptions. Then, the students choose the minimal of these 𝛿s to implicate that 𝑓 fulfils the 𝜀-𝛿-definition of continuity in line 5 (lines 4-6). This is a building-with action on contextual layer since taking the minimum of the 𝛿s is quite a standard technique in analysis. However, if the function is not "nice enough" between 𝑥 -𝛿 1 and 𝑥 + 𝛿 12 No matter how small the neighbourhood 𝑈 𝜀 (𝑓(𝑧)) is chosen, the values of 𝑓(𝑎) ∈ 𝑈 𝜀 (𝑓(𝑧)) for 𝑎 ∈ 𝑈 𝛿 (𝑧). 13 ⇒ Γ 𝑓 ′ = {(𝑎, 𝑓(𝑎)): 𝑎 ∈ (𝑧 -𝛿, 𝑧 + 𝛿)} ⊂ 𝑈 𝛿 (𝑧) × 𝑈 𝜀 (𝑓(𝑧))

14 Since ℝ is connected, the neighbourhoods are also connected. 15 ⇒ It exists a path 𝛾: [0,1] → Γ 𝑓 16 with 𝛾(0) = (𝑥, 𝑓(𝑥)) 𝑇 𝛾(1) = (𝑦, 𝑓(𝑦)) 𝑇

The solution starts with the definition of the graph of 𝑓 and the students choose two points on the graph (lines 7-8). They want to show the existence of a path in Γ 𝑓 (line 9), presumably that links the two points, although they do not state it. This is contextual recognition since the path is adjusted to the concrete setting and formal recognition would be hypothetical because the definition of path-connectedness is not completely adapted correctly to the given problem. Afterwards, they recognise the definition of continuity of 𝑓 at some point 𝑧 between the first components of the given points on the graph (lines 10-11) on the formal layer (independent of a concrete 𝑓). Next, they recognise on the formal layer a topological version of continuity via neighbourhoods (line 12) and built-with on the contextual layer the implication that a part of the graph lies inside the product of the neighbourhoods 𝑈 𝛿 (𝑧) and 𝑈 𝜀 (𝑓(𝑥)) (nevertheless, one has to mention that the neighbourhoods have never been instantiated because 𝜀 and 𝛿 only appear within quantifiers) (line 13). Afterwards, the recognition of ℝ as a connected space is formal (proven property in the lecture) but the deduction of the connectedness of the neighbourhoods (likely those in lines 12-13), or their product, is not justified (line 14) (possibly, the students also mixed up connectedness with pathconnectedness here). As a last step, the group now directly concludes that there exists a path in Γ 𝑓 joining the points on the graph chosen in the beginning (lines 15-16). It can be interpreted that the students believe that subspaces of path-connected sets are pathconnected and apply this to Γ 𝑓 ′ ⊂ 𝑈 𝛿 (𝑧) × 𝑈 𝜀 (𝑓(𝑧)) without making clear that the neighbourhoods are path-connected, not only connected. Under this interpretation, the building-with action would be propositional, but since the group's claimed implication does not seem to be logically connected to their previous proof steps, their thinking cannot be ascertained.

Case study: Group 2 additional information of ℝ is used) (line 17) and that they have to find a path between two arbitrary points in the graph which is indicated by the declaration of two points in Γ 𝑓 (line 18). On the propositional layer, they build (which means they state its existence) a path 𝛾 between the first components of the chosen points using that ℝ is connected (in fact, they should have used that it is path-connected; it is not clear whether this is just a notational error since they recognised before that ℝ is pathconnected) (line 19). Next, the students recognise on the propositional layer that compositions of continuous functions are continuous and build-with this fact on the propositional layer that the product function 𝑔 = (𝛾, 𝑓 ∘ 𝛾), contextually recognised as a product of continuous maps, is continuous (their argument lacks the fact that the continuity of the components implies the continuity of the product map) (line 20).

Recognising that they have to plug in 0 and 1 for verification (formal layer of a part of the definition of path-connectedness), they conclude that 𝑔 is in fact a path between the initial points (lines 21-23) (singular/contextual layer in lines 22-23 because the concrete form of 𝑔 is used). The last line contains again a propositional building-with action since the graph of 𝑓 is (presumably) shown to have the property that any two of its points can be linked with a path (even though not explicitly stated).

Case study: Group 3

25 "⇐" Approach: If one can show, let it be supposed that there exists a path between two points of the graph of a function, then there also exists an injective path, it follows (*) 26 Suppose, 𝑓 not continuous at 𝑥 ∈ ℝ ⇒ ∃𝜀 > 0: ∀𝛿 > 0 ∃𝑥 ̃∈ ℝ: 𝑥 ̃∈ 𝑈 𝛿 (𝑥), 𝑓(𝑥 ̃) ∉ 𝑈 𝜀 (𝑓(𝑥)) 27 By assumption there is a path 𝜇: [0,1] → Γ 𝑓 with 𝜇( 0 The group begins a proof by contradiction in line 26 and therefore the students recognise the negation of the 𝜀-𝛿-definition of continuity of 𝑓 at some 𝑥 on the formal layer, relying on the notation with neighbourhoods from the lecture. The students builtwith on the contextual layer a path from two points on the graph whose first coordinates 𝑥 ̃ and 𝑥 ′ surround 𝑥, the point where the function is assumed to be discontinuous (lines 26-28) (however, 𝑥 ′ is not explicitly declared). Taken for granted that one can construct an injective path linking two points given any path between these two (line 25)admittedly, the group does neither argue how this may work nor state explicitly that this injective path has to have the same start and end point or domainthe students use such a path 𝜇 ̃ and compose it with the above 𝜏 which is recognised on contextual level as continuous (line 29), to perform a building-with action deducing that 𝜋 2 ∘ 𝜇 ̃∘ 𝜏 is equal to 𝑓 restricted to the interval [𝑎, 𝑏] (lines 29-30). The map 𝜏 is used here to transform the path defined on the unit interval to a path 𝜇 ̃∘ 𝜏 in Γ 𝑓 defined on [𝑎, 𝑏] which shall function as the domain where 𝑓 can be applied (likely, the students actually meant 𝑎 = 𝑥 ̃ and 𝑏 = 𝑥′). In lines 30ff., the students try to justify that the second component of 𝜇 ̃∘ 𝜏 is 𝑓 |[𝑎,𝑏] . Here, the students believe to recognise the first component of 𝜇 ̃∘ 𝜏 as the identity on [𝑎, 𝑏] because there shall be only one possibility for an injective path between two real numbers (lines 32-33). This is wrong. However, this mistake could formally be resolved by performing a "velocity change of paths" which makes the first component of 𝜇 ̃∘ 𝜏 equal to the identity on [𝑎, 𝑏]. The second equal sign in line 32 is then only justified by this erroneous recognition of the only injection [𝑎, 𝑏] → [𝑎, 𝑏] being the identity. Obtaining 𝜋 2 (𝑥, 𝑓(𝑥)) as solution of the calculation in line 32 is a building-with action on the singular/contextual layer; singular here refers to the special situationthe assumption in line 25the students find themselves in. As a last step of the proof, I hypothesise that the students recognise 𝑓 |[𝑎,𝑏] (they write 𝑓 in line 33 though, which clearly agrees with 𝑓 on [𝑎, 𝑏]) to be continuous (at 𝑥) (a contradiction to their assumption in line 26). Their justification is however not directly observable; they may have used the composition of the continuous maps 𝜏, 𝜇 ̃ and 𝜋 2 . This is recognised as a contradiction to the assumption of discontinuity of 𝑓 at 𝑥 on formal layer (line 33). Finally, the end of the proof is obtained as the result of the propositional building-with action that finding a contradiction to the hypothesis of the contraposition of the statement to prove is equivalent to the original statement.

DISCUSSION & CONCLUSION

The notion of propositional layer of objects refines the three layers of objects Schäfer (2010) used to analyse the epistemic action of recognising. In particular, this new layer describes the building-with action of applying a proposition about relationships between objects on the formal layer. In the case studies, it turned out that the recognition and building-with actions usually succeeded when the prerequisites of a definition or theorem in use had been successfully recognised. However, the recognition of the non-satisfaction of necessary conditions for the application of a theorem failed several times because of insufficient mental imagery of continuity (e. g. "local niceness" in the "only if"-proof of Group 1) or paths (e. g. injectivity of paths with group 3) and wrong properties attributed to the objects to be acted on. Nevertheless, subsequent building-with actions on the propositional layer were often carried out coherently based on these wrong assumptions.

  is path-connected, there exists 𝛾: [0,1] → Γ 𝑓 3 with 𝛾(0) = (𝑥 -𝛿 1 , 𝑥 -𝜀) 𝑇 and 𝛾(1) = (𝑥 + 𝛿 2 , 𝑥 + 𝜀) 𝑇 4 Choose 𝛿 = max {𝛿 1 , 𝛿 2 }, then it holds that 5 |𝑥 -𝑦| < 𝛿: |𝑓(𝑥) -𝑓(𝑦)| < 𝜀 𝑦 ∈ ℝ 6 and thus 𝑓 is continuous because 𝜀 arbitrary.

  2 , for example monotonically increasing on [𝑥 -𝛿 1 , 𝑥 + 𝛿 2 ] as indicated by the students' figure, then the interval (𝑥 -𝛿, 𝑥 + 𝛿) is not necessarily mapped into the interval (𝑓(𝑥) -𝜀, 𝑓(𝑥) + 𝜀). Hence, the building-with action does not lead to the needed conclusion in line 5. Even if there were preimages of 𝑓(𝑥) ± 𝜀 (only to the left or right of 𝑥 according to the sign of 𝜀), one had to choose 𝛿 1 = inf{δ > 0: 𝑓(𝑥 -𝛿) = 𝑓(𝑥) -𝜀} and 𝛿 2 = inf{δ > 0: 𝑓(𝑥 + 𝛿) = 𝑓(𝑥) + 𝜀}, and would need to show that these are different from 0. Since functions often encountered are "nice enough" or monotonically increasing, this wrong argumentation might originate in students' "met-befores"[START_REF] Mcgowen | Metaphor or met-before? The effects of previouos[!] experience on practice and theory of learning mathematics[END_REF]. 7 "⇒" Let Γ 𝑓 = {(𝑥, 𝑓(𝑥)): 𝑥 ∈ ℝ} ⊂ ℝ 2 be the graph of the continuous function 𝑓. 8 Let (𝑥, 𝑓(𝑥)) 𝑇 , (𝑦, 𝑓(𝑦)) 𝑇 ∈ Γ 𝑓 (wlog 𝑥 < 𝑦) 9 To show ∃𝛾: [0,1] → Γ 𝑓 path. 10 Let 𝑧 ∈ [𝑥, 𝑦]. Since 𝑓 is continuous, it holds that 11 ∀𝜀 > 0 ∃𝛿 > 0 ∶ |𝑧 -𝑎| < 𝛿: |𝑓(𝑧) -𝑓(𝑎)| < 𝜀 (𝑎 ∈ ℝ)

  ) = (𝑥 ̃, 𝑓(𝑥 ̃)), 𝜇(1) = (𝑥 ′ , 𝑓(𝑥 ′ )). 28 Wlog 𝑥 ̃< 𝑥 < 𝑥 ′ . 29 By (*) there exists an injective path 𝜇 ̃. Now let 𝜏:[𝑎, 𝑏] → [0,1], 𝜏(𝑥) ∶= (𝑥 -𝑎)/(𝑏 -𝑎), is continuous! 30 ⇒ 𝜋 2 ∘ 𝜇 ̃∘ 𝜏 = 𝑓 |[𝑎,𝑏], where 𝜋 2 is the projection of the second component. 31 Namely, for 𝑥 ∈ [𝑎, 𝑏] it holds: 32 𝜋 2 (𝜇 ̃(𝜏(𝑥))) = 𝜋 2 (𝜇 ̃((𝑥 -𝑎)/(𝑏 -𝑎)) = 𝜋 2 (𝑥, 𝑓(𝑥)), since there is only one possibility for an injective continuous map in the first component of 𝜇 ̃ in 33 question. 𝜋 2 (𝑥, 𝑓(𝑥)) = 𝑓(𝑥). // ↯ to 𝑓 not continuous

"⇒" Let 𝑓: ℝ → ℝ be continuous. Note that ℝ is path-connected.

Now let (𝑎, 𝑓(𝑎)) and (𝑏, 𝑓(𝑏)) ∈ Γ 𝑓 .

Since ℝ is connected, there is a path 𝛾 such that 𝛾(0) = 𝑎, 𝛾(1) = 𝑏.

Then, 𝑔 = (𝛾, 𝑓 ∘ 𝛾) is continuous because the composition of continuous maps is continuous.

𝑔 is also a path from (𝑎, 𝑓(𝑎)) to (𝑏, 𝑓(𝑏)) because:

𝑔(0) = (𝛾(0), (𝑓 ∘ 𝛾)(0)) = (𝑎, 𝑓(𝑎))

𝑔(1) = (𝛾(1), (𝑓 ∘ 𝛾)(1)) = (𝑏, 𝑓(𝑏))

thus, Γ 𝑓 is path-connected since 𝑔 is continuous.On the formal layer, the students recognise that ℝ is path-connected (note that no

ACKNOWLEDGEMENT

I would like to thank the German Federal Ministry of Education and Research (BMBF) for its support in the subproject "Spotlights Lehre" within the project "Schnittstellen gestalten" at the University of Bremen.

NOTES

1. Of course, in definitions previously defined concepts are usually specialised, thus definitions may also be seen as objects on propositional layer. Seeing what makes up a definition or whether something satisfies a definition is regarded as a recognition action here, and objects which are recognised to satisfy a definition will be on formal layer.

2. The graph and the image of arg: 𝑆 1 → ℝ, 𝑒 𝜃𝑖 ↦ 𝜃 (with 0 ≤ 𝜃 < 2𝜋) are path-connected but arg is not continuous.


Erik Hanke
email: erik.hanke@uni-bremen.de

Ingolf Schäfer
email: ingolf.schaefer@uni-bremen.de

Learning complex analysis in different branches -Project Spotlight-Y for future teachers

Keywords: Novel approaches to teaching, teaching and learning of specific topics in university mathematics, specialised content knowledge, teacher training, complex analysis

At the University of Bremen in teaching complex analysis, we split the last part of the lecture into two branches according to profession: While future mathematicians deepen their understanding in a branch for them, future teachers take a branch of the lecture where they prepare a task with dynamical geometry software for pupils which is based on phenomena of complex analysis. Here, we describe the design of the course, some general aims and first results obtained from the branch for future teachers.

INTRODUCTION

More than one hundred years ago, Felix [START_REF] Klein | Elementary mathematics from a higher standpoint[END_REF] acknowledged that there is a "double discontinuity" in teacher education in mathematics. When a pupil enters university he or she does not see the connection of elementary mathematics taught in school with the formal mathematics in university, and when teaching in school he or she does not see how the university maths informs his or her practice [START_REF] Hefendehl-Hebeker | Doppelte Diskontinuität oder die Chance der Brückenschläge[END_REF][START_REF] Vollstedt | Framework for examining the transformation of mathematics and mathematics learning in the transition from school to university[END_REF]. Within mathematics courses at university, it is thus necessary to highlight connections between the mathematics future teachers are taught in lectures at university and the mathematics they will teach in schools [START_REF] Prediger | Unterrichtsmomente als explizite Lernanlässe in fachinhaltlichen Veranstaltungen[END_REF].

The project "Spotlight-Y" aims for an institutional link of mathematics and mathematics education and at providing students with the experience to relate content from the lecture in complex analysis to their future teaching practice. By helping them to design tasks for pupils with mathematics from a "higher standpoint" [START_REF] Klein | Elementary mathematics from a higher standpoint[END_REF], the students get the opportunity to try out these tasks with pupils from local schools.

THEORETICAL CONSIDERATIONS [START_REF] Ball | Content knowledge for teaching: What makes it special[END_REF] proposed to distinguish the mathematical content knowledge (CK) of teachers further into common content knowledge (CCK) and specialised content knowledge (SCK). The first is mathematical knowledge that does not depend on a specific profession and is known to pure mathematicians as well as teachers and others, and the latter is mostly useful only for teaching mathematics. So for a teacher, not only pedagogical content knowledge (PCK) is important and unique for the profession but also SCK.

The international study TEDS-FU established that PCK is the most important factor to judge the didactical problems and opportunities of a classroom situation and that CK is the most important factor to assess the utterances of pupils [START_REF] Blömeke | Von der Lehrerausbildung in den Beruf -Fachbezogenes Wissen als Voraussetzung für Wahrnehmung, Interpretation und Handeln im Unterricht[END_REF]. Thus, teaching mathematics requires an interplay of different kinds of knowledge for the teacher and it seems worthwhile to try to specifically address SCK when doing teacher education programs at university. [START_REF] Winsløw | Klein's double discontinuity revisited[END_REF] proposed to study the phenomenon of the second discontinuity by the study of different mathematical praxeologies between university and high school. In their paper, they are able to identify certain challenges for this transposition. For university students, one of these challenges is the establishment of a school praxeology for a problem typically solved with a university maths praxeology (their example is the method of least squares for simple linear regression). The authors claim that the main related difficulties lie in spawning capacities for students' autonomous research and handling (ir-) relevant literature. This is where Spotlight-Y is situated. Our approach can be seen as complementary: Instead of focusing on how the mathematical praxeologies may change and what challenges may hinder, we try to focus on how the didactical praxeologies are combined with the mathematical praxeologies from university.

Research question

In this paper, we want to foster the understanding of the combination of PCK and SCK by the Y-model that structured the lecture (see next section). We try to address the following question: "In what ways do students combine specialised and pedagogical content knowledge within the preparation and implementation of their learning environments?" For this purpose, we will describe example projects.

OVERVIEW OF THE PROJECT

At the University of Bremen the lecture in complex analysis is the final course in mathematics for future teachers. Historically, many teacher students in Bremen are known to view this as a course you have to pass to get through your studies, but which is thought of having no direct connection to what is taught in school. From the standpoint of the designer of the curriculum of teacher training in mathematics, complex analysis was seen as a course which brings different aspects of mathematics together and may help to get a more holistic view on various areas of the subject, e. g. polynomials, trigonometric functions, power series or conformal mappings. As such, it should be a good starting point to get a "higher standpoint" as [START_REF] Klein | Elementary mathematics from a higher standpoint[END_REF] put it.

"Spotlight-Y" is a design research project within the project "Schnittstellen gestalten" at the University of Bremen, funded by the German Federal Ministry of Education and Research (BMBF) that started in fall 2016. Its general aim is the interlock of the scientific disciplines of mathematics and mathematics education. Our students shall see that elements of school mathematics, in particular all relevant classes of functions, can be understood from the complex setting which explains more than the real picture. In three design cycles we develop 1) the structure of the lecture on complex analysis, 2) the specific branch for future teachers with a focus on identifying phenomena of complex analysis relevant for teaching mathematics in upper classes of secondary schools and creating exploratory learning environments, and

3) a day for eXperimental Mathematics (XMaSII) which local pupils from upper classes of secondary school (Sekundarstufe II) attend to work on the learning environments by the teacher students.

It was mandatory for our students to use the free dynamical geometry software (DGS) GeoGebra (https://www.geogebra.org/) for implementation because we assumed that this would make difficult concepts easier to present and provide a suitable technology to really be able to explore a phenomenon. Besides, our students get an opportunity to create authentic materials, plan and structure group work with pupils from secondary schools in 11 th or 12 th grade and gain experience in working with them. In particular, they get experience with pupils outside internships and their practical semester teaching mathematics for rather gifted learners.

Some of the general research questions in the project are: How do students combine mathematics from the complex analysis course, their general knowledge on mathematics education taught earlier in their studies and simultaneously to the course on complex analysis (a seminar on task design)? In which ways do they handle their learning environments and the material for the pupils? How do the future teachers see mathematics education, do they consider it as a scientific discipline or rather a collection of methods to teach school mathematics?

After about two years, we will report on our design principles, experiences and empirical data we collect during the design project to establish a transfer package for other mathematics lectures. In the second year, we perform a first transfer to the stochastics lecture in the Bachelor for future mathematicians and mathematics teachers.

Structuring a mathematics lecture: The Y-model

In "Spotlight-Y" we adopt a Y-shaped model for the course in complex analysis. The course takes place in the fall semester and is attended by pure/applied mathematics students in the Bachelor as well as teacher students in their first year of the Master programme. The course takes one semester and the participants are split up around Christmas after two thirds of the semester (see Figure 1a). The part for everyone covers the introduction of holomorphic and conformal mappings, line integration and Cauchy's theorem up to the residue theorem. The specific branch for future mathematicians continues with more advanced topics like invariance of integrals with respect to homotopy or homology of paths, analytic continuation, the Riemann mapping theorem or the prime number theorem etc., depending on the lecturer. In the branch for teachers we cover geometric properties of complex functions and an introduction to Riemann surfaces. Furthermore, we give the students time to work on their learning environments. The course is finished with a written or an oral exam. In fall 2016/2017, the first cycle in the course on complex analysis started and the next semester was used to start analysing data we gathered during the first implementation. In fall 2017/2018, the second design cycle lead to improvements with respect to the profession-specific branch for the future teachers. In fall 2018/2019, the last cycle will start: A third run during the course on complex analysis will be devoted to the final answers to our research questions and in summer 2019 a second transfer to the course on stochastics will be performed.

Elements of the task design process

In order to prepare the tasks for the pupils, the students have to bring input from three different components together (cf. Figure 1b). They make use of the mathematical CK from the general branch of the lecture, they activate their SCK from the branch for future teachers and bring those branches together with their PCK, i. e. heuristics on task design from their seminar on mathematics education.

Methodology

For the whole project we collected several data. First, the students had to write a "preflection", a short reflection which is not after an action on an action but rather the anticipation of actions, aims and sequences: They should fix which phenomenon they want to discuss in their learning environment, what the pupils are supposed to discover, the planned sequence of events, expected difficulties and the mathematics from the lecture that is directly used. Immediately after the implementation of XMaSII, the students filled in a questionnaire named "Ad-hoc notes" to write down own executed actions, recapitulate the execution and match it with expected or anticipated occurrences. In order to pass the course, the students also needed to write a reflection on their project and their learning. They had to discuss their topic and how they found it, describe detailed the phenomenon to be explored by the pupils, the organisation of their tasks, the schedule of the implementation etc. In a final section, we explicitly asked whether there were certain elements of the course on complex analysis that changed the relationship of the students to mathematics as a science and mathematics as a school subject. In total, 19 students participated in the study. We also conducted guided interviews with two students of four groups each to get more insight to the aspects above (and some more). We omit details here since this paper deals mainly with the learning environments the student groups created.

EXAMPLE PROJECTS

We describe two case examples. Group A created a learning environment called "Differentiation as linearisation" and Group B worked on "Polynomials of infinite degree". Two members of Group A have also been interviewed. Two other groups created tasks for spherical geometry, another one introduced complex numbers and a sixth group also worked on power series. As mentioned above, the tasks themselves focus on phenomena that are rooted in complex analysis and can be explored by the pupils from high school by the means of a DGS.

"Differentiation as linearisation"

Group A looks at real differentiation, i. e. differentiable functions defined on (subsets) of the real numbers to the real numbers. The students concentrate on the image that a differentiable function locally looks like a straight line by magnifying the graph around a given point, say (p, f(p)). The students emphasise that the derivative at a point can be interpreted as the slope of the function, respectively of the tangent at the graph of the function, at this point (which is CCK). They create a sequence of tasks and subtasks within their learning environment that aim at magnifying the graph of a function (use of PCK) and lead to answer the question which exponential function has itself as a derivative. However, the students do not explicitly clarify what this has to do with linearisation besides the optical appearance. These two "Grundvorstellungen" ("basic mental models", Greefrath, Oldenburg, Siller, Ulm, & Weigand (2016, p. 101)) of the derivative at a point, "tangent slope" and "local linearity", are usually distinguished in mathematics education literature (see e. g. [START_REF] Greefrath | Aspects and "Grundvorstellungen" of the concepts of derivative and integral[END_REF] who also discuss "local rate of change" and "amplification factor"). However, we do not see this as a lack of understanding of different Grundvorstellungen with the students. Rather, group A focuses on a geometric point of view and many of their tasks focus on the idea of zooming into the graph of a function to get a more and more straight line (PCK, SCK) on the computer screen (e. Remember the slope of a secant)." (own translation)). In other subtasks, group A provides the pupils with GeoGebra worksheets they have prepared before, where they had plotted several functions and their "slope function", and wants the pupils to make observations about these. In another task at the end of the task sequence, they display an exponential function f(x) = a x and its slope function with a parameter a which can be changed with a slider and ask for which a these functions coincide.

The definition of derivative at a point with limits of difference quotients is not stressed in detail but the GeoGebra worksheets nicely implements that when zooming into the graph two points on it approach the point (p, f(p)) (one from left, another from right) to show that the slopes of the secants through these points eventually yield the slope of the tangent, no matter if one approaches from left or right. However, since the students do not provide counterexamples, it does not seem very likely that the concept of derivative can be fully understood, since the class of functions the students used are pretty standard and "friendly" in the way that they all really look locally straight (e. g. s given by s(x) = x²•sin(1/x), s(0) = 0, is differentiable at 0 but oscillates very much around the origin and therefore hardly is locally straight).

An additional task in the learning environment is about rotation-dilation, which is an interpretation of the complex derivative discussed in detail in the first third of the lecture and exercise classes (SCK). A hexagon and two points are displayed in the Cartesian plane and pushing around the points changes the hexagon according to the multiplication by the complex number associated to the points (see Figure 2a, SCK). However, such a multiplication is not made explicit in the task. Rather, the pupils shall measure distances and angles, and the task seems rather unrelated to differentiation. Not even a function is in play. The complex interpretation of derivative is transformed into a task about rotation-dilation, but not related to the previous tasks on differentiation. This is not surprising since the idea of magnifying pursued in the real approach does not shed any light on dilation-rotation. In the interview with two of the three group members we asked for an intuitive, vivid meaning of a holomorphic function. The students were unable to argue and did not even respond with the keyword of (local) rotation-dilation.

Thus, in terms of our framework, the connection of SCK and PCK did not happen as was hoped for. One problem seems to be that the necessary SCK for the complex derivate could not be utilised by the students, while they argue very well about prerequisites the pupils should have for their learning environment such as secant, tangent, linear functions etc. (SCK for the real derivative). The students of group A did not take part in the parallel seminar on task construction. Therefore, we hypothesise that they followed an alternative strategy to construct tasks: They scrolled through mathematics education literature to find a suggestion on how to fill the aspect of local linearity with life. In the interview, one of the students said, roughly speaking, "From the earlier lectures [in maths education] one knew at least where to research, because it is clear that one cannot have everything stored [in one's mind] and fetch it up immediately, so one needs to do research" (own translation).

From a real point of view in terms of concept formation, the learning environment seems rather unproblematic. We believe this is the case since didactic aspects of real functions and derivatives are very present in the courses on mathematics education the students in Bremen are required to take. However, there does not seem a thorough idea on how to relate the complex interpretation of the derivative of a complex function to the real setting. Thus, intuition of differentiation is not coherently transported from one to the other setting. Also, group A does not argue whether their real images of derivatives find their counterpart in the complex setting.

"Polynomials of infinite degree"

Group B has two general aims with its learning environment about power series which they steadily call "polynomials of infinite degree". Firstly, in school polynomial functions form a very frequently used class of functions (PCK) and power series seem a quite direct generalisation of it (CCK). Secondly, nearly all functions that appear in school can be represented by power series (even though this is usually not made explicit in schools, PCK). For a motivation why power series can be useful for pupils, group B argues that using power series one may be able to explain easily why the derivative of the exponential function is itself and the derivative of the sine function is the cosine function etc. (PCK, SCK).

As necessary mathematical background from the complex analysis course group B recognises holomorphic functions, power series and their derivatives, entire and nonentire functions, radii of convergence of power series and the power series expansion theorem. They relate all these ideas in a correct manner to each other. Nevertheless, the language the students use in their written reflection and on the task sheets for the pupils is quite tenuous: The words polynomial and power series are not clearly differentiated. It may be possible that the students wanted to use the same word "polynomial" with the addition of "finite degree" or "infinite degree" because it is known to the pupils. Group B recognises that the distinction of entire and non-entire functions is relevant for school and illustrates this with examples like -ln(1-x) and (x+1) 1/2 (see Figure 2b, PCK and SCK).

In their preparation and reflection the students raise the question to what extend convergence issues of power series or why Taylor approximation works (CCK) should be covered (PCK), but they consider this too difficult and not anymore relevant for the discovery of the phenomenon for the pupils that previously known functionsand basically all encountered in schoolcan be expressed differently. In fact, the group recognises that arguing in whatever way that Taylor expansion requires differentiation and defining elementary functions by their series expansions may lead to circular reasoning (CCK).

The mathematics background is well explained with a few mistakes mostly in language and notation. Group B's learning environment really allows to discover one of the fundamental phenomena of complex analysis, namely that every holomorphic is at least locally a power series, in the real and school-related setting. In the GeoGebra environment, several functions and their Taylor polynomials of varying degree, adjustable with a slider, are presented. This shows a creative part in the work of our student group, since it visualises very nicely the approximation of a function by its Taylor polynomials. However, the distinction of entire and non-entire functions is mathematically deep and relevant for school (SCK). The question of why the representation of a function by a power series is relevant for pupils is answered with a deepening of their insights into higher mathematics and the application to calculate derivatives similar to the procedure with polynomials. Unsurprisingly, as described above, the problem of convergence is left aside.

Nevertheless, the task sheets for the pupils lack rigor and do not provide consistent language at certain places. For example, in a "remember box" group B writes "A function f(x) is an entire function if it is a polynomial of degree n or is a function which can be written as a polynomial of infinite degree P(x) for every x∈ℝ [grammatical errors deleted, E.H.]", and in another box they write "A function f(x) is a non-entire function if it is NO polynomial of degree n but can be approximated by a polynomial of infinite degree. However, this polynomial of infinite degree approaches the function f(x) only in some interval. It [the function] is not defined for all x∈ℝ or does not have a tangent everywhere [slight grammatical adaptions, E.H.]". The absence of the domain of f is usual in school and probably left out for this reason. It could have been stressed more detailed that a power series expression for an entire function is globally valid, i. e. that P equals the given function on the whole of ℝ. In the definition of non-entire functions, on the other hand, the local approximation by a power series is mentioned correctly (assuming that the group meant that the sequence of Taylor polynomials approximates the original function) but it is not clarified that this power series actually equals the function in some interval containing more than one point. Also, it should have been stressed that this local phenomenon is valid for every point in the domain of the non-entire function but the corresponding power series may change. Another problem is the mentioning of tangent. Tangents do never show up in the learning environment nor in the students' reflection again and it is very unclear what the students wanted to say.

In this case the connection between SCK and PCK seems to be working much better, although there are some flaws in the formulation of the definition. These problems may be rooted in misconceptions in the CCK.

FINDINGS

The learning environments differ heavily in mathematical rigour, creativity and recognisability of complex phenomena in either a real setting or else. In the section on the example projects we described that Group A showed little understanding of the geometric meaning of the derivative of holomorphic functions (SCK) (or, at least, they did not implement this well) but otherwise provided a nice example driven and exploratory environment. Group B, on the other hand, translated the notion of power series representation of entire and non-entire functions to real examples and made a complicated phenomenon accessible to pupils (PCK and SCK). Due to the complexity, arguments on why and where power series converge and how to find Taylor series expansions have not been dealt with (CCK). The written report is mostly correct, however the material for the pupils needs improvement with respect to consistency in language, mathematical exactness on the level of pupils and correctness from a formal point of view.

Still in evaluation phase, i. e. coding the interviews, we are nevertheless already able to state first results regarding our design process:

1. Most students seemed more engaged in creating the tasks than in the lecture.

2. One has to make connections between SCK and PCK explicit, many students fail to see them on their own.

3. It has been evident from the interviews that basic mathematical content knowledge in complex analysis was not available two months after the exam, but the content of the tasks they created was.

4. One needs to clarify and guide how to implement a topic from complex analysis when complex numbers are not available for pupils or shall not be introduced.

5. A possible improvement for XMaSII could be to restructure the day into different sessions or workshops: 1) Introduction to complex numbers and geometry of the complex plane either by the lecturers or some student groups and then 2) working groups dealing with different phenomena from complex analysis.

Figure 1 :

 1 Figure 1: a. Lecture with Y-model and seminar on task design (left), b. The different components that support the task design process (right)

  g. "Plot with GeoGebra the function f(x) = x³ -0.75x² -9x + 1. Then draw the point A(1, f(1)) […] Zoom in the neighbourhood of point A into the drawing. What do you notice? Try to describe your observation." In a later task: "Describe with this magnified image how to calculate the slope in point A approximately (Hint:

Figure 2 :

 2 Figure 2: a. Dilation-rotation of a hexagon (left, from group A's GeoGebra worksheet), b. -ln(1-x) and its sixth Taylor polynomial at 0 (right, from group B's GeoGebra worksheet).

ACKNOWLEDGEMENT

We would like to thank the German Federal Ministry of Education and Research (BMBF) for its support in the subproject "Spotlights Lehre" within the project "Schnittstellen gestalten" at the University of Bremen.


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

Theoretical and empirical description of phases in the proving processes of undergraduates

Keywords: proof construction, proving process, phases, proving cycle

In the presented study we adopt a process-oriented perspective on proving in order to gain further insights into relevant actions and typical obstacles in undergraduates' approaches to proving. The primary aim is to theoretically and empirically describe different phases, understood as bunches of intentionally closely related actions of proving. Therefore, we suggest a theoretical model of the proving process and confirm empirically that it can be used as an analytical tool for proving approaches. Based on this model, several proving processes have been analysed. In this paper we present first findings regarding the contribution of each phase to proof construction as well as the general structure of the proving process.

THEORETICAL BACKGROUND

It is well known that undergraduates commonly have to deal with great difficulties in constructing proofs (e.g. [START_REF] Weber | Student difficulty in constructing proofs. The need for strategic knowledge[END_REF]Moore, 1999). In order to gain additional insights into the main obstacles in proving at university level, we focus on the processes of proof construction. Preparing further projects, the presented study serves as a pilot study, which initially aims at investigating the general structure of proving processes on a macroscopic level. Therefore, we first give an overview about the main characteristic of the proving process in contrast to its product. Based on this, we deduce a theoretical model from literature in order to use this model as an analytical tool for undergraduates' approaches to proving.

The process of proving

Talking about proof construction, there is a consensus that a proof, especially at university level, is a line of reasoning, which is strictly deductive and exclusively based on theorems and axioms. In contrast to its result, the process of proving contains not only deductive reasoning, but also includes inductive and explorative processes (e.g. [START_REF] Hersh | Proving is convincing and explaining[END_REF]. Thus, developing a key idea and transferring it into a formal proof is a complex and highly demanding process, which occurs in the tense atmosphere of conviction and explanation as well as intuition and formality [START_REF] Hemmi | Students' encounter with proof: the condition of transparency[END_REF]. In the present paper we will discuss on a theoretical and empirical basis, how these different perspectives and processes interact in constructing proofs. [START_REF] Mejia-Ramos | Argumentative and proving activities in mathematics education research[END_REF] distinguish between three distinct types of construction activities, which emerge from different external conditions. Although all of these activities aim at constructing an appropriate proof, each is guided by a specific goal: Exploration of a problem consists of working on an open-ended question in order to discover and at least prove a new statement. However, estimation of truth starts with an already prepared conjecture and justification begins with a statement estimated to be true. In these cases, proof construction is rather aimed at determining or verifying the truth-value of a statement instead of inferring it. [START_REF] Meyer | A logical view for investigating and initiating processes of discovering mathematical coherences[END_REF] differentiates activities associated with mathematical proof construction in a similar way based on the theory of abduction, induction and deduction. According to his approach, each kind of inferring is related to a different type of construction activity. Thus, external conditions and, especially, task designs have a relevant influence on proving processes by stimulating different types of inferring and construction activities. Combining both frameworks, we assume that the activities described above are independent for the reason that they consist of different types of processes and pursue different goals. In particular, justification is not a part of exploration, which follows producing a conjecture, but requires specific cognitive processes.

In order to describe those cognitive processes, various models have been developed, which summarise the relevant actions and demands associated with proof construction. These models mainly focus on processes related to the exploration of a problem. As proving tasks at university level often consist of a statement estimated to be true, these models only seem to be partially suitable for analysing undergraduates' proving processes. Hence, we suggest a model of proving, which is mainly following existing models, but focusses on justification.

Models of proving processes

To analyse the cognitive processes of mathematical proof construction, there is a need for abstraction. The complex structure of the process has to be reduced to the relevant actions and, therefore, transferred onto a macroscopic level. Doing so, most process-oriented models use the unit of episodes or phases in order to subsume closely related actions in service of the same goal under a generic activity. Thus, phases are "macroscopic chunks of consistent behaviour" (Schoenfeld, 1985, p. 292), which summarise the relevant processes associated with proof construction.

Existing models of the proving process mainly differ in their amount of suggested phases. The most cited models have been presented by [START_REF] Stein | Beweisen [Proving[END_REF] and [START_REF] Boero | Argumentation and mathematical proof. A complex, productive, unavoidable relationship in mathematics and mathematics education[END_REF]. Both models focus on the activity of problem exploration, which means that the proving process is based on an open-ended problem area. In the first phase of proving this problem area is explored regarding relevant conditions and regularities in order to produce a conjecture. When the conjecture has been formulated as a statement, the proving construction contains three phases: Identifying arguments for the correctness of the statement, linking them into a deductive chain and formulating an appropriate proof. As [START_REF] Stein | Beweisen [Proving[END_REF] model focusses on proving approaches of students at secondary level and [START_REF] Boero | Argumentation and mathematical proof. A complex, productive, unavoidable relationship in mathematics and mathematics education[END_REF] describes the proof construction of mathematicians, quality and formalisation of the proofs intended in the models differ according to the mathematical standards shared in the particular context. [START_REF] Boero | Argumentation and mathematical proof. A complex, productive, unavoidable relationship in mathematics and mathematics education[END_REF] even adds a further phase, in which mathematicians approach a formal proof. Apart from this last step, which primarily seems to be relevant for experts, the phases of proof construction described by [START_REF] Stein | Beweisen [Proving[END_REF] and [START_REF] Boero | Argumentation and mathematical proof. A complex, productive, unavoidable relationship in mathematics and mathematics education[END_REF] have been taken as a basis for several research projects in mathematics education (e.g. [START_REF] Reiss | Learning to prove. The idea of heuristic examples[END_REF]. According to this, we suppose the phases described in both models to be relevant for undergraduates' proving processes as well. However, proving tasks in the initial phase of studies rather initiate justification instead of exploration. Due to this, we suggest the following variation of Stein's and Boero's model:

In contrast to the existing models, the proving cycle starts with a proving task consisting of a statement estimated to be true. Although this statement can be similar to the produced conjecture in other models, it plays another role in the proving process. In case of problem exploration the statement is connected to insights from conjecturing and exploration, but in case of justification no further information about the statement is given. Hence, it is necessary to analyse the given statement, to clarify the terms and conditions and to access previous conceptual or strategic knowledge. Leaning on the approach of a situation model [START_REF] Kintsch | Understanding and solving word arithmetic problems[END_REF][START_REF] Reusser | From text to situation to equation: Cognitive simulation of understanding and solving mathematical word problems[END_REF], we suppose undergraduates to develop a mental representation, which refers to their individual understanding of the given statement and guides further processes. Based on this mental representation we suggest similar phases as presented in the models of [START_REF] Stein | Beweisen [Proving[END_REF] and [START_REF] Boero | Argumentation and mathematical proof. A complex, productive, unavoidable relationship in mathematics and mathematics education[END_REF]. The second phase aims at exploring the statement area and discovering key ideas. By (re-)constructing relations and objects or applying theorems reasons for the validity of the statement can be identified. If some key ideas have been found, it is necessary to select promising ideas, work out their details and structure single arguments in an appropriate deductive order. This phase results in a proof outline, which contains the main aspects of the proof, but can be fragmentary or difficult to understand for someone else. In order to prepare a final proof, which meets the mathematical standards of one's community, one has to fill gaps and revise the linguistic and formal arrangement of the proof. As a last phase the model includes validating activities, which can be compared to a certain extent to Pólya's stage of looking back [START_REF] Pólya | How to solve it[END_REF]. In this phase the final proof is reviewed regarding content, structure and linguistics. Beyond that, one can consider further (shorter or more elegant) proofs in this phase or reflect on the proving process, as, for example, thinking about key ideas, difficulties and their solutions.

The single phases in the model are arranged as a cycle. In accordance with [START_REF] Stein | Beweisen [Proving[END_REF] and [START_REF] Boero | Argumentation and mathematical proof. A complex, productive, unavoidable relationship in mathematics and mathematics education[END_REF] considerations, we assume that proof construction does not necessarily proceed in a linear way. Instead, the proving process is shaped by interruptions and revisions that cause transitions between different phases.

RESEARCH QUESTIONS

The primary aim of the presented study is to develop an analytic tool for proving processes, which makes it possible to describe and compare students' approaches to proof construction on an individual, macroscopic level. Therefore, a proving cycle has been derived from literature consisting of five phases estimated to occur in undergraduates' proving approaches. The unit of a phase seems to be well-suited to describe proving activities in a clear and abstract way without neglecting specific details and differences in the proving processes. Accordingly, the proving cycle may be used to analyse proving approaches by describing the frequency, the duration and the order of different phases occurring in the process. The presented study aims to provide evidence of the utility of the proving cycle as an analytical tool andif soto gain more detailed information about the process of proof construction at the level of phases. In detail, the aims of the study described above lead to the following research question:

1. Is the proving cycle an appropriate tool for analysing proving processes? That means, is it possible to reconstruct the different phases stated in the proving cycle empirically? Do further activities exist, which do not fit the theoretical description?

2. How can the process of proof construction be described in general? That means, which phases are taking a relevant share in the process? In which order do undergraduates go through the different phases? Can the cyclic nature of the proving process be confirmed?

METHOD

In accordance with the open-ended character of the research questions, an explorative laboratory study has been designed. In this study proving processes of undergraduates are initiated, observed and finally analysed. The concept of the study and its conditions are described in more detail below.

Sampling and data collection

The study focuses on undergraduates and pre-service mathematics teachers (high school) attending their first year of studies. Performing an informal unstructured interview, participants are encouraged to work on proving tasks in the field of real analysis. Doing this, they are told to prepare a joint solution that satisfies the requirements of a proof in the initial phase of studies and that would be accepted by a tutor or a lecturer. To encourage the participants to talk about their ideas and approaches, the working processes are organised in pairs. In order to secure the same conditions for all participants and, therefore, the comparability of the observed processes, the interviewer offers no support. Instead, a commonly used textbook of real analysis is provided. For investigation such proving tasks have been chosen that require a one-or two-step proof and can be solved by applying a prominent theorem of real analysis like the intermediate or mean value theorem. In order to explore the proving cycle's applicability, the proving tasks contain universal as well as existential quantifications and can be proved directly or by contradiction.

While working on the proving tasks, the participants are videotaped. That means, their proving approaches are recorded in sound and vision. Additionally, we collected the final solutions as well as the notes of each pair. During the pilot phase of the study seven pairs of undergraduates took part in the interviews and worked each on one or two proving tasks. Processing time for a single task varies from 30 to 75 minutes. However, not all proving approaches were successful. While some participants gave a proof, which was not completely correct, but contained useful approaches, other students could not achieve any solution.

Preparation and analysis of data

To prepare the observed proving approaches for analysis, we transcribed each videotape entirely. For precise investigations the transcripts of the dialog are expanded by further information like non-verbal activities and notes. Combining the transcription of natural conversation and written approaches, the protocols of the proving processes are finally encoded according to [START_REF] Mayring | Qualitative Content Analysis. Foundation, Basic Procedures and Software Solution[END_REF] structuring content analysis. In accordance with the research questions a deductive category formation with nominal categories has been chosen, which is closely related to the proving cycle described above. The aim of the coding is to identify changes in the participants' behaviour in order to describe the structure of their proving process as a sequence of transitions between different phases. Leaning on [START_REF] Schoenfeld | Mathematical problem solving[END_REF] method of protocol analysis the coding consists of two steps: First, the proving process is parsed, that is, making decisions regarding dividing lines of phases. Once a proving process is partitioned into phases, each phase is characterised as one of the theoretical stated phases in the proving cycle. The coding results in a macroscopic description of the students' proving processes that combines closely related actions into phases and provides a summary of relevant activities.

RESULTS

In this section the results of analysing data from nine proving processes is presented.

For the moment, a case study is introduced to demonstrate the methodological approach as well as possible results on an individual level. Based on this, findings regarding the proving cycle being an appropriate analytic tool are discussed in general. Beyond that, we present first assumptions concerning the frequency, the duration and the order of different phases in undergraduates' approaches to proving.

Case study of Michael and Leon

Michael and Leon are working on the following task, which can be solved by applying the intermediate value theorem:

Partitioning their working process and characterising the identified phases, the proving process of Michael and Leon can be described by the sequence of phases presented in Figure 2.

After reading the task, Michael and Leon start their working process with a brief brainstorming concerning useful theorems. Because the given function does not meet the required preconditions, they experience difficulties applying these theorems, which makes them reading the given statement a second time more carefully. Doing so, they express confusion about the term fixed point and, hence, try to clarify its meaning by looking it up and making a drawing. Based on an enhanced understanding, Michael and Leon review their ideas and add new ones by leafing through the book. Sooner or later, each of these ideas proves to be inadequate. While Michael is reading in the book searching for applicable theorems, Leon makes further drawings in order to visualise the preconditions and assertions given in the task. Thereby, he remembers a strategic approach used in a proof before. Leaning on this approach, he tries to construct an auxiliary function, which turns the original problem of the existence of a fixed point into a similar one of a zero. Michael and Leon work together on this approach until they recognise they have different concepts regarding the connection between the domain and the range of the given function. They resolve this disagreement by referring to the drawings and regarding some exemplary points of the function. For more than half an hour Michael and Leon have been working out the details of the proof now. At times they stop working on new ideas and summarise previous insights in order to structure their arguments and to identify gaps or inconsistencies in the proof outline. One of this structuring activities sends Michael having doubts about the correctness of their formulation. Leon is able to convince Michael of their approach, but has to admit that some modifications are necessary like regarding an additional case. Later, another phase of validating occurs as a result of identifying arguments. This time they recognise that their auxiliary function is the identity function and that they can prepare their arguments in a more elegant way. Michael and Leon are now able to construct a satisfying proof by formulating and structuring alternately.

During the whole process of proof construction Michael and Leon are progressing continuously until they have achieved a satisfying proof. They only return to a previous phase, when their proving process comes to a temporary halt and they feel there is a need for reviewing some steps done before in order to specify or improve their mental representation, their key idea or the proof outline. Michael and Leon go through a phase of validating twice, because one of the student voices doubts about the previous considerations being correct. They do not review their final proof to check or improve details.

As displayed in figure 2, at times it has been inevitable to encode the same period of time with two different categories. This kind of double-coding is necessary, if both students work on their own in service of different goals or if some related activities from different phases are done contemporaneously. The analysis of the proving process presented in this section is quite typical for the sample. In the following section similarities and differences between the individual proving processes are discussed in general.

General observations

The analysis of data from nine interviews shows that the categories are applicable to the proving protocols in a satisfactorily objective and reliable way. Interrater reliability in coding is quite high (𝜅 = .73-.93). Therefore, the proving cycle seems to be an appropriate tool for describing and analysing undergraduates' approaches to proof construction on a general macroscopic level. Regardless of the proving tasks and a direct or non-direct approach to proving, each of the suggested phases could be empirically confirmed in at least five of nine cases (Figure 4). In those cases, where the phases identifying arguments and selecting and structuring arguments are missing, participants have not been able to establish any serious approach due to a poor mental representation or a lack of key ideas. However, quitting the working process without any solution was only observed twice. More frequently it happens that even in successful proving processes the phases of formulating and validating are omitted. Instead of formulating a precise and clear proof, some participants content themselves with quick but fragmentary solutions. Here, formulating the proof according to mathematical standards does not seem to be as important as gaining insight into the key ideas of a proof and verifying the statement for oneself. This kind of view might be a general attitude towards mathematical formalism, but could also be caused by the laboratory setting. Analysing the phase of validating, we differentiate two kinds of action: Validating activities in service of reviewing and enhancing a final proof could only be observed in one single case. In contrast, activities like checking details and analysing suggestions occur continuously and are connected to several other activities in the proving process. Although validating activities are only listed in five of nine cases, it is reasonable that more validating takes place without being encoded on a macroscopic level because of no sufficient impact on the proving process. Comparing the percentages of different phases, it emerges that in nearly all cases the phase of identifying arguments as a highly creative and demanding action takes a large share in the proving process and, therefore, seems to be one of the most significant parts of mathematical proof construction. In contrast, the percentage of understanding and exploring the given assertion varies greatly. While some participants spend a lot of time on drawings or clarifications, others start with a brief glimpse on the task and continue with applying theorems immediately.

In regard to the composition of proving processes, it has been assumed that the underlying structure is a cycle. Due to that, there must be a high amount of transitions and revisions in the proving approaches. In fact, only one third of the encoded transitions is linear in that way that students move forward to the next phase in the proving cycle. Though, the proving processes does not proceed as cyclic as suggested. As illustrated in the case study of Michael and Leon a large shape of non- linear transitions is made by transitions between consecutive phases, that is, moving backwards to the phase before. Wider leaps from one phase to another are quite rare. Most of these transitions between non-consecutive phases contain an interaction with validating activities as students switch from any phase to validating and backwards. This finding supports the assumption that validating is an activity, which is closely related to other phases of the proving process. Only in a very few cases the key idea is rejected at some point in the proving process and the participants restart identifying arguments and exploring the given statement cyclically. Accordingly, we suggest proof construction to be less cyclic than a linear process, which is interrupted by several mini-cycles between consecutive phases.

DISCUSSION

In this paper a model for the process of mathematical justification at university level has been derived from literature to develop a macroscopic analytical tool, which describes a proving process as a sequence of phases. In a sample of nine processes the proving cycle has proved suitable for describing undergraduates' approaches to mathematical proof construction. Each of the suggested phases could be empirically confirmed. Analysing similarities and differences between individual proving processes, there have been two key findings: Undergraduates' proof construction mainly proceeds on a straight line basis, which is interrupted at times by transitions into immediately preceding phases in order to specify or improve considerations done before. An exception to this are validating activities. Questioning, reviewing and reflecting seem to be processes, which are rarely performed at the end of proof construction, but are closely connected to other phases of the proving cycle. Hence, initial results indicate that validating is not confined to the final proof, but relates to the mental representation as well as the key ideas and the proof outline.

The presented study prepares a larger project by providing an analytical framework for students' approaches on proof construction. Based on the proving cycle, we intend to analyse a larger sample including participants, who differ in progress and performance. Comparing first-year and advanced students as well as successful and non-successful proving processes might provide new ideas for fostering programs in the introductory phase of studies. On the one hand investigations will remain on the macroscopic level of phases in order to identify effective and less effective patterns of proving processes. Therefore, the occurrence and the duration of a phase in a proving process are compared with the quality of the corresponding proof. On the other hand, further investigations are intended, which gain more inductive insights on a microscopic level. Therefore, we aim at describing typical actions of each relevant phase in detail. By doing so, frequent difficulties and potential obstacles of an individual phase can be identified as well as effective and non-effective proving strategies to overcome these obstacles. This information can help arranging effective, process-oriented fostering programs.

Figure 1 :

 1 Figure 1: Proving cycle as a model for proof construction

Figure 2 :

 2 Figure 2: Parsing of the proving process of Michael and Leon

Figure 3 :

 3 Figure 3: Proving process of Michael and Leon, validating activities are represented by stars

Figure 4 :

 4 Figure 4: Percentages of the different phases occurring in proving processes