7.3.1 Individual strategies and activities during conjecturing and proving
Conjecturing can be considered as a creative work including different experimental activities such as investigating examples and counter-examples, discovering new logical relations between previously unrelated ideas and arguments as well as drafting and formulating conjectures (Yang, 2012). These activities seem to be crucial for a broad range of disciplines, but still they differ across domains. Dissimilarities include, for instance, what counts as a valid conjecture and as supporting evidence (Lin, F. L. et al., 2012). In contrast to other domains such as biology, medicine or politics, the ideal evidence in university mathematics is a chain of deductive arguments based on axioms and definitions (Fischer et al., 2014). Mathematicians agree that empirical, intuitive and authoritative arguments have strong limitations (Weber et al., 2014). Following Stylianides, G. J. et al. (2017), proving is the process of constructing a sequence of arguments for or against a mathematical conjecture that is characterized by using only previously accepted statements, theorems and definitions, valid forms of reasoning and adequate forms of notations. What can be regarded as ‘valid’ or ‘adequate’ is defined by the respective mathematical community and is partially dependent on the specific context of the proof construction process. The proof construction process itself has been described by different researchers in the form of models that are based on theoretical assumptions (e.g., Boero, 1999) or on self-reports (e.g., Schwarz et al., 2010).
Boero (1999) created a process model of proof consisting of various phases that start from exploring the problem situation to formulate a mathematical conjecture and end up in writing down a proof in a readable way that corresponds to the sociomathematical norms. This expert model refers to the assumption of cognitive unity (Garuti et al., 1998), which outlines strong relations between the activities of conjecturing and proving. Schwarz et al. (2010) suggested that three main activities related to proof construction – “enquiring”, “proving” and “inscribing proof” – should be differentiated. The first activity concerns making sense of the problem, establishing conjectures and intermediate hypotheses (subgoals) for the proof (cf. Heinze, Cheng, Ufer, Lin, & Reiss, 2008a). Developing a deductive chain of theory-based arguments that connects the prerequisites with the claim of the conjecture represents the second activity. The third activity includes checking the logical integrity of the proof, and communicating it with formal precision.
These frameworks give some indications about which activities may occur during the proof construction process, and how an ideal proof construction process may look like. Since empirical-inductive and formal-deductive steps are incorporated in both models, we conclude that exploratory activities, checking the consistency between the mathematical concepts
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related to the proof problem and one’s own arguments, as well as unpacking the logical structures of statements are crucial within conjecturing and proving processes.
Exploratory activities such as generating examples (e.g., Koedinger, 1998), reflecting on familiar problems, and associating similarities between them (Selden, A. & Selden, 2013a) can help students to understand the problem, find a conjecture or devise a plan for solving the task (e.g., Mills, 2014; Polya, 1945). Sandefur et al. (2013) the strategy of creating examples in university students’ efforts to prove or refute a mathematical conjecture. In their study, students worked in small groups on number theoretic tasks. Although most groups tried to find examples that would provide some conceptual insight into the structure of a statement, example use varied among the groups within problems. The authors assumed that the use of example- based reasoning strategies depends on students’ experience, their personal example space, and the way problems are presented. Drawing diagrams can be considered as a further exploratory activity that might help discovering new ideas or getting empirical evidence for the truthfulness of an argument (Gibson, 1998). For some mathematicians, the exploration of examples or visual arguments is an essential part in coming to understand new concepts and to justify new theorems, while others construct proofs that are entirely based on the manipulation of symbols within the representation system of the given problem (Alcock & Inglis, 2008). Certainly, it is an advantage to be able to use both strategies, but many students and also some professionals obviously prefer one type of reasoning (Zazkis et al., 2015).
The consistency between the mathematical concepts that are related to the given proof problem and one’s own ideas constitutes a central aspect within the proof construction process (Mariotti, 2006). This means that students have to connect the formal definitions of the concepts to the instantiations they use for their argumentation. The mathematical objects that make up their arguments may only have properties that conform with the formal theory (Weber & Alcock, 2004). To achieve the required consistency, activities such as using different representations (Boero, 1999; Ufer et al., 2009) or operable definitions (Selden, J. et al., 2014) may be helpful. These types of activities may depend on students’ individual conceptual understanding that, in turn, enables them to monitor their performance and to identify their own impasses (Ohlsson & Rees, 1991).
Selden, A. et al. (2010) claimed that the enactment of “behavioral schemas” (p. 205), which are partly procedural knowledge, affect the use of logical structures. Unpacking the logical structure of a statement can be considered as a first important step within the proof construction process, since “the logical structure of a mathematical statement is closely linked to the overall structure of its proof” (Selden, A. & Selden, 2008, p. 105). Inferring from their observations of 61 students participating in a university preparatory course, they suggest that it may be a promising strategy to unpack the conclusion by writing down a proof framework
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before getting started with the problem-centred part of a proof (Selden, J. & Selden, 1995). Other researchers emphasized the importance of analysing and using the structure of mathematical arguments in the context of learning proofs (Knipping, 2008; Pedemonte, 2007), as well.
Though all of these individual-mathematical activities and aspects may be useful for students to find a conjecture and to establish the validity of it, they do not necessarily lead to an interesting conjecture and an accepted proof. Until now, there is still limited knowledge about which activities actually predict the quality of the resulting proof.
7.3.2 Collaboration in mathematical conjecturing and proving
It is a common approach to treat proving as a cognitive activity employed by individuals with the aim to verify the correctness of a mathematical statement or to gain insight into why it is true is a common approach (e.g., Villiers, 1999). However, generating conjectures and developing proofs can also be considered from a social-discursive perspective. This is consistent with the view that conjectures are provided for the reflection of other mathematicians (and learners), sharing ideas and discussing arguments for or against them (Alibert & Thomas, 1991). What is accepted as a proof depends on the social context (Thurston, 1994) and on different criteria defined by the mathematical community (Stylianides, G. J., Sandefur, & Watson, 2016) and to be acquired in mathematical discourse. Within this perspective, proof might be seen as a “means of convincing oneself whilst trying to convince others” (Alibert & Thomas, 1991, p. 215). Participating in argumentative dialogues requires the ability to justify and explain a claim to peers on the one hand, as well as to interpret ideas of the speaker and to give feedback for correctness on the other hand. One challenge is to develop a “common ground” (Clark, H. H. & Schaefer, 1989). For instance, interacting with peers demands establishing a shared perception of what is recognized as a claim, an inferential rule or a given fact (Yackel & Cobb, 1996), building common frames of reference and resolving discrepancies in understanding (Barron, 2000). A very basic indication of successful collaboration is that students’ conversational turns built upon each other (e.g., Chi & Wylie, 2014; Roschelle & Teasley, 1995; Vogel et al., 2016). These so-called transactive (Teasley, 1997) or interactive (Chi, 2009) activities are attributed a high potential for fostering domain-general argumentation skills and deepening conceptual knowledge (Asterhan & Schwarz, 2009). According to Blanton and Stylianou (2014), interactive reasoning might also be seen as a discourse tool by which students can improve their proof understanding as well as their strategic knowledge in constructing proofs. Especially, criticizing or integrating the learning partner’s utterances are considered as interactive activities that are likely to trigger deep cognitive processes (Vogel et al., 2016). Therefore, the extent to which students monitor each other’s utterances, integrate divergent interpretations and, finally, make decisions together can be seen as an influential
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factor that may explain the variability of outcomes in collaborative math-problem-solving tasks (Barron, 2000; Clark, K., James, & Montelle, 2014).
Several studies have demonstrated that, on average, collaborative work may lead to better learning outcomes than individual engagement (cf. Barron, 2000; Cohen, 1994; Johnson, Johnson, & Smith, 2007), but this is not happening automatically. In unstructured learning environments (without any guidance), collaborators often tend to engage in low-level argumentation processes (e.g., Kollar et al., 2007), for instance, they rarely relate explicit evidence to their explanations (e.g., Sandoval, 2003). We conclude that students may benefit from collaboration and that learning occurs through interaction with peers when an atmosphere that enhances productive collaboration can be reached. Furthermore, we take the view that mathematical proving is at least partially a social activity and thus, are interested in understanding the relation between, and effects of, social-discursive and individual- mathematical characteristics of conjecturing and proving processes.