Educational Studies in Mathematics

, Volume 96, Issue 2, pp 145–167 | Cite as

Analysis of the cognitive unity or rupture between conjecture and proof when learning to prove on a grade 10 trigonometry course

Article

Abstract

We present results from a classroom-based intervention designed to help a class of grade 10 students (14–15 years old) learn proof while studying trigonometry in a dynamic geometry software environment. We analysed some students’ solutions to conjecture-and-proof problems that let them gain experience in stating conjectures and developing proofs. Grounded on a conception of proof that includes both empirical and deductive mathematical argumentations, we show the trajectories of some students progressing from developing basic empirical proofs towards developing deductive proofs and understanding the role of conjectures and proofs in mathematics. Our analysis of students’ solutions is based on networking Boero et al.’s construct of cognitive unity of theorems, Pedemonte’s structural and referential analysis of conjectures and proofs, and Balacheff and Margolinas’ cK¢ model, while using Toulmin schemes to represent students’ productions. This combination has allowed us to identify several emerging types of cognitive unity/rupture, corresponding to different ways of solving conjecture-and-proof problems. We also show that some types of cognitive unity/rupture seem to induce students to produce deductive proofs, whereas other types seem to induce them to produce empirical proofs.

Keywords

Ck¢ model Cognitive unity of theorems Conjecture-and-proof problems Learning conjecture and proof Structural and referential analysis Toulmin scheme Trigonometry 

1 Introduction

In some national curricula, an objective of upper secondary school mathematics is to start learning deductive proof. Numerous studies report interventions aimed at teaching proof at secondary school (Harel & Sowder, 1998; Mariotti, 2006). Several frameworks are used to analyse the processes of learning to prove. Duval (1991) identifies a cognitive rupture between argumentation and proof that would explain the difficulties students have in understanding and making deductive proofs. Other researchers (Balacheff, 1988; Harel & Sowder, 1998; Marrades & Gutiérrez, 2000) focus on identifying types of empirical and deductive proofs produced by students that allow students’ progress in learning to prove to be assessed.

Some authors identify and explain the reasons for students to be or not to be able to complete deductive proofs of their conjectures (Arzarello, Micheletti, Olivero, & Robutti, 1998; Boero, Garuti, Lemut, & Mariotti, 1996; Pedemonte, 2002). Researchers also focus on analysing the learning of specific types of deductive proofs (Antonini, 2003; Antonini & Mariotti, 2008; Stylianides, Stylianides, & Philippou, 2007). Detailed discussions of these issues can be found in Mariotti (2006), Reid and Knipping (2010), and Hanna and De Villiers (2012).

Conjecture-and-proof problems are a key element in learning to prove. Their solution is divided into two parts: first, to state a conjecture, and second, to prove that the conjecture is true. Several researchers have analysed the processes of solving these problems to understand the relationship between students’ reasoning in both parts. Boero et al. (1996) propose the construct of the cognitive unity of theorems to identify cognitive aspects that come into play during the production of conjectures and the construction of proofs. Pedemonte (2005) proposes referential and structural analysis of argumentations and proofs, and an analytical tool based on a combination of the cK¢ model (Balacheff & Margolinas, 2005) and the Toulmin scheme (Toulmin, 2003). We describe these constructs and the way we have networked them to analyse students’ productions “from different theoretical perspectives as a method for deepening insights into the phenomenon” (Bikner-Ahsbahs & Prediger, 2014, pp. 119–120).

In this paper, we present a teaching experience fitting the requirements stated by Stylianides and Stylianides (2013) for a classroom-based intervention (research based, in ordinary classrooms, with close collaboration between teachers and researchers, addressing problems of learning mathematics, producing solutions to those problems, explaining how/why things worked successfully). The intervention took place in a Colombian grade 10 classroom (students aged 14–15 years) whose aims were: first, to design and administer a teaching unit of trigonometry, and second, to provide students with a suitable environment to begin learning proof. The intervention had a problem-solving style and was based on a dynamic geometry software (DGS) environment in which students conjectured and proved new properties.

Research has demonstrated that DGS environments are helpful for teaching the different areas of school geometry and in teaching proof (e.g., Laborde, Kynigos, Hollebrands, & Strässer, 2006). Pratt and Noss (2010) conclude that the design of activities for teaching mathematics based on DGS environments can be made in terms of several design heuristics, one of which, particularly pertinent to our research, is to “enable the testing by children of their personal conjectures” (p. 95). Several authors have analysed the role of dragging in students’ discovery of properties or relationships and the production of proofs (Arzarello, Micheletti, Olivero, Robutti, Paola, & Gallino, 1998; Leung, 2011). We agree with Mariotti (2001) that dragging also induces a specific criterion for the validation of conjectures, namely that a property or relationship is true when it is stable under a dragging test in the figure on the screen. This powerful characteristic of DGS and dragging may be an obstacle to students in moving from empirical to deductive proofs; thus, researchers have experimented with teaching methodologies to avoid such obstacles. One way of doing it, that we have used in our classroom-based intervention, is to ask students to use theoretical statements of properties or definitions as part of their proofs (Marrades & Gutiérrez, 2000).

The DGS environment was a key element in the intervention we designed, as described in Sect. 3. In this paper, however, we are not focusing on its role during the classroom-based intervention, but on the types and structures of conjectures and proofs produced by students and the relationships between their conjectures and proofs.

The specific research objective of this paper is to analyse the evolution of the proofs produced by two pairs of students throughout the classroom-based intervention. To accomplish such an analysis, we have looked for information on two interweaved critical questions on conjecture-and-proof problem solving:
  1. 1.

    In what way does the kind of connection (or lack of it) between the production of a conjecture and the construction of its proof affect the type of proof (empirical or deductive) produced?

     
  2. 2.

    How can we assess whether there is an evolution in the types of proofs produced by students throughout the classroom-based intervention?

     

To organise information to answer question 1, we present a model for analysing such connections based on an integration of the construct of cognitive unity (Boero et al., 1996) into the analytical tool proposed by Pedemonte (2005) and Pedemonte and Balacheff (2016). To identify a possible evolution in the types of students’ proofs, we have used categories of proofs elaborate on by Marrades and Gutiérrez (2000).

2 Theoretical framework

In this section, we define the components of the theoretical background of our research. Conjectures raised and proofs produced by the students are the central elements, alongside the discourses elaborated to reflect upon conjectures (argumentations) and to convince of their truth (proofs).

2.1 Argumentations and proofs

An argumentation is a discourse consisting of a sequence of verbal statements based on mathematical elements (definitions or properties, results of experiments, observations, etc.), organised with the aim of explaining how a conjecture was identified or convincing that it is plausible. Pedemonte (2002) differentiates two types of argumentations corresponding to these two aims: constructive argumentations, which contribute to the statement of conjectures, and structuring argumentations, that contribute to justifying the plausibility of conjectures previously raised.

In this paper, like Balacheff (1988), Harel and Sowder (1998), and Maher (2009), we consider a proof to be any mathematical argumentation raised to convince oneself or others of the truth of a mathematical statement. In common with those authors, we differentiate between:
  • Empirical proofs, characterised by showing that the conjecture is true only in one or a few examples taken from a larger set of examples and assuming that the conjecture is also true in all other examples in the set.

  • Deductive proofs, consisting of chains of logical implications connecting the hypothesis to the statement of the conjecture, and characterised by the decontextualisation of the ideas presented.

To analyse the whole process of solving conjecture-and-proof problems, it is necessary to relate argumentations and proofs produced by students. Then, as for proofs, we differentiate between:
  • Empirical argumentations: are based on the observation or manipulation of particular cases to identify a property or fact (the conjecture) that, after having been inductively assigned to the whole set of cases, could be worded as a general statement that is an answer to the problem.

  • Deductive argumentations: consist of some form of deductive chain that, unlike deductive proofs, can be stated by using natural language and may not be supported by a mathematical theory.

2.2 Types of proofs

We have used the types of proofs in Marrades and Gutiérrez (2000) to identify the proofs produced by our students and to assess the evolution of their abilities of proving. The types of proofs produced by our students are:
  • Naïve empiricism: show that a conjecture is true in a few examples, usually selected without specific criteria.

  • Generic example: show that a conjecture is true in a specific example presented as a characteristic representative of its class. The proofs include attempts to transform properties observed in the example into abstract properties of the whole class.

  • Thought experiment: consist of a chain of deductive statements organised with the help of specific examples.

  • Formal: consist of a chain of deductive statements organised without the help of specific examples and based solely on the hypothesis of the problem, axioms, definitions or accepted theorems.

These types of proofs are useful for assessing students’ progress in learning proof as they form a hierarchy, because, to move from one type to a higher one, students must achieve a higher level of internalisation and decontextualisation of reasoning (Balacheff, 1988).

2.3 Cognitive unity of theorems

The solutions of conjecture-and-proof problems may be seen to consist of two phases, conjecturing and proving. These phases are not necessarily sequential, because sometimes they appear interweaved. Boero et al. (1996) propose the construct of the cognitive unity of theorems to explain possible reasons for the success or failure of students at writing deductive proofs: there is cognitive unity when the argumentations used to produce and validate a conjecture help to construct its deductive proof. Otherwise, there is a cognitive rupture. In Boero et al. (1996), “proof” means deductive proof.

When students produce empirical argumentations about a conjecture, to include them as parts of the chain of a deductive proof, students must decontextualise the argumentations to transform them into abstract deductive argumentations (logical implications). Our students had never had contact with deductive proofs before this classroom-based intervention; thus, we cannot expect them to produce deductive proofs from the very beginning. During the first classes, they were only able to produce empirical proofs showing that the conjecture was true in specific examples. If we only consider deductive proofs, we should discard many empirical proofs and we would lose rich information about the factors that helped or hindered our students to begin to understand deductive proof and to learn to produce their first deductive proofs. Therefore, we have expanded the meaning of cognitive unity to empirical proofs: There is cognitive unity when the argumentations used during the conjecturing phase help students to construct a proof, either empirical or deductive. This extension made it possible to analyse the relationships between conjectures and empirical proofs and identify reasons for students having failed to convert empirical argumentations into deductive proofs. This extension of meaning makes it possible to analyse the process of learning to prove from the very beginning.

Some researchers identify the emergence of abductive reasoning in the transition from the conjecturing phase to the proving phase (Boero, Douek, Morselli, & Pedemonte, 2010; Pedemonte, 2002). This has not been the case for our students, because the problems posed included all necessary hypotheses; thus, students did not have to look for other reasons for their conjectures to be true.

2.4 A tool for analysing the cognitive unity

Pedemonte (2005) identifies several components of students’ activity that are central to understanding why there is cognitive unity or rupture: the structures of argumentations and proofs, and the mathematical systems of reference used by students. Pedemonte proposed two kinds of analysis of construction of an argumentation about a conjecture and a proof of that conjecture:
  • Structural analysis: refers to the link between the structures of statements used in argumentations and in proofs. There is structural cognitive unity when statements used in the argumentation are also used in the proof. Otherwise, there is structural cognitive rupture.

  • Referential analysis: refers to the systems of reference used in argumentations and in proofs, that is, the systems of signs (drawings, calculations, algebraic expressions, etc.) and systems of knowledge (definitions, theorems, etc.) used. There is referential cognitive unity when some systems of signs or knowledge are used both in the argumentation and the proof. Otherwise, there is referential cognitive rupture.

This analysis needs to take into consideration the different elements used by students to state and verify conjectures and to produce proofs, which are, among others, students’ mathematical knowledge (concepts, properties, relationships, systems of signs) used as warrants for their argumentations and proofs, logic operators connecting elements of the argumentation, and systems of representation used to express their productions. These elements must be backed by a coherent structure (Pedemonte & Balacheff, 2016). The cK¢ model (Balacheff & Margolinas, 2005) may be such a structure, as it has been successfully used in making referential analysis (Pedemonte, 2002, 2005).

The cK¢ model defines a conception as a vector C = (P, R, L, Σ), where P is a set of problems, R is a set of operators that allow processing the problems, L is a system of representation that allows problems and operators to be represented, and Σ is a structure of control organising decisions, choices, value judgements and suitability of actions (for instance, dragging). The operators help students to transform the problems. Typical operators in conjecture-and-proof problems are “if … then” and other expressions characteristic of deductive proofs. The system of representation contains the different systems of signs used by students to express their productions; the most frequent systems of signs in secondary mathematics are verbal, algebraic, numerical, and geometrical. The structure of control may be concrete, like a sketch on paper, a Cabri drawing or a dragging action, or theoretical, like a set of axioms, definitions and properties.

The Toulmin scheme (Toulmin, 2003) was created to represent a single deductive step, although empirical or abductive steps may also be represented (Boero et al., 2010; Fiallo, 2011). In this paper, we have tried to advance over previous ways of using the Toulmin scheme by representing into a single scheme a sequence of students’ utterances, i.e., a sequence of steps in their argumentations of conjectures or their proofs. This provides researchers with a synthetic view of the whole solution. To perform a complete analysis of students’ answers, it is necessary to include in the scheme information about the warrants and the backing used by students, in particular, about the systems of reference of their argumentations or proofs, which is offered by the cK¢ model. The conception C = (P, R, L, Σ) is the backing, because it includes the different elements used by students to identify and validate conjectures and to develop proofs. Figure 1 shows the Toulmin scheme integrating the cK¢ model (Pedemonte, 2002, 2005). When there is cognitive unity, the operators in the conception C are the warrant, because, if they are incorporated into the empirical argumentations, they ensure the transformation of an empirical argumentation into deductive steps.
Fig. 1

The Toulmin scheme integrating the cK¢ model

The networked framework (Bikner-Ahsbahs & Prediger, 2014) that we present in this paper is aimed at opening the construct of cognitive unity to empirical proofs by presenting a more realistic conception of the process of learning to prove, and a more useful analytical tool for mathematics education researchers. This way of combining cognitive unity, the analytical tool proposed by Pedemonte, and the use of a single Toulmin scheme to represent the whole phase of conjecturing or proving, allows researchers to make a detailed analysis of students’ solutions and obtain fine-grained information, even when they produce empirical proofs.

3 Research methodology

In this section, we describe the main components of the classroom-based intervention, namely the content of the teaching unit, the role of teacher and researcher, the sample of students, and our procedure for analysing students’ productions.

3.1 The teaching unit

The Colombian curriculum (MEN, 2006) includes the teaching of trigonometry in grade 10. The curriculum does not specify a methodology for teaching trigonometry nor does it suggest ways to connect the different representations of the trigonometric concepts, and most teachers use textbooks that emphasise teaching trigonometry by memorisation and procedures. Our classroom-based intervention was based on the content specified by the Colombian curriculum and a problem-solving approach supported by a DGS environment. The objectives were teaching the prescribed contents of trigonometric ratios and promoting the understanding of proof (Fiallo, 2011). To attain these objectives, from the very beginning, the teacher asked students to justify the veracity of their conjectures, and to write proofs for them. The teaching unit consisted of 32 conjecture-and-proof problems, organised into four topics: ratios in a right-angled triangle, ratios of standard angles in the unit circle, reference angles and other special angles in the unit circle, and the Pythagorean identity. After each topic, the teacher led a whole-group discussion of some students’ solutions and institutionalised new knowledge.

The teaching methodology was guided-discovery, based on conjecture-and-proof problem solving supported by a DGS environment. The statement of each problem explicitly asked the students to state a conjecture and to write a proof for it. Each problem included a Cabri II file containing a figure with a geometric representation of the statement. Students had to neither modify the Cabri figures nor create new ones. Their interaction with the DGS consisted of opening the files and dragging the points marked in the figures to modify angles and then visualise, explore and analyse trigonometric relationships and properties to, eventually, raise and state conjectures that could answer the questions stated in the problems (Fig. 2, 5, 10). Students could also use the DGS to verify the validity of their conjectures and to produce their proofs. Therefore, the DGS was a facilitator, because the figures were designed to make explicit the relationships or properties being studied.
Fig. 2

Cabri figure for problem 4

3.2 The sample

The sample was a group of 17 grade 10 students (aged 14–15 years) in a secondary school in Floridablanca (Santander, Colombia). The sessions took place during the ordinary classroom schedule, and were conducted by the teacher of the group. The students worked in pairs (and a group of three) with a computer running Cabri for each group. The intervention took place over 4 months, with two 90-min sessions per week in the computer room and a 45-min session per week either in the regular classroom or in the computer room. The first author was present in the classroom as a participant observer, noting students’ activity, answering their questions, or inquiring about their work. The teacher had taught these students in grade 9 with Cabri; thus, teacher and students had good previous knowledge of the software. Students’ previous knowledge of geometry included the sum of the angles of triangles, similarity and congruence of triangles, and the Pythagorean and Thales theorems.

To collect information on students’ productions and their activity during the solution of the problems, two pairs of students (G1 and G2) were video-recorded during all the sessions. These students were chosen because they had an average academic level, were very participative and used to ask questions, and they consented to being recorded. Group G1 included two girls, Diana and Mapa, and group G2 included two girls, Mabe and Cata (all names are pseudonymous). The classroom was video-recorded during the whole-group sessions and the worksheets delivered by all groups were collected.

3.3 Analysis of the characteristics of students’ solutions

We followed these steps to analyse students’ solutions:
  1. 1.

    We identified operators, systems of representation, structures of control, types of argumentations, and types of proofs. We synthesised this information in Toulmin schemes representing students’ elaboration of conjectures or proofs.

     
  2. 2.

    Referential analysis: we compared the components of the cK¢ model put to work by the students in the conjecturing and proving phases. We specified, for each phase, the components mentioned in step 1, so that we could identify the referential unity or rupture.

     
  3. 3.

    Structural analysis: we looked at the type of argumentation made to get the conjecture and the structure of the conjecture itself, and we also identified the type of proof. By comparing the data about conjectures and their proofs, we could identify the structural unity or rupture.

     

4 Analysis of students’ productions

Section 4 presents a detailed analysis of students’ behaviour, represented by two pairs of students and four of their solutions corresponding to the different methods of cognitive unity or rupture that we have identified and described. In Sect. 4.14.4, we analyse the four solutions that we have selected. The first case is characterised by the structural and referential cognitive unity between an empirical conjecture and a naïve empiricism proof. The second case is characterised by the structural cognitive unity and the referential cognitive rupture, due to a change in the system of reference. The third case is characterised by the structural rupture and the referential unity, because students stated a conjecture by empirical analogy with the solution to another problem, but then they tried to produce a deductive proof. The fourth case is characterised, like the first one, by the structural and referential cognitive unity, but now the unity is between a structuring argumentation of the conjecture, supported by deductive chains of statements, and the construction of a thought experiment proof. In Toulmin schemes, we use dashed/continuous boxes to visually distinguish between empirical and deductive argumentations or proofs.

In Sect. 4.5, we present a summary of the solutions to all the problems we have analysed produced by the two pairs of students selected. This allows us to identify their trajectories of learning to prove along the classroom-based intervention. Although a pair of students advanced towards the consistent production of deductive proofs, the other pair was not able to make the same progress, as they produced empirical proofs in most of the problems.

4.1 Case 1: empirical cognitive unity

This problem is based on the Cabri figure shown in Fig. 2. Students can modify ∠A and leg BC by dragging point B along line CB, and modify legs AC and BC by dragging point C along ray AC. The first problems posed asked students to “explain why” instead of “prove it” as an introduction to the meaning of the term “prove”.

4.1.1 Search for a conjecture

The numerical information provided by dragging actions (D 1) allowed G1 to observe that ∠B decreases when ∠A increases. The students stated this relationship as a conjecture (C 1) justified by a reasoning, leading to a constructive argumentation.

[1] G1: [after dragging for a while] When the angle A is decreasing, the angle B is increasing, and the same will happen to all.

[2] Diana: So, A and B are inversely proportional, that is, when one increases, the other decreases. Let’s write it.

Students’ production of this conjecture is presented in Fig. 3. It shows a numerical–perceptive conception (B 1) characterised by an operator (W 1) based on the values of ∠A and ∠B displayed on the screen (D 1), which warrants, for G1, the truth of the conjecture. The operator (R) used in the conjecturing phase was a generalisation of data observed in Cabri. Their systems of representation (L) were the Cabri figure and natural language. G1’s conjecture is true (although it was wrongly verbalised), and it was supported by a constructive argumentation. The control (Σ 1) was exerted by dragging.
Fig. 3

Toulmin scheme for G1’s argumentation of their conjecture in problem 4

4.1.2 Construction of a proof

After having stated the conjecture, students spent some time looking for ideas to organise a proof:

[4] Diana: So when BC is zero, the angle …, is at 89 … I mean, look, B is 89.8 …

[10] Diana: When BC increases, angle A increases and angle B decreases.

[11] Diana: When AC is constant, then angle A must be linked to this side and when A is greater than the value, I mean BC, it causes the value of A to increase, then A will be greater than B.

[15] Diana: No, look, when BC was smaller than AC, the … angle B is greater than A.

[16] Mapa: Angle B is equal to angle A [she meant when AC = BC] … Hey, that was the conjecture, wasn’t it?

[17] Diana: I do not know if this is conjecture or proof, but it is like a proof.

[18] G1: Look, the relationship is like … when BC is smaller, as AC is constant, … when BC is smaller than AC, then angle B will be greater than A.

[19] G1: When they are equal … umm … look, then … the angles are equal.

[20] G1: And, when BC is greater than the value of AC, because it is constant, then angle A becomes greater than angle B, and it starts to decrease.

[21] [The students write in their worksheet]: The relationship between sides AC and BC and anglesA andB is that when side BC is smaller than side AC, ∠B is greater thanA. However, when the value of BC is greater than AC, angleA is greater thanB.

G1’s production of this proof is presented in Fig. 4. Students were not able to correctly prove the conjecture, but their attempt had the style of a naïve empiricism proof, because it consisted of explanations based on numerical values of sides and angles observed in non-systematic examples in Cabri. G1 expressed their observations by using geometric terms, but they did not feel that they had proved the conjecture ([16]–[17]) and they experienced difficulties in verbalising coherent statements ([18]–[19]); thus, the qualifier (Q 1) is weak. The warrant consisted of W 2 to W 5, resulting from numerical data observed in Cabri. The reference framework (B 2) backing the proof was a numeric–perceptive conception. The operators (R) used in the proving phase were a generalisation of the data observed in Cabri and statements of relationships by using “when … [then] …” wording. The systems of representation (L) were observation of the Cabri figure and natural language. The controls were dragging (Σ 1) and checking the conjecture numerically (Σ 2).
Fig. 4

Toulmin scheme for G1’s proof in problem 4

Comparing the Toulmin schemes in Figs. 3 and 4, we observe that there was cognitive unity: there was empirical referential unity, because backing, operators, systems of representation, and controls used in the conjecturing phase were also present in the proving phase. There was also empirical structural unity, because both the argumentation for explaining the conjecture and the proof were empirical. Students tried to write a sequence of valid properties, but they could not prove the conjecture.

4.2 Case 2: referential rupture and empirical structural unity

This problem is based on the Cabri figure shown in Fig. 5. ∠A and ∠(−A) are modified by dragging point P around the circle. G2’s solution is quite complex, because it includes three attempts to prove the conjecture. This shows the power and usefulness of the analytical tool we are using.
Fig. 5

Cabri figure for problem 15

4.2.1 Search for a conjecture

After having dragged point P for a while, one of the students noted that ∠A and ∠-A have the same cosine. The students stated this relationship and verified it by the empirical checking of some examples.

[2] Mabe: I don’t know. Cosine doesn’t change neither in the first nor the fourth [quadrants].

[6] Mabe: Look, make cosine of 30 and then cosine of minus 30, and you will see what you get, because, if you have 30 here and 30 here [pointing to angles of 30° and −30°], the cosine is positive here and here. So it is x over r, and it is x here, let’s say 150 and minus 150 …

[8] Mabe: What is the conjecture?

[11] Cata: Sorry. Cosine of A is equal to cosine of minus A. Let’s see another example, this is the same, but with cosine and without the minus. Cosine of A is equal to cosine of minus A.

Figure 6 represents the argumentation of the conjecture. Warranted by a generalisation of the examples explored in Cabri (W 1), and recalling the definition of cosine [6], G2 raised their conjecture (C 1) and justified it with a constructive argumentation. The operator (R 1) used was generalisation of the data observed in Cabri. They used algebraic language (L 1). The control was numerical (Σ 1), based on the examples observed, and theoretical (Σ 2), based on the data (D 1) and the definition of cosine (D 3).
Fig. 6

Toulmin scheme for G2’s argumentation of their conjecture in problem 15

4.2.2 Construction of a proof

This analysis is divided into three parts, because students made three attempts to write a proof. A main reason for this behaviour is the students’ lack of confidence in their own productions.

[13] Cata: I don’t know why. How is that of …? It is x over r, right? [She wrote on the worksheet: \( \begin{array}{c}\hfill \cos\ \left(\mathrm{A}\right)= \cos \left(-\mathrm{A}\right)\hfill \\ {}\hfill \frac{\mathrm{x}}{\mathrm{r}}=\frac{\mathrm{x}}{\mathrm{r}}\hfill \end{array} \)]

[14] Mabe: That cannot be the proof. The proof, I think it is … because … um …

This attempt at proof is presented on the left side of the Toulmin scheme in Fig. 7. It was a deductive proof, because it is based on previous content knowledge. However, the force of their argumentation (Q 1) was weak, because students were not convinced of their proof ([14]); thus, they continued working:
Fig. 7

Toulmin scheme for the three parts of G2’s proof in problem 15

[24] Researcher (Res.): Right. How did you realize that [the conjecture was this relationship]?

[25] Mabe: As the angles start in the positive x-axis, going clockwise or against, then the cosine is positive in the first and fourth quadrants. Let’s say, if you have an acute angle of 70 [degrees], then it [its cosine] will be positive, and if you have one measuring minus 70 in this quadrant, its cosine will be positive too.

[27] Mabe: Because the x is on the same side.

[28] Res.: What if the angle is greater than 90?

[29] Mabe: It’s the same. Both will be negative because here [pointing to second and third quadrants on the screen] are the negative cosines.

G2’s second attempt at proof is presented in the centre of Fig. 7. It is based on examples, and students felt more confident than before, although their conversation with the researcher led them to a third proof:

[31] Cata: Is it so? [pointing to the worksheet, see [13]]. She [the teacher] told me that it could not be verified in this manner. The other one is very obvious.

[32] Res.: What is obvious?

[33] Mabe: Cata doesn’t know why in the first and fourth quadrants cosine is positive and, in the two others, it is negative, so …

[34] Cata: It is due to the x coordinate, because in these two quadrants [pointing to quadrants I and IV], the x is positive, and in these two [pointing to quadrants II and III], the x is negative.

The third attempt at proof is presented on the right side of Fig. 7. The force of the argumentation (Q 2) was weak because students were not sure of their proof. A global view of the Toulmin diagram shows that warrant W 2 was theoretical, based on the definition of cosine, and warrants W 3 to W 8 were perceptive, based on the data observed in Cabri (W 5 is wrong, because it is only valid for ∠A < 180°). The operators of the conception (R 2), expressed in algebraic and natural languages, consisted of dragging and the calculation of examples. The reference framework backing the proofs (B 1 to B 3) changed along the construction of the proofs. G2 used algebraic language (L 1) and natural language (L 2). The control was theoretical (Σ 2) for the first attempt at proof, but it moved to a numeric perceptual control (Σ 3) based on Cabri. The overall process of proving was a generic example proof, because students tried to get a deductive proof from an inductive generalisation of properties identified in Cabri.

Comparing the Toulmin schemes in Figs. 6 and 7, we note that there was empirical structural unity because group G2 mainly worked empirically all the time. However, there was a referential cognitive rupture, owing to the use of new operators of the conception (L 2) and a change in the control (Σ 3) in students’ attempts to produce a deductive proof, although they only produced a generic example proof. Anyway, this proof was clear progress in relation to the proofs G2 had produced in previous problems (see Table 2).

4.3 Case 3: structural rupture and referential unity

This case presents group G1’s solution to problem 15 (Fig. 5). We can see that these students approached the solution in a way that was very different from group G2 (case 2 above).

4.3.1 Search for a conjecture

The students had solved problem 14, which asked for a relationship between sin(A) and sin(−A). Now, after reading the statement of problem 15, they immediately, without paying attention to the computer, state a conjecture:

[1] Mapa: What is the relationship between cos(A) and cos(−A)? Isn’t it the same?

[2] Diana: It is the same, but with x, the same. We use absolute values.

[3] G1A: [they wrote in their worksheet] They have the same absolute value. That is, |cos(A)| = |cos(−A)|.

​Figure 8 represents the argumentation of the conjecture. The students obviated the steps of exploration of the DGS figure, because the statement and figure of problem 15 reminded them of those of problem 14; thus, they resorted to an analogy with the former problem, which was their warrant (W 1), and they implicitly used an empirical argumentation by generalising that process of solution. They had a theoretical control (Σ 1) and used algebraic language (L 1).
Fig. 8

Toulmin scheme for G1’s argumentation of their conjecture in problem 15

4.3.2 Construction of a proof

The phase of proof was very short, based on the definition of cosine for ∠A and ∠(−A):
Fig. 9

Toulmin scheme for G1’s proof in problem 15

[4] G1A: Cosine of A, is it x over r? … x over r, then, cosine of (−A) is -x over r.

[5] G1A: If cosine of A is -x over r, then cosine of -A is x over r. The same explanation? [The values of] x for angles A and -A are inverse.

[6] G1A [They wrote on the worksheet]: If cos(A) = \( \frac{x}{r} \) ⇒ cos(−A) = \( \frac{- x}{r} \)

[7]If cos(A) = \( \frac{- x}{r} \) ⇒ cos(−A) = \( \frac{x}{r} \)

[8]x for angle A and angle -A are inverse.

Students' proof is presented in Fig. 9. As the conjecture was raised by analogy with procedures and results from the previous problem, without any empirical exploration in the DGS figure provided, there was referential unity. There was also a structural rupture because there was a change from an empirical argumentation (W 1) in the phase of conjecture to a deductive argumentation (W 2 to W 4) in the phase of proof. However, this rupture did not result in a correct proof, because the students based their proof on incorrect warrants (W 2 to W 4).

Although the proof that the students wrote down was wrong, it was like a thought experiment, because they had in mind the examples handled in the previous problem, but it was detached from specific examples. Group G1 continued using the algebraic system of representation (L 1), although it was combined with natural language (L 2). They also continued exercising a theoretical control (Σ 1), characterised by the use of the definition of cosine and the algebraic structure of the proof. Group G1’s solution to this problem showed that they were able to carry out simple deductive processes, although they still did not have enough control of the theoretical elements of the proof. Thus, in this case, the structural rupture made by the students favoured the writing of a deductive chain of implications, which, with the help of some rebuttals by the teacher, could have been converted into correct deductive proof.

4.4 Case 4: deductive cognitive unity

This problem is based on the Cabri figure shown in Fig. 10. Students can modify ∠A, ∠(90-A) and ∠(A-90) by dragging point P around the unit circle.
Fig. 10

Cabri figure for problem 17

4.4.1 Search for a conjecture

After having dragged the figure for a while, G1 said:

[1] Mapa: Look, sin(A) = cos(90-A) = −cos(A-90).

[2] Res.: Yes, that’s right. How did you realize it?

[3] Mapa: From the drawing [pointing at the screen], because, let’s say that the triangle was like this [she copied the screen onto her worksheet (Fig. 11)]
Fig. 11

Mapa’s drawing

[4] Mapa: Then, the triangle of A, let’s say this one [pointing at A in her drawing], and then 90-A would be this one [pointing at 90-A in her drawing].

[5] Diana: Equal to B.

[6] Mapa: So 90-A would be equal to B. Then, this is the reason for sin(A) = cos(B).

[7] Mapa: So, sin(A) = cos(90-A) [pointing at angles A and 90-A].

[9] Mapa [writing on the worksheet while speaking]: And, if cos(90-A), let’s say that it were cos(B), then cos(A), or cos(B) = −cos(−B), would equal to -cos(−(90-A)) [she wrote sin(A) = cos(90-A) = −cos(−(90-A))], and that is the relationship.

[10] Mapa: Wait, sin(A) = cos(90-A) = cos(−(90-A)). Which is the question, cos(A-90) is …?

[13] Diana: Equal to cos(90-A), which is equal to cos(A-90) [they wrote sin(A) = cos(90-A) = cos(−(90-A)) = cos(A-90) [note a change with regard to the conjecture verbalised].

Students’ production of this conjecture is presented in Fig. 12. When G1 started working, they were reminded of a relationship from a previous problem; thus, they stated a first conjecture (C 1) by an analogy to that problem. This conjecture had an error, which initially was perceived by neither the students nor the researcher.
Fig. 12

Toulmin scheme for G1’s argumentation of their conjecture in problem 17

Thanks to the structuring argumentation G1 were doing, they realised their error [10] and stated a new conjecture (C 2), whose warrant was the equality between cosines of inverse angles (W 4). The backing was first perceptive (B 1), but shifted to theoretical (B 2). The operators were, first, the figure shown on the computer screen (R 1 ) and, for the second conjecture, the equality between cos(90-A) and cos(A-90) (R 2 ). The systems of representation were geometric drawings (L 1) and algebraic language (L 2). It is important to note that a connection began to be established among geometrical, algebraic and natural systems of representation, which contributed to helping students move from a geometric control (Σ 1) based on the Cabri figure to a theoretical control (Σ 2) based on definitions and properties.

4.4.2 Construction of a proof

After the students wrote their second conjecture, they started working to prove it. The process of producing this proof has two clearly different parts. G1’s first attempt at proof is presented on the left side of Fig. 13.
Fig. 13

Toulmin scheme for the two parts of G1’s proof in problem 17

[14] Mapa: Why?

[15] Diana: Because sin(A) = \( \frac{y}{r} \) [she wrote the expression on her worksheet].

[16] Mapa: Cos(A-90) is … Which one is A-90? This one? [she refers to the labels of the angles on the Cabri figure]. Then cosine is adjacent, x, over …

[17] Diana: Hypotenuse.

[21] Diana: Then, adjacent over hypotenuse. As they are isosceles [she meant that the three triangles in Fig. 10 are congruent], they have the same hypotenuse.

G1’s second attempt at proof is presented on the right side of Fig. 13.

[22] Mapa: Wait, A is blue, right? sin(A) is y over r, cos(90-A) is …

[23] Diana: It is AC over A.

[24] Mapa: So it is x over r, but …, although this one is the same as this one, right?

[25] Mapa: And cos(A-90). This triangle here, A-90, but A-90, they look like equal triangles, that is, A-90 and A are equal [she meant that red and blue triangles (Fig. 10) are congruent].

[28] Diana: So they [the red and yellow triangles (Fig. 10)] are equal.

[29] Diana: Then cos(A-90), as this one, the leg adjacent to A-90, is equal to 90-A.

[30] Mapa: Do you know how is it easier? [she drew on her worksheet the drawing in Fig. 13] Imagine this triangle here, and this triangle, the same, but this one, okay?

There is a connection between the (algebraic) trigonometry of the unit circle (B 3) and the (geometric) trigonometry of the right-angled triangle (B 4). G1 recognised and justified graphically that the triangles in Fig. 10 are congruent, and they used this property to connect the trigonometric ratios of ∠A, ∠(90-A) and ∠(A-90) (W 9, W 10). They used theoretical operators and controls based on definitions (R 5, R 6, R 8, and Σ 3), geometrical properties of right-angled triangles displayed on the screen (R 7, R 9 to R 11, and Σ 4), and on the drawing (Fig. 13) made on their worksheet (W 11). The system of representation was characterised by the connection among geometric representations (L 1), algebraic representations (L 2), and natural language (L 3).

The students made a thought experiment proof producing deductive argumentations based on the triangles shown on the screen and the worksheet and abstract properties learned in previous sessions; this helped them to transform the problem into theorems already proven ([1] and [9]). A key aspect of students’ progress in understanding proofs is that they were concerned with questioning, justifying and proving the claims that they used as operators ([14], [24], [30]).

In this solution, there is referential unity, because the operators, systems of representation, and structures of control are equivalent in both the conjecturing and the proving phase. There is also structural unity, because, in both phases, the students began to make an empirical argumentation that they turned into a deductive argumentation. The structuring argumentation made in the conjecturing phase allowed G1 to produce a deductive argumentation based on theoretical operators, enabling them to make connections between theoretical properties. Thus, there is cognitive unity in this solution to problem 17. This cognitive unity, and the fact that the students produced deductive argumentations to validate their conjecture, led to the production of a deductive proof. This result confirms the issues raised by Pedemonte (2002, 2005), who claims that the cognitive unity favours the construction of proofs.

4.5 Synthesis of the data gathered from groups G1 and G2

We used the same procedure shown in previous pages to analyse the solutions to all the problems solved throughout the classroom-based intervention, but we have selected G1’s solutions to six problems and G2’s solutions to seven problems, which represent the different types of proofs each group produced and allow us to observe their varying progress in learning to prove. The problems are different because each group’s pattern of progress was different. Tables 1 and 2 summarise the data on these solutions and inform on the structures used in the conjecturing and proving phases, and on the types of proofs produced.
Table 1

Summary of solutions by group G1 representative of their pattern of progress

 

Problem 4

Problem 14

Problem 15

Problem 16

Problem 17

Problem 30

Conjecture

Proof

Conjecture

Proof

Conjecture

Proof

Conjecture

Proof

Conjecture

Proof

Conjecture

Proof

Structures

Empirical

Empirical

Empirical

Empirical

Empirical

Deductive

Deductive

Deductive

Deductive

Deductive

Deductive

Deductive

Proofs

 

Naïve

empiricism

 

Generic

example

 

Thought

experiment

 

Formal

 

Thought

experiment

 

Thought

experiment

Table 2

Summary of solutions by group G2 representative of their pattern of progress

 

Problem 1

Problem 14

Problem 15

Problem 16

Problem 17

Problem 25

Problem 30

Conjecture

Proof

Conjecture

Proof

Conjecture

Proof

Conjecture

Proof

Conjecture

Proof

Conjecture

Proof

Conjecture

Proof

Structures

Empirical

Empirical

Empirical

Empirical

Empirical

Empirical

Empirical

Empirical

Deductive

Deductive

Empirical

Empirical

Deductive

Deductive

Proofs

 

Naïve

empiricism

 

Generic

example

 

Generic

example

 

Generic

example

 

Formal

 

Generic

example

 

Formal

We see in Table 1 that G1 progressed in learning deductive proof. These students began the classroom-based intervention producing empirical proofs, but, after several weeks, they began to produce deductive proofs, sometimes helped by the teacher or researcher. It is interesting to note that G1 had a shift in problem 15, because they moved from an empirical argumentation about the conjectures to a deductive proof based on the use of the figure in the DGS to recall properties.

Table 2 shows that group G2’s argumentations were mostly empirical, although very close to deductive argumentations because, in most problems, they produced generic example proofs. Balacheff (1988) highlights the importance of generic example proofs as a way of overcoming the epistemological discontinuity between empirical and deductive proofs, noting that students overcome such discontinuity when they are able to look at mathematical statements in an abstract way.

The progress throughout the intervention of students in improving their ability to produce proofs is perceived by shifts in the ways in which they used data, operators, systems of representation and controls, that moved from purely empirical to deductive (partially for G2). The progress is also shown by the types of proofs produced, going from naïve empiricism proofs in the first problems to generic example proofs and, for G1, to thought experiment and formal proofs. Such progress was not constant (particularly for G2), because it was dependent on the difficulty of some of the problems.

5 Discussion and conclusions

We have presented a detailed analysis of several solutions of conjecture-and-proof problems throughout a classroom-based intervention to teach trigonometry. It has led us to understanding some students’ success or failure in solving this type of problem by close scrutiny of the reasons for their actions and decisions during the solutions. The students whose work was analysed in this paper started producing naïve empiricism proofs and ended up producing deductive proofs. Two relevant factors facilitating this improvement were the DGS environment, which helped students to discover and verify conjectures, and the teaching methodology, which promoted discussion and asked students for justifications of their answers.

The analysis of their solutions has been based on a theoretical framework in which we have reconceptualised the construct of cognitive unity of theorems and used it to analyse both empirical and deductive proofs. In fact, we have shown the cases of a solution with empirical cognitive unity and another one with deductive cognitive unity. The analysis of the cognitive unity in the solutions examined shows the difficulties experienced by students while learning to prove. Structuring argumentations on the conjectures were more likely to produce deductive proofs, but constructive argumentations, based on perceptual elements of figures in the DGS, were more likely to produce empirical proofs. The most relevant difficulty in solving the first problems of the intervention was the students’ inability to connect operators and representations in the geometric, algebraic, and numeric contexts, which prevented them from achieving theoretical control and from connecting the right-angled triangle and Cartesian plane representations for trigonometry, which is necessary for constructing deductive proofs.

Through our analysis of the cognitive unity of the solutions, we have identified four categories of cognitive unity/rupture, and we have showed an example of each category, adding a different point of view to research on learning to prove. The categories are:
  • Empirical cognitive unity: empirical, structural and referential unities (e.g., case 1). Using examples does not favour the structural rupture necessary to move from perceptual argumentations about conjectures to deductive proofs.

  • Referential rupture and empirical structural unity: rupture of the system of reference without structural rupture. Sometimes students have difficulties in constructing deductive proofs because generalisations made from the observed data do not become theoretical knowledge, hindering the structural rupture. However, this kind of cognitive unity/rupture may lead students to produce generic example proofs (e.g., case 2) and favour the emergence of deductive argumentations because, in the generic example proofs, students have a theoretical control over the process of proof.

  • Referential unity and structural rupture: unity of the systems of reference because of the use of analogy with previous problems (e.g., case 3). Students tend to repeat previously used claims or statements, that lead to the construction of chains of statements that look like deductive proofs, but are incorrect. For this reason, there is a structural rupture, but it does not guarantee the production of a formal proof.

  • Deductive cognitive unity: deductive structural and referential unities (e.g., case 4). There is a close relationship among the representations in a right-angled triangle, the unit circle and the Cartesian plane. Students’ empirical actions are intended to check the accuracy of the conjectures, but argumentations and proofs are based on abstract general properties.

The classroom-based intervention presented here is specific and we do not intend to generalise our results to other students or other mathematical topics, but it shows the application of an original research framework to make a detailed and consistent analysis of the learning trajectories of secondary school students when learning to prove while studying trigonometry. Our intention has been to present an integration of different constructs that had previously been used independently by other researchers (types of proofs, cognitive unity of theorems, structural and referential analysis, the cK¢ model and the Toulmin scheme) into a multifaceted network that has allowed us to trace the progress of students while learning to prove, by identifying different ways of connecting the steps of formulating a conjecture and writing a proof for it. We have not used this analytical network in the context of other mathematical topics; thus, this might be a possible objective for future research. We anticipate that it could be successfully applied in other topics.

Notes

Acknowledgements

The authors are grateful to the anonymous reviewers of this paper and the editors of the special issue for their thorough revision and many valuable suggestions that helped us to improve earlier versions of the paper. We are also grateful to the teacher of the Floridablanca school and his pupils for agreeing to collaborate in this experience.

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Copyright information

© Springer Science+Business Media Dordrecht 2017

Authors and Affiliations

  1. 1.Escuela de MatemáticasUniversidad Industrial de SantanderBucaramangaColombia
  2. 2.Dpto. de Didáctica de la MatemáticaUniversidad de ValenciaValenciaSpain

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