Educational Studies in Mathematics

, Volume 96, Issue 2, pp 129–144

# Fostering empirical examination after proof construction in secondary school geometry

Article

## Abstract

In contrast to existing research that has typically addressed the process from example generation to proof construction, this study aims at enhancing empirical examination after proof construction leading to revision of statements and proofs in secondary school geometry. The term “empirical examination” refers to the use of examples or diagrams to investigate whether a statement is true or a proof is valid. Although empirical examination after proof construction is significant in school mathematics in terms of cultivating students’ critical thinking and achieving authentic mathematical practice, how this activity can be fostered remains unclear. This paper shows the strength of a particular kind of mathematical task, proof problems with diagrams, and teachers’ roles in implementing the tasks, by analysing two classroom-based interventions with students in the eighth and ninth grades. In the interventions, the tasks and the teachers’ actions successfully prompted the students to discover a case rejecting a proof and a case refuting a statement, modify the proof, properly restrict the domain of the statement by disclosing its hidden condition, and invent a more general statement that was true even for the refutation of the original statement.

### Keywords

Proof Empirical examination Refutation Proof problem with diagram Task design Teacher role

## 1 Introduction

Proof and proving play a crucial role in the discipline of mathematics and should be an essential component of mathematical learning from at least the secondary school level. This study aims to enhance student activity of empirical examination after proof construction leading to revision of statements and proofs. I use the term empirical examination to refer to the use of examples or diagrams to investigate whether a statement is true or a proof is valid. Empirical examination is not sufficient to prove the truth of a general statement because it considers only a subset of all cases of the statement. Nevertheless, empirical examination facilitates the process of proving, and typical research in mathematics education has focused on the process from empirical examination to proof construction, analysing how example generation can support, or hinder, the subsequent production of valid proofs (e.g., Iannone, Inglis, Mejia-Ramos, Simpson, & Weber, 2011; Pedemonte, 2007; Sandefur, Mason, Stylianides, & Watson, 2013). The study reported in this paper differs from these studies in that it addresses the reverse order, namely the process from proof construction to empirical examination.

Empirical examination after proof construction is significant in school mathematics for two reasons. First, this activity leads to cultivating students’ critical thinking and attitude because it entails inspecting the truth of given statements and the validity of students’ own proofs. As the Common Core State Standards Initiative (2010) emphasises, students’ ability to critique their own reasoning and that of others should be developed as one of the standards for mathematical practice. Second, empirical examination after proof construction leading to revision of statements and proofs enables students to experience authentic mathematical practice that mirrors development of mathematical knowledge in the discipline (Lampert, 1992). In particular, this activity is relevant to the fallibilism in mathematical philosophy proposed by Lakatos (1976), who rationally reconstructed actual mathematical history to argue that “informal, quasi-empirical, mathematics does not grow through a monotonous increase of the number of indubitably established theorems but through the incessant improvement of guesses by speculation and criticism, by the logic of proofs and refutations” (p. 5).

To foster empirical examination after proof construction leading to revision of statements and proofs, this paper addresses mathematical tasks and teachers’ roles in implementing the tasks. Although tasks and associated teacher actions are influential in students’ mathematical learning (e.g., Henningsen & Stein, 1997), no research to date has developed tasks to enhance this mathematical activity. The relevant student behaviour has been analysed in many studies, where some revealed that few students scrutinised the validity of given proofs with examples (Alcock & Weber, 2005; Ko & Knuth, 2013), while others showed the process where students engaged in empirical examination after proving to improve their conjectures (Komatsu, 20102016; Larsen & Zandieh, 2008). However, the purposes of these studies resided in the description of student behaviour, rather than facilitation of the behaviour by task design. There are researchers who designed tasks to enhance particular aspects of proving (e.g., Buchbinder & Zaslavsky, 2011) and investigated how teachers enacted tasks in textbooks in their classrooms (Bieda, 2010; Sears & Chávez, 2014). The activity these studies dealt with was, however, different from empirical examination after proof construction; for instance, the task sequence developed by Stylianides and Stylianides (2009) was relevant to the process prior to proof construction where students recognised the limitation of empirical arguments and a need for proofs.

This study concentrates on Euclidean geometry as a mathematical domain because Euclidean geometry is the main area for teaching and learning of proof in school mathematics. This implies that with rich tasks in that area, it becomes possible to introduce mathematical activity targeted by the tasks into more classrooms. This paper analyses two classroom-based interventions implemented in geometry classes in lower secondary schools to examine how a particular kind of task, together with teachers’ roles in enacting the tasks, can be used to enhance empirical examination after proof construction leading to revision of statements and proofs.

## 2 Theoretical background

### 2.1 Proof problems with diagrams

This paper focuses on the mathematical tasks named as proof problems with diagrams, previously developed by Komatsu and Tsujiyama (2013) and Komatsu, Tsujiyama, Sakamaki, and Koike (2014), where statements are described in reference to particular given diagrams with symbols; there is typically one diagram per problem. An example is shown in Fig. 1. I adopt the specific interpretation where the tasks are regarded as questions of whether the statements are true for certain general classes to which the given diagrams belong, rather than only for the diagrams. In the example in Fig. 1, the statement, BD = CE, is considered for all shapes of isosceles triangles ABC where AB = AC (e.g., Fig. 2a).1

There are two characteristics in proof problems with diagrams. First, in contrast to universal geometrical propositions that are described in general terms and without any diagrams or mathematical symbols (e.g., “two pairs of opposite parallel sides in a parallelogram are equal in length”), statements in proof problems with diagrams are described with specific reference to given diagrams and symbols, as illustrated in Fig. 1 (Herbst & Brach, 2006). This feature leads to the second characteristic: the conditions of some (but not all) of the tasks are ambiguous because hidden assumptions may exist within the diagrams (Lampert, 1992; Stylianides, 2007). For instance, the statement in Fig. 1 depends on the given diagram implicitly assuming that the perpendicular lines from points B and C intersect with sides AC and AB, respectively, with the points of intersection defining D and E.

The second characteristic would enable students to engage in empirical examination and improvement of statements by producing various diagrams. In the example in Fig. 1, if solvers prove the statement and then draw different shapes of isosceles triangles ABC, they would discover the case shown in Fig. 2b, noticing that the statement includes the above implicit assumption.2 They may modify the sentence of the statement by articulating this assumption. They may then extend sides AB and AC to produce the intersection point (Fig. 2c) and could prove BD = CE even in this case. Thus, by altering “sides” into “lines” in the original statement in Fig. 1, they would create a more general statement that is true for all cases.

The tasks addressed in this study may appear typical tasks in proof learning in school geometry where students are usually given proof problems for which diagrams and symbols have already been prepared (Herbst & Brach, 2006; Komatsu et al., 2014). However, what is distinctive in this study is the capitalisation on a subset of proof problems with diagrams where hidden assumptions exist, in order to enhance empirical examination after proof construction leading to revision of statements and proofs. In addition, students may be given only problem sentences without diagrams and then asked to draw diagrams and construct proofs and/or counterexamples/ non-examples. However, this sequence may not lead to the mathematical activity aimed for in this study because students may discover counterexamples/ non-examples before proof construction.3 Thus, I focus on asking students to prove statements described with diagrams and then to consider the cases where the shapes of the given diagrams are changed.

The theoretical underpinning of proof problems with diagrams resides in Proofs and Refutations (Lakatos, 1976), which dealt with mathematical history regarding the Descartes–Euler conjecture on polyhedra. This conjecture included ambiguity in the meaning of polyhedra, making it possible to propose various polyhedra and counterexamples. This stimulated production of new proofs that were valid for cases invalidating earlier proofs, restriction of the conjecture to exclude its counterexamples, and invention of more general conjectures that were true even for these counterexamples. Although Lakatos’ work was relevant to mathematical philosophy, Larsen and Zandieh (2008) and Ohtani (2002) argued that his research could form a basis for the design of classroom teaching. Proof problems with diagrams are the same as the conjecture of Lakatos (1976) regarding the ambiguity of the domains of statements because, as illustrated above, given diagrams may include hidden assumptions. Thus, changing the shapes of the given diagrams would enable students to engage in mathematical activity similar to the process described by Lakatos (1976).

### 2.2 Teachers’ roles in implementing proof problems with diagrams

Not only tasks but also teacher actions associated with the enactment of the tasks are essential for quality mathematical learning (e.g., Henningsen & Stein, 1997; Stylianides & Stylianides, 2009). Proof problems with diagrams, in particular, require teacher actions because, as described in Fig. 1, the tasks ask only for students to “prove the statements” without suggesting the subsequent process. Below, I describe my hypotheses about teachers’ roles in implementing the tasks after student proof construction for fostering empirical examination leading to revision of statements and proofs.

First, students are likely unfamiliar with empirical examination by producing various diagrams. Herbst and colleagues indicated that students often expected diagrams in proof problems to be presented by teachers or textbooks, not generated by themselves (Herbst & Arbor, 2004; Herbst & Brach, 2006). Hence, it seems necessary for teachers to prompt students to draw their own diagrams by which they can inspect the truth of statements and the validity of proofs. Alternatively, teachers may be required to present diagrams that result in rejection of statements or proofs if it is complicated for students to draw such diagrams by themselves.

Second, students would have little experience of instances where statements and proofs are denied in spite of the construction of valid proofs, and so may be upset and lose the direction of their work when faced with rejection by empirical examination. For instance, like the illustration in “Proof problems with diagrams” section, proof problems with diagrams require students to revise problem sentences by examining hidden assumptions, but this activity would be rare for students, as indicated again by Herbst and colleagues (Herbst & Arbor, 2004; Herbst & Brach, 2006). This implies the necessity for teachers to direct students’ attention towards revision of statements and proofs.

Third, during student individual work and small-group discussion, some students may find ideas that are worthwhile to be pursued by the whole class. In this case, it is often beneficial for teachers to purposefully select these students and ask them to share their ideas with other students in whole-class discussion (Stein, Engle, Smith, & Hughes, 2008; Stylianides & Stylianides, 2014). This practice makes it possible to both respect student contributions in lessons and steer their progression towards important mathematical ideas (Stein et al., 2008). For this practice to be successful, it is vital for teachers to circulate around classrooms to carefully monitor students’ thinking during their individual work and small-group discussion (Stein et al., 2008). Circulating around classrooms to monitor student work and then purposefully selecting students for whole-class discussion is common practice in Japanese problem-solving lessons (Stigler & Hiebert, 1999).

In summary, this study hypothesised the following teachers’ roles (TR) in implementing proof problems with diagrams after student proof construction:
1. (TR1)

Prompting students to draw diagrams different from given diagrams, or presenting these diagrams, so that students can inspect the truth of statements and the validity of proofs.

2. (TR2)

Posing questions that direct students’ attention towards revising the statements and proofs.

3. (TR3)

Selecting particular students whose ideas are worthwhile to be examined and inviting the students to share the ideas with the whole class.

I acknowledge that TR1 and TR2 each contain two different types. The one type is actual teacher actions such as presenting diagrams and posing questions. The other type is part of the mathematical activity that teachers are helping students to engage in, e.g., inspecting and revising statements and proofs. However, it would not be beneficial to separate the two types as different teacher roles because it is significant to consider teacher actions together with their purposes. For instance, what is important is not whether teachers pose questions but rather what kind of student activity teachers aim to foster and what questions teachers should pose for that purpose. Thus, this study proposes TR1 and TR2 as described above.

### 2.3 Research question

Although my earlier work (Komatsu et al., 2014) demonstrated the strength of a proof problem with a diagram to foster student process of proofs and refutations, the study had three limitations. First, there was only one teaching experiment with a single task. It remains necessary to investigate the usefulness of different proof problems with diagrams by implementing further classroom-based interventions. Second, the teaching experiment focused on a limited part of the activity addressed in this paper; the students drew diagrams to refute a statement and produced a generalisation that was true for this refutation. It is necessary to determine whether proof problems with diagrams can be used to foster more extensive aspects of the activity that this study aimed to enhance, including not only empirical examination of statements but also empirical examination of proofs that leads to invalidation and modification of the proofs. Third, although the teacher in the teaching experiment played several roles in implementing the task, his actions were not scrutinised in detail. Consequently, this paper addresses the following research question: “How can proof problems with diagrams, together with the teachers’ roles in implementing the tasks, be used to enhance student activity of empirical examination after proof construction leading to (i) invalidation and modification of proofs and (ii) refutation and improvement of statements?”

## 3 Methods

This paper analyses classroom-based interventions (Stylianides & Stylianides, 2013) implemented as: (1) a part of curriculum development for explorative proving (Miyazaki et al., 2016), and (2) research into developing a set of a task sequence and associated teacher actions to foster student engagement with proofs and refutations (Komatsu & Tsujiyama, 2013; Komatsu et al., 2014). In my study, an intervention consists of a teaching experiment in a classroom where a developed task is put into practice over two or three lessons. I developed four tasks in the form of proof problems with diagrams, and secondary school teachers implemented six teaching experiments in their classrooms to investigate the value of these tasks. Two of the four developed tasks were implemented once, and the other two tasks were implemented twice.

The teaching experiments were designed, implemented, and reviewed in close collaboration with the teachers involved. This collaboration is described here only briefly as it is beyond the focus of this paper. For each experiment, I first drafted lesson plans that included a task and teachers’ roles in enacting it. After discussion with the teacher, the plans were modified to fit with the teacher and his classroom ethos. The teacher then carried out lessons, which I observed as a non-participant. We held post-lesson discussions to enable the improvement of the lesson plans, which aimed to develop the aforementioned set of a task sequence and associated teacher actions. The description of this development is beyond the scope of this paper because it was relevant to the research cycle subsequent to that reported in this paper.

Three video cameras were used to record each teaching experiment: two cameras were fixed in the front and back of the classroom to record the whole classroom, and one camera was mobile to record events relating to individual student work and interaction with others. Transcripts were made and the students’ worksheets were collected. During the teaching experiments, I circulated around the classrooms to take field notes about the students’ individual work and discussion with their peers. I also took field notes about the atmosphere of the classrooms during whole-class discussion, such as whether the students agreed or disagreed with opinions uttered in the discussion. The data for analysis included these videos, transcripts, student worksheets, and field notes.

Each of the six teaching experiments in this study constitutes a unit of analysis. The procedure for data analysis consisted of three phases. In phase 1, videos and transcripts of the first three cases were analysed. Each case was split into several segments, which were tentatively coded. Based on the codes, I devised the single-path model that consisted of a statement, planning a proof, constructing the proof, empirical examination of the statement, and production of a new statement or generalisation, in order to describe events occurring in each of the three cases. In phase 2, these tentative codes and model were scrutinised through attempts to use them to describe the remaining three cases step by step. New codes were assigned when there were segments that the initial codes could not explain. Some segments were unified with a new code, while others were reconsidered as subsegments. This modification of codes led to re-examination of the tentative single-path model, which was finally modified into the branched type depicted in Fig. 3, so that all the six teaching experiments could be described. In phase 3, I paid particular attention to whether the used tasks and the teachers’ actions enabled the students to engage in empirical examination leading to revision of statements and proofs. The students’ worksheets and field notes were added for analysis to determine whether the students independently achieved this activity and whether the students speaking in whole-class discussion were representative of these students. If a specific part of this activity was achieved by only some students, all data were scrutinised to determine whether, in the subsequent whole-class discussion, the successful students were able to offer explanations to help their classmates understand more fully.

This paper reports two teaching experiments, Cases 1 and 2, as case studies. The rationale for selecting these is as follows. First, the two cases represent other teaching experiments in that the paths of the two cases were similarly observed in the other four teaching experiments, one of which followed the upper path in Fig. 3 and three of which followed the lower path in Fig. 3. Second, these two cases are different from each other in that they are relevant to each of the two processes depicted in Fig. 3.

Cases 1 and 2 involved students in the ninth and eighth grades (14–15 and 13–14 years old respectively), both with considerable experience working on geometric proofs. Japan has a national curriculum that prescribes proof-oriented geometry in lower secondary school mathematics. Students in the eighth grade learn the meaning and structure of proof and how to plan and construct proofs based on the congruence of triangles. They then prove various statements related to isosceles triangles, right-angled triangles, and parallelograms. Students in the ninth grade learn proofs using the conditions for similar triangles and the inscribed angle theorem. The participating students in Cases 1 and 2 followed this curriculum. Further information about the classes, as well as the tasks used, will be described in the following sections. English translations of the problem sentences, student and teacher utterances, and proofs are given from the original Japanese. All the students’ names used here are pseudonyms.

## 4 Case 1: Empirical examination and modification of proof

The first teaching experiment was implemented in a classroom consisting of 30 students in a public lower secondary school in Japan. We used the task in Fig. 4, where this study considers a general class to which the diagram belongs, namely many possible locations of lines m and n. We chose this task because, as shown later, some diagrams different to Fig. 4 invalidate a proof but do not refute the statement. Proof modification in the task requires the inscribed quadrilateral theorem; no student could succeed in the modification without the knowledge of this theorem. The participating students learned the theorem, with which they had worked at problems to find the unknown degrees of angles. However, they had not experienced any problems in proving statements by using the theorem. The mathematical capabilities of the students were above average.
Although the teaching experiment was implemented over two lessons (50 minutes per lesson), the first half of the first lesson was not relevant to this study because the students practised routine problems involving proofs of the similarity of triangles with the inscribed angle theorem. In the latter half of the first lesson, the students tackled the task in Fig. 4, where most of them independently constructed correct proofs during individual work. The following is the proof written on the blackboard by the teacher through eliciting responses from the students.
• For ΔACD and ΔAEF,

• From arc AB of circle O, ∠ACD = ∠AEF (inscribed angles) … (1)

• From arc AB of circle O’, ∠ADC = ∠AFE (inscribed angles) … (2)

• From (1) and (2), since two pairs of corresponding angles are equal, ΔACD ∼ ΔAEF

Given the focus of this paper, I now start from the description of the second lesson.

### 4.1 Empirical examination of proof and articulation of the domain where the proof is invalidated

The second lesson started with a review of the proof constructed in the first lesson. After this review, the teacher used GeoGebra to move line n and present the diagram in Fig. 5a, saying: “Now, I moved [line n]. Is this proof still valid?” Students Yuto and Sota answered that each reason for the congruence of angles ACD and AEF and that of angles ADC and AFE was preserved. Mei stated that their answers verified the applicability of the initial proof to the case in Fig. 5a.
Next, the teacher further moved line n to present the diagram where point E was located on arc AB as shown in Fig. 5b (TR1). Many students were surprised and confused by this. The teacher then prompted the students to inspect the validity of the initial proof, specifically encouraging them to investigate which part of the proof was denied by the diagram (TR1). After student individual work and discussion with their neighbours, the proof was scrutinised in detail through whole-class discussion. Yusei voluntarily stated that angles ADC and AFE remain congruent, explaining that they are still inscribed angles corresponding to arc AB of circle O’. His explanation persuaded the entire class, including students who had initially considered this congruence denied. Next, the congruence of angles ACD and AEF, which most students thought invalidated, was examined. Riku voluntarily said:

153. Riku: [The proof] states that A, angle ACD, … and angle AEF, … are equal because they are inscribed angles of arc AB of circle O, but now angle AEF is not an inscribed angle of arc AB, so [it is] wrong.

Riku here explained that the part of the initial proof on the congruence of angles ACD and AEF is no longer applicable because angle AEF is not an inscribed angle corresponding to arc AB of circle O. Yuto additionally argued that angle AEF is not an inscribed angle of circle O’ as well because point E is not on the circle. Although his additional argument was not necessary in a mathematical sense, the argument further convinced the class that the initial proof was denied. Afterwards, the teacher additionally moved line n on GeoGebra to question when the proof became invalid, and Sosuke answered that it was when point E was on arc AB. Thus, the students engaged in empirical examination of their proof by looking at the diagram shown in Fig. 5b to discover the invalidity of the proof in this case.

### 4.2 Modification of proof

Some students might regard the case shown in Fig. 5b to be a counterexample to the statement, doubting the similarity of triangles ACD and AEF. However, the teacher selected a particular student, Koki, and asked him to publicly share his idea that he (and several other students) had conceived during his former work (TR3):

191. Teacher: A while ago, Koki, Haruto, and Hinata first wrote that it held, and then they said it did not hold in that sense. Koki, what did you think held?

192. Koki: Triangle ACD is similar to triangle AEF.

193. Teacher: You conjecture that the similarity is still true.

194. Koki: Yes.

195. Teacher: What should we do for this [proof]?

196. Koki: Adding a reason.

197. Teacher: Yeah, he says that if we change this reason, add, and modify this, we may have success.

In the above dialogue, Koki anticipated that despite the invalidity of the initial proof, triangles ACD and AEF would still be similar. He further proposed modifying the proof, particularly the reason for the congruence of angles ACD and AEF. The other students began to investigate his proposal through individual work and discussion with their neighbours, and then Hinata voluntarily explained his idea to the whole class:

204. Teacher: I want someone to explain in the front … What properties do you use?

205. Hinata: The property of inscribed quadrilaterals.

[The teacher asked short questions to clarify Hinata’s thought and Hinata answered them.]

213. Hinata: The opposite angle of this [angle ACD] is here [angle AEB]. That’s all.

214. Teacher: That’s all? The opposite angle is there, then?

215. Hinata: [Angle AEF is] the exterior angle [of angle AEB], so [angles ACD and AEF are] equal.

Hinata referred to part of the inscribed quadrilateral theorem: an interior angle is equivalent to the exterior angle of the opposite angle (lines 205–215). Although Hinata clearly understood the problem and his approach, he struggled to fully verbalise his thinking. Prior to his explanation, only some students successfully modified the proof by themselves because the class’ experience with the inscribed quadrilateral theorem was solely in solving problems to find the unknown degrees of angles; the students had not previously worked on proof problems that required the theorem. However, Hinata’s explanation and the students’ subsequent verification with their neighbours convinced them that by modifying their initial proof with the theorem, they were able to prove the similarity of triangles ACD and AEF even in the case of Fig. 5b.

## 5 Case 2: Empirical examination and improvement of statement

The second teaching experiment took place in a classroom of 38 students in a lower secondary school affiliated to a Japanese national university. We decided to implement the task in Fig. 1 because, in contrast to Case 1, it leads to refutation of the statement. Like the illustration in “Proof problems with diagrams” section, we expected the students to empirically examine the statement by drawing various shapes of isosceles triangles ABC, to restrict the domain of the statement by disclosing the hidden condition, and to invent the more general statement. Similar to Case 1, the mathematical capabilities of the students were above average, and two lessons were devoted to the teaching experiment.

### 5.1 Empirical examination and restriction of statement

As for Case 1, I describe the students’ proving in the problem in Fig. 1 only briefly. The first lesson began with the proving, where most of the students succeeded by themselves. Yuma wrote the following proof on the blackboard, while Koharu wrote another proof on the blackboard based on the congruence of triangles BCE and CBD (omitted in this paper for brevity).
• For ΔABD and ΔACE,

• From the hypothesis, AB = AC … (1) and ∠ADB = ∠AEC = 90 ° … (2)

• From the common angles, ∠BAD = ∠CAE … (3)

• From (1), (2), and (3), since the hypotenuses and a pair of acute angles of the right triangles are equal, ΔABD ≡ ΔACE

• Since the lengths of corresponding sides in congruent figures are equal, BD = CE

After this, the teacher questioned the whole class (TR1), “We proved BD = CE … Now … this triangle ABC is like this shape [Fig. 1]. If [triangle ABC is] any shapes of isosceles triangles, can we still maintain BD = CE?” The lesson moved to student individual work where they investigated this question by producing various diagrams in discussion with their neighbours. Many students independently discovered the cases as shown in Fig. 6a, in which perpendicular lines from points B and C did not intersect with sides AC and AB, respectively. They were surprised by their discovery, typified by the dialogue between Soma and Airi. Soma said, “This is a big problem … I can’t draw [the perpendicular lines]” and Airi answered, “What? … Oh, I got your problem … too flat [laughing].”
In the subsequent whole-class discussion, Haruki first mentioned the case of right isosceles triangles, arguing that proving the congruence of triangles was not required to show BD = CE because these two segments coincide with sides AB and AC that are the same length from the problem condition. Then, the class inspected the case shown in Fig. 6a, where the teacher selected Asahi to present his idea to the whole class (TR3):

80. Teacher: What do you think? Asahi also said ‘um’.

81. Asahi: I tried the case where the top angle of an isosceles triangle was obtuse. In that case, when I tried to draw perpendicular lines inside the figure, I was not able to connect the sides and angles.

82. Teacher: You can’t connect the angles and sides by perpendicular lines … How did you try to draw the perpendicular lines?

83. Asahi: Well, since I can’t draw inside the figure, I extended the two segments constructing the top angle to connect these sides and the bottom angles [cf. Figure 6b taken from Asahi’s worksheet].

84. Teacher: I see. You thought in that way, but you can’t draw under the original situation … [The teacher saw Minato’s worksheet.] Oh, Minato also wrote “impossible”. Yeah, impossible.

In this discussion, Asahi mentioned the reason why the case where angle A was obtuse was not possible (line 81). In addition to his and Minato’s comments (line 84), the fact that many students discovered the cases as shown in Fig. 6 corroborates that the students engaged in empirical examination of the statement by producing diagrams to discover the case where segments BD and CE were not constructed.

At the end of the first lesson, the teacher questioned the whole class (TR2), “What case does the statement, BD = CE, hold for?” Hina answered, “The statement, BD = CE, holds for the case where the top angle is not greater than 90 degrees, and it does not hold for the case where the top angle is greater than 90 degrees”. Other students agreed with her answer. Hence, the students revised the statement by articulating the hidden condition regarding the degree of angle A and restricting the domain of the statement to the case where the angle was acute or right.

### 5.2 Generalisation of statement

Although Asahi referred to the possibility of generalising the statement by extending sides AB and AC (line 83), only some students drew relevant diagrams in their worksheets. To direct the class’ attention towards this generalisation, the teacher began the second lesson on the next day by questioning (TR2), “What should we do to produce points D and E when angle A is greater than 90 degrees?” Haru responded by drawing a diagram on the blackboard where he extended sides AB and AC, but he put points D and E on lines AB and AC, respectively (Fig. 7a). Although some students agreed with his diagram, Sakura objected, claiming that the places of points D and E should be reversed because the original problem in Fig. 1 let point D be the intersection point between the perpendicular line passing through point B and side AC (Fig. 7b). Haru’s mistake was derived from the appearance of the original diagram (Fig. 1), where points D and E were on the right and left sides, respectively.
Based on Sakura’s idea, the students started their individual work proving BD = CE in the case of Fig. 7b. Some students considered the congruence of triangles ABD and ACE, while others focused on the congruence of triangles BCE and CBD. At that time, the teacher selected Koharu and asked her to publicly present her observation that these triangles were identical to the triangles that the students had considered during the first lesson (TR3):

157. Teacher: Koharu said, “By yesterday’s triangles”. Are these [triangles BCE and CBD] the same as yesterday’s triangles?

158. Koharu: The labels are the same.

159. Teacher: I see … how about … ABD and ACE? How about the labels?

160. Koharu: Same.

161. Teacher: Well … since the labels seem to be the same as yesterday, let’s remind ourselves which [triangles] you proved yesterday, and prove [BD = CE] from the same triangles.

Most of the students constructed valid proofs by themselves. Hayato wrote the following proof on the blackboard, while Ren wrote another proof showing the congruence of triangles BCE and CBD (omitted in this paper for brevity).
• For ΔABD and ΔACE,

• From the hypothesis, ∠BDA = ∠CEA = 90 ° … (1) and BA = CA … (2)

• From vertical angles, ∠DAB = ∠EAB [correctly, ∠EAC] … (3)

• From (1), (2), and (3), since the hypotenuses and a pair of acute angles in the right triangles are equal, ΔABD ≡ ΔACE

• Corresponding sides of congruent figures are equal.

• Therefore, BD = CE

These proofs could have been efficiently constructed if the students had used the original proofs of the statement in Fig. 1. When focusing on triangles ABD and ACE, the students only needed to change the reason for the congruence of angles BAD and CAE from the identity of the angles to the equality of vertical angles. No modification was necessary in the case of another proof focusing on triangles BCE and CBD. Nevertheless, most of the students did not refer to their original proofs, constructing new proofs from the beginning. Although Koharu had publicly mentioned that these triangles were the same between the first and second lessons (lines 158 and 160), her comment remained superficial and few students attempted to use the proofs in the first lesson.

At the end of the second lesson, the teacher questioned (TR2), “In order to state BD = CE … including this case [where angle A is greater than 90 degrees], … how should we modify … the original problem sentence?” Haruki proposed altering sides AC and AB to lines AC and AB or rays CA and BA, and other students agreed with his idea. Thus, the students generalised the original statement to include the case where angle A was obtuse.

## 6 Discussion

This paper shows that the classroom-based interventions implemented in this study enabled students to engage in empirical examination after proof construction leading to revision of the statement and proof. Although what was occurring in the classrooms was clearly joint activity by the students and teachers via the tasks, I focus the following discussion on the research question of this paper that involves the tasks and the teachers’ roles in implementing the tasks.

Cases 1 and 2 demonstrate the strength of proof problems with diagrams, which builds on my previous study (Komatsu et al., 2014) in two senses. First, in the previous teaching experiment, a proof problem with a diagram enabled the students to refute a statement by producing various diagrams and to create a generalisation that was true even for the refutation. This result was reconfirmed by Case 2 where the same generalisation was observed with another task (see “Generalisation of statement” section). Second, this paper shows that the tasks can be used to foster mathematical processes that are different from the generalisation of statements: (1) Discovery of hidden conditions to appropriately restrict the domains of statements as shown in Case 2 (“Empirical examination and restriction of statement” section); and (2) Empirical examination of proofs as shown in Case 1, where the students were presented with the case invalidating their proof (“Empirical examination of proof and articulation of the domain where the proof is invalidated” section) and modified it to cope with this invalidation (“Modification of proof” section). Therefore, it can be concluded that proof problems with diagrams are very helpful in achieving extensive aspects of empirical examination after proof construction leading to revision of statements and proofs. Note that not every proof problem with diagram can be successful; diagrams given in the tasks need to involve hidden assumptions so that changing the shapes of the diagrams can lead to denying statements or proofs.

It was the existence of implicit assumptions in proof problems with diagrams that made the teaching experiments successful. Regarding the intersection of assumptions and proving, Stylianides (2007) argued that rich mathematical activity could be generated by tasks with three features. The first feature is that the conditions of tasks are ambiguously stated and are subject to different legitimate assumptions, while the third feature is that the conclusions of proofs that students construct based on each assumption appear to be contradictory. Proof problems with diagrams satisfy the first feature as stated in “Proof problems with diagrams” section. However, the third feature was not seen in this study as, for example, the statements in Case 2 were not contradictory but rather had a particular-general relationship (the statement restricted to 0° < ∠A ≤ 90° or the statement generalised to all cases, 0° < ∠A < 180 °). The difference between this feature and the teaching experiments in this study arises from different mathematical backgrounds. Stylianides (2007) was oriented towards the coexistence of Euclidean and non-Euclidean geometries where contradictory results are obtained owing to the difference in the parallel-line postulates. In contrast, the current study is oriented towards Lakatos’ (1976) fallibilism that described the process of improvement of conjectures with counterexamples, e.g., restriction and generalisation of conjectures. In future, it will be worthwhile to explore what features of mathematical tasks in general, not just proof problems with diagrams, can generate empirical examination after proof construction leading to revision of statements and proofs.

Not only the tasks but also the teachers’ roles in implementing the tasks, namely TR1, TR2, and TR3 described in “Teachers’ roles in implementing proof problems with diagrams” section, were crucial in the teaching experiments. TR1 was indispensable in both Cases 1 and 2 because no students independently started examining or drawing diagrams different from the given diagrams without the teachers’ prompting. TR1 enabled the students to notice the existence of the case invalidating their proof and to discover the refutation of the statement. The subsequent student activity was facilitated by TR2 and TR3. For example, in Case 1, only a few students noted the idea of modifying their proof at first. However, the other students were subsequently able to engage in the proof modification through TR3 where the teacher selected a student, Koki, and asked him to share his idea with the whole class (Stein et al., 2008). In Case 2, the teacher’s questionings (TR2) enabled the students to revise the statement by articulating its hidden condition and then generalising the statement. Although my earlier study (Komatsu et al., 2014) did not consider teacher actions, the current paper shows the importance of both proof problems with diagrams and the teachers’ roles in implementing the tasks.

Before concluding this paper, I would like to mention two noteworthy tendencies regarding student behaviour. The first tendency was the difficulty the students in Case 2 experienced in drawing diagrams that differed from the given diagram. Many students confused the positions of points D and E when the degree of angle A was larger than 90 degrees (Fig. 7a). In Case 1, the diagram shown in Fig. 5b was presented by the teacher. If the students had drawn such diagrams by themselves, many students would have failed because the problem condition and the diagrams in Case 1 were more complicated than those in Case 2. This difficulty experienced by the students was partially because diagrams are usually given by teachers and textbooks, and students rarely draw diagrams by themselves (Herbst & Arbor, 2004; Herbst & Brach, 2006). Thus, it would be important for teachers to more frequently have students draw diagrams that satisfy problem conditions. Second, during the phase of generalisation of the original statement in Case 2, the students did not use their original proofs that would have made it more efficient to prove the generalised statement. It may be possible that proving the generalised statement was so simple that the students did not feel the necessity of reflecting on the original proofs. However, the tendency not to use initial proofs seems to be common in student activity because the same behaviour as in Case 2 was observed in another setting (Komatsu et al., 2014) where proving the generalised statement was longer and more complicated. Further research is required to investigate how to resolve these student tendencies.

## 7 Conclusion

In contrast to existing research that has typically focused on the process from example generation to proof construction, this study aimed at fostering empirical examination after proof construction leading to revision of statements and proofs. This mathematical activity is significant in school mathematics in terms of cultivating students’ critical thinking and achieving authentic mathematical practice. By analysing two classroom-based interventions, this paper showed that a specific kind of mathematical task, proof problems with diagrams, and three teachers’ roles in implementing the tasks were highly valuable for achieving the following activity in secondary school geometry: discovery of cases that reject proofs and cases that refute statements, modification of the proofs, disclosure of hidden conditions to properly restrict the domains of the statements, and invention of more general statements that are true even for the refutation of the original statements. Although two cases are not enough to derive any general conclusion, processes similar to these cases were observed in four other teaching experiments using either the same tasks as Figs. 1 and 4 or tasks different to them as stated in “Methods” section. Therefore, the conclusion of this paper has potential beyond the two featured cases. In future, studies in various contexts, such as in different school levels and countries, should be conducted to collect further evidence regarding the value of the tasks and the teachers’ roles in enacting the tasks.

## Footnotes

1. 1.

Although I acknowledge the mathematical difference between lines and segments, segments BD and CE, not lines, are drawn in the diagram because this is enough to solve the initial problem.

2. 2.

Note that Figure 2b is not a counterexample but a non-example. For details, see Komatsu et al. (2014).

3. 3.

I do not intend to undervalue this sequence. It would be worthwhile to investigate in future whether this sequence can lead to the activity aimed for in this study or another valuable activity.

## Notes

### Acknowledgments

This paper was finalised during my visit to the University of Southampton as a visiting fellow. I would like to express my thanks to Keith Jones for his continuous support. I am grateful to Keisuke Makino and Isao Ohira for their cooperation in conducting classroom teaching experiments. I also thank the editors and the anonymous reviewers for their helpful comments on earlier versions of this paper. This study is supported by the Japan Society for the Promotion of Science (Nos. 15H05402, 16H02068, and 26282039).

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