Int J of Sci and Math Educ DOI 10.1007/s10763-015-9624-7

A Qualitative Research on Example Generation Capabilities of University Students Yasemin Sağlam & Şenol Dost

Received: 1 December 2013 / Accepted: 14 January 2015 # Ministry of Science and Technology, Taiwan 2015

Abstract Examples which are used in exploring a procedure or comprehending/ concretizing a mathematical concept are powerful teaching tools. Generating examples other than conventional ones is both a means for research and a pedagogical method. The aim of this study is to determine the transition process between example generation strategies, and the factors affecting success of the students in generating examples in a Real Analysis course. The participants of the study consisted of 27 undergraduate mathematics students. At the end of the study, it was observed that some of the participants used especially the trial and error strategy as an effective step in the transition to the transformation strategy. Definitions were used by participants as a trigger for example generation and to reflect on concepts during this process in order to reduce cognitive demand. Keywords Example generation . Example space . University students . Real Analysis The primary purpose of teaching activities is to enhance the conceptual comprehension of students to the highest possible level. To achieve this aim, different teaching methods have been developed in numerous studies conducted in the field of education. When the concepts of mathematics and comprehension are considered together, both mathematicians and mathematics educators agree that mathematical examples play an important role in helping students understand concepts. Examples, a crucial part of the learning process, are an effective communication tool between teachers and students (Bills, Dreyfus. Mason. Tsamir. Watson & Zaslavsky, 2006). Examples can be used to help students understand a given text or can be the first step in solving a problem. They can also be used by students in order to explore a certain property after solving a problem (Morselli, 2006). In addition, counterexamples play a functional role in reasoning skills. Standard examples are typically used to present a mathematical idea or concept whereas counterexamples are used to negate incorrect claims or to refute propositions

Y. Sağlam (*) : Ş. Dost Secondary Mathematics Education, Hacettepe University, Beytepe, Ankara, Turkey e-mail: [email protected]

Y. Sağlam, Ş. Dost

(Leung & Lew, 2012), and serve to refine distinctions and deepen comprehension of mathematical concepts (Zodik & Zaslavsky, 2008). Although the examples generated by the teachers are used as instructional tools, we can ask students to generate examples in order to understand their comprehension. Apart from the ordinary uses of examples in mathematics courses, generating examples by students has brought forth a new perspective as discussed below.

Example Generation, Example Generation Strategies, and Example Space Different definitions of example generation have been proposed by researchers. Watson & Mason (2005) defined examples generation (or exemplification) as ‘to describe any situation in which something specific is offered to represent a general class with which the learner is to become familiar—a particular case of a generality’. Example generation can also be described as a problem solving activity in which an individual can develop different strategies (Zaslavsky & Peled, 1996). These two statements highlight the interconnected dimensions of example generation: while generating examples, learners try to find appropriate examples for given situations by using different strategies. In recent studies (Zazkis & Leikin, 2007; Furinghetti, Morselli, & Antonini, 2011), we come across example generation both as a pedagogical method and a research tool. Dahlberg & Housman (1997) found that example generation was a more effective pedagogical method than reformulating or memorizing a definition in the early stages of learning a new concept, because students can perceive the effects of their changes and reflect on the underlying mathematical structure while generating examples (Watson & Shipman, 2008). On the contrary, Iannone, Inglis, Mejia-Ramos, Siemons & Weber (2011) found that example generation was not significantly efficient in some cases (e.g. proof production) compared to other methods (e.g. reading worked examples). However, they concluded that there may be different underlying reasons for that finding. Consequently, the following crucial research questions arise: how example generation is used as a teaching strategy, in which cases does it becomes successful, and what kind of strategies students develop in the context of example generation. Accordingly, Antonini (2006) categorized the strategies used by mathematicians in example generation under three headings: Trial and Error. The example is sought among some recalled objects; for each example, the subject only observes whether it has the requested properties or not (p. 58). Transformation. An object that satisfies part of the requested properties is modified through one or more successive transformations until it is turned into a new object with all the requested characteristics (p. 59). Analysis. Assuming the constructed object [exists], and possibly assuming that it satisfies other properties added in order to simplify or restrict the search ground, further properties are deduced up to consequences that may evoke either a known object or a procedure to construct the requested one (p. 59).

Example Generation Capabilities of University Students

The analysis and transformation strategies are more advanced than trial and error (Iannone et al., 2011). However, deductions that students have made may not be always correct (Edwards & Alcock, 2010). Therefore, in addition to Antonini’s (2006) strategies, Edwards & Alcock (2010) indicated that the transformation and analysis strategies can be subdivided with regard to deductions into two types: correct and incorrect. Antonini (2006) reported that expert mathematicians occasionally use more than one example generation strategy at once, and switch from one strategy to another. For example, he reported that the transformation strategy is used when the object employed at the beginning of an example generation activity can be transformed properly through several transfer operations. On the other hand, the analysis strategy is used when none of the other strategies work. Iannone, Inglis, Mejia-Ramos, Siemons, & Weber (2009) compared the strategies used by expert and novice mathematicians. They concluded that novice mathematicians (i.e. university students) used the trial and error strategy rather frequently, that the analysis strategy was rarely used, and that university students were incompetent in switching from one strategy to another. Although transitions between strategies seem to have been used more frequently by expert mathematicians, there has not been much research on the use of transitions by novice mathematicians. One of the aims of this study is to understand how novice mathematicians switch from the commonly used trial and error strategy to other strategies. By understanding how novice mathematicians use the trial and error strategy, we can help them transition between strategies, which is considered to be the indicator of being an expert. One of the important factors of example generation is example space. The example space comprises examples that perform a special function (Watson & Mason, 2005). Therefore, the example space is not static but has an evolving structure (Goldenberg & Mason, 2008). According to Watson & Mason (2005), learning mathematics involves exploring the example space and forming relationships among its examples. It also involves gaining proficiency in using these examples. Therefore, example spaces do not just consist of any examples put together, but of examples that are equipped with a certain structure (Arzerello, Ascari, & Sabena, 2011). The example space that is explored through proper guidance improves flexibility in the thought process and contributes to the strengthening of new concepts. However, forming one’s own examples requires cognitive skills that are different from those required when working on examples provided in a textbook or by a teacher (Moore, 1994). In areas where students have limited knowledge, their use of examples and their capacity to form examples is limited (Moore, 1994). Therefore, Zazkis & Leikin (2007) suggested that the example spaces of students must be examined in terms of accessibility, correctness, richness, and generality. Watson & Mason (2005, p. 76) also classified example spaces according to accessibility and formation properties: Situated (local), personal (individual) example spaces, triggered by a task, cues, and environments as well as recent experience. Personal potential example space, from which a local space is drawn, consisting of a person’s past experience (even though not explicitly remembered or recalled), which may not be structured in ways that afford easy access. Conventional example space, as generally understood by mathematicians and as displayed in textbooks, into which the teacher hopes to induct his or her students.

Y. Sağlam, Ş. Dost

A collective and situated example space, local to a classroom or other group at a particular time that acts as a local conventional space. Another factor in generating examples is concept definition. A concept definition is included as a component in student comprehension schemes related to a particular concept (Moore, 1994). Students are expected to understand concept definitions and use the concepts in the most appropriate manner by enriching images related to the concept definitions. Watson & Mason (2005) stated that this is possible only when students continually enlarge their example spaces of a concept and use the definitions learned in courses to construct concept images. Definitions are also used to provide a basis for reasoning (Furinghetti et al., 2011). However, Bills & Tall (1998) proposed the idea of operability of definitions, namely, that some definitions are more operable depending on definitions, experiences of individuals, and cognitive demands on individuals, which are included in a problem situation (e.g. example generation in this study).

The Study The Aim In this study, the transition process among example generation strategies in a Real Analysis course is investigated. To accomplish this goal, we first identified the strategies used by students in the example generation process. The factors that affect the example generation process were also studied. Participants Twenty-seven undergraduate mathematics students enrolled in a Real Analysis course at a public university participated in the first phase of the study. In the subsequent phases, 14 students were selected from the 27 students to participate in semiconstructed interviews. These 14 students were selected on the basis of factors such as their attempts to solve questions in the study, strategies used, examples generated, unusual approaches to questions, and mathematical abilities. Thus, it was aimed to uncover all the possible strategies of example generation and to elaborate on the cases which contain relatively more details. Pseudonyms were used instead of the real names of the participants. Context The Real Analysis course focuses on analysis of metric and normed linear spaces. Generalization of mathematical concepts such as open and closed sets, convergence, continuity, and Cauchy sequences were covered in the course. The content of the course was arranged in a system generally used in university-level mathematics courses with the definitions, theorems, and proofs in succession. The course was taught by one of the researchers. Before the commencement of the study, 42 class hours of lectures were given.

Example Generation Capabilities of University Students

Students successfully completed Calculus I and II in the fall and spring semesters of their freshman year. Further, they completed Advanced Calculus I and II in the fall and spring semesters of their sophomore year, respectively. Calculus I and II cover concepts such as functions, limits, integrals, and sequences. Advanced Calculus I and II cover topologies of ℝ, ℝ2, ℝ3 spaces, limits and continuity of multivariable functions, and cover partial differentiation and multiple integrals. In the Real Analysis course, the students are expected to internalize the given definitions and use theorems for solving problems and constructing proofs. Furthermore, this course is directly related to the Calculus courses mentioned above, and it requires frequent use of the concepts in these courses. Therefore, this study focused on the Real Analysis course because it is strong and rich in terms of both practice potential and example generation activities. Data Sources In the first phase, the participants were asked six questions related to the basic topics of the course after 42 h of lectures. Each question was framed as an example generation activity. The students were reminded of the definitions of basic concepts related to the questions during the example generation activity. The students were not subjected to any time limit for solving the questions; however, they completed the tasks in 90 min (two course hours). Their answers were gathered and examined by the researchers. In the second data phase, 14 students who solved most of the questions and who were successful in the midterms were selected. Semi-constructed interviews were conducted with these 14 students regarding their example generation process. During the interviews, the students were asked to explain their answers. They were asked to explain how they attained the solutions. We asked prompting questions such as the following: How did you decide to start and why? What did you think? How did you know that? They were also asked to explain how they verified whether their answers were correct and what they thought of the question. At the end of all the interviews, the participants were asked to express their views on the example generation process. The students were asked if they faced difficulty in solving the problems, and if they did, what were their reasons, and their views on the effect of these questions on the evaluation of their knowledge on the topic. Each of these interview sessions were recorded and lasted an average of 60 – 75 min. The interviews were conducted 1 day after the written test and lasted 3 days. Questions Used in the Study 1. If possible, give examples of non-equivalent metric spaces which are defined on X={1,2,3}. 2. In a metric space, every open ball is an open set. Give an example of a metric space in which the open sets are not open balls. 3. Let A be a subset of a metric space X. Let Ā denote the closure of A and let ∂(A) denote the boundary of A in X. Then, ∂(Ā)⊆∂(A). Give an example of a metric where ∂(Ā)⊂∂(A). 4. Let (X,d) be a metric space. For any r > 0 and xϵX, let B(x,r) denote an r ball centered at x and let δ (B(x,r)) denote the diameter of the ball B(x,r). Then, δ (B(x,r))≤2r. Give an example of a metric space where the above inequality is strict, namely, δ (B(x,r))<2r.

Y. Sağlam, Ş. Dost

5. Give an example of a continuous, bijective function that is not open. 6. Give an example showing that boundedness is not invariant under uniform continuity.

Data Analysis The data of this study were gathered from three sources: answer transcripts of the students, interview records, and interview transcripts. First, the answer transcripts of the students were examined to determine the number of types of their metric spaces. Frequency tables were created by determining the spaces (such as discrete and Euclid metric spaces) they used within the scope of questions. The recorded interviews were then transcribed, and the content was analyzed using open coding (Strauss & Corbin, 1998). In the coding process, new findings and previously used definitions in previous studies on example generation strategies were used. The strategy employed by the students and the emerging strategy triggers are included in the findings section. For internal consistency, the data set was encoded by both researchers of this study. The intercoder reliability was determined using the SPSS 17.0 program, and the Kappa coefficient was calculated. The correlation between the coders was determined to be 0.76. In cases of differences between the coders, the researchers worked together and reached a common decision. The coding scheme of the data is shown in Table 1.

Findings Strategies Used by the Participants In this section, the strategies used by the participants are discussed. The frequencies of these strategies are also presented in Table 2. Trial and Error As reported in previous studies (Edwards & Alcock, 2010; Antonini, 2006; Iannone et al., 2009), trial and error was the most frequently used strategy (Table 2). Murat used this strategy while working on the second question as follows: M: I wanted to try this on ℝ. I thought we should check already known metrics. So I should try frequently used metrics to see whether they work or not. It doesn’t work in ℝ. I tried it in ℝ2. Actually, I could have tried [it] in ℝ3, too. Murat has tried to reach the solution by using already known metrics. He knows ℝ the best. However, when it cannot lead him to the solution, he moves on to ℝ2. He cannot find the solution with this one, either. He says he can even try ℝ3. Murat tries to reach the solution by trial and error, by trying it on the metrics he knows well.

Example Generation Capabilities of University Students Table 1 Coding scheme of data Themes

Subcategories

Examples of codes

Trial and error

•I accidentally obtained it 1. Random trial 2. Change metric space worked on for •Let me come up with an already known thing again a new trial 3. Change set worked on for a new trial •I tried the ℝ space; let me try it like this •Can we, for example, do this? •Maybe it can be obtained in this metric space

Transformation 1. Set limitations on the chosen set 2. Add/remove required features on the example 3. Transform the rule of function 4. Transform the domain of function

•Let us change the domain (or rule) of function a bit •Let us make delta (radius of the ball) less than 1, I know it will work then •Now there will be an interval (selecting the set in the form of an interval)

Analysis

1. Make generalizations 2. Consider all possible situations on set and metric space 3. Synthesize theorems

•If the inverse of a function is not continuous, it is not an open •I should work on the closed interval, only in that case will the continuous function be a uniformly continuous •I should select discrete metric space as the domain of the function, and then it will be a continuous independent of the function rule

Writing definition

1. Need to have the definition in written form 2. Read the definition multiple times during the solution process 3. Apply the definitions to the questions

•Let me first remember the definition of diameter of a set •I am going to write the definition •Let me have the definition in front of me so that I can adapt [the situation] to it •I will simulate it to the definition

Transition between strategies

1. Express why the generated example did not work 2. Eliminate sets or metric spaces worked on after trial and error

•In discrete metric space, open sets are either a single point or the space itself. In other words, I cannot express a set with three elements as a ball in the discrete metric space •Interval-type sets do not satisfy the requirement; I must select a different set

Transformation The transformation strategy can be observed in Emir’s answer for the fifth question. E: I first thought of the function x2 + 2. This function is continuous; however, the one-to-one condition does not hold. Therefore, I used the one-to-one function f(x) = x + 2. I have not specified the space I worked in; however, I worked in the known ℝ space. Because the function I chose is fairly simple, this function is clearly continuous and surjective. However, I was unable to show that it is an open function. Emir preferred to work with familiar functions. The first function he chose provides some of the properties asked for in the question; however, it is not one-to-one.

Y. Sağlam, Ş. Dost

Therefore, he transformed the function into a one-to-one function preserving the other properties. Some of the participants try to reach the solution by adding/removing some properties to the first examples they have examined. When the operations they perform on the example at the start do not lead them to the solution, they change the structure of their example completely and continue working on a different example. M: Let us consider the Euclid space on ℝ and A = (2, 5). The closure of A is [2, 5], and the bound set of the closure of A is {2, 5}. Then, what is the bound of A? That would also be {2, 5}. However, give an example of a metric space example where this equation does not hold. [He repeats the question.] Instead, let us consider the discrete space. What will be the bound of A in this space[?] It is an empty set. Let me chose a different set in ℝ (again). Let us consider the set B = {1, 2, 3, 4, 5}. What will be the closure of B? It is the set itself. The example selected by the student may be a feasible one for the solution. However, because of students’ ability to think in other dimensions (structure of the used example, the chosen space, etc.) and because of the limited strategies used by them, transformations performed on the selected example cannot help them to attain the solution. Thus, students move onto an example that is completely different in structure. According to Antonini’s (2006) definition of strategy, although alterations made on the example look structurally like the transformation strategy, conceptually, they resemble the trial and error strategy. Analysis The analysis strategy was the least frequently used strategy (Table 2). Ahmet used this strategy for the sixth question. He explained how he generated the example as follows: A: From the studies we have done during the semester, if we restrict the domain of a continuous function, we turn it into a uniformly continuous one. Therefore, I chose the f(x) (f:(0,1] → ℝ, f(x) = 1/x) function and the set A = (0,1]. Actually, I first considered the function f(x) = x; however, I could not obtain the set ℝ from the interval (0,1]. Most of the conclusions drawn using the analysis strategy were not always clear. However, from the interview with Ahmet, we can conclude that he used this strategy. Another example of using the analysis strategy was given by Tufan. In the second question, Tufan used the analysis strategy through the deductions he made while determining the appropriate example for the question: T: When we consider a ball in the ℝ2 space, I think of open disks. [He repeats the question.] Is every open set a ball? [Thinks.] For example, A={(x,y)∈ℝ2|x>0} is an open set. It is not an open ball; however, it can be written as a union of open balls. Then, this question pops in my head: Can the union of open balls be an open ball in a metric space? [Thinks.] For example, in the ℝ space, the (0,1) and (1,2) sets are open balls but (0,2) – {1}, which is a union of the two, is not an open ball.

Example Generation Capabilities of University Students

Tufan put forth a claim through reasoning, and he refuted it by finding a counterexample. In addition to these strategies, some of the participants started solving the questions by writing all the definitions of related concepts. They initiated the example generation process by recollecting acquired knowledge. They used the trial and error strategy first after recalling the definitions. For instance, Mert wrote the definition of the closure of a set and its boundary points before starting on the solution for the third question. M: This is the definition of the boundary points. R: Why did you feel the necessity to write this? M: Why did I feel the necessity to write it? Well, so as not to have this thing while examining the discrete metric space. So that I can see [it] clearly. Now, I am going to do the calculations here. Instead of visualizing this in my head, I write it. I can follow it clearly from here. I wrote it to ensure that I do not do anything wrong. For example, I would not want to miss this ∀ here. Therefore, I felt the necessity to write it. Moreover, in exams, I write definitions next to some questions or at the beginning of the question so that I can use it or follow its lead. Now I have seen that the required property has been provided for a closed set. I also think that it can be provided for a set in a normal metric space. For example, I immediately think of the ℝ2 metric space, I think of a circle. Let me write the things around the circles like this. For example, let this be an open set. As can be clearly seen in Table 2, not all participants write definitions in all questions. Most of the participants preferred to write definitions for the fifth and sixth questions. In fact, the participants who did not write the definitions used or manipulated them mentally. T: For a ball in the ℝ2 space, I think of open disks. [He repeats the question.] Is every open set a ball? [Thinks.] For example, the set A={(x,y)∈ℝ2|x>0} is an open set. It is not an open ball; however, it can be written as a union of balls. As can be seen in Tufan’s excerpt, he considered an open set. He did not write the definition of an open set or a ball; however, he was able to easily apply the definition on the set. Although beginning the process of obtaining the solution process by writing definitions is not considered to be a strategy, it acts as a trigger for the trial and error strategy or simply as a starting point for generating examples for complex questions, which may contain more than two concepts or less familiar concepts. Table 2 Frequencies and ratios of example generation strategies

a

Writing definition before any strategy

Strategies

Frequencies

Ratio in total

Writing definitiona

22

22/172 (13 %)

Trial and error

78

78/172 (45 %)

Transformation

46

46/172 (27 %)

Analysis

26

26/172 (15 %)

Y. Sağlam, Ş. Dost

The Factors Affecting Example Generation An analysis of the questions used in the study indicates that participants must select an appropriate metric space to find an example that satisfies the given qualities. Therefore, the participants first examined the example spaces related to the topic of metric space without considering how they can transit between the used strategies. The participants have a very limited example space related to the concept of metric space, as can be seen in Table 3. During the study, all participants worked with either the discrete metric or Euclid spaces (Table 3). Therefore, these spaces formed the accessible example spaces of the students. The fact that the students took courses from the same instructor in similar class environments may have affected the scope of their example space; which is classified as a personal potential example space and a collective and situated example space by Watson & Mason (2005). Although the participants share a similar culture, their individual activities outside the class affected the limits and characteristics of their example spaces. For instance, in the first question, when Zehra could not achieve satisfactory results with the discrete metric or Euclid spaces, as was the case for most of the participants, she worked with the d(x,y) = |(1/x) − (1/y)| metric function. She recollected this space from her studies outside the class. Similarly, Deniz formed the function d(x,y) = max{x,y} − min{x,y} in the first question from his previous knowledge. D: When I could not achieve anything with these choices, I worked with different functions. I could easily determine if the function d(x,y) = max{x,y} − min{x,y} fulfilled the metric criteria. As mentioned earlier, individual differences such as available strategies, mathematical skills, and ability to think in different dimensions may affect the example generation process. By employing unusual methods, some participants were able to solve the questions efficiently. For example, Tufan used function graphics (Fig. 1) with certain properties for the example he tried to generate in the first phase of his solution. He then determined whether the function satisfied the last property from these graphics. Therefore, he could easily solve some parts of the problem. During the interviews, most of the participants indicated that the questions were difficult, and that they needed more time to solve them. They mentioned that the questions were unlike any they had seen so far. They also mentioned that example generation activities require a more comprehensive thought process. They found that the example generation activities enabled them to assess what they have learned.

Table 3 The frequency table of the metric space types used by students according to questions 1. Question

2. Question

3. Question

4. Question

5. Question

6. Question

Discrete metric space

12

13

4

12

10

4

Euclid metric space

13

7

14

2

13

11

Other spaces

8

2

–

–

–

–

Example Generation Capabilities of University Students

Fig. 1 A part of Tufan’s solution

F: I think I was more productive when solving these problems. I really thought hard about the concepts I know, and I tried to mentally visualize the pictures. … Z: Such questions do not guarantee that you exhibit what you have learned. Some students may have studied but may not have been able to generate examples. There should be more time; these questions are difficult. I am exhausted.

Transition Between Strategies It was observed that the transition from the trial and error strategy, which was used most frequently by the participants, to other example generation strategies happens in two different ways. When students could not attain what they wanted to achieve with the trial and error strategy, they were either stuck completely (case 1) or they headed toward other strategies (case 2). In case 1, the students used the trial and error strategy only to find an appropriate example exhibiting the properties asked in the question. The correctness of the example was checked. If a mistake was detected, a few more attempts were made, resulting in either a correct solution or the abandonment of the problem. E: First I wrote the definition of the bound. I made some trials. In order to work in the natural metric of ℝ [absolute value metric], I chose the set [−5,3]. I substituted it for the definition of the bound; however, it did not work. Therefore, I repeated [it] for the set (−5,3). Because it did not work either, I gave up. I worked with natural numbers, [and] this time, I chose the set (2,8). However, it was also wrong…

Y. Sağlam, Ş. Dost

For this question (third question), the student tried to find a metric space in which a specific property is not provided. However, the student conducted the trials by considering different sets from the same metric space. When trying different examples, instead of understanding why the first example did not work, the student continued to work with similar sets. He only changed the set but not the space he worked in. Therefore, he abandoned the question. Here, the example generation strategy that the student used is limited. Thinking only in terms of sets and not in terms of other dimensions, he gives up continuing with the solution after his trials. When the trial and error strategy is used as indicated in the first case, it stands out as a strategy which is dependent on the width of the example space, available strategies, and mathematical skills of the participant. Moreover, this strategy has a minimal effect on conceptual comprehension, which is consistent with the general characteristics of the trial and error strategy. However, in this study, some students employed the trial and error strategy in a different manner. In this case 2, the trial and error strategy was used to explore the characteristics of the existing problem and was followed with a new strategy. In each trial, the students determined the factors that hindered their solution. Therefore, the students gathered data for the new strategy. For instance, Mert conducted some trials on the ℝ space for the fourth question and determined that δ(B(x, r))≤2r provided in the question satisfies the inequality. However, he could not obtain the required δ(B(x, r))< 2r inequality from the examples he tried. M: Let me re-examine it through familiar things. For example, let us assume that (ℝ, |.|) and a=3. I want to find out its diameter. Its diameter is this. With 3 centers and 1 radius, namely, this is my ball. What is its diameter? It is 2r; I mean 2 again.

Mert’s Trials With (ℝ,|.|), we always obtain δ(B(x,r))≤2r M: Let me consider the discrete metric on the ℝ. Assuming a ∈ ℝ, a is, let us assume that r < 1. B(a,r). I know that the 1 neighborhood of 3 consists only of 1. Therefore, the diameter of B (3,1) is 0 because it is a single point. I got it. Let me assume that r = 12, and 2r did not work. It was less than 1. Further, 2⋅ 12 is equal to 1. R: Was that coincidental? M: That did not happen accidentally. I already tried ℝ. There is no problem with ℝ. We have been dealing with examples since high school. The diameter is always this. Now, I am going to find a metric space such that, you know, like this. The ball will be smaller than this thing I have chosen, namely, smaller than r. I am going to select a point, and I will select an r neighborhood of this point. That point will produce a completely different result. It will not contain everything as in ℝ. I will select a or b; however, I will be confronted with a completely different result. If I obtain the same result, it is already equal to 2r as in ℝ. Some discrete metric functions are known to provide this criterion.

Example Generation Capabilities of University Students

Consequences of Mert’s Trial 1. In (ℝ, |.|), the diameter is always equal to 2r (as studied in high school). 2. He has to find a diameter different from 2r. 3. It occurs in the discrete metric space. Mert realized that he could not reach the inequality in ℝ he was looking for. It is because the concept of diameter in ℝ yields the same solution as the concept of diameter in high school curriculum, i.e., the diameter is defined as 2r. However, he should have reached a different solution. Later, he decided to use the discrete metric space which he knew, would meet this quality. R: What made you do this transition? M: I thought that if I take r smaller than 1 in the discrete metric spaces, the ball is a single point, and that point is on 0 … Therefore, the point does not have a diameter. This actually occurred to me immediately. R: How did it occur to you? M: I considered the function. I must find a metric function that will result in a diameter other than 2r. I wanted it to be smaller. It is already a single point when r < 1. Zero is smaller than anything you may consider. Participants who used the second case of the trial and error strategy typically used the transformation strategy after the trial and error strategy. Through trials, they obtained the information of the properties for the example they were trying to generate. The students who had information about some or all of the properties were able to either add or extract properties, or draw conclusions from these properties and generate the desired example. Another participant, Tufan, started his trials for the sixth question with function f(x) = x2. Nevertheless, he could not generate an appropriate example because he could not make a choice independent from the point. To attain the correct solution, he is required to consider the concept of boundedness given in the question. He realized from his trials with the function f(x) = x2 that he must use a bounded set. T: I must think about the shape and retry. What I remember from uniform continuity is that because I could not make the selection independent of the point in the function f(x)=x2 (he draws the graphic), it did not work. Therefore, if I connect it to the boundedness … However, as far as I know, uniform continuity is something of which we use the boundedness. When we think of the boundedness of a function, something comes to my mind. For example, the function tan(x) is something like this [draws the graphic]; it changes from a bounded interval to an unbounded one. However, is it uniformly continuous? I guess it is not … Then, if I think of the function f(x) = 1/x, which is defined from ℝ to the bounded interval (0,1], its image will not be bounded. However, it is still not uniformly continuous. Boundedness is defined as the distance between points; therefore, I can change

Y. Sağlam, Ş. Dost

the metric. If I consider a discrete metric on [0,1], the function becomes uniformly continuous.

Tufan’s Trials 1. The function f(x) = x2 is unbounded. 2. The function f(x) = tanx is defined on a bounded interval; however, the function itself is unbounded.

Consequences of Tufan’s Trials The function should be both bounded and defined on a bounded interval (connecting with uniform continuity). Although f(x) = x2 and f(x) = tanx can also be defined on a bounded interval, they are not uniformly continuous. By analyzing the data he obtained from the functions he tried, he decided to work with a discrete metric space on the (0,1] interval. As can be seen, when compared with the other examples, this last example Tufan used was most likely to yield the correct answer. Therefore, he used the data obtained from the trial and error strategy for transitioning to the transformation strategy. Although transitions between the transformation strategy and the analysis strategy are not very conspicuous, Ahmet’s excerpt under the heading ‘analysis strategy’ is an example of this transition. Ahmet indicates that he starts with f(x) = x which partly satisfies the criteria asked in the question. However, he realizes that the image of a bounded interval under this function is not bounded, and thus he changes the function. Ahmet’s Trial The function f(x) = x is defined from the open interval (0,1) to an unbounded set. Consequences of Ahmet’s Trials To make the image set bounded, the function is required to change. Using this result, Ahmet starts to work with f(x) = 1/x. Therefore, he can define this new function from the interval (0,1] to IR. In fact, Ahmet restricted the space he worked on by selecting continuous functions and employing the knowledge he has obtained throughout the semester. Thus, he reached the example he had been looking for. Depending on the findings described above, transition of students between strategies can be modeled with Fig. 2. In case 1, the trial and error strategy includes the process of reaching the desired example by trying different examples. In case 2, the reason for the failure of the examples was investigated. Therefore, the trial and error strategy was used to obtain an example with certain characteristics. These characteristics are required in the transformation and analysis strategies.

Example Generation Capabilities of University Students

Fig. 2 Transition between strategies

Conclusion and Discussion Examples used to comprehend or concretize a mathematical concept or explore a procedure are powerful learning tools. Example generation activities are a pedagogical method and a research tool that reflects the proficiency or a lack of proficiency in student comprehension of mathematical concepts. However, because of the difficulties arising from the structure of a mathematical concept and the availability of strategies for use in example generation or broadness of example space, it is not always easy to generate many examples for a concept. In this study, the number of examples in the example spaces and the examples created by the participants for metric spaces were limited. Because the participants shared the same classroom culture (e.g. class environment, instructor, courses completed earlier), the accessibility of their example spaces have some commonality. Moreover, the fact that the examples related to the concepts taught within this course were usually of a beginner level and the fact that the reference examples given were in relation to the Euclid metric on ℝ and ℝ2 could be the reasons why these metric spaces came first in example spaces of students. The discrete metric space was also frequently used in this research. The students were aware from the Real Analysis course that the discrete metric space can be used to generate a counterexample and to verify the correctness of a property obtained through geometrical intuition. Occasionally, when the instructor, who is also one of the researchers, wanted to provide a spontaneous example, he worked primarily with Euclid and discrete metric spaces. Therefore, the Euclid and discrete metric spaces are preferred at the moment of decision making (Zodik & Zaslavsky, 2008)—instead of a pre-planned one. All these reasons constituted the basis for the frequent use of the aforementioned spaces in the example spaces of the students. These results show that the dominant example space within the Real Analysis course can be considered to be a personal potential example space and a collective and situated example space, as classified by Watson & Mason (2005, p. 76). The accessibility of examples depends on factors such as clues, association, need, and individual differences. Although the example sources of the students were affected by the knowledge they acquired in this or other courses, their experiences resulting from out-of-class learning activities extended the scope of their example spaces. Deniz and Zehra generated new examples with the help of examples they encountered in outof-class studies. In addition, individual differences such as mathematical fluency/ flexibility enable the learners to easily skip some time-consuming parts and keep motivated for the remaining parts of example generation activity. These skills are also crucial for successful example generation, as can be seen in the case of Ahmet, Tufan,

Y. Sağlam, Ş. Dost

and Mert. These participants stand out by thinking differently. Their success in the transitioning process may be attributed to their mathematical skills and available strategies. As was indicated in the section entitled BFindings,^ the most frequently used strategy by students is trial and error. The transformation strategy would be the second most frequently used strategy. Although participants can make use of transformation by making the right deduction, they either could not reach the solution because of their example space, available strategies, or mathematical skills, or their tendency toward structurally different examples. While the strategy used in this case appears to be a transformation, it is only superficially so. Conceptually, it is much closer to the trial and error strategy. Edwards & Alcock (2010) indicated that the results of deduction during use of the transformation and analysis strategies can be divided into two sub-sections: namely correct and incorrect. Additionally, transformations are not always sufficient to reach the correct answer even when they are based on correct deductions. In this case, transformations performed as a result of correct deductions can be divided into two sub-sections in terms of their adequacy to reach the example. University students and expert mathematicians differ in the manner in which they transit between example generation strategies. In previous studies (Antonini, 2006; Iannone et al., 2009), it was determined that expert mathematicians tend to transit more between example generation strategies compared with novices. Students are not very skilled in these transitions (Antonini, 2006; Iannone et al., 2009). However, one of the striking findings of the study was that some of the participants made transitions between strategies, and used the trial and error strategy as a step while transitioning to the transformation and analysis strategies. Understanding the reasons why the examples used during the trial and error strategy were not satisfactory and the consequent generation of new examples enables the transition to different strategies. Such transitions improve the effectiveness of the trial and error strategy and enable the use of strategies that require more advanced skills. These transitions that activate the trial and error strategy can improve the example generation skills of students. In addition to being the first step of the transition process, the trial and error strategy itself is a strategy in which the correct answer can be derived accidentally. Students started with examples they were familiar or comfortable with and then moved onto contradictory examples. However, any example that was not helpful was considered to be useless, and the students proceeded to the next example. Such use of the trial and error strategy was ineffective for transitioning to other strategies. Moreover, some students felt the need to write the definitions of concepts used in the given questions before they even used the trial and error strategy. This can be considered a preparatory step for the trial and error strategy or to generate an example. We can also interpret that the conceptual comprehension of these students is not at a level that would enable them to mentally manipulate the concepts in a given question. However, students felt the necessity to write the definitions mostly in the fifth and sixth questions. The fifth question contains the concepts of continuity, bijectivity, and open functions whereas the concepts of boundedness and uniform continuity are included in the sixth question. Participants had to think of more than one concept concurrently, which may be complicated for a novice mathematician. Bills & Tall (1998) explain this problem as the struggle to make definitions operable. They proposed that when the cognitive demand is too great for students, they cannot use

Example Generation Capabilities of University Students

definitions effectively. Further efficient usage of definitions often depends on the students’ mathematical ability and experience. Therefore, writing definitions is considered to be a preparation step, instead of a strategy, in the example generation process. Example generation activities can aid in developing cognitive awareness, a key quality that instructors prefer in their students. Using the trial and error strategy, as in the transition process, allows the learners to deepen their understanding of concepts and realize deficiencies in their comprehension. Moreover, during example generation, the learners move from specific cases (examples) to general cases, or vice versa. Therefore, examples—possibly intuitively—indicate an invitation to the process of generalization (Bills & Rowland as cited in Watson & Mason, 2005, p. 120). Hence, teachers may use example generation activities as a teaching tool. However, it still remains to be a research gap. Teachers may need appropriately designed example generation tasks, explicit teaching plans for example generation, and more nuanced implementation suggestions (Iannone et al., 2011). Further, example generation can be used as an assessment tool, because assessors can make sense of comprehension skills of students by evaluating their generated examples (Watson & Mason, 2002). During the process of generating examples, writing definitions may provide information about the cognitive demand that the concept creates, and consequently indicate the comprehensibility level. It may also provide significant clues for the teachers about the range of the example space depending on the topic. However, students found example generation exhausting and very time consuming because it requires a strong conceptual understanding and it was the first time that students encountered example generation questions. The questions in the study required manipulation of more than one concept at the same time. In other words, more than one concept must be combined to create a single example. Actually, when the concepts in the questions were considered separately, students did not have difficulty finding an example for the concept. However, finding an example which carries all the properties requested in the question is more cognitively demanding and time consuming. Consequently, some participants stated that these kinds of questions may be insufficient in expressing their understanding and learning.

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