This study examines the role of middle school students' example spaces in generation and determination of two parallel and perpendicular line segments.
THE ROLE OF LEARNERS’ EXAMPLE SPACES IN EXAMPLE GENERATION AND DETERMINATION OF TWO PARALLEL AND PERPENDICULAR LINE SEGMENTS Fadime Ulusoy Kastamonu University This study examines the role of middle school students’ example spaces in generation and determination of two parallel and perpendicular line segments. Data was collected from 83 middle school students in grades 6 and 7 via two tasks having items on the example generation and determination of parallel and perpendicular two line segments in the grid paper. Data analysis indicated that many of students could not provide fully complete and correct responses when generating and determining parallelism and perpendicularity of two line segments because of limited example spaces under the influence of prototypicality and overgeneralization and undergeneralization errors. This study proposes a catalogue on common limitations in students’ example spaces about parallelism and perpendicularity of line segments. THEORETICAL BACKROUND Mathematics educators and mathematicians agree that the use of examples in teaching and learning as a communication tool between learners and teachers is very useful in helping students comprehend mathematical concepts (e.g. Bills et. al, 2006; Watson & Mason, 2005). In this sense, Zaslavsky and Zodik (2014) define example space as “the collection of examples one associates with a particular concept at a particular time or context” (p. 527). Example space has been used as a similar term with the concept image (Tall & Vinner, 1981). Concept image is the set of all the mental representations associated in the students’ mind with the concept name. The image might be nonverbal and implicit. According to researchers, if students are encountered limited examples having common figural features of a geometric concept in school or other context, these examples lead to prototypes phenomenon. The prototype examples are usually the subset of examples that had the “longest” list of attributes all the critical attributes of the concept and those specific (noncritical) attributes that had strong visual characteristics” (Hershkowitz, 1990, p. 82). By the influence of prototypical examples and non-examples, learners begin to exhibit two types of common errors as undergeneralization and overgeneralization (Klausmeier & Allen, 1978). Undergeneralization error occurs when examples of a concept are encountered but are not identified as examples. For example, if a learner does not admit a rotated square as an example of square and he or she take this rotated square as an non-example in square set, which indicates he or she makes an undergeneralization error. On the other hand, overgeneralization error occurs when examples of other concepts treated as members of target concept (Klausmeier & Allen, 1978, p. 217). For example, if a learner treats 2016. In Csíkos, C., Rausch, A., & Szitányi, J. (Eds.). Proceedings of the 40th Conference of the International Group for the Psychology of Mathematics Education, Vol. 4, pp. 299–306. Szeged, Hungary: PME. 4–299
Ulusoy a regular hexagon as an example of parallelogram without considering the number of sides, he or she makes an overgeneralization error. Researchers suggest that it is important to detect all details of the limitations in students’ examples spaces in order to develop effective examples and tasks when teaching mathematical concepts. In this regard, asking students to generate examples of a specific concept and to determine examples of the concept among a set of examples that involves both examples and non-examples can get more details about students’ comprehension about specific mathematical concepts. By this way, it can be possible to assert limitations in students’ example spaces because example generation can be seen as an indicator of example space (e.g. Sağlam & Dost, 2015). Moreover, example generation and example determination activities help teachers and educators in order to understand less accessible and more accessible examples in students’ mind (Zaslavsky & Zodik, 2014). Such activities allow entering learners’ “personal example spaces” that constitute a collection of examples in learners’ mind when facing a particular task (Watson & Mason, 2005). Thus, considering students’ personal example spaces and their accessibility of the examples can give big chance to the teachers in terms of developing a didactic way when choosing of examples in their teaching activities in order to construct and enrich learners’ examples spaces. Many of mathematical concepts depend on lower order concepts (Skemp, 1971) or subconcepts. In high school or universities, teachers assume that learners know and understand these lower order concepts and sub-concepts. However, among the learning domain of mathematics, students are generally exposed prototypical examples of the concepts in the instruction of geometrical concepts and textbooks rather than encountering non-prototypical examples or less-accessible examples. As a result, studies indicate that students have limited knowledge about the different forms of geometric concepts and their use of examples is limited (e.g. Moore, 1994). As basic geometric concepts, parallelism and perpendicularity of line segments have critical importance in terms of developing correctly and completely students’ conceptions about the concepts of the altitude, perpendicular bisector, median, angle, slope and the subjects of quadrilaterals, coordinate system, and three-dimensional figures, as well as developing students’ proficiencies in proof and argumentation. Many of research revealed both teachers and students have difficulties in some geometric concepts like altitude of triangle (e.g. Gutierrez & Jaime, 1999) and trapezoid (e.g. Ulusoy, 2015) because of inadequate knowledge about parallelism and perpendicularity of two line segments. As a reasonable argument, Zazkis and Leikin (2007) proposed that students’ example spaces should be examined in terms of different perspectives such as accessibility, correctness, richness and generality. However, in the literature, there is limited study that directly focused on students’ examples about parallelism and perpendicularity of two line segments. Considering the importance of parallel/perpendicular line segments in geometry and the influences of students’ example spaces in comprehension of geometric concepts, I decided to investigate the role of middle school students’ example spaces in example generation and 4–300
PME40 – 2016
Ulusoy determination of two parallel/ perpendicular line segments in the current study. In line with this purpose, I tried to answer following research question: What is the role of middle school students’ example spaces in example generation and determination of two parallel and perpendicular line segments? As a concluding remark, as stated in the literature, examining students’ example spaces is crucial to present a catalogue of responses showing the characteristics of students’ limitations in example spaces of two parallel/perpendicular line segments. For this reason, I mostly concentrated on students’ partial and incorrect examples to prepare a catalogue that shows the role of students’ limited example spaces in example generation and example determination of two parallel/perpendicular line segments. METHODOLOGY The school, in which participants were selected, was chosen in Ankara, Turkey with regard to easy accessibility to the researcher. The students were average-income families’ children. In this school, 83 middle school students in Grade 6 and 7 (ages 11 to 13) were determined as the participants of the study. There were 40 students in Grade 6, 43 students in Grade 7. Studies dealing with concept formation highlight the role of carefully selected examples and non-examples in supporting the distinction between critical and non-critical features and the formation of rich concept images and example spaces (e.g. Watson & Mason 2005; Zodik & Zaslavsky, 2008). For this reason, I made a great effort in preparation of examples and non-examples in the tasks by focusing on the studies related to exemplification and basic geometric concepts. In this sense, I prepared two tasks as “example generation task” and “example determination task”. The first task included 10 example generation items and the second task included 11 example determination items related to parallel and perpendicular two line segments. In the example generation task, there are two sections. In the first section, there are two items that ask students to generate two parallel/perpendicular line segments in the grid paper. These items were prepared to understand how students generate examples of perpendicular and parallel line segments. In the second part of example generation task, there are eight items to understand the role of prototypical and non-prototypical position of a line segment in a grid paper. These items requested students to generate a parallel or perpendicular line segment to the given another line segment in the grid paper (see fig. 1). For example, while “item3”, “item4”, “item7”, and “item8” can give information about students’ example spaces of parallel/perpendicular two line segments in terms of prototypicality, remaining items in fig. 1 can provide information about students’ example space in terms of non-prototypicality. Item3
Item4
Item5
Item6
Item7
Item8
Item9
Item10
Figure 1: Item3-6 for the construction of a perpendicular line segment and Item7-10 for the construction of a parallel line segment to the given another line segment
PME40 – 2016
4–301
Ulusoy On the other hand, example determination task includes 11 items that asked students to determine whether given two line segments in the grid paper are perpendicular/parallel or not (see fig. 2). Items were arranged randomly in the task; however, I arranged and named them as in fig. 2 to provide clear explanation about their characteristics. While “item11” and “item16” are prototypical examples of parallel and perpendicular line segments, “item12”, “item14” and “item17” were added as the main non-prototypical examples. “Item13” was prepared to evaluate students’ example space in terms of verticality and perpendicularity. Furthermore, “item14 and “item21” were prepared to understand the role length of line segments on their example spaces about parallelism and perpendicularity of line segments. “Item15”, “item18”, and “item19” can give idea about students’ limited conceptions. Finally, “item20” was added to the task to understand students’ example spaces in terms of perpendicularity and perpendicular bisector. Before conducting data, the suitability of all items was asked two mathematics teachers and a mathematics educator who makes research on geometric concepts. Finally, I piloted all items in both tasks with sixteen seventh grade students in a different school by making semi-structured interviews.
Item12
Item11
Item16
Item17
Item18
Item13
Item19
Item15
Item14
Item20
Item21
Figure 2: Items on determination of perpendicularity/parallelism of two line segments Example generation task firstly implemented to the classrooms. After they finished responding to the task, I started to apply example determination task. The tasks took totally 40-45 minutes in each classroom. In both tasks, students asked to explain and justify the reason why they think these two line segments are parallel/ perpendicular or not. For the data analysis, all student-generated examples and written responses reflecting their decisions and justifications were analyzed in terms of correctness and completeness for each item. Then, common limitations in example generation and determination items were grouped in order to present a catalogue of responses that shows the characteristic of students’ example spaces involving partial or poor concept images on parallel and perpendicular line segments. Finally, I made themes for the common limitations in students’ example spaces. RESULTS The role of students’ example spaces in example generation task Student-generated examples in the example generation task showed that most of the students generally provided prototypical and more-accessible examples of both parallel and perpendicular two line segments in first two items of the task. On the other hand, 4–302
PME40 – 2016
Ulusoy while 20 students (24%) generated incorrect parallel line segment examples, 29 students (35%) made incorrect perpendicular line segment examples. In students’ incorrect perpendicular line segments examples, it was observed the negative influence of mixing verticality and perpendicularity because they generated examples involving only a line segment or two vertical parallel line segments as an example of perpendicular two line segments. Besides, the most common two limitations in students’ examples of parallel line segments were observed as generating only one inclined line segment or disregarding the properties of grid paper when generating inclined two parallel line segments. Furthermore, students’ examples to “item3 to 10” supported the idea of most of students’ examples spaces constitutes only prototypical examples of two parallel/perpendicular line segments. The role of students’ limited example spaces about parallelism of two line segments in example determination task Limitation to see intersection of line segments by extension. A huge number of students (n=59) decided the example in “item15” as an example of two parallel line segments without considering the meaning of parallelism. They partially focused on the information that two lines on a plane which never meet. However, they made an overgeneralization error because they could not consider two line segments in “item15” can eventually cross over each other when extending both straight line segments. Their constructions in example generation task for especially “item9” and “item10” also supported the students’ limitations to generate parallelism in two vertical parallel line segments because they generated two line segments that cross each other in case any extension. Limitation to see parallelism in two vertical parallel line segments. Students (n=32) generally admitted the example in “item13” as a non-example of two parallel line segments although “item13” constitute an example of two parallel line segments. Instead, they treated this example as a member of perpendicular line segments. In written explanations, students made similar comments like in the following: “These line segments are not parallel. They are perpendicular because they are vertical to the base”. In this regard, students’ responses indicated their confusion between vertical line segments and perpendicularity of two line segments. These incorrect responses showed the presence of undergeneralization errors in students’ example spaces. Moreover, such errors in learners’ example spaces can be evaluated as an indicator students’ inadequate knowledge about the meaning of parallelism. Considering length of line segments as a critical factor. Some students (n=17) considered the length of line segments as a critical factor when determining the parallelism of two line segments in some examples like “item14”. For example, these students made following explanations about the example: “These line segments cannot be parallel because they are not same length” or “One is short and another one is long, so they are not parallel”. At this point, they could not establish a relation between a line and the concept of parallelism. Instead, they merely focused on the visual PME40 – 2016
4–303
Ulusoy appearance of line segments in the grid paper. Consequently, students’ statements clearly showed the students’ limited example spaces that were formed under the influence of undergeneralization errors. Thus, they treated an example of two parallel line segments as if they were non-examples because they could make a distinction between critical and non-critical features of an example. The role of students’ limited example spaces about perpendicularity of two line segments in example determination task Seeing enough the presence of crossing two line segments at an angle close to 90˚. A number of student’s (n=35) example spaces involved a concept image about perpendicularity like in “item19”. They made similar written explanations like in the following: “Because these line segments are crossing each other, they are crossing perpendicular“. These students were aware of the requirement of crossing of two line segments for perpendicularity. However, they disregarded the crossing of two line segments at right angles, which indicated the possible influence of students’ limited example space on the examples formed by overgeneralization errors. Verticality vs. perpendicularity. Students’ determination of perpendicularity for the example in “item13” and their written explanations revealed that a case of student (n=27) mixed concepts of vertical line segments and perpendicular line segments. This confusion can be the reason of limited concept image in students’ mind. For this reason, they overgeneralized perpendicularity situation by admitting a non-example in “item13” as if it is an example of perpendicular two line segments. They did not consider perpendicularity of two line segments requires crossing at right angles to each other. As a result, they treated non-crossing vertical line segments as an example of perpendicular line segments. On the other hand, when I analyzed students’ decisions for the example in “item18”, I realized that a case of student found enough the intersection of a vertical line segment at any angle to another line segment. One of them made following explanation: “Perpendicularity of two line segments requires a vertical line segment and crossing of two line segments. In this example, there is a vertical line segment and another one cross it. So, they are perpendicular.” Considering length of line segments as a critical factor. Similar to the situation in parallelism of line segments, some students (n=7) considered the length of line segments as a deterministic factor for perpendicularity of two line segments. Although the example in” item21” is a member of the set of two perpendicular line segments, students did not admit the example in “item21” as perpendicular because of the nonequal length line segments. This situation showed that they treated an unnecessary condition as if it is a necessity for the perpendicularity under the influence of partial concept image, which case an undergeneralization error. Perpendicularity vs. perpendicular bisector . A few students’ (n=5) example spaces involved a pell-mell about the concepts of perpendicularity and perpendicular line segments. For this reason, they thought that perpendicular two line segments have to form perpendicular bisector. As a result, they treated the example in “item20” as a non4–304
PME40 – 2016
Ulusoy example of perpendicular line segments. In such situations, the students do not have a wrong concept image of perpendicularity, but they made an unnecessary restriction due to the limitations in their example spaces. CONCLUSION This study aimed to examine the role of students’ example spaces on their example generation and example determination of two parallel and perpendicular line segments. The results of this study revealed that there are different limitations in students ‘examples spaces related to parallel and perpendicular line segments. For instance, student-generated examples generally showed the striking influence of students’ limitations arising from partial concept images based on prototypical examples. Besides, students’ responses in example determination task mostly allowed seeing students’ limitations originating from overgeneralization and undergeneralization errors. For example, many of middle school students are unable to see parallelism in two vertical parallel line segments, and to see crossing of non-parallel line segments when making an extension. Another important result showed the role of students’ limitations in example spaces of perpendicular line segments because they generated incorrect or partial correct examples related to the perpendicularity of two line segments due to the mixing of perpendicularity and verticality. They generally tended to treat examples of perpendicular line segments as that of non-examples due to the inadequate knowledge about the meaning perpendicularity, median, and perpendicular bisector. Some results resemble similarities with Gutierrez and Jaime’s (1999) study in which they examined preservice primary teachers’ understanding of the concept of altitude of a triangle. Students’ limitations in example spaces can be related to their mathematics teachers’ choices of examples in the instruction of the concepts. Since teacher choices of example either facilitate or impede students’ example spaces, I recommend that future studies should concentrate on teachers’ choices and treatment of examples related to perpendicularity and parallelism. Furthermore, the catalogue I prepared to show students’ limitations in example spaces of parallelism and perpendicularity of two line segments can be utilized in prospective teacher education programs to show the boundaries of students’ example spaces about parallelism and perpendicularity. Educators can give opportunities prospective teachers to analyse students’ examples. This kind of an analysis can provide prospective teachers with insights when they become teachers with the responsibility to teach these concepts to their students. Thus, they can have a chance to expand and enrich their students’ example spaces beyond the prototypical and more-accessible examples to more sophisticated examples (Zaslavsky & Zodik, 2014) by purifying students’ overgeneralization and undergeneralization errors (Zodik & Zaslavsky, 2008). Additionally, further studies can examine learners’ determination process of properties of quadrilaterals or slope of lines and in making proof and argumentation processes by selecting participants who have limited example spaces of perpendicularity/parallelism by referencing the catalogue. Finally, I suggest that it may be useful to ask students compare their examples in the classroom to enrich their example spaces. PME40 – 2016
4–305
Ulusoy References Bills, L., Dreyfus, T., Mason, J., Tsamir, P., Watson, A., & Zaslavsky, O. (2006). Exemplification in mathematics education. In J. Novotná, H. Moraová, M. Krátká, & N. Stehlíková (Eds.), Proceedings of the 30th conference of the international group for the psychology of mathematics education (Vol. 1, pp. 125-154). Prague, Czech Republic. Gutiérrez, A., & Jaime, A. (1999). Preservice primary teachers' understanding of the concept of altitude of a triangle. Journal of Mathematics Teacher Education, 2(3), 253-275. Hershkowitz, R. (1990). Psychological aspects of learning geometry. In P. Nesher, & J. Kilpatrick (Eds.), Mathematics and Cognition (pp. 70–95). Cambridge: Cambridge University Press. Klausmeier, H.J. & Allen, P.S. (1978). Cognitive Development of Children and Youth: A Longitudinal Study. New York: Academic Press. Moore, R. C. (1994). Making the transition to formal proof. Educational Studies in Mathematics, 27, 249– 266. Sağlam, Y., & Dost, Ş. A qualitative research on example generation capabilities of university students. International Journal of Science and Mathematics Education, 1-18. Skemp, R. R. (1971). The psychology of learning mathematics. Harmondsworth, UK: Penguin Books, Ltd. Tall, D. O., & Vinner, S. (1981). Concept image and concept definition in mathematics, with special reference to limits and continuity. Educational Studies in Mathematics, 12(2), 151– 169. Ulusoy, F. (2015). A meta-classification for students’ selections of quadrilaterals: the case of trapezoid. Paper presented at the meeting of the 9th Congress of the European Society for Research in Mathematics Education (CERME-9), Prague, Czech Republic. Watson, A., & Mason, J. (2005). Mathematics as a constructive activity: Learners generating examples. Mahwah, NJ: Lawrence Erlbaum. Zaslavsky, O., & Zodik, I. (2014). Example-generation as indicator and catalyst of mathematical and pedagogical understandings. Y. Li et al. (Eds.), Transforming Mathematics Instruction: Multiple Approaches and Practices, Advances in Mathematics Education, (pp. 525-546). Springer International Publishing. Zazkis, R., & Leikin, R. (2007). Generating examples: From pedagogical tool to a research tool. For the Learning of Mathematics, 27(2), 15–21. Zodik, I., & Zaslavsky, O. (2008). Characteristics of teachers’ choice of examples in and for the mathematics classroom. Educational Studies in Mathematics, 69(2), 165-182.
4–306
PME40 – 2016
