A meta-classification for students' selections of quadrilaterals: The case of trapezoid

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A meta-classification for students’ selections of quadrilaterals: The case of trapezoid

Fadime Ulusoy

To cite this version:

Fadime Ulusoy. A meta-classification for students’ selections of quadrilaterals: The case of trapezoid.

CERME 9 - Ninth Congress of the European Society for Research in Mathematics Education, Charles University in Prague, Faculty of Education; ERME, Feb 2015, Prague, Czech Republic. pp.598-604.

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A meta-classification for students’ selections of quadrilaterals: The case of trapezoid

Fadime Ulusoy

Middle East Technical University, Faculty of Education, Ankara, Turkey, fadimebayik@gmail.com

This study aimed to propose a meta-classification for middle school students’ selections of quadrilaterals in terms of trapezoid. Data were collected from thirteen seventh grade students via a trapezoid selection instru- ment and semi-structured interviews. Data analyses were executed by using thematic coding of qualitative research methods by synthesising past and current the- ories about teaching geometry (figural concepts, concept image-definition, prototypical phenomenon etc.). Meta- classification was characterized considering the types of students’ selections, concept images, and errors through seven special categories. This meta-classification shad- ed light on how students conceive a shape as trapezoid, what they notice their selection procedure and what influences their selections for trapezoids.

Keywords: Concept images, figural concepts, prototypes, quadrilaterals, trapezoid.

INTRODUCTION

NCTM (2000) implies the importance of analyzing characteristics and properties  of two-and three-di- mensional geometric figures and developing math- ematical arguments about geometric relationships.

In this regard, one of the basic topics of geometry is quadrilaterals which involve the concepts of rectan- gle, square, rhombus, parallelogram, kite and trape- zoid. Among these 2D-figures, there have been two different definitions of trapezoid in geometry text- books. While one is that a quadrilateral with exactly one pair of parallel sides, another is that a quadrilat- erals with at least one pair of parallel sides. As the for- mer one is an example of exclusive definitions (e.g.

parallelograms are not also trapezoids), latter one is a type of inclusive definitions (e.g. parallelograms are also trapezoids). Since the trapezoids high up in the hierarch of quadrilaterals, the choice of the defi- nition influences the derivation of properties both of

trapezoid and parallelograms, rectangle, rhombus- es, and squares (Usiskin & Griffin, 2008). Despite the importance of the concept, there are a few studies specifically focused on trapezoid (Manizade & Mason, 2012; Nakahara, 1995; Türnüklü, 2014). Among them, Türnüklü (2014) conducted a qualitative study to determine middle school students’ and prospective teachers’ concept images regarding trapezoid. She reached that individuals used non-critical proper- ties in non-formal and incorrect definitions and they made overgeneralizations. When considering con- ducted other studies, it can be claimed that scattered and limited literature related to trapezoid does not give a complete picture about students’ conceptions.

From this perspective, the aim of this study was to pro- pose a comprehensive meta-classification model for students’ selections regarding trapezoid. Thus, this study will put forth more complete and well-coordi- nated structure shedding light on how students select a shape as trapezoid, what they notice their selections and what influences their selections for trapezoids.

THEORETICAL BACKROUND

In this study, it is necessary to clarify the meaning of essential terms that used when classifying and ex- plaining students’ selections, statements, and draw- ings. All the required terms within the current study were described in the following.

Concept Image/Definition: Concept image is the set of

all the mental representations associated in the stu-

dents’ mind with the concept name. The image might

be nonverbal and implicit. On the other hand, concept

definition constitutes a form of words which are used

to specify the concept (Vinner, 1991). According to this

framework, suitable and robust interactions between

concept definition and concept image might guaran-

tee the conceptual learning rather than instrumental

ones. Unfortunately, learners do not make sense to

A meta-classification for students’ selections of quadrilaterals: The case of trapezoid (Fadime Ulusoy)

599 link between the two elements because there might

be irrelevant properties about the concept evoking in students’ mind specifically. For instance, the re- sults of some studies indicate that many of students at different grade levels have a concept image of equilateral triangle having a right angle or slanted sides of equal length (Burger & Shaughnessy, 1992;

Clements & Battista, 1992). In this sense, if students are encountered limited examples having common figural features of a geometric concept in school or other context, these examples lead to prototypes phe- nomenon (Hershkowitz,1989).

Prototypes Phenomenon: The prototype examples are usually the subset of examples that had the “longest”

list of attributes all the critical attributes of the con- cept and those specific (noncritical) attributes that had strong visual characteristics” (Hershkowitz, 1990, p. 82). Students often see figures in a static way rather than in the dynamic way that would be necessary to understand the inclusion relations of the geometri- cal figures (de Villiers, 1994). For instance, students receive square is not a rectangle because of their mis- conception about the length of the opposite sides of rectangle. Consequently, a contradiction between con- cept images and concept definitions emerges, which may elicit misconceptions in students’ mind when classifying quadrilaterals. In the current study con- text, although a student has learned trapezoid as quad- rilaterals with at least one pair of parallel sides, she/

he may not admitted parallelogram, rhombus, square, and rectangle as a trapezoid, which clearly asserts the influence of prototype examples on relationship between concept image and concept definition that was structured in student’s mind.

Personal/formal figural concepts: Geometrical con- cepts are characterized as having double nature by two aspects: figural and conceptual (Mariotti & Fischbein, 1997; Fischbein, 1993) similar to the concept image and concept definition (Vinner, 1991) respectively.

While figural aspect involves spatial properties like shape, position, and magnitude; conceptual aspect involves abstract and theoretical nature as ideality, abstractness, generality and perfection. According to Fischbein (1993), figural aspect is generally more dominant than conceptual one. For example, paral- lelograms do not look like a trapezoid, but they are formally trapezoids considering the formal exclusive definition of trapezoid in our context. Based on these ideas, Fujita and Jones (2007) proposed the ideas of

personal and formal figural concepts. Formal figural concepts involve formal concept images and defini- tions in Euclidian geometry. However, personal fig- ural concepts were constituted through individuals’

own geometry learning experiences about geometric shapes. For instance, “rectangle is a parallelogram with four right angles” is a formal figural concept definition. Besides, the expression of “a rectangle is a quadrilateral with only opposite sides congruent and four 90

angles” in a student’s mind reflects student’s personal figural concept.

Undergeneralization and Overgeneralization: Two types of common errors that are exhibited by students have been described in the literature as undergener- alization and overgeneralization (Klausmeier & Allen, 1978). Undergeneralization occurs when examples of a concept are encountered but are not identified as examples. It results when the examples provided in instruction are not sufficiently different from one another in the variable attributes (Klausmeier &

Allen, 1978, p. 217). In the context of this study, for example, a student who has experienced only right trapezoids having exactly one pair of parallel sides may not identify trapezoids not having right angle even it has exactly one pair of parallel sides. On the other hand, overgeneralization occurs when examples of other concepts treated as members of target con- cept (Klausmeier & Allen, 1978, p. 217). In the current context, a quadrilateral having no parallel sides and non-equal length of sides or a polygon having more than four sides may be treated as trapezoid because of some reasons as omitting key properties of the con- cept or focusing only language-related factors.

METHOD

In this study, basic qualitative design methods were utilized to classify students’ selections of trapezoids such as semi-structured interviews for data collection and thematic coding for the data analysis.

Participants

Firstly, an elementary school located in the capital city of Turkey was selected in order to determine the par- ticipants. At the school, there were two seventh grade classes with 47 students in total. After determining the school, I had an interview with mathematics teachers of the both classes to get information about students’

mathematics grades and personal characteristics (e.g.,

talkative). Furthermore, each class was observed for

four hours in order to monitor students’ behaviors.

Based on the maximum variation sampling, I conduct- ed semi-structured interviews with 13 seventh grade students aged thirteen who were enumerated from S1 to S13. Their achievement levels were categorized according to their average math note belonging to the first and second semester. Semester notes were cate- gorized as 5-5 was high; 5-4, 4-5 4-4, 3-4 and 4-3 were middle; and 3-3 and lower ones were low. According to semester notes, three students (S1, S2 and S3) were attended low achievement level. Five students (from S4 to S8) were attended middle achievement level and five students (from S9 to S13) were attended to high achievement level.

Instrument and data collection procedure To understand how students select trapezoids in var- ious polygons, researcher generated an instrument of “Which figures are trapezoid?”(see Figure 1). The instrument was organized in a manner of containing prototype and non-prototype polygons having differ- ent sizes and orientations. While creating the instru- ment, similar questionnaires about parallelogram in the literature were analyzed (Fujita, 2012; Nakahara, 1995; Okazaki, 1995). Then, a preliminary study was conducted with 86 seventh grade students to under- stand what they think about trapezoid concept. In the preliminary study, they only defined trapezoid and drew three different trapezoids in a grid paper. After examining the results of the preliminary study, sev- en possible selection categories were formed (Note:

Details were explained in the heading of characteris-

tic features of meta-classification). According to seven possible categories, figures were added into the data collection tool. Finally, data collection tool was con- trolled by experts and pilotted with five seventh grade students throughout the interview sessions. Before the selection procedure, participants also asked to make definition and different drawings for trapezoids to support data coming from students’ selections of trapezoid.

Data were collected via semi-structured individual in- terviews for in-depth analysis. Average time of an in- terview was twenty minutes. During interviews, par- ticipants explained their selections and the reasons why they select a figure as a trapezoid. Furthermore, they made definition and drawing of trapezoids. All interviews were videotaped and transcribed. To an- alyse the data, the researcher carefully examined the students’ selections, drawings, and definitions. Then, thematic analysis was used to identify, analyse and report the themes in the data. For this purpose, all data were examined by taking account the phases of familiarization with data, generating initial codes, searching for themes among codes, reviewing themes, defining and naming themes, and producing the final report (Braun & Clarke, 2006).

Characteristic features of meta-classification To propose a comprehensive meta-classification for students’ selections of quadrilaterals in terms of trapezoid, I formed seven specific categories after the thematic analysis (see Table 1). The types of cate- gories in meta-classification model were characterized considering the types of errors, and concept images with the correctness of students’ se- lections, drawings, and definitions.

To be more precise, the categories show the way in which students made to discriminate trapezoids in various polygons. Error types reflect whether students’ errors have a character of undergeneralization or overgeneralization for each selec- tion category. For instance, students either made undergeneralization by focusing only right trapezoid and exclusive selections or they made overgeneralization by selecting quadrilaterals with no parallel sides and irregular polygons. Since types

Figure 1: The instrument of “Which figures (1–16) are trapezoid?”

A meta-classification for students’ selections of quadrilaterals: The case of trapezoid (Fadime Ulusoy)

601 of the students’ concept images changed throughout

the categories, it was necessary to differentiate the concept images for each category. All other details were given in the following paragraph.

Category 1 reflects learners’ correct selections of trap- ezoid according to hierarchical (inclusive) relation among quadrilaterals . This category indicates an ide- al situation including a harmony between the two aspects of a figural concept (Fischbein, 1993). In other words, learners’ personal figural concepts were com- pletely and correctly consistent with formal figural concepts (Fujita & Jones, 2007). In category 2, learn- ers can connect the correct relations between some quadrilaterals; however, they make their selections by considering partial hierarchical relations (e.g. they only select parallelograms as a trapezoid rather than selecting rhombus, rectangle etc.) even they know the geometric concept in terms of conceptual aspect.

This situation supports the idea about the domination of figural aspect (Fischbein, 1993) which can be a re- sult of encountering prototypes in learning process (Hershkowitz, 1990). Category 3 reflects students’ pure prototypical images because they select only all proto- typical examples of trapezoids according to exclusive definition even they know the inclusive definition of trapezoid in written or verbal form. As a result, they do not select parallelogram, rhombus, rectangle and square as a trapezoid, which reflects undergen- eralization error. According to the students, a square does not figurally a trapezoid, which reflects a conflict that may appear between the figural and the formal constraints as indicated in Fischbein and Nachlieli’s study (1998). In a similar vein, category 4 reflects stu- dents’ partial prototypical concept images about trap- ezoid because they think only prototypical examples such as isosceles/right trapezoids, which leads the error of undergeneralization. In summary, catego-

ry 2-3-4 show students’ personal figural concepts of trapezoid consist of correct concept definition and limited images. In category 5, learners make incorrect selections considering only similarity of the shapes in terms of position and visual appearance without thinking formal definition and critical attributes of the figure. This situation causes overgeneralization error because they treat irrelevant figures as a trap- ezoid. Category 6 reflects students’ language-based images because they think the meaning of the word of “trapezoid” in Turkish ordinary language with the meaning of “oblique”. As a result, they image trape- zoid as a figure having more than 4 sides. In other words, they treat some non-examples as examples by extending their knowledge to another context in an inappropriate way, which indicates overgeneraliza- tion error. Finally, category 7 reflects the consistency of students’ selections based on the contradiction in their between definition, drawings and selections. To sum, category 5-6-7 indicate students’ personal figur- al concepts of trapezoid consist of incorrect concept image and concept definition.

RESULTS AND DISCUSSION

In a general sense, results of the current study indi- cated that although higher level students generally selected shapes according to exclusive relations of quadrilaterals nobody made a selection on inclusive relations without a mistake. In other words, there were no students classified in category 1&2 in which learners require connecting even partially hierarchi- cal relations among quadrilaterals.

Category 3&4: Generally higher level students (S7- S12-S13) selected trapezoids based on exclusive rela- tions of quadrilaterals and they attended in category 3. As a result, they did not think that parallelogram,

Category Correctness Image Types Error types

Cat-1 Correct Hierarchical No error

Cat-2 Correct and incomplete Partial hierarchical Undergeneralization

Cat-3 Pure prototypical

Cat-4 Partial prototypical

Cat-5

Incorrect

Visual Overgeneralization

Cat-6 Language-based

Cat-7 Contradictive

Table 1:* Classification of students’ selections for trapezoid

*This table reflects the selections of students who learn inclusive definition of trapezoid in their lessons.

rhombus, rectangle and square were also a trape- zoid, which showed undergeneralization that stu- dents made with pure prototypical concept images.

Students explained the selection procedure as below:

S12: (By referring prototype trapezoid shape) we drew trapezoid as a quadrilateral having only two parallel sides in our lessons.

S13: I remembered trapezoid as a shape hav- ing different angles and sides but top and bottom sides must be parallel.

On the other hand, two middle level students’ an- swers (S5 and S8) were placed in the fourth category.

Students chose only figures of 1 and 14 as trapezoids.

They stated that there must be one pair of parallel sides in a trapezoid and they added unnecessary con- ditions as having two right angles. To understand their pre-existing concept images, I asked them to construct and to define trapezoid before the selec- tion procedure. Their definitions and drawings also indicated that they had a concept image of trapezoid as a right trapezoid, which was the presence of the undergeneralization due to the restricted prototyp- ical concept image development.

Specifically, the selections attended category 3&4 showed students’ limited understanding based on prototypical concept images of trapezoid (Fujita, 2012; Hershkowitz, 1990). Yet, their selections were interestingly based on exclusive definitions although they learned inclusive definition of trapezoid in their lessons. As a result, students tried to explain the con- cept based on critical attributes less than required.

In other words, they made undergeneralization. The reason may be associated with the influence of giv- ing prototype examples of trapezoid in the classroom and textbooks. In the textbooks, while trapezoid was defined a quadrilaterals with at least one pair of par- allel sides, prototype trapezoid shapes were heavily given to illustrate the concept. Additionally, when the

researcher examined students’ notebooks to under- stand how their teachers explained trapezoid, it was received that teachers gave only prototype trapezoids.

To prevent the formation of prototype concept images, it is recommended that teachers need to focus on the definitions by giving examples (e.g. quadrilaterals with at least one pair of parallel sides) and non-exam- ples (e.g. five-sided shape or a quadrilateral with no parallel sides) of the concept (Bills et al., 2006; Petty

& Jansson, 1987).

Category 5: Different from the situation in category 3&4 students (S9, S10 and S11) having high achieve- ment level selected the quadrilaterals with no par- allel sides as trapezoid in addition to all prototypes trapezoid shapes in category 5. Students noticed only similarity of the shapes in terms of position and visual appearance rather than having at least one pair of parallel sides. Additionally, when researcher asked them to explain the reasons why they selected that shapes as trapezoid, their explanations indicated that they also perceived trapezoid as a quadrilater- al having non-equal sides. Students’ concept images were limited only visual representation of trapezoid with personal definitions rather than a formation of formal definition and properties. In this way, they reached overgeneralization for special cases by ex- tending their information to special cases in an inap- propriate way rather than considering the parallelism property as a critical attribute. At this point, Chazan (1993) proposed various reasons for overgeneraliza- tion in mathematics such as use of an insufficient set of examples, prevalent concept images and beliefs.

To overcome students’ overgeneralizations, the defi- nition and critical attributes of the shape should be stressed. In this regard, using geometry software might be useful since they involve dynamic manip- ulations such as dragging that preserves critical at- tributes of the shape in the hierarchical perspective (Erez &Yerushalmy, 2006).

Category 6: Lower (S2-S3) and middle (S4-S6) level stu- dents selected an irregular polygon (the shape of 16)

Figure 2: Students’ drawings for the trapezoid in category 6

A meta-classification for students’ selections of quadrilaterals: The case of trapezoid (Fadime Ulusoy)

603 as a trapezoid with language-based concept images,

which was supported by the results of similar nation- al studies (Erşen & Karakuş, 2013; Türnüklü, 2014).

Students in this category were not able to identify attributes of a trapezoid. Students’ drawings were given in Figure 2 to reflect their concept images about trapezoid.

Furthermore, students written and verbal definitions indicated that they did not know the formal defini- tion and properties of trapezoid (e.g. the number of sides) of a trapezoid. In Turkish language, the word of

”yamuk” is used instead of “trapezoid” in all textbooks and teachers’ instruction. However, “yamuk” is syn- onym and also means “oblique” in Turkish ordinary language. As a result, they made drawings under the influence of linguistic factors rather than focusing on definition and properties of the concept. Because lan- guage influences students’ knowledge, ability and im- ages about mathematical concepts (Monaghan, 2000;

Silfverberg & Matsuo, 2008) teachers and curriculum developers must be more careful about whether they use both necessary and sufficient mathematical and linguistic structure for definitions of the concepts.

Category 7: This category showed the complexity of a lower level student’s (S1) concept image and concept definition of trapezoid. Student’s selections, defini- tion and drawing of trapezoid contradicted them- selves. Although student appeared to make a partial connection with inclusive relations by selecting par- allelograms and rectangles as trapezoid, her drawing and definition of trapezoid revealed the imperfect nature of concept images in her mind.

As seen in Figure 3, although student stated there are no parallel sides of trapezoid, she drew a figure having a pair of parallel sides, but | AB | ∕∕ | DC | , and selected all parallelograms and rectangles as trapezoids, which showed student’s insufficient knowledge about pre- requisite concepts such as parallelism of two lines.

Moreover, she drew a figure having four sides, but she selected irregular polygons having more than four sides. As a result, it can be inferred student had contradictive concept images and she did not know

what trapezoid and its properties are. This result has reflected the importance of having prior knowledge about basic geometric concepts. Students’ inadequate knowledge on geometric concepts may influence their concept images about quadrilaterals because many properties of quadrilaterals are based on basic geo- metric concepts such as parallelism and perpendic- ularity (Monaghan, 2000). For this reason, teachers should give more attention to whether their students obtained all required prior knowledge before intro- ducing a new mathematical concept. As a final point, in-depth analysis of this study was limited to exami- nation of thirteen seventh grade students’ trapezoid selections. However, the idea of meta-classification of learners’ selections can be applied and extended to other classes of quadrilaterals. Thus, the similarity and difference between the structures of meta-clas- sification model may be analysed on the basis of dif- ferent quadrilateral concepts. Furthermore, it will be more interesting to compare methods of teaching quadrilaterals classification in different mathematics curriculum.

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