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672Unlu & HorzumPrism is a concept that individuals meet a lot in their daily lives, it has an important place among geometricconcepts (Gökbulut & Ubuz, 2013). However, teacher candidates have difficulties in giving definitions (Bozkurt& Koç, 2012; Ubuz & Gökbulut, 2015; Gökkurt & Soylu, 2016), distinguishing critical characteristics (AlkışKüçükaydın & Gökbulut, 2013; Ubuz & Gökbulut, 2015; Gökkurt & Soylu, 2016) and naming (Çetin et al.2012; Horzum & Ertekin, 2018) according to the researches where their knowledge about prism is investigated.In their studies Bozkurt and Koç (2012) shows that MTCs make mistakes when they give definition for prismconcept. Gökbulut and Ubuz (2013) have found that MTCs do not know necessary and enough logicalprinciples for definition of prism, they cannot choose necessary and sufficient prism characteristics to form adefinition and they cannot distinguish its critical characteristics. Horzum and Ertekin (2018) indicate that MTCscall a square prism which stands on its square surface on a desk as square prism. But they call the same prism arectangular prism when it stands on its rectangle surface. Besides, MTCs have misconceptions about prism(Alkış-Küçükaydın & Gökbulut, 2013; Altaylı et al., 2014). Altaylı and her colleagues (2014) asked MTCswhether the new object is a prism or not when a prism is bended and the results show that 76.6% of MTCsfound out that new figure was a prism even if the angles between the corners were right and 12% of MTCssuggested that the new figure was not a prism if the angle was not right. On the other hand, Çetin and hiscolleagues (2012) has investigated the conceptions of rectangular prisms of MTCs with the help of models andthe results show that they have difficulty in naming the figures in skeleton shape and recognizing thecharacteristics about these models.In addition, although pyramid is a concept with a definition and samples that individuals meet often in theirdaily lives, it is difficult to be perceived (Ubuz & Gökbulut, 2015). In related investigations teacher candidateshave difficulty in giving definitions and distinguishing its critical characteristics. Ubuz and Gökbulut (2015)indicate that teacher candidates have difficulty in giving a mathematical definition about pyramid concept andtheir pyramid drawings are generally limited to prototype samples. Also Altaylı and her colleagues (2014) havefound out that teacher candidates have misconceptions in definitions of prism, pyramid and cone concepts andthey do not know the concepts of truncate cone, truncate pyramid at all. Çakmak and her colleagues (2014) havehandled MTCs’ subject matter knowledge of three dimensional solids (prism, pyramid, cylinder, cone, sphere)conceptually and have investigated three dimensional solids under drawing, defining, sampling, reorganizingand spatial thinking categories. The results show that MTCs have some difficulties in defining and recognizingproblems in their subject matter knowledge about three dimensional solids. While defining them, MTCs made ageneral definition instead of giving a mathematical definition and they gave visual-oriented answer forrecognizing solids’ characteristics. Moreover Marchis (2012) stated that some students can’t recognize basicgeometrical shapes and solids.As a result of literature review, prism and pyramid concepts are so important that it is necessary to investigatethe difficulties about these concepts. The person who will teach these concepts has to learn them perfectly inorder not to cause any misconception or contradiction in terms for learners (Çakmak et al., 2014). Therefore,investigating the differences between prism and pyramid concepts definitions, which is not mentioned in theliterature, and how MTCs define them will reveal their content knowledge on prism and pyramid concepts. It isthought that it is important to know the difference between these two concepts and not to be confused. BecauseMTCs need to know the critical characteristics of the concepts and to think about the definitions so that they candetermine the difference between these two concepts. Therefore trying to determine how the MTCs handle thedifference between these two concepts and how they define these concepts offers a unique advantage to thefield. Besides, knowing MTCs’ ideas about prism and pyramid concepts and the differences between them cangive clues about how they will teach in future.Problem of the ResearchThis study analyses how the mathematics teacher candidates understand the “prism” and “pyramid” concepts ingeometry. Therefore, the following questions are investigated:1. How do MTCs define a prism?2. How do MTCs define a pyramid?3. How do MTCs define the differences between prism and pyramid?

Int J Res Educ Sci673MethodResearch DesignIn the current study, the case study method based on qualitative approach was used. A case study investigates agroup or event deeply (Merriam, 2013). In this research, the case study method was chosen because MTCs’definitions of prism and pyramid was investigated by focusing in depth and the case was the selected group ofMTCs to determine their definitions related to prism and pyramid.ParticipantsCriterion sampling method, which is one of the purposeful sampling methods, was used in the research. In thismethod, researchers choose participants according to specific criterions (Patton, 2002). The acceptable criterionin this research is that MTCs should have taken “Geometry” and “Instructional Technologies and MaterialDesign” courses which were taken for 14 weeks. According to this, MTCs took the Geometry course at theirfirst academic year, and Instructional Technologies and Material Design course at their second academic year.Geometry course includes activities and theories about the line, plane and space in Euclid geometry, angles,triangles and quadrilaterals, circle, solid objects, and geometric place. Instructional Technologies and MaterialDesign course includes activities and theories about learning and teaching technology, teaching tools andmaterials, concept maps, computer based training taking into account the mathematical concepts. The thirdcriterion was that MTCs volunteered to participate the study and their names were not used, but instead S 1, S2,S3, .S92 pseudonyms were used. The reason of choosing this MTCs is the conjecture that they have enoughknowledge as institutional and agreeable with geometry concepts investigated. In this sense, this research wascarried out in 2017-2018 academic year with 92 MTCs, 47 of them were 3rd graders and 45 of them were 4thgraders in mathematics teaching department of a university in Central Anatolian Region in Turkey.Data Collection ToolA test for data collection was developed by the researchers. Test was used to reveal definitions of MTCs aboutprism and pyramid concepts. This test consists of 3 open-ended questions which can be answered conceptuallyby the participants. The questions in this test were checked by an expert with PhD in mathematics education interms of correctness of the language and pedagogical-academic accuracy and then, necessary corrections weremade. So, the first open-ended question became as “Define a prism”, the second became as “Define a pyramid”and the third became as “Explain the differences between prism and pyramid”. After data analysis, semistructured interviews were held with 11 MTCs who were randomly selected or who had ineligible expressions intheir definitions. In the interviews, the question “What did you want to say here? Can you explain?” was askedto MTCs who had ineligible points in their definitions. Besides they were also asked to give examples and drawthree different prisms and pyramids with the questions such as “Can you draw 3 different prisms/pyramids?Which peculiarities make them prisms/pyramids?”In the current study open-ended questions were used as the main source of data to determine participants’definitions of prism and pyramid concepts. The documents obtained from the drawings related to the prism andpyramid concept and semi-structured interviews were used as supporting data. And through this datatriangulation structural validity of the current study was obtained.Data Collection and AnalysisDuring data collection process MTCs were told that this test was not an examination and they were asked togive detailed and clear answers. Besides, there was no time limitation when they were answering. MTCs werealso asked to write their names as they would be interviewed in future. Besides, they were told to use theirknowledge in their minds and they were not allowed to use any sources such as internet, book etc. Before thedata analysis, documents, interview audio records and answers of open-ended questions were numberedseparately for each participant. Thus, the relations between the data (obtained from open-ended questions,documents, interviews) could easily be seen and also reliability and validity of data analysis could be controlled.Besides, definitions of prism and pyramid in literature were investigated and in this direction, criticalcharacteristics of prism and pyramid concepts were determined. The following prism and pyramid definitionswere used in this study. A prism is defined as “a polyhedron with two parallel, congruent polygon bases, and

674Unlu & Horzumlateral faces that are parallelograms” (Schwartzman, 1994, p.172). A pyramid is defined as “a polyhedronobtained by connecting each vertex of a polygon to a single point outside the plane of the polygon; the slopingsides of a pyramid are therefore triangles” (Schwartzman, 1994, p.177). According to this, the criticalcharacteristics for prism were determined as follows; PR1. Identical (two) bases, PR2. Parallel bases, PR3.Polygon bases, PR4. Parallelogram lateral faces. The critical characteristics for pyramid were determined asfollows; PY1. Polygon bases. PY2. Triangular lateral faces, PY3. It has an apex. And also the criticalcharacteristics between prism and pyramid concepts were defined according to the critical characteristics ofthese two concepts. The criterias were determined as follows; a. The number of bases: Pyramid has 1 and prismhas 2 (identical) bases; b. Bases are parallel: Prism has, but there is not for pyramid because it does not have twobases; c. Characteristics of lateral faces: Lateral faces are triangle on pyramid and parallelogram on prism; andd. Existence of an apex: Pyramid has an apex but prism does not have it.The answers of MTCs were investigated according to correctness and generalization criteria (Gökbulut & Ubuz,2013; Ubuz & Gökbulut, 2015; Zaskis & Leikin, 2008) and obtained data were arranged and interpreted inaccordance with the research questions. The categories for defining prism and pyramid were investigatedthrough both correctness and generalization criteria. However, the category of explaining the differencesbetween prism and pyramid was investigated through only correctness criterion. Descriptive analysis wasperformed about definitions and explanations of MTCs. The data were classified as definition having one, a fewor all critical characteristics. And then, they were categorized according to correctness and generalizationcriteria. The contents of these criteria are as the followings:Correctness criterion: In the analysis performed with this criterion which belongs to Zazkis and Leikin (2008).When examining the correctness of the definitions provided by the MTCs, it was distinguished as correctanswer, insufficient answer and wrong answer as in the study of researchers (Gökbulut & Ubuz, 2013; Ubuz &Gökbulut, 2015) for prism, pyramid and difference between these concepts. Then the definitions were classifiedas correct answer that had all critical characteristics and included necessary and sufficient conditions. In thisstudy, the correct answer for the prism that include all critical characterics couldn’t be found. On the other hand,the definition for the pyramid as “three dimensional figure with a base in polygon shape, with triangle lateralfaces whose upper corners unite in one point” was evaluated as the correct answer.Besides the definition handled by MTCs were classified as insufficient answer that had some missing criticalcharacteristics or even non-critical characteristics and included logical structure of necessary but insufficient orsufficient but unnecessary conditions. For example, the definition for prism as “a closed three dimensionalgeometric shape whose base and ceiling (meaning the top base) consist of similar geometric forms and lateralfaces are rectangle” has non-critical characteristics and some missing critical characteristics. So this definitionwas coded as insufficient answer.Lastly in the correctness criterion, the definitions handled by MTCs were classified as wrong answer that haddefinition without any critical characteristics and included logical structure of neither necessary nor sufficientconditions. For example, the definition for pyramid as “three dimensional object having corners” does notinclude any necessary and sufficient conditions.Generalization criterion: When examining the generality of the definitions provided by the MTCs, it wasdistinguished as in some researches like special answer, general answer (Gökbulut & Ubuz, 2013; Ubuz &Gökbulut, 2015; Zaskis & Leikin, 2008) and quasi-special answer (Gökbulut & Ubuz, 2013) for prism, pyramidconcepts. Accordingly, this criterion was used to investigate whether a definition comprises all criticalcharacteristics or it was a general definition comprising all other concepts. The definition with any criticalcharacteristics were classified as “general answer”. The following examples can be given for the generalanswer. For example, the definition for the prism as “three dimensional geometric figure composed of width,depth and height” and the definiton for the pyramid as “three dimensional objects which have height, volume,side, corner” can be any geometric object. Besides, the definitions which define prism/pyramid or comprisesome critical characteristics were classified as “quasi-special answer”. The following examples can be given forthe quasi-special answer. For example, the definiton for prism as “three dimensional geometric figures whosebottom and top bases are same polygons and four lateral faces are rectangle” and the definition for pyramid as“a structure whose bottom base consists of regular quadrilaterals and which has an apex” have some criticalcharacteristics. Lastly all definitions comprising all critical characteristics of prism/pyramid were classified as“special answer”.To achieve the reliability for data analysis coding processes were performed by two researchers separately. Thenthe points of “agreement” and “disagreement” were discussed and necessary changes were made. Then the first

Int J Res Educ Sci675and the second coding results were united and consistency percentage through Miles and Huberman (1994)Realibility Agreement/ (Agreement Disagreement). At the end 88% agreement was achieved. The finalcategorizations were produced by resolving the disagreement through consulting with two researchers who havePhD in mathematics education. In order to increase external reliability of the study, all the processes of the studywere detailed and supported with the quotations of the MTCs. In order to increase internal reliability of thestudy, findings were presented directly without any interpretation.FindingsIn this section, according to the data analysis, the definitions of MTCs were sampled according to determinedcategories and interpreted through frequency table.Definitions of MTCs about Prism ConceptDefining prismThe definitions of MTCs about prism concept were investigated through both correctness and generalizationcriteria. For the correctness criterion, the definitions were classified taking into account the criticalcharacteristics for prism defined as identical two bases (PR1), parallel bases (PR2), polygon bases (PR3), andlastly parallelogram lateral faces (PR4). In this context, prism definitions were classified under the titlesincluding one or more of these critical characteristics (Table 1).Table 1. Definitions about prism conceptCriteriaThemeCorrect answerPR1 PR2 PR3 PR4PR1PR2PR3PR4CorrectnessInsufficient answerPR1 PR2PR1 PR3PR2 PR3PR1 PR4*PR1 PR2 PR3Wrong answerSpecial answerGeneralization Quasi-special answerGeneral 1.0913.041.0945.6554.3545.65*MTCs touch upon the point that lateral faces of a prism are quadrilateral or rectangle.According to correctness criterion in Table 1, it can be seen that none of the MTCs could define prism correctly.However, 54.34% of MTCs mentioned one or a few critical characteristics of prism in their definitions that wereaccepted as insufficient answer. In these definitions, MTCs touched upon “identical bases and rectangular orquadrilateral lateral faces” (13.04%) criterion most. What attracts attention on this theme is that most of theMTCs touched upon lateral faces being rectangle and the rest emphasize the lateral faces being quadrilateral.For example, S55 defined prism as “It is a closed shape whose base (bottom) and ceiling (meaning the top base)have the same polygon and lateral faces are quadrilateral” and S52 defined prism as “It is a closed threedimensional geometric shape whose base and ceiling consist of similar geometric forms and lateral faces arerectangle”. In interview, S52 was asked to give example to represent her prism definition and she drew triangle,square and rectangular right prisms that stood on their bases as shown in Figure 1.Figure 1. Prism drawings of S52

676Unlu & HorzumThe second insufficient answer was the criterion “identical and polygon bases” (11.96%) that MTCs touchedupon. Through this theme some MTCs emphasized that bases were generally polygon but some showed theyhad a limited prism knowledge specifying the bases as regular polygons. For example, S75 indicated that “Prismis a three dimensional figure whose base and ceiling are identical regular polygon”. Third, MTCs were handledequally (7.61%) the citeria “identical bases” and “identical and parallel bases” as the insufficient answer. Forexample, S82 defined prism as “a three dimensional figure whose base and top base have the same shape” andS72 defined it as “objects whose bottom and top bases are parallel and in identical forms”. Fourth, some MTCs(6.52%) emphasized in their prism definitions that only bases are polygon. One of the MTCs, S24, emphasizedthat the bases were polygon on prism with her definition as “an object having polygonal base and ceiling, andheight”. As the fifth insufficient answer, few MTCs (3.26%) mentioned in their definitions that only lateralfaces were parallelogram (especially rectangle). For example, S 32 saying that “a three dimensional figure that

call a square prism which stands on its square surface on a desk as square prism. But they call the same prism a rectangular prism when it stands on its rectangle surface. Besides, MTCs have misconceptions about prism (Alkı-Küçükaydın & Gökbulut, 2013; Altaylı et al., 2014). Altaylı and her colleagues (2014) asked MTCs