Vol.:(0123456789)1 3

ZDM (2018) 50:659–673 https://doi.org/10.1007/s11858-018-0964-y

ORIGINAL ARTICLE

Geometry knowledge test about triangles: evidence on validity and reliability

Behiye Ubuz1  · Utkun Aydın1,2

Accepted: 20 June 2018 / Published online: 26 June 2018 © FIZ Karlsruhe 2018

AbstractIn the present study we aimed to develop a multidimensional test assessing high school students’ knowledge about trian-gles, and then to determine the validity evidence for it based on the internal structure and relations to another variable and its reliability. The test developed was administered to 557 tenth grade students. To assess the validity evidence based on the internal structure, the data were analyzed using confirmatory factor analysis, inter-dimension correlations and two-way MANOVA across gender and school type. For validity evidence based on relations to another variable, the test scores were associated with previous semester geometry grades at the tenth grade level. Reliability was assessed using Cronbach alpha and item-total correlations to report internal consistency. The resulting instrument including 24 questions showed adequate validity evidence based on the internal structure and relations to another variable, as well as good reliability. This indicates that the results produced by the instrument are valid and reliable. So, the geometry knowledge test about triangles is well-suited for use in research and classrooms.

Keywords Declarative knowledge · Conditional knowledge · Geometry · Procedural knowledge · Reliability · Validity

1 Introduction

Triangles are polygons with three vertices and three sides. They are the basic building blocks of all two dimensional polygons. Understanding triangles helps students understand all other polygons because they can be broken down into triangles. It also helps students understand other topics in mathematics (e.g., trigonometric functions, unit circle) and science (e.g., Fourier method, force). With that understand-ing, humans can have a good grasp of how something in the world works. The full development of trigonometry, to which triangles lead to, is used to explain what we hear and see, as well as how electrons, molecules, and atoms work. Because of their rigidity, we can construct bridges, roof frames, and so forth. Thus, triangles are one of the most fre-quently addressed topics in the mathematics curriculum (see

Miyakawa 2017; Otten et al. 2014); and they are addressed frequently in studies on geometry teaching and learning.

Currently, there are available instruments used to col-lect data on students’ learning of diverse geometrical topics including triangles. Although they are assumed to provide credible and important information about what students know, there is no instrument sufficiently addressing the types of knowledge and the validity and reliability of score-based inferences. For an instrument to be scientifically robust, it must be formally developed and psychometrically tested. Thus, it is necessary to investigate its validity (i.e., whether it measures what it intends to measure) and reliability (i.e., whether it produces consistent data). This can be done by following the series of steps to secure psychometric proper-ties, laid out in the Standards for Educational and Psycho-logical Testing (AERA, APA, NCME, 2014). The Stand-ards provides five sources of validity evidence (those based on content, response processes, internal structure, relation to other variables, and test consequences) in addition to reliability.

In this study we develop a multidimensional Geometry Knowledge Test about Triangles (GKT-T) measuring high school students’ knowledge of triangles and then determine the validity evidence for it, based on internal structure and

* Behiye Ubuz ubuz@metu.edu.tr

1 Department of Mathematics and Science Education, Middle East Technical University, 06800 Ankara, Turkey

2 Present Address: The American University of the Middle East, Kuwait City, Kuwait

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relations to another variable and its reliability. Two basic aspects of internal structure are: dimensionality and meas-urement invariance. To examine dimensionality, confirma-tory factor analysis and inter-relationships among the dimen-sions are common practice among researchers (Rios and Wells 2014). For measurement invariance, it is suggested that evidence be provided that the item (e.g., item diffi-culty) characteristics are comparable across groups (Rios and Wells 2014). From this perspective, the main concern is to determine groups. Students’ knowledge in geometry seemed connected to gender. Gender differences, however, could be the result of having various advantageous features such as school types (Batyra 2017a). Validity evidence based on relation to other variables refers to whether the instru-ment scores have consistent linkages to external variables with a similar or dissimilar construct (e.g., whether geom-etry knowledge relates to prior grade in geometry). Valid-ity evidence based on test content and response processes, both of which are central to the question generation and the refinement of the test questions, are presented in the methods section of the current study.

GKT-T could be valuable for teachers and researchers as it provides them with an expedient tool for gathering information on students’ knowledge of triangles. Through the completion of this instrument, teachers can monitor stu-dents’ knowledge/thinking. This monitoring in turn could lead to modifications in the learning environments such as that involving a hypothetical learning trajectory (Simon 2014), and in teacher knowledge. Discovering what type of knowledge is deficient is a necessary first step for plan-ning remedial geometry instruction (Schunk 1996) as the deficiencies in geometrical knowledge hinder learning. For researchers, the GKT-T can be used in correlation and inter-vention studies. Correlation studies could be about investi-gating the relationships among different kinds of knowing (see Aydın and Ubuz 2010), or relationships between dif-ferent kinds of knowing and other constructs such as meta-cognition (see Aydın and Ubuz 2010) or others (e.g., spatial ability). For intervention studies, it could be used to evalu-ate the effectiveness of educational programs designed to improve geometry knowledge of students on triangles (see Ubuz and Erdogan 2018).

2 Conceptual framework and prior research

2.1 Knowledge of geometry

Research studies on students’ understanding of geometry investigated how they defined concepts and/or stated/iden-tified their properties (e.g., Burger and Shaugnessy 1986; Chinnappan et al. 2012; Fujita and Jones 2007; Fujita 2012; Gutiérrez and Jaime 1998; Mullis et al. 2000; Senk 1989;

Ubuz 2017; Ubuz and Ustun 2004; Usiskin 1982); iden-tified figures (e.g., Burger and Shaugnessy 1986; Fujita 2012; Gutiérrez and Jaime 1998; Mullis et al. 2000; Senk 1989; Tsamir et al. 2008; Ubuz 2017; Ubuz and Ustun 2004; Usiskin 1982; van Hiele 1986); drew figures or graphs (e.g., Burger and Shaugnessy 1986; Fujita and Jones 2007; Fujita 2012); sorted figures (e.g., Burger and Shaugnessy 1986; van Hiele 1986); found angles in figures (e.g., Biber et al. 2013; Ubuz 1999); classified figures hierarchically (e.g., Fujita and Jones 2007; Ubuz 2017); and stated/identified relational-rules about general objects or proved relational-rules about general objects or particular objects (e.g., Burger and Shaug-nessy 1986; Chinnappan et al. 2012; Fujita and Jones 2007; Gutiérrez and Jaime 1998; Herbst and Brach 2006; Marrades and Gutiérrez 2000; Senk 1989; Ubuz 2017; Usiskin 1982; van Hiele 1986) using mainly constructed response ques-tions taken mostly with paper- and-pencil or rarely posed in face to face interviews, depending on the number of partici-pants and questions. Constructed response questions elimi-nate random guessing and let students demonstrate their in-depth knowledge of specific content (Bridgeman 1992).

The early studies concerning geometry were mostly framed within the van Hiele theory (van Hiele 1986) and/or SOLO taxonomy (Biggs and Collis 1982), proposing a series of cognitive levels/modes through which every student passes while solving geometry tasks. Students in the iconic mode (SOLO) passing through Level 1 visualization (van Hiele) recognize two-dimensional figures by their appear-ance (e.g., “It’s a square”) and associate abstract shapes with familiar objects (e.g., “It’s like a box”). Students in the concrete symbolic mode (SOLO) passing through Level 2 analysis and Level 3 abstraction (van Hiele) are capable of identifying a single property (e.g., “a square has four sides”) or several geometric properties (e.g., “a square has four equal sides”), and, further, use the relationships between these properties (e.g., “square is a rectangle”). Students in the formal and postformal mode (SOLO) passing through Level 4 deduction and Level 5 rigor (van Hiele) are able to reason formally within the context of a complete geometric system (e.g., “proving that the three bisectors of the angles of a triangle have a point in common”). Progressing from one level to the next is dependent upon knowing-that and knowing-why. Hereby, the tasks used in the studies based on van Hiele theory (e.g., Burger and Shaugnessy 1986; Fujita and Jones 2007; Gutiérrez and Jaime 1998; Senk 1989; Usis-kin 1982) or SOLO taxonomy (e.g., Pegg and Davey 1989) were about knowing-that and/or knowing-why.

Studies of similar underpinnings of students’ know-ing about geometry can be undertaken in the cognitive domains–knowing, applying, and reasoning–included in international comparative studies. The items aligned with these domains mainly covered knowing-that (e.g., “Of the following, which is not true for all rectangles?” or “Two

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of the triangles below are similar. Which two triangles are similar?”) (TIMSS Repeat 1999) and knowing-how (e.g., “The line m is a line of symmetry for figure ABCDE. The measure of the angle BCD is”) (TIMSS Repeat 1999) in multiple content domains: number, algebra, geometry, and data and chance. In these studies, such as TIMSS, it was reported that gender seems to play an important role in the overall geometry score, favoring male (e.g., Cheng and Seng 2001; Dindyal 2008) or female (e.g., Cheng and Seng 2001; Dindyal 2008) students, or with negligible small difference between them (e.g., Louis and Mistele 2012).

As proving is an integral component of geometry for understanding the ground or reason for the acceptance of a relational rule, students’ proving abilities have been studied mainly in the context of proof schemes (see Harel and Sow-der 1998 for the definition of proof scheme). For example, while responding to the question “Is segment CB the angle bisector of ∠ACD?” (see Marrades and Gutiérrez 2000), students’ justifications were based on knowing-that (e.g., knowing properties of isosceles triangles, alternate interior angles, congruency, side angle side postulate for congru-ent triangles; identifying figures) and knowing-how (e.g., drawing lines, figures; measuring angles). Knowing-that and knowing-how seem to constitute significant components of proof schemes.

More recently, numerous studies in geometry have started to focus on students’ perception or understanding of geomet-ric figures (e.g., Gagatsis et al. 2010), particularly based on four apprehensions of a geometrical figure framework: per-ceptual, sequential, discursive and operative (Duval 1995). Each has its specific laws of organization and processing of the visual stimulus array. As geometrical knowledge is related to spatial knowledge characterizing shapes, positions, and movement (Soury-Lavergne and Maschietto 2015), knowledge related to spatial knowledge is also applicable to a wider variety of types of geometrical knowledge and can appear in any of them.

Within all those aforementioned studies students’ perfor-mances on the tasks are drawn on knowing-about (de Jong and Ferguson-Hessler 1996; Mason and Spence 1999; Ryle 1949), that is, knowing-that, -how, and-why, or domain/con-tent specific knowledge (Chinnappan et al. 2012), particu-larly from the perspective of knowledge-in-use as proposed by de Jong and Ferguson-Hessler (1996). Knowing-that (declarative), knowing-how (procedural), and knowing-why (conditional) are empirical in nature and centered on school-related domains (Alexander and Judy 1988; Alexan-der et al. 1991). Since conceptual knowledge is comprised of what discrete pieces of information are (Declarative Knowledge), how they function or operate, and the condi-tions under which they are used (Conditional Knowledge) (Rittle-Johnson et al. 2001; Ryle 1949 cited in; Alexander

et al. 1991), declarative and conditional knowledge together are considered as conceptual knowledge.

2.2 Declarative, procedural, and conditional knowledge

Declarative knowledge refers to factual information about mathematics (Alexander et al. 1991; Smith and Ragan 2005). Facts about geometry are rooted in experience that requires defining geometrical objects and identifying their figures. Defining in a classroom teaching environment involves not only stating a definition but also revising or refining a defi-nition with regard to what wording should be used, what the criteria for judging the adequacy and acceptability of a definition should be, and what the concept itself should be (Zandieh and Rasmussen 2010). Through definitions, the objects of the theory are introduced definitions express the properties that characterize them and relate them within a net of stated relations (Mariotti and Fischbein 1997). So, definitions are a basic component of geometrical declarative knowledge. For stating definitions, conventions and sym-bols, which are also factual information, are used. Actually, definitions are deeply rooted in geometrical figures. A fig-ure constitutes the external and iconical representation of a concept or a situation in geometry (Mesquita 1996). Geo-metrical definitions yield classification or identification of geometrical objects according to well-defined specific geo-metrical properties (Mariotti and Fischbein 1997). Let us consider the following example: identify the figures that rep-resent a triangle (see Question 2 in Appendix). The identifi-cation of triangles among various objects can be carried out according to the properties of triangles or as visual gestalts.

Procedural knowledge refers to how to use factual infor-mation (e.g., rules, algorithms, procedures) to solve math-ematical tasks (Alexander et al. 1991; Hiebert and Lefevre 1986). Procedural knowledge is used to solve computational problems as well as word or real world problems (Miller and Hudson 2007). To find the measure of the interior angle of an isosceles triangle divided into two triangles (see Ques-tion 23 in Appendix), students need to apply the properties of an isosceles triangle, the theorem (a measure of an exte-rior angle of a triangle is equal to the sum of the measures of the two non-adjacent interior angles) about the exterior angle measures of a triangle, and interior angle-sum formula algorithm of a triangle. Although it is possible to solve it in different ways, all these declarative knowledge elements collectively help students to find the unknown angle in an isosceles triangle.

Conditional knowledge involves knowing the ground or reason for the acceptance of a relational rule (Ryle 2009). Relational rules are typically expressed in an if-then form, although other ways of stating generality are also accept-able (Smith and Ragan 2005). Learning the ground for

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the relational rules can be achieved through participating in reasoning-and-proving activities (see Miyakawa 2017; Weber 2005) that provide opportunities for students to make, refine, and justify or validate mathematical relational rules. To understand whether students have learned the reason or ground for a relational rule, for example, related to the Isos-celes Triangle Theorem (See Question 10 in Appendix), they could be asked to form a logical justification demonstrating that a given rule is true or not. Proving this theorem can proceed as follows: let the triangle be ABC with AB = AC. Construct the angle bisector of angle BAC and extend it to meet BC at X. Hence, AB = AC and AX is equal to itself. Furthermore angle BAX = angle CAX, so, applying side-angle-side, BAX and CAX are congruent. It follows that the angles at B and C are equal in measure. As noticed, declarative and procedural knowledge collectively help stu-dents to provide the ground for the acceptance of the theo-rem. Studies investigating the nature of proof in geometry textbooks and curricula (e.g., Miyakawa 2017; Otten et al. 2014) also revealed how declarative and procedural knowl-edge bring students to the understanding and constructing of proofs of different kinds, of general, particular, and gen-eral with particular instantiation mathematical statements or relational-rules. Additionally, Ufer et al. (2008) showed that declarative and procedural knowledge significantly predicted students’ geometry proof performance.

The literature further highlights the reciprocal relation-ship that exists among declarative, conditional, and proce-dural knowledge of triangles (Aydın and Ubuz 2010).

2.3 Validity arguments on geometry tests

Within validity arguments, validity is gathered through judgmental evidence (e.g., evidence based on test content) and/or empirical evidence (e.g., evidence based on internal structure). The aforementioned studies on geometry gathered validity evidence of the instruments used in their studies mainly through judgmental evidence (i.e., expert evaluations of the test content, expert decisions on the faulty items, and expert reviews on the excerpts from the transcripts).

A few studies focusing on developing a geometry test (e.g., Chinnappan et al. 2012; Mullis et al. 2000; Usiskin 1982) utilized also judgmental evidences (i.e., expert exami-nations of the content of the items, expert discussions on the generalization of the items to students’ knowledge and understandings, expert reviews of the test framework) and to a lesser extent empirical evidence (i.e., statistical analyses based on classical or modern test theory) along with reliabil-ity estimates. Usiskin (1982) reported KR-20 reliabilities for the five van Hiele levels as ranging from 0.10 to 0.49 or 0.26 to 0.56. Duatepe-Paksu and Ubuz (2009) reported KR-21 reliability coefficients of the pre- and post-implementation of the test developed by Usiskin as 0.39 and 0.57. These

low reliability values, however, are of concern. Chinnappan et al. (2012) designed a proof-type geometry problem solv-ing test and geometry content knowledge test. The geometry content knowledge test aimed to measure students’ acqui-sition of declarative knowledge (knowledge of properties of diagrams, knowledge of conventions and symbols that are associated with such diagrams, knowledge of theorems) that is required for the solution of geometry proof problems. In the development stage of the tests they used judgmental evidences as well as field testing along with reliability esti-mates. They reported high internal consistency for the tests (Cronbach’s α > 0.80).

The TIMSS benchmarking report (see Mullis et al. 2000) documented Cronbach’s alpha reliability coefficients of the mathematics test ranging from 0.69 to 0.94 across 38 partici-pating countries without reporting reliability coefficients at content domain levels (i.e., number, algebra, geometry, and data and chance). Although it is strongly recommended to accumulate appropriate judgmental and statistical evidence that supports the meaningful interpretations of the data (McCoach et al. 2013, p. 244), mainly judgmental evidence was reported.

3 Method

3.1 Question generation and the refinement of the GKT‑T

To construct a closed constructed response question pool in the light of learning objectives on triangles (Milli Eğitim Bakanlığı 2006) the learning objectives were first catego-rized according to the knowledge types to which they refer: declarative, procedural, and conditional knowledge. Upon the completion of this process, 34 questions altogether were generated either by taking questions from the existing instruments or by developing new ones that fit the knowl-edge types.

To refine the questions, expert evaluations and a devel-opmental field test were conducted. For expert evaluation, the 34 questions in GKT-T were submitted to four staffs members in universities and one high school mathematics teacher, in order to review them for their readability, clarity, and congruity together with the corresponding geometrical knowledge construct. The university staff members were conducting research on the teaching and learning of geom-etry and had significant mathematics teaching experience at the university level. The mathematics teacher had mathemat-ics and geometry teaching experience over 30 years. Based on their feedback, three declarative, three procedural, and four conditional knowledge questions were eliminated and thus the instrument for the next stage contained 24 questions (see Appendix).

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For the developmental field test, two female students out of two male and three female middle-achieving students (geometry grades 3 or 4 out of 5) agreed to participate in the study to test the 24 questions in GKT-T. To prevent poten-tial Types 1 and 2 errors, middle-achieving students were selected. One was requested to complete it within one class period (45 min) in order to determine the suitability of the time limit, while the other student was interviewed to syn-thesize her thought processes while solving each question. By observing these processes as explicitly as possible, the clarity and intelligibility of the questions were ensured. Both students did not make any comments about the questions except in the case of one question. They indicated that they encountered some difficulties while identifying the pairs of congruent/similar triangles in the given figures (declarative knowledge question 7). The difficulty was in part due to students’ desire to know the angle measures and side lengths of the triangles given. As the students are required to iden-tify figures by their holistic appearance, the question was retained.

The GKT-T questions’ content relevant to the knowledge types is presented in Table 1. Questions 1–7 measure declar-ative knowledge, where students are required to provide the definition of a concept or identify its figure. Figures pro-vided in question 2 about identification of triangles fall into two main categories: atypical examples and non-valid exam-ples (Satlow and Newcombe 1998). Non-valid examples fall into two categories: intuitive non-example and non-intuitive non-example (Tsamir et al. 2008). Atypical examples are presented based on two ways of modification: the optic (i.e., making the figure larger or narrower) and the place way (i.e., changing the figure’s position or orientation) (Duval 1995, 1999 cited in; Gagatsis et al. 2010). In question 7 about iden-tifying congruent and similar triangles, triangles fall into two categories: typical and atypical examples (Satlow and Newcombe 1998). As in question 2, atypical examples are presented based on the two ways of modification. Triangles which are not similar or congruent can be called intuitive non-examples.

Table 1 Content of the GKT-T questions relevant to knowledge types

Knowledge type Question Objective

Declarative knowledge 1 Define types of triangle (equilateral, right, and isosceles triangles)2 Identify triangles3 State the sides and interior angles of a triangle using symbols4 Define congruency of triangles5 Define similarity of triangles6 State the congruent and similar triangles in the given triangle, formed by assemblage of two triangles that

are touching, using symbols7 Identify congruent and similar triangles from the given triangles

Procedural knowledge 9 Apply Angle–Side–Angle congruency postulate11 Apply Angle–Angle–Side congruency postulate14 Apply Thales Theorem17 Apply Pythagorean Theorem and the similarity postulate19 Apply Menelaus Theorem21 Apply special right (i.e., 30–60–90) and equilateral triangle properties23 Apply isosceles triangle properties24 Apply Euclidean propositions

Conditional knowledge 8 Justify the truthfulness/falsity of the statement relevant to equilateral triangle10 Justify the truthfulness/falsity of the statement relevant to isosceles triangle12 Justify the truthfulness/falsity of the statement relevant to similar triangles13 Justify the truthfulness/falsity of the statement relevant to the interior angles sum of a triangle15 Justify the truthfulness/falsity of the statement relevant to the relationship between equilateral and isosceles

triangle16 Justify the truthfulness/falsity of the statement relevant to the construction of congruent triangles in an

isosceles triangle18 Justify the truthfulness/falsity of the statement relevant to the relationship between congruency and similar-

ity of triangles20 Justify the truthfulness/falsity of the statement relevant to the construction of similar triangles in a right

triangle22 Justify the truthfulness/falsity of the statement relevant to the congruency of triangles

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Questions 9, 11, 14, 17, 19, 21, 23, and 24 measure pro-cedural knowledge. All these questions except question 9 are accompanied by figural representations which are formed by mereologic modification (Duval 1988 cited in; Gagatsis et al. 2010) on triangles. Question 9 includes two separate triangles (one typical and one atypical with place way modi-fication), questions 11, 19, 21, and 24 include the superposi-tion of two triangles, and the rest (questions 14, 17, and 23) include a triangle divided into two parts. Questions 9 and 11 were taken from the Trends in International Mathematics and Science Study 1995 and 2003 (Beaton et al. 1996; Mar-tin et al. 2004), respectively; and the rest were adapted from geometry test banks (see Hancerliogullari and Alan 2006) with minor changes in their numerical values.

Questions 8, 10, 12, 13, 15, 16, 18, 20, and 22 meas-ure conditional knowledge, where students are required to know relationship between concepts or within a concept. Relational-rules given are about general objects as math-ematics is unique in its concern for general claims (Otten et al. 2014). Relational-rules about general objects are also common in the curricula of other countries curriculum such as France, Japan, and USA (see Miyakawa 2017; Otten et al. 2014).

3.2 Test administration

The 24 question GKT-T was administered to 557 tenth grade students (293 female and 264 male). Participants were from one Private (n = 43), two Public General (n = 223), and three Public Anatolian (n = 291) high schools. Although Public Anatolian and Private high schools selected students accord-ing to their Secondary Education Entrance Examination scores, geometry was taught in similar ways, using the same geometry curriculum aligned with the national standards, and within the same time frames during the academic year. Every student who completes the eighth year of education can go to a Public General high school.

The GKT-T was administered by the students’ mathemat-ics teachers during their mathematics classes. On the day of the administration, the second researcher was also present to ensure that the test was properly administered. Before proceeding to statistical analyses, the scores of students who missed the questions toward the end of the GKT-T and/or responded to less than four questions on either of the knowl-edge types were discarded. Accordingly, the average per-centage of missing data in each student test was 3.9%.

At the time the study took place, the triangle topic was first introduced to the students at the primary school (Grades 3–5). Throughout these years, the emphasis was placed on defining triangles, stating the properties of triangles, defin-ing triangle types, stating the properties of triangle types, classifying triangle types, labeling the triangles, identifying the vertices, sides, and interior/exterior angles of triangles,

and classifying the triangles with respect to the side length measures (declarative knowledge). Drawing on this knowl-edge, at the middle school (Grades 6–8) the emphasis was placed on the application of procedures and linking rela-tionships between concepts. The application of procedures was about finding the interior/exterior angle measures of a triangle, evaluating the side length measures of a triangle, computing the area of a triangle, implementing Pythagoras’ Theorem and Triangle Inequality, constructing the median, angle bisector, and hypotenuse in a triangle, and using con-gruency/similarity symbols, and evaluating the congruency/similarity ratio of the triangles (procedural knowledge). Linking relationships between concepts were about connect-ing the sum/difference of two sides of a triangle with the measure of the third side, connecting the side lengths with the angle measures opposite these sides, making transforma-tions within and among the median of a triangle and its angle bisectors or vice versa, and building relationships between congruency and similarity of triangles (conditional knowl-edge). When students moved towards high school (Grades 9–12), the emphasis was on defining the triangle types, list-ing properties of triangle types, stating the Euclidean rela-tions, congruency/similarity postulates and congruency/similarity theorems (declarative knowledge); applying the congruency/similarity postulates, implementing Euclidean relations and Pythagoras’ Theorem, employing congruency/similarity ratio of triangles to calculate the unknown angle measure, side length, or the area (procedural knowledge); making transformations within and among the properties of a triangle/triangle types, building links within and among congruency (e.g., Side–Side–Side, and Side–Angle–Side) and similarity (e.g., Angle–Angle and Side–Side–Side) pos-tulates for triangles, justifying whether the given triangles are congruent or similar, and explaining the congruency/similarity theorems (conditional Knowledge).

The triangles content coverage listed above is not a coverage that is met only by Turkish students. Miyakawa (2017), for example, reported that middle school students in France (grades 6–9) and Japan (grades 7–9) learn triangles with some variations on what is taught and at what age. While French students learn about triangles, right triangles, Pythagorean Theorem, and Thales Theorem, Japanese stu-dents learn about triangles, Pythagorean Theorem, congru-ent and similar figures. Some relational rules available for proving in the Japanese curriculum are two properties of interior and exterior angles of a triangle, corresponding parts of congruent triangles are congruent, and three conditions for congruent triangles (SSS, SAS, and ASA). With regard to similar triangles, the secondary mathematics curriculum in Hong Kong requires students to understand the sufficient conditions for triangles to be similar, identify the corre-sponding sides and angles, and solve for unknowns using the common ratio found (Poon and Wong 2017).

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3.3 Scoring and generalizability analysis

The holistic scoring scheme developed by Lane (1993) that incorporates three interrelated components—mathematical knowledge, strategic knowledge, and communication—was adapted in order to score the questions in the GKT-T by the two authors taking into account goals and objectives of the 10th grade mathematics curriculum. Examples of grading are included in Tables 2, 3 and 4 for declarative, procedural, and conditional knowledge questions, respectively. Upon the completion of this process, the second researcher and a high school mathematics teacher with 3 years of experience inde-pendently scored 37 randomly selected students’ responses to the test to establish the extent of consensus on the use of the holistic scoring scheme. Following that, the second researcher evaluated the rest of the students’ test results.

To investigate how different sources of variability influ-ence the generalizability of the scoring scheme, generaliza-bility analysis (Brennan 2001) was conducted to examine the

consistency across raters, questions, and students. Table 5 presents the results of generalizability analyses for student x question x rater (s × q × r). The PASW Statistics 18 soft-ware was used to execute Analysis of Variance (ANOVA) for partitioning variance attributable to each source. The substantial agreement between raters was reflected in the amount of variance accounted for by the r facet, and the s × r and q × r interaction facets (0.03, 0.00, and 0.01, respectively). The variance due to raters (0.03) was small. This result suggests that there was no systematic variation between raters. The variance components for the interac-tion between students and raters (0.00) and between question and raters (0.01) were smaller than the variance that was due to raters alone. These interactions showed that raters agreed somewhat in their ratings of the students’ responses and questions.

The variance due to questions (3.4% of the total vari-ance) was larger than that due to raters. This result suggests that there was variation between difficulty of questions. The

Table 2 Specimen grading for the declarative knowledge question 5

Score Description Example of response

4 Provides a definition of a concept by covering its basic properties with a clear expression

Triangles are similar if (1) all three pairs of corresponding angles are the same—angle angle angle (AAA), (2) all three pairs of corresponding sides are in the same proportio—side side side (SSS), and (3) two pairs of sides in the same proportion and the included angle of each pair is equal in measurement—side angle side (SAS)

3 Provides a definition of a concept by covering its basic properties with no clear expression

Triangles are similar as long as they imply one of the postulates: SSS, AAA, or SAS

2 Provides a definition of a concept by covering some basic proper-ties with a clear expression

Triangles are called similar if they both have the same shape and proportional sides

1 Provides a definition of a concept by not covering its basic proper-ties

Triangles are similar if they have the same shape, but can be dif-ferent sizes

0 Provides a definition of a concept with irrelevant facts or gives no response

Similarity means being similar

Table 3 Specimen grading for the procedural knowledge question 14

Score Description Example of response

4 Provides completely correct solution with explicitly detailed correct approach

As [GH]∕∕[EF], D̂GH = D̂EF and D̂HG = D̂FE because they are corresponding angles. Triangles DEF and DGH are similar as their angles are equal. In similar triangles, corresponding sides are in the same ratio. The similarity ratio algorithm is |DG|

|DE|=

|GH|

|EF|=

|DH|

|DF| , 812

=6

x , 2x = 18 . Thus, x = 9

3 Provides nearly correct solution with explicitly/nearly detailed cor-rect approach or completely correct solution with nearly detailed correct approach

Triangles DEF and DGH are similar. Hence, |DG||DE|

=|GH|

|EF|=

|DH|

|DF| ,

8

12=

6

x , and thus x = 3

2 Provides correct or incorrect solution with explicitly/nearly correct approach including serious computational errors

If 812

=1

3 , then |EF| = 3.|GH| , |EF| = 3x . Hence, x = 9

1 Provides incorrect solution with partly correct approach 8

4=

6

x⇒ x = 3

0 Provides incorrect solution with unclear, inappropriate, and/or unreasonable approach or no response

8

6=

4

x⇒ x = 3

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variance due to students (1% of the total variance) was larger than that due to raters. This result indicates that there was a variation amongst students. The interaction between students and questions (9.2% of the total variance) was large, indicat-ing that students’ responses differed considerably from one question to another.

4 Results and discussion

4.1 Dimensionality: Confirmatory factor analysis and inter‑relationships among the factors evidences for internal structure

Confirmatory Factor Analysis (CFA) using LISREL 8.7 (Jöreskog and Sörbom 1993) was conducted to provide sup-portive evidence for the factor structure of the GKT-T. To test which model was best suited to the data, four competing CFA models were tested. The common factor model was specified such that all questions loaded on one factor called geometrical knowledge of triangles. This model proposes that declarative, procedural, and conditional knowledge are not conceptually or statistically distinct. The two-factor model was specified as conceptual and procedural knowl-edge. Considering that conceptual knowledge is a connected web of knowledge, in which the linking relationships are as prominent as the discrete pieces of information (Ander-son 2015; Hiebert and Lefevre 1986), and considering that knowing pieces of information constitutes a lower level, and constructing relational rules, making an inference from pieces of information, constitutes a higher level (Anderson 2015), conceptual knowledge was distinguished as declara-tive and conditional knowledge in the three-factor model. The null model specified that all questions were uncor-related, proposing that each question in the GKT-T—in itself—constitutes a single factor.

CFA results shown in Table 6 display that the three-factor model fits the data better according to the following criteria: The Chi square to the Degrees of Freedom ratio (χ2/df) less than or equal to three, the root mean square residual (RMR)

Table 4 Specimen grading for the conditional knowledge question 8

Score Description Example of response

4 Explains the relation between concepts with strong logically sound justification

This triangle is an equilateral triangle. Drawing a perpendicular bisector (AH) from the vertex A to the opposite side BC, the resulting smaller triangles ABH and ACH are equal according to the Pythagorean Theorem. So, angle ABC is equal to ACB. Drawing a perpendicular bisector (BL) from the vertex B to the opposite side AC, again the resulting smaller triangles are equal and angle BAC is equal to ACB. So all angles are equal

3 Explains the relation between concepts with nearly logically sound justification

This triangle is an equilateral triangle. By the Pythagorean Theo-rem, we can utilize the perpendicular bisector (AH) from the vertex A to the opposite side BC. Thus, triangles ABH and ACH are similar. [i.e., without emphasis on the use of repetitive steps about drawing a perpendicular bisector from the relevant angle to its opposite side]

2 Explains the relation between concepts with insufficient justifica-tion

If perpendicular bisectors are drawn, Side Angle Side (SAS) postulate is satisfied. Hence, all the angles are equal

1 Explains the relation between concepts with logically unsound justification

In an equilateral triangle all angles are equal. Correct

0 Explains the relation between concepts with incorrect justification, or gives no response

This statement is incorrect

Table 5 Sources of variability and relative magnitude

S students, r raters, q questionsa Close to zero estimated variance component set equal to zero

Estimated variance component ( �2)

% of total variance

s × r × q s × r × qns = 37 ns = 37

Source of variation nr = 2 nr = 2nq = 24 nq = 24

s 0.10 1r 0.03a 0a

q 0.34 3.4s × r 0.00a 0a

s × q 0.92 9.2r × q 0.01a 0a

s × r × q, e 0.29 2.9

667Geometry knowledge test about triangles: evidence on validity and reliability

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below 0.05, the Root Mean Square Error of Approximation (RMSEA) from 0.06 to 0.08, goodness-of-fit index (GFI) above 0.90, adjusted-goodness-of-fit index (AGFI) above 0.90, and Comparative Fit Index (CFI) above 0.95 indicate adequate fit (Kline 2005).

Given the adequate fitness of the three-factor model, max-imum likelihood estimations were calculated for the three-factor model. The results revealed that maximum likelihood estimations were higher than 0.30 and appeared between 0.36 and 0.97 with all t-values significant at p < .05 (see Table 7). As seen in Table 7, squared multiple correlation (R2) of individual questions were substantial in size (> 0.50)

ranging from 0.62 to 0.90. These results indicated that each question can be explained by the corresponding knowledge types: declarative knowledge (questions 1, 2, 3, 4, 5, 6, and 7), procedural knowledge (questions 9, 11, 14, 17, 19, 21, 23, and 24), and conditional knowledge (questions 8, 10, 12, 13, 15, 16, 18, 20, and 22).

Statistically significant bivariate correlations among the dimensions declarative and procedural knowledge (r = 0.30, p < 0.01), declarative and conditional knowledge (r = 0.24, p < 0.01), procedural and conditional knowledge (r = 0.27, p < 0.01) provide further divergent validity evidence. According to Field (2005), a moderately strong association

Table 6 Model fit indices for the four models

Model �2 df �2∕df RMSEA RMR GFI AGFI CFI Δχ2 Δdf

Three-factor 512.22 199 2.5 0.05 0.03 0.93 0.91 0.98Common factor 908.93 203 4.4 0.07 0.06 0.88 0.82 0.96 396.71 4Two-factor 600.61 202 2.9 0.06 0.05 0.90 0.88 0.94 88.39 3Null 682.99 245 2.7 0.08 0.07 0.82 0.78 0.93 170.77 6

Table 7 Reliabilities, standardized estimates and t-values of the questions in the GKT-T*

*t-values indicating the ratio between the standardized estimate and its standard error are presented in parenthesis. t-values exceeding 1.96 indicate that the corresponding parameter is significant and that the corresponding question is statistically significantly related to its relevant knowledge type (Jöreskog and Sörbom 1993)

Question R2 Declarative knowledge Conditional knowledge Procedural knowledge

1 0.72 0.40 (9.72)2 0.65 0.43(7.76)3 0.70 0.36 (8.10)4 0.74 0.49 (11.00)5 0.82 0.50 (12.62)6 0.70 0.69 (8.87)7 0.62 0.39 (6.66)8 0.84 0.47 (7.29)10 0.78 0.41 (10.16)12 0.85 0.70 (13.87)13 0.69 0.97 (14.08)15 0.72 0.87 (11.07)16 0.88 0.72 (12.78)18 0.71 0.90 (14.93)20 0.71 0.74 (16.14)22 0.66 0.81 (15.12)9 0.74 0.74 (7.03)11 0.86 0.46 (9.27)14 0.80 0.42 (14.43)17 0.69 0.49 (18.31)19 0.76 0.41 (20.61)21 0.88 0.63 (13.07)23 0.68 0.36 (13.44)24 0.90 0.66 (12.27)

668 B. Ubuz, U. Aydın

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between factors is acceptable. Total variance explained by the three dimensions was 30.51% for the declarative, 37.43% for the procedural, and 30.43% for the conditional knowledge.

These findings provide empirical support for the dimen-sionality of geometrical knowledge in that those dimensions capture the components of geometrical knowledge as sug-gested by previous researchers (e.g., de Jong and Ferguson-Hessler 1996; Mason and Spence 1999; Ryle 1949).

4.2 Gender and school type differences: Measurement invariance evidence for internal structure

Means and standard deviations of the dimensions of the GKT-T, split by school types and gender, are provided in Table 8. The two-way Multivariate Analysis of Variance (two-way MANOVA) was conducted to understand whether there were differences in students’ declarative, procedural, and conditional knowledge based on school type and gen-der. The results revealed that the gender and school type interaction (F(6, 1098) = 4.216, p = 0.00; Wilks’ Λ = 0.955, η2 = 0.023) and both the school type (F(6, 1102) = 43.506, p = 0.00; Wilks’ Λ = 0.653, η2 = 0.192) and gender (F(3, 549) = 3.236, p =0 .02; Wilks’ Λ = 0.983, η2 = 0.017) main effects are significant on the combined dependent variables.

The partial eta square (η2) values indicate the rela-tive degree to which the variance that was found in the MANOVA was mainly associated with the effect of school type (19.2%) rather than gender (1.7%) or interaction effect (2.3%). To understand which schools were significantly dif-ferent from each other, Dunnett Post Hoc t-tests were per-formed. The results indicated that Anatolian high school students’ declarative (p = 0.00 and p = 0.01), procedural (p = 0.00 and p = 0.03), and conditional (p = 0.00 and p = 0.00) knowledge were significantly better than that of General and Private high school students, respectively. Differences in geometrical knowledge occurred, favoring particularly the Anatolian high schools because the best schools in Tur-key are Public Anatolian and Science high schools (Batyra 2017b). Admission to these schools is based upon competi-tive placement examinations.

Looking at the means and standard deviations at the fac-tor (see Table 8) and item levels indicates that (a) although male students’ conditional knowledge was higher than their female counterparts, Anatolian high school female students performed equally well in three conditional knowledge questions (questions 8, 10 and 13) and Private high school female students performed better in five conditional knowl-edge questions (questions 8, 10, 15, 16, and 18) compared to their male counterparts; (b) although General and Anato-lian high school male students’ declarative and procedural knowledge were higher than their female counterparts,

Anatolian high school female students performed equally well in two procedural questions (questions 9 and 11); (c) although Private high school female students’ declarative and procedural knowledge were higher than their male coun-terparts, male students performed equally well (questions 2 and 7 in declarative; questions 9 and 24 in procedural), or better (questions 11 and 14 in procedural), in two declara-tive and four procedural knowledge questions than did their female counterparts; and (d) the consistent gender differ-ences, though, occur in General high school students favor-ing male students. In sum, similar patterns of gender differ-ences in declarative, procedural, and conditional knowledge

Table 8 Descriptive statistics for gender and school type

Knowledge School type Gender Mean SD

Declarative General (n = 223) Female (n = 122) 2.04 0.58Male (n = 101) 2.60 0.57Total 2.29 0.64

Private (n = 43) Female (n = 33) 2.56 0.56Male (n = 10) 2.36 0.81Total 2.51 0.62

Anatolian (n = 291) Female (n = 138) 2.70 0.65Male (n = 153) 2.89 0.62Total 2.80 0.64

Total (n = 557) Female (n = 293) 2.41 0.68Male (n = 264) 2.75 0.63Total 2.57 0.68

Conditional General Female 1.53 0.69Male 2.16 0.72Total 1.81 0.77

Private Female 2.19 0.77Male 2.21 0.92Total 2.20 0.80

Anatolian Female 2.56 0.79Male 2.71 0.69Total 2.64 0.74

Total Female 2.09 0.89Male 2.48 0.76Total 2.27 0.85

Procedural General Female 1.62 0.67Male 2.16 0.70Total 1.86 0.73

Private Female 2.74 0.88Male 2.55 1.17Total 2.70 0.94

Anatolian Female 2.85 0.916Male 3.17 0.85Total 3.02 0.89

Total Female 2.33 1.01Male 2.76 0.94Total 2.53 1.00

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exist for General high schools but not for Private and Ana-tolian high schools.

4.3 The relationship between prior geometry achievement and geometry knowledge: Evidence based on relationships to other variables

The studies (e.g., Aydın and Ubuz 2014; Kyttälä and Björn 2010; Post et al. 2010) investigating the association between prior mathematics achievement and later mathematics achievement or performance led us to predict that students’ prior semester geometry grades in the 10th grade and their geometrical knowledge are significantly correlated. The relationship between prior semester geometry grade and geometry knowledge (r = 0.37, p < 0.001) shows promis-ing convergent and divergent validity. The strength of the relationship was greater for procedural knowledge (r = 0.29, p < 0.001) than for declarative (r = .21, p < 0.001) and condi-tional knowledge (r = 0.20, p < 0.001). This result might be due to the fact that the geometry teaching in the high schools mainly focuses on procedural knowledge. The correlation between previous semester geometry grades and the GKT-T dimensions may have been attenuated due to the characteris-tics of geometry grades and the topics covered in the previ-ous semester. Grades for each course are calculated based on examination scores, teacher opinions/observations, and so forth. Accordingly, we recommend further investigation of the relationship of scores from the GKT-T with the in-class examination scores. The topics covered in the previous semester at the 10th grade were as follows: basic elements in plane geometry (Euclid’s postulates) and proof forms (two-column, paragraph and flow proof); point, line and vectors in a plane; coordinate systems. So, it is important to take into account the differences between the previous semester geometry content and the test content while interpreting the significant but small or medium relations.

4.4 Reliability: internal consistency

Cronbach’s alpha coefficients (α = 0.91 for the entire test; α = 0.70 for the declarative knowledge; α = 0.85 for the pro-cedural knowledge; and α = 0.86 for the conditional knowl-edge) indicated high internal consistency. Item-total correla-tions ranged from 0.33 to 0.51 in the declarative knowledge, from 0.36 to 0.69 in the procedural knowledge, and from 0.43 to 0.67 in the conditional knowledge dimensions. Correlations greater than 0.30 (Crocker and Algina 1986) indicate that questions belong on the test. Cronbach alpha coefficients and the item-total correlations denoted question homogeneity in relation to the dimensions and the entire test.

5 Conclusion

The need to gather evidence that supports the validity and reliability of score-based inferences is imperative from sci-entific, ethical, and legal perspectives (Rios and Wells 2014). In this current study we provided four sources of validity evi-dence (based on content and response processes in the meth-ods part, internal structure and relation to other variables in the results part) in addition to reliability for the GKT-T. The methods used for each source of validity and reliability will assist researchers in gathering evidence to strengthen the validity and reliability of intended score-based inferences.

For analyzing test dimensionality, which is one of the basic aspects of internal structure, this paper focused on CFA and the inter-relationships among the dimensions. Although CFA is the most comprehensive means for com-paring hypothesized and observed test structures, a further study could use further factor analytic methods available (e.g., multidimensional item response theory) for analyzing test dimensionality.

For analyzing measurement invariance, which is another basic aspect of internal structure, means and standard devia-tions together with inferential statistics were examined at the construct- and/or question-levels across gender and school types. Based on the results we could say that gender differ-ences must be interpreted cautiously because of different pattern of results in different school types. The gender ineq-uity particularly in General high schools, to the advantage of males, may be due to some other reasons such as students’ (e.g., spatial visualization, logical reasoning, etc) character-istics and those of their parents (e.g., father’s and mother’s education level and employment status, parental emotional support, etc). Battista (1990), for example, reported that spatial visualization and logical reasoning were important factors in geometry achievement and geometry problem solving for both male and female students. Batyra (2017a) indicated that there were no gender differences between male and female Turkish students’ mathematics achievement in TIMSS 2015 if there were no big differences in their advan-tageous properties. As a result, future research may examine more variables regarding students’ characteristics and those of their parents to explain the variance in the geometrical knowledge based on gender. School types seemed to have contributed more to differences in geometrical knowledge because school type in some way is a measure of socioeco-nomic background and achievement levels.

In the process of obtaining validity evidence based on relations to other variables, it was not possible to compare the GKT-T results with those of another instrument because there are no instruments assessing the same construct that are valid and reliable for high school students. Thus, a test with a similar approach to that of the GKT-T needs to be prepared and evaluated for its psychometric properties.

670 B. Ubuz, U. Aydın

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As mentioned previously, we expect that this valid and reliable GKT-T will encourage further research in examining the effect of specific educational programs on students’ geo-metrical knowledge, and its relations with different variables (e.g. academic achievement, attitude), or the interrelations among them. The use of the GKT-T for evaluating educa-tional programs, improving instruction, and investigating relational variables will enable support for other source of validity evidence, testing consequences. We also hope that this test will provide a foundation for mathematics education researchers in structuring tasks relevant to other topics in

geometry and other domains of mathematics (e.g., algebra) in a way that elicits types of knowing.

Although the GKT-T was corroborated as a valid and reli-able test, the findings could be verified through interviews with students.

Appendix

Geometry knowledge test about triangles (GKT-T).

1) Define the following triangles: A) Equilateral triangle: B) Right triangle: C) Isosceles triangle: 2) Identify which one of the following figures represents a triangle:

3) Given below the triangle ABC, identify and write the sides and interior angles in the symbolic form.

4) Define congruency of triangles. 5) Define similarity of triangles. 6)

Given the figure above: A) Examine the congruent triangles, and write the congruency relation(s) in symbolic form. B) Examine the similar triangles, and write the similarity relation(s) in symbolic form.

8) “If triangle ABC has three equal sides, then the angles at the sides A, B, and C are equal in measure.” Is this statement true? Justify your answer.

9)

Given above the side lengths and angle measures of two congruent triangles, find x?

10) “If two sides of a triangle are equal in length, then the angles opposite these sides are equal in measure.” Is this statement true? Justify your answer. 11)

Given above the congruent triangles ABC and DEF, |AC|=|DF|. Find the angle measure DGC?

12) “If triangles ABC and DEF are similar, then triangles ACB and DFE are similar.” Is this statement true? Justify your answer.

13) “A triangle can have two right angles.” Is this statement true? Justify your answer.

7)

A) Which of the triangles given above are congruent? B) Which of the triangles given above are similar?

14)

Given triangle DEF, [GH]//[EF], |DG|= 8 cm, |GE|= 4 cm , |GH|= 6 cm. Find x?

I H

ED C B A

F G

C A

B

A

H C B

(7) I II III IV

VI VIII

VII V

4

8

H

F E

G

D

6

x

671Geometry knowledge test about triangles: evidence on validity and reliability

1 3

15) “An equilateral triangle is an isosceles triangle.” Is this statement true? Justify your answer.

16) Given below PRS, . “If I draw an altitude to SR, then I divide triangle PRS into two congruent triangles.” Is this statement true? Justify your answer.

17)

Given a right triangle DEF,

| | = 9 . Find | |? 18) “If two triangles are congruent, then they are similar.” Is this statement true? Justify your answer.

19)

Given the figure above, ,

, and . Find ?

20) “If an altitude is drawn to the hypotenuse of a right triangle, then the two triangles formed are similar.” Is this statement true? Justify your answer.21)

Given the equilateral triangle ABC, and . Find

| |?

22) “If triangle ABC is congruent to triangle ACB, then triangle ABC is an isosceles triangle.” Is this statement true? Justify your answer.

23)

Given above the triangle ABC, ,

and . Find x?

24)

Given above the right triangles ABC, ADF, and AEF. cm, cm, and

cm. Find ?

RP SP=

[ ] [ ]ED FD⊥ [ ] [ ]EFGH ⊥ DG mcEG 6=

cmEF 5=

cmED 4= DCAD =BCFB

DA BE FB= cmBC 24=

BA CA=

DB CB= °∧

=( ) 81ABDs

4=AE 5=EC 6=BC

DEEF

672 B. Ubuz, U. Aydın

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