Definitions are part of teachers’ subject matter knowledge (Delaney, 2012), and they have implications for curricular (Usiskin & Griffin, 2008) and pedagogical decisions (Zazkis & Leikin, 2008) that teachers make. Therefore, definitions play a key role in mathematics learning, teaching, and curriculum development. Teachers’ understanding of definitions gives us clues about their understanding of the concepts associated with them. Geometry is a field where definitions play a significant role and research has shown that teachers need additional support in this area (Clements & Sarama, 2011). Hence, research aiming to investigate teachers’ understanding of definitions in geometry is necessary. In this study, we focus on definitions of quadrilaterals, in particular trapeziums/trapezoids, and preservice teachers’ understanding and use of these definitions.  

Quadrilaterals are rich constructs to study definitions. Quadrilaterals’ richness can be attributed to their being foundational in measurement, studying geometric properties as well as understanding the addition of complex numbers and vectors (Usiskin & Griffin, 2008). On the other hand, it is internationally recognised that defining and classifying quadrilaterals can be challenging for learners (e.g., pupils, preservice teachers) due to their reliance on prototypical shape images rather than definitions based on geometric properties (Fujita & Jones, 2007). This challenge influences preservice teachers (PTs), impacting their school practices (Jones et al., 2002). This study aims to explore PTs’ use of inclusive and exclusive definitions of quadrilaterals, using trapezium as a context. The research question we pursue is: What characterises preservice teachers’ use of inclusive and exclusive definitions of the trapezium in relation to other quadrilaterals?  

Individuals interpret mathematical problem situations with their concept images and/or concept definitions. Concept images are “something non-verbal associated in our mind with the concept name […] a visual representation […] a collection of impressions or experiences” (Vinner, 2002, p.68), whereas concept definitions are the mathematical definitions of concepts specific to an individual. Furthermore, “referring to the formal definition is critical for a correct performance on given tasks ([…] identification of examples and non-examples of a given concept”) Vinner (2002, p.80). In making sense of PTs’ use of definitions, especially in geometry, we focused on two types of definitions: exclusive and inclusive (Usiskin & Griffin, 2008). Exclusive definitions of geometric figures are the definitions that lead individuals to understand those figures in isolation (e.g., rectangles are not part of the parallelogram set). In contrast, inclusive definitions allow the defined figures to include others (e.g., rectangles are part of the parallelogram set). 

The use of inclusive definitions in teaching geometry is advantageous as it allows learners to better understand the interrelations among quadrilaterals (e.g., squares are part of the rectangle set) by focusing on how properties of one quadrilateral set satisfy the properties of another set. Interestingly, however, Usiskin and Griffin (2008) investigated mathematics textbooks published in the USA during the 1833-2008 period and found that only about 10% of these textbooks (n=8) used inclusive definitions and 90% (n=76) used exclusive definitions. Hence, if teachers align their teaching with the latter type of textbooks, they are more likely to miss the opportunity to help their pupils learn about the aforementioned interrelations when teaching geometry.   

To what extent teachers are aware of different types of definitions and their advantages or disadvantages in teaching geometry is not known well in the research literature (Sinclair et al., 2016) even though it is valued universally. Our research focuses on PTs’ use of inclusive and exclusive definitions in the context of defining trapeziums. Although this study took place in Scotland, it will provide insights into teachers’ use of definitions that can inform mathematics education researchers in Europe and other countries.            

The study participants were from a tightly structured one-year-long initial teacher education programme at the University of Glasgow. The programme prepares PTs to teach in Scottish primary schools. These teachers have a first degree (not necessarily in mathematics), as Scottish primary teachers are generalists and are expected to teach all curricular subjects. One hundred forty PTs from the 2021-2022 cohorts were invited to complete an anonymous online questionnaire before they started their teacher education programme. The response rate was 50% (71 students). The questionnaire consisted of two sections; one focused on participants’ beliefs about and attitudes toward mathematics, and the other was on their understanding of the subject. This paper focuses on the latter section, which asks PTs to use the formal definitions of the trapezium taken from Usiskin and Griffin (2008) and to choose and justify which quadrilaterals (parallelogram, rectangle, rhombus, square, isosceles trapezium) fit those definitions: Definition #1: A trapezium is a quadrilateral with exactly one pair of parallel sides. Definition #2: A trapezium is a quadrilateral with at least one pair of parallel sides. a) If we accept Definition #1 which one of the following figure(s) would be considered as trapeziums? Please explain why. b) If we accept Definition #2 which one of the following figure(s) would be considered as trapeziums? Please explain why. c) If you were to use one of these definitions to teach students in your maths classes, which definition would you use? Please explain why. We checked through participants’ responses for open-ended questions. We then generated categories out of those answers (e.g., no answer, correct, incorrect, partially correct, prototypical concept image, other) and then converted them into numeric codes for quantitative analysis (e.g., no answer=0, correct=1). We then used cross-tabulation (i.e., frequencies and percentages across relevant questions) and performed the Chi-square Test of Independence to explore the relationships among the answers given to the questions, using IBM SPSS 24.

We found that 65% of the participants matched the exclusive definition of trapezium solely to isosceles trapezium and 57% linked the inclusive definition to all geometrical figures. Furthermore, 20% of the participants linked the exclusive definition and 29% linked the inclusive definition of trapezium to multiple geometrical figures. On the other hand, 15% could not connect the exclusive definition and 13% could not connect the inclusive one to any other geometric figure. 44% of the participants explained their choices correctly when applying the inclusive definition, whereas this proportion moves up to 64% for those providing correct justifications concerning the exclusive definition of a trapezium. There was a statistically significant association between the participants’ selection and justification of geometric figure(s) as a trapezium with an exclusive definition (χ^2(4, 94) = 170.16, p < .001 with a strong effect size of Cramer’s V = .95 (Cohen, 1988)) and with an inclusive definition (χ^2(8, 94) = 108.72, p < .001 with a strong effect size of Cramer’s V = .76 (Cohen, 1988)). Most participants (64%) justified why the given shape(s) would be considered trapeziums by operating from a prototypical image as follows: PT52: Only the trapezium and the isosceles trapezium have one set of parallel lines as the left, and right lines would meet if they were to continue on. This suggests that their judgments are impacted by a prototypical (concept) image of an (isosceles) trapezium for exclusive definition. To conclude, participants were more likely to operate with the exclusive definition of a trapezium when analysing the given geometric figures, rather than the inclusive definition requiring a concept definition geared by geometric properties. Our findings support previous research (e.g., Fujita, 2012) indicating that most learners, even high-achievers, rely heavily on the prototypical examples of quadrilaterals and thus fail to understand the inclusion relations.

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