Conference Agenda

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.

This study aims to understand the relationships between mathematics anxiety, attitude, experience and choice of subject specialism among primary student teachers in the first year of their initial teacher training (ITT). During an ITT course at a London University, elements of anxiety and negative attitude surrounding mathematics were observed by tutors among trainee teachers. For example, some demonstrated a reluctance to actively participate in mathematics-related sessions or expressed concerns about their ability to teach mathematics, particularly with older and/or more able primary-aged students. Furthermore, at the end of the first year, students appeared to prefer other subjects to mathematics when they were required to select a subject specialism. Such observations are not unusual or new. Since the 1980s (Beilock et al., 2010; Cockcroft,1982; Ernest,1988; Hembree,1990), student teachers’ fear, anxiety, and negative attitudes towards mathematics have been highlighted as problematic with recommendations that training institutions should help students develop positive attitudes to and confidence in the subject (Cockcroft,1982).

To help prospective primary school practitioners develop positive attitudes and confidence during their ITT course, it is important to understand the learning experiences of students undertaking the course before their arrival on the ITT programme and their existing mathematics anxiety levels and attitudes to mathematics. During the programme, their attitude to mathematics as both teacher and learner may change; this research aims to critically analyse students’ experiences and mathematics anxiety. At the time of writing this, limited research exploring mathematics anxiety and the teaching of mathematics among primary student teachers is available; this research, therefore, seeks to contribute to understanding in this area.

Global interest in mathematics anxiety has been evident in the volume of research conducted and published over the last three decades. Such research often provides evidence that mathematics anxiety is associated with various negative cognitive and emotional outcomes (e.g. Ashcraft & Krause, 2007; Espino et al., 2017). Impact on cognitive outcomes is evidenced in low performance and achievement, mainly because anxiety interferes with the ability to maintain and manipulate information and resources within the mind, which is necessary for arriving at solutions in arithmetic calculations and problem-solving (Ashcraft & Krause, 2007). Emotionally, mathematics anxiety is associated with disliking and avoiding and possessing an unfavourable attitude towards mathematics, which, in turn, may impact achievement and eventually mathematical progress as well as career and course-related decisions that shape people’s future (Espino et al., 2017). Although most studies agree that there is no difference in the prevalence of mathematics anxiety between teachers and non-teachers (Barroso et al., (2021), there is evidence suggesting it is more prevalent among primary student teachers than in other sectors (Hembree, 1990). Research also indicates that formal mathematics instruction experiences influence this anxiety (Brady & Bowd, 2005). Generally, teachers anxious about mathematics have poor attitudes and perpetuate their anxiety and negative attitudes towards mathematics among their students (McAnallen, 2010). However, the relationships described above are not simple. Recent research, for example, suggests that the relationship between mathematics anxiety and mathematics performance is not unidirectional but rather bidirectional. Mathematics anxiety has a negative impact on performance, but poor mathematics performance may also increase anxiety (Carey et al., 2017; Maloney, Risko, Ansari, & Fugelsang, 2011). These are important observations as an established bidirectional relationship that concerns performing or teaching mathematics will exacerbate any impact and adverse outcomes for both ends, turning it into a vicious cycle.

The importance and value of formative assessment for improving learning is widely accepted. This is true for student learning (Black & Wiliams, 2009) and for in-service teacher continued growth and development (Yorke, 2003). Formative assessment with teachers creates opportunities for teachers to explore student understanding (Yorke, 2003). Such opportunities are centred on the idea of teachers as reflective practitioners, enabling them to better tailor their own teaching practice to make progressive improvements (López-Pastor and Sicilia-Camacho, 2017). The bringing together of formative feedback and reflective practice is now well established as useful for teacher development, as is the use of video to support teacher reflection.

Video has been used in teacher education since the 1960s in a variety of ways. Examples include using video for lesson analysis (Santagata, 2014), for teacher professional development (Sherin, 2004), as a prompt for discussion between teachers (Borko et al., 2008), and to create professional learning communities (Sherin, 2004; van Es, 2012). Video can be viewed as both a ‘representation of practice’ and a powerful tool for the ‘decomposition of practice’, that is, breaking the teaching into parts to enable others or oneself to focus on particular elements (Grossman et al., 2009 p. 2064). A key enabler of video as a tool for reflection in teaching is that enables the educator to become more analytical in their reflections. It does so by removing the cognitive overload of ‘in-the-moment’ decision making (Rich & Hannafin, 2009) freeing the educator to reflect on the teaching and learning interactions. This confirms Girardet’s (2018) finding that video reflection can support the development of both the analytical and reflective abilities of teachers.

In our research we bring together the elements of formative assessment, teacher reflection, and video as a tool for reflective practice to explore ways pre-service teachers (PSTs) can be supported to better understand their own practice (O’Keeffe & White, 2021, 2022). Girardet (2018), among others, reminds that video observation of teaching practices (observation of oneself or others) can support the development of analytical and reflective abilities. Detailed and systematic observation of specific practices of the teacher in classroom promotes the development of the so-called ‘professional vision’ (Goodwin, 1994), that is, the ability to notice and interpret significant features of classroom interactions.

While much video-reflection research to date has been conducted with ‘regular’ flat video, we use 360-degree video (Balzarettia et al, 2019). We argue that 360-degree video mediates the process of teacher reflection more effectively than regular video because it provides greater capacity for the user (pre-service or in-service teacher) to focus and re-focus their reflection as needed (by panning around the screen/recording) while watching back or reviewing a recording. 360-degree video allows PSTs to experience video-recorded lessons from an immersive 360-degree perspective, providing a greater understanding of the entire context in which an interaction or an action is situated. For example, an interaction mis-remembered or not observed can be reviewed from multiple perspectives after the event —nothing is ‘off-camera’.

Introduction and research question

Pedagogical content knowledge (PCK), as coined by Shulman (1986), pays attention to the way of thinking about teacher knowledge and making a school subject understandable to others. An important concept within the PCK community is the notion of Big Ideas, which involves recognising fundamental principles that underpin the subject. Loughran et al. (2004) emphasised the significance of identifying Big Ideas as a crucial component of articulating one’s PCK. Additionally, Hurst (2019) emphasised that teachers must not only grasp Big Ideas but also understand how these selected concepts interconnect. Furthermore, Chan et al. (2019) highlighted appropriate selection, connection, and coherence of Big Ideas as the first-ordered rubric for measuring the quality of PCK. This paper answers researchers’ calls for further exploration of the fundamental principles underlying the teaching of statistics (Watson et al., 2018). The research aim is to examine how preservice teachers (PTs) select and connect Big Ideas when designing a lesson sequence dedicated to statistics instruction in primary school.

According to Charles (2005), “A Big Idea is a statement of an idea that is central to the learning of mathematics, one that links numerous mathematical understandings into a coherent whole” (p. 10). This definition can be applied to the domain of statistics: A statistical Big Idea represents a statement of an idea central to learning statistics, and connects different statistical concepts and methods into a coherent structure. The findings in this paper are drawn from an ongoing multi-year educational design research aimed at supporting the development of PTs’ PCK in teaching statistical inference (Blomberg, 2022). One research question that has been explored refers to the impact of a reflection and planning tool on the outcome of the PTs’ lesson planning. In the search for potential design principles, I conjectured that predefining Big Ideas in the reflecting and planning tool would enhance the quality of the PTs’ outcomes, particularly regarding their understanding and interconnections of Big Ideas.

Conceptual framework

A recent addition to the PCK research field is the Refined Consensus Model (RCM), developed by Carlson et al. (2019). Unlike previous PCK theories, RCM embraces a more dynamic perspective that acknowledges multiple dimensions of PCK and the exchange of knowledge between these dimensions. Central aspects of teachers’ professional knowledge encompass collective, personal, and enacted PCK, and their interconnections with professional knowledge bases such as content knowledge, curricular knowledge, and pedagogical knowledge. For instance, enacted PCK encompasses activities like instructional planning, teaching, and reflection on teaching practices and student outcomes. To frame the present research and discern the elements under investigation, RCM has served as a background theory.

Within the discipline of statistics education, like the field of mathematics education, there is a noticeable range of perspectives regarding essential elements and ideas for statistical thinking and statistical literacy, and changes in practice are continually influencing them (Zieffler et al., 2018). In this study, I have employed a developed framework as a foreground theory to discern and characterise the outcomes of statistical Big Ideas among PTs. The framework is a combination of data modeling (Lehrer & Schauble, 2004) and informal statistical inference (ISI) (Makar & Rubin, 2009). The components of the statistical inference modeling framework (SI modeling) can be summarised as follows: (a) posing statistical questions within meaningful contexts, emphasising variability through real-world problems; (b) generating, selecting, and measuring attributes that exhibit variation in relation to the posed questions; (c) collecting first-hand data, prompting students to make decisions about investigation design; (d) representing, structuring and interpreting sample and sampling variability; and (e) engaging in informal inferences based on these processes and the interconnectedness of these ideas.