Teaching and Learning Spatial Thinking with Young Students: the Use and Influence of External Representations

Previous research suggests spatial thinking is fundamental to mathematics learning (Bronowski, 1947; Clements & Sarama, 2007, 2011), and acts as a predictor for future mathematical achievement levels (Battista, 1990; Gunderson et al., 2012). However, research with regard to spatial thinking is almost non-existent in early years mathematics classrooms (Bruce, Moss, & Ross, 2012; Clements & Sarama, 2011; Newcombe & Frick, 2010; Sarama & Clements, 2009, 2011; Stipek, 2013), and how to teach it in these contexts has received little attention. Fewer studies again have focused on the use of virtual manipulatives in influencing young students’ spatial thinking (Highfield & Mulligan, 2007; Ng & Sinclair, 2015). Despite a recent surge in studies exploring the influence of virtual manipulatives in mathematics classrooms, little is known about how these manipulatives compare to physical manipulatives, especially in regard to the changes that occur in the social interactions between teacher and students during the learning process. To date, there has been no comparative study conducted that explores the influence of different external representations (e.g., physical manipulatives and virtual manipulatives) on both the teaching and the learning aspects within mathematics classrooms. The purpose of this research is to explore the use of external representations (i.e., physical manipulatives as compared to virtual manipulatives) in the mathematics classroom and how these representations support young, disadvantaged students’ spatial thinking. The use of manipulatives is a common starting point for the teaching and learning of spatial thinking.

Previous research on manipulative use (both physical and virtual) in mathematics education has yielded positive results with regard to student learning (Clements, 1999; Heddens, 1997; Highfield & Mulligan, 2007; Riconscente, 2013; Siemon et al., 2011; Warren, 2006; Warren & Miller, 2013). Recent studies indicate that these newer digital technologies promote interactions between visual and kinaesthetic learning, which have been shown to support the teaching and learning of spatial thinking (Battista, 2008; Bruce, McPherson, Sabeti, & Flynn, 2011; Clements & Sarama, 2011; Highfield & Mulligan, 2007; Jorgensen & Lowrie, 2012; Sinclair, de Freitas, & Ferrara, 2013; Sinclair & Moss, 2012). However, results from comparative studies between physical manipulatives and virtual manipulatives have been varied (e.g., Brown, 2007; Olkum, 2003; Suh, 2005). It is proposed that different types of manipulatives influence the teaching and learning of spatial thinking in different ways. By viewing the learning of spatial thinking through a sociocultural perspective, aspects of the teaching and learning of spatial learning in mathematics classrooms can be scrutinised.
A review of the literature generated two research questions that informed the research design of this study. These were:

1. What influence do different external representations (e.g., physical manipulatives and virtual manipulatives) have on young students’ learning of spatial thinking?
2. What changes occur in the teaching and learning of spatial thinking when using different external representations (e.g., physical manipulatives and virtual manipulatives)?

Given that the study focused on exploring students’ spatial thinking as they construct their knowledge from the interactions they experience with external representations, an interpretive paradigm was an appropriate epistemological, ontological and methodological stance adopted for the research. Vygotsky’s (1978) sociocultural theory provided a lens to interpret the interaction between teacher and students. Practical application of this theory permitted a narrowing lens to pinpoint particular aspects of the teaching of spatial thinking and students’ learning of spatial thinking. Within this study, these practical applications included the use of Anghileri’s “hierarchy of scaffolding practices” (2006) and Sfard’s “commognitive approach” (2008). The methodology for the study included teaching experiments. Data collection methods incorporated the use of pre-test, post-test and post post-testing using spatial testing material and observations of lessons from a teaching experiment (n = 68) comprising six lessons (three based on spatial orientation concepts and three based on spatial visualisation concepts).
Findings from this study provide further insights into the teaching and learning of spatial thinking. First, the use of manipulatives (either physical or virtual) appears to be important to students’ learning of spatial thinking. Furthermore, the use of virtual manipulatives increases the communicative functions used by students, thus benefiting their spatial thinking. Second, teachers need to be able to instantaneously access deep content and pedagogical knowledge in order to maintain their role as “more knowledgeable other” and continually contribute to the teaching and learning of spatial thinking. Finally, teaching and learning appears to be positively influenced when both the teacher and students are major contributors to the classroom discourse.

This study contributes to the understanding of how different external representations influence the teaching and learning of spatial thinking. Theoretical contributions to new knowledge include a hypothesised theory on the interaction between teacher, student and manipulatives type. Implications for future classroom practice include placing importance on the use of manipulatives and communication in mathematics classrooms. Furthermore, teachers need to be aware that their ability to instantaneously access deep levels of content and pedagogical knowledge to further develop students’ spatial thinking is essential and that for optimum learning to occur, both the teacher and students need to be major contributors to the teaching and learning process.