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

Educational interventions involving physical manipulatives for improving children's learning and development: A scoping review

Physical manipulatives (PMs) are concrete objects used during hands-on learning activities (e.g., building blocks, fraction tiles, counters), and are widely used in primary-school teaching, especially during maths instruction. This scoping review collated studies that have examined the effectiveness of educational PM interventions with pre-primary and primary-age children. A total of 102 studies met the inclusion criteria and were synthesised in the review. Most studies included a sample of children aged 4–6 years and were conducted in a school setting. They spanned 26 different countries, but almost all took place in high- or middle-income contexts, mainly in the USA. Interventions were grouped into three main learning domains: maths, literacy and science. Considerable heterogeneity was identified across the review studies in terms of the PMs and hands-on activities used (e.g., block building, shape sorting, paper folding, enactment with figurines). Evidence relating to effectiveness of the intervention programmes was synthesised, with the most promising findings identified in the maths domain. Benefits to children's spatial, literacy and science skills were also reported. Overall, however, the evidence was mixed: other studies found that PMs were not associated with learning benefits, and many were hindered by methodological shortcomings. This calls for caution when drawing conclusions about the overall effectiveness of PM interventions. Nevertheless, the findings illustrate the many ways hands-on PM activities can be incorporated into children's early learning experiences. Recommendations for further research and for using PMs in practice are made

An Examination of Pre-Service Mathematics Teachers\u27 Integration of Technology into Instructional Activities Using a Cognitive Demand Perspective and Levels of Technology Implementation

Technology has changed every aspect of our lives such as communication, shopping, games, business, and education. Technology has been used for decades in the teaching and learning environment in K-12 education and higher education, especially in mathematics education where the use of instructional technology has great potential. Today\u27s students have grown up in the technology era, so our education system should consider this situation before developing curriculum and instructional strategies. Technology can increase the quality of mathematical investigations, portray meaningful mathematical ideas to students and teachers from multiple perspectives, and change traditional ways of doing mathematics (NCTM, 2000). According to NCTM\u27s Principles and Standards for School Mathematics (2000), technologies not only increase students\u27 understanding and learning of mathematics but also help teachers make instruction more effective and meaningful for students. The purpose of this study is to explore how pre-service teachers design mathematics lesson activities that integrate technology. Specifically, the level of cognitive demands of the mathematics tasks in the technology activities: 1) created by pre-service teachers (PST) for an assignment in their elementary, middle level, and/or secondary mathematics methods course, and 2) created and used by secondary mathematics PSTs during student teaching were examined. PSTs designed technology-based instructional activities with high-level cognitive demands, and the mean scores were increased for Described Implementation and Student Response. In these instances, elementary and secondary level PSTs were able to select technology-based tasks with high cognitive demands in greater percentages than middle level PSTs. The mean scores for Described Implementation and Expected Student Response were higher than the means for Potential of the Task for all grade levels. However the means scores for Expected Student Response were lower than the means for Described Implementation for the elementary and middle levels, and the means scores are same for the secondary level. The results also indicated that PSTs were doing very well with their own personal computer use, troubleshooting, identification of instructional practices that reflected a learner-based curriculum design, and effectively technology implementation. By the analyzing relationship between IQA rubrics scores (Potential of the Task, Describe Implementation, and Expected Student Response) and each LoTi-Digital Age levels (PCU, CIP, and LoTi), the researcher discovered that the LoTi Digital-Age scores did not correlate with the IQA scores

An Inquiry into the TUI Design Space for Parent-Child Math Engagement at Home

Preschoolers’ early-math development is vital for their later math and academic achievement. Tangible user interfaces (TUI) may support early math as they feature physical objects imperative to math development and multimedia to support engagement. As a potentially meaningful context for TUIs, developmental studies highlight the need to support the home math environment (HME) that covers math-related interactions among parents and children. Therefore, we focus on HME as a design space that has not been investigated in TUI literature. We conducted an observational study involving physical-object based math activities and semi-structured interviews with 13 parent-child dyads. Our findings revealed the multifaceted nature of the HME, where children's agency is valued and providing lasting materials is challenging. Also, we realized that parents juggled their child's demands and the object-based physical activity at once. By reflecting on these findings, we propose design directions for supporting the home-math environment with TUIs

Using Effective Teaching Strategies and Personality Type to Enhance the Mathematics Classroom: A Handbook for Intermediate Math Teachers

This project addressed the need for more insightful, current, and applicable resources for intermediate math teachers in Canadian classrooms. A need for a handbook in this division seemed warranted by a lack of government resource support. Throughout an extensive review of the literature, themes and topics for the handbook emerged. The handbook was designed to not only provide educators with examples of effective teaching strategies within the mathematics classroom but to also inform them about the ways in which their personal characteristics and personality type could affect their students and their own pedagogical practices. Three teaching professionals who had each taught in an intermediate math class within the past year evaluated the handbook. The feedback received from these educators was directly applied to the first draft of the handbook in order to make it more accessible and applicable to other math teachers. Although the handbook was written with teachers in mind, the language and format used throughout the manual also make it accessible to parents, tutors, preservice education students, and educational administrators. Essentially, any individual who is hoping to inspire and educate intermediate math students could make use of the content within the handbook

The Use of virtual reality in learning descriptive text at SMA Al-Islam Krian

Regarding to the development of the technology, many schools begin to adapt technology-based media in purpose of facilitating the learning process and attracting students' interest in learning, especially in language learning. However, the use of a computer-based media commonly called CALL must be attended by an understanding of how to adapt the specific media so that the teacher can apply the media in the classroom without causing negative effects to students or teachers. This study aims to examine how the implementation of using VR as a CALL media in the classroom and obtaining the students’ perceptions toward the use of this media in learning English. This research was conducted in SMA AL-ISLAM Krian at 10 ips 1 class only. In analyzing the data, the researcher wrote the implementation VR clearly and used questionnaire to obtain the student’s perception toward the use of VR by using Technology Acceptance Model (TAM) theory form Davis (2000). The result of the research showed that the teacher implemented 5 among 9 concepts implementing VR in CALL. Whilst the concept which were not implemented by the teacher were the reflection of awareness of self and awareness of other, interacting with native speaker, and manipulation in virtual environment. The reason why the teacher did not implement those concepts was because it required an advanced technology. Besides, concerning about the student’s perceptions toward the use of VR, many of students considered it as interesting, comfortable and useful to use in English language learning

How Design Features in Digital Math Games Support Learning and Mathematics Connections

Current research shows that digital games can significantly enhance children’s learning. The purpose of this study was to examine how design features in 12 digital math games influenced children’s learning. The participants in this study were 193 children in Grades 2 through 6 (ages 8-12). During clinical interviews, children in the study completed pre-tests, interacted with digital math games, responded to questions about the digital math games, and completed post-tests. We recorded the interactions using two video perspectives that recorded children’s gameplay and responses to interviewers. We employed mixed methods to analyze the data and identify salient patterns in children’s experiences with the digital math games. The analysis revealed significant gains for 9 of the 12 digital games and most children were aware of the design features in the games. There were eight prominent categories of design features in the video data that supported learning and mathematics connections. Six categories focused on how the design features supported learning in the digital games. These categories included: accuracy feedback, unlimited/multiple attempts, information tutorials and hints, focused constraint, progressive levels, and game efficiency. Two categories were more specific to embodied cognition and action with the mathematics, and focused on how design features promoted mathematics connections. These categories included: linked representations and linked physical actions. The digital games in this study that did not include linked representations and opportunities for linked physical actions as design features did not produce significant gains. These results suggest the key role of mathematics-specific design features in the design of digital math games

The role of perception and movement in mathematics learning. Italian and Australian teachers’ beliefs and practices.

The research project is a mixed-method study on the implementation of active learning activities involving students' bodies and movement in mathematics teaching-learning practices. Although the role of the body and movement in mathematics learning and teaching strategies implicating active, experiential student engagement are central themes in mathematics education research, it is nevertheless complex to understand the extent to which these perspectives have penetrated school contexts, which are often rooted in transmissive teaching. The study involves Italian and Australian researchers in Mathematics education and Mathematics primary and secondary school teachers to investigate the reasons for this possible gap. Hindering and fostering factors were identified, highlighting differences due to cultural and contextual characteristics