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# Part I Commentary 3: Proposing a Pedagogical Framework for the Teaching and Learning of Spatial Skills: A Commentary on Three Chapters

## Abstract

Education, generally, and mathematics education specifically, have long-held associations with the field of psychology. Schoenfeld (1987) and Mayer (1992) both described the connections between the two fields and indeed, many educational theories of development evolved from psychology. To this point, one of the longest running groups in mathematics education derived from the field of cognitive psychology, namely, The International Group for the Psychology of Mathematics Education (IGPME). IGPME was established in 1976 under the guidance of Efraim Fischbein, a cognitive psychologist. Initially, the focus was, as the name suggested, on the developmental and psychological complexities of learning various mathematical concepts and processes. However, over the years, the organization has broadened to include new ways of thinking about mathematics learning that go beyond the purely cognitive aspect. In fact, very few cognitive psychologists attend the annual conference these days. Although the direct insights and engagement of cognitive psychology researchers are not commonplace, some overlap remains.

Education, generally, and mathematics education specifically, have long-held associations with the field of psychology. Schoenfeld (1987) and Mayer (1992) both described the connections between the two fields and indeed, many educational theories of development evolved from psychology. To this point, one of the longest running groups in mathematics education derived from the field of cognitive psychology, namely, The International Group for the Psychology of Mathematics Education (IGPME). IGPME was established in 1976 under the guidance of Efraim Fischbein, a cognitive psychologist. Initially, the focus was, as the name suggested, on the developmental and psychological complexities of learning various mathematical concepts and processes. However, over the years, the organization has broadened to include new ways of thinking about mathematics learning that go beyond the purely cognitive aspect. In fact, very few cognitive psychologists attend the annual conference these days. Although the direct insights and engagement of cognitive psychology researchers are not commonplace, some overlap remains.

In an article published recently in the journal *Educational Studies in Mathematics* , Bruce and colleagues (Bruce et al., 2017) outlined influences and pathways that cognitive psychologists and mathematics educators have followed, many independent seemingly from one another. Their network analysis revealed a number of factors inhibiting transdisciplinary connections including discipline-based validity and outcome expectations and unawareness of work outside researchers own fields. Nevertheless, they advocate that the two fields have reconnected through the work being undertaken in spatial reasoning. Across mathematics education and cognitive psychology, there has been a focus on identifying the ways in which spatial reasoning is linked to mathematics learning and how spatial thinking provides the foundational support for mathematics reasoning. Three of the chapters drawn from the cognitive psychology section of this book provide great insight into the work being done to understand the relation between spatial thinking and mathematics reasoning. Collectively, Casey and Fell, Jirout and Newcombe, and Young, Levine and Mix considered the effects of training spatial skills and how such skills can be predictive of and influence later mathematics performance. The three chapters focus on young children through to middle school and begin to draw out causal relationships between spatial thinking and mathematical reasoning, as studies move away from correlational data. As a collective, these chapters identify and define the underlying cognitive processes and skills that are common across spatial thinking and mathematics reasoning. This body of work, and related studies within the cognitive psychology domain, provide insights into learning seldom addressed by the mathematics education community. By way of example, studies that map curriculum content to the specific structures of spatial skills, provide opportunities to align skill development to school curricula experiences (Mix et al., 2017).

Young, Levine, and Mix point to the fact that both spatial ability and mathematics ability are multifaceted and complex, yet little research has “examined mathematical measures broadly enough to reveal separate skills. Yet this examination is vital because mathematics is frequently divided by differences in content rather than skills” (p. 121). Their studies have found that spatial processing and mathematics processing are separate latent factors that are highly correlated, through the early to late elementary school. A few factors indicated cross-loadings, with specific links found between mathematics and mental rotation at Kindergarten—possibly because it helps with encoding, imagery, and the transformations of quantities important for arithmetic problems—and visuospatial working memory and visuo-motor integration in sixth grade. They also highlight literature that indicates that the teaching of spatial constructs affords opportunities for spatial tools to be used more purposefully in teaching—with tools such as gesture and the use of spatial language helpful in mathematics learning, especial when students are exposed to novel situations.

Casey and Fell consider students’ strategy use across spatial and mathematical processing. They argue that spatial reasoning can be improved through embedding a variety of spatial strategies within different mathematics content areas, namely: fractions; word problems; and geometry. They highlight the links between being able to visualize and generate mental images and the use of spatial strategies on mathematics problems. Research has suggested that “children with higher spatial skills have greater ability to draw on spatial as well as analytical strategies when solving mathematical problems” (p. 55). This approach provides greater flexibility in mental processing strategies that may be beneficial as students move into more complex mathematics.

Jirout and Newcombe describe research associated with relative magnitude and spatial-relational reasoning when solving number line, fraction, and proportional reasoning problems using external representations, such as diagrams and models. They argue that correct interpretation and use of external representations relies on spatial scaling and understanding the relative magnitude of the representation. They identify that “interventions that promote relational reasoning and use spatial representations seem to have positive impacts on relative magnitude understanding” (p. 19), where the interventions have occurred across several different areas such as linear spatial relations; approximate number system acuity; and number lines. Jirout and Newcombe conclude that relative magnitude may rely more heavily on spatial-relational reasoning than exact number magnitude and that “education should consider ways of explicitly prioritizing relative magnitude learning along with more traditional whole-number knowledge and arithmetic processes” (p. 21).

From an education perspective, all three chapters present a range of suggestions on how to improve children’s spatial reasoning skills. There is a tension here though, which warrants further investigation. Young, Levine, and Mix highlight the success training spatial skills separately has at an undergraduate level and the play-based approach to spatial thinking in the early years. It seems they advocate for a segregated approach, where training spatial skills are taught separately from mathematics. Elsewhere, in a well-cited and highly influential article, Cheng and Mix (2014) provided evidence that spatial training could improve children’s mathematics performance. This worked had been replicated, although with somewhat moderate effect sizes (Mix & Cheng, 2018). Alternatively, Casey and Fell, and Jirout and Newcombe suggest that spatial thinking should be taught explicitly for the type of mathematics skills and thinking it relates to and promotes. In fact, Casey and Fell suggest that different types of spatial thinking need to be aligned to the most appropriate mathematics content via focused interventions. It could be argued there is a need for a synergy between the two approaches. There is great potential for cognitive psychology to have a large and meaningful impact in classrooms, beyond its influence to date. To do so, there needs to be more of a focus on context and pedagogy, with connections between curriculum and classroom practices.

Many of the intervention programs coming out of cognitive psychology have been implemented by the members of the respective author’s research teams. By way of example, Uttal et al.’ (2013) meta-analysis described training programs delivered via video games; course training, usually at undergraduate level; and spatial task training, predominantly undertaken in laboratory settings. Fewer programs are implemented by classroom teachers in situ (e.g., Bruce & Hawes, 2014; Casey et al., 2008). Perhaps this is understandable given the nature of experimental design and the associated fidelity measures required in the field. However, as we investigate how spatial training programs relate to, and improve, mathematics understanding and skills, closer attention needs to be paid to the classroom settings where most mathematics learning takes place and to those charged with educating children, teachers, and educators.

Building on these ideas, this commentary proposes a way of moving forward in the spatial reasoning literature by connecting the cognitive psychology training to the mathematics education practices through a pedagogical framework that provides a structure for classroom-based interventions.

## ELPSA Framework

Our classroom-based intervention research to date has tried to incorporate high levels of fidelity (where at all possible) into intervention programs that promote spatial training through classroom activities that were both connected to curriculum and distinctively skill based (Lowrie, Logan, & Ramful, 2017). In parallel, we have developed a pedagogical tool that could be embraced by classroom teachers, one which utilized a framework that drew on well-regarded sociological and psychological understandings of learning (Adler, 1998; Cobb, 1988; Lerman, 2003)—the Experience-Language-Pictorial-Symbolic-Application (ELPSA) learning framework (Lowrie & Patahuddin, 2015). ELPSA was used to design the lessons for the spatial reasoning intervention program and explain how students developmentally understood concepts within the respective spatial reasoning constructs. The framework promotes learning as an active process where individuals construct their own ways of knowing (developing understanding) through discrete, scaffolded activities and social interactions. Each step of the framework is critical for establishing sense making, and the sequence provides a logical structure to scaffold, reinforce, and apply knowledge and concepts.

The first element of the learning framework (Experience) draws on the knowledge that students possess. In this stage, the teacher should determine what the students know and what new information needs to be introduced to scaffold their understanding. In this first phase, students are encouraged to make connections between their own spatial practices and specific spatial forms (e.g., how they orientate a map to determine which direction they should navigate). Pedagogically, this phase also provides opportunities for the classroom teachers to understand “what individuals know.” The second component of the framework (Language) outlines how specific terminology is used to promote understanding—that is, being explicit about spatial features and intrinsic connections. This stage of the process is also associated with particular pedagogy practices, since it is important for teachers to model appropriate terminology and encourage students to use this language to describe their understandings in ways that reinforce their knowledge and promote discourse with others. The third component of the learning framework (Pictorial) is characterized by the use of spatial and concrete representations to exemplify ideas and concepts (Burte, Gardony, Hutton, & Taylor, 2017; Pillay, 1998). Such representations could be constructed by the teacher (including shared resources and artifacts) or students (including drawing diagrams or visualizing). The fourth component (Symbolic) is aligned to the formalization of ideas or concepts. This stage draws on students’ capacity to represent, construct, and manipulate analytic information with flexibility and a degree of fluency (Stieff, 2007). In this phase, capable spatial thinkers are encouraged to go beyond visual forms of reasoning, particularly when automation is possible. The final component of the learning framework (Application) highlights how symbolic understanding can be applied to new situations. This is evident in students’ ability to transfer their knowledge to novel situations.

An example of the ELPSA framework in action is described below, accompanied by student work samples and anecdotes aligned to the teachers’ pedagogy. The lesson focused on lines of symmetry and visualizing symmetry. Symmetry is part of the Australian Curriculum Mathematics. The concept is introduced around Grades 2 and 3 and elaborated on through all grades to Grade 7. The lesson was designed to encourage the students to visualize horizontal, vertical, and diagonal lines of symmetry (or reflection) and attempt to discover a pattern in the way images were represented when reflected. Symmetry and reflection are integral aspects of mathematical and scientific thinking (e.g., Hargittai & Hargittai, 2009; Livio, 2006) and as such, children require a solid foundation in understanding the spatial concept. Throughout the lesson, the teachers reinforced the need to visualize by engaging students in a cyclic process of Visualize, Predict, Experiment, Check. This cycle encouraged students to undertake the mental process of imagining what the reflection or symmetrical image would be, then attempting to describe or represent that prediction, experiment through undertaking the task, then compare their predictions to their results.

### Experience

*What is*

*symmetry*

*?*The teachers began the lesson from the viewpoint of what students knew about the topic and encouraged active engagement through contextualized whole-class discussions. Students were asked to design a symmetrical design using geometric pattern blocks or on a geometric pattern block app (see Fig. 8.1). This gave students the opportunity to illustrate to teachers their understanding of symmetry. Students also completed a task where they were required to draw the other side of a symmetrical image, in this case, a leaf (see Fig. 8.2). This was completed with varying degrees of success.

### Language

visualize→predict→experiment→check; reflection, reflective symmetry, line symmetry, reflection line, horizontal, vertical, diagonal, inclined, reflect, translate, upside down, sideways.

### Pictorial Reasoning

In the Pictorial phase, the teachers modeled symmetry concepts through diagrams, and encouraged students to do the same, aiding the transition from concrete and diagrammatical representations to more sophisticated visualization strategies. Students began with reflections of more familiar letters and symbols along the *y* and *x* axes. They were then asked to consider reflections of similar letters and symbols along the diagonal axis. During the pictorial phase, the visualize, predict, experiment, check process was used to help students from the concrete to the visual, encouraging the students to rely less on the materials.

### Symbolic Reasoning

The symbolic stage of the cycle requires analytic thinking . Typically, this form of reasoning involves the appropriate use of symbolic tools and representations. When content is spatial in nature, symbolic reasoning is associated with pattern noticing and a capacity to interpret spatial demands in a more automatic manner, often without the concrete or visual demands typically required to decode novel spatial tasks. In this phase, the classroom teachers encouraged students to reason analytically, as a transition beyond representing information “in the mind’s eye” or concretely. This symbolic reasoning was evident in the development of rules such as the orientation of objects after a diagonal reflection.

*x*and

*y*axes, horizontal stays horizontal and vertical stays vertical. However, with diagonal reflections, horizontal moves to vertical and vertical moves to horizontal. See Figs. 8.5, 8.6, and 8.7 for the symbolic concept of perpendicularity.

These diagrammatical representations are more than concrete representations or drawings of spatial information, since the pattern noticing affords opportunities for analytic reasoning. Thus, the representations become analytic thinking (see Fig. 8.7).

### Applications

## Conclusion

In Chap. 5 of this manuscript, Young, Levine, and Mix maintain that a range of spatial tools should be used to promote spatial thinking, beyond the specific content and skills typically used in intervention programs. They acknowledge that spatial tools are especially effective when students encounter novel problems, advocating that “by providing rich spatial information in multiple ways, educators can help students create a lexicon of spatial relations, terms and connections to mathematics” (p. 140). Jirout and Newcombe describe the importance of providing students with a variety of representations and advocate that certain types of spatial representations may provide explicit mathematics concepts in ways that extrapolate information more purposefully (e.g., continuous proportional representations). We argue that the ELPSA framework encourages mathematics ideas to be represented in different ways, encouraging classroom teachers to re-represent spatial ideas to consolidate student’s understanding. Each component of the ELPSA framework provides a distinct pedagogical approach to foster a repertoire of representations and encourages teachers to present information across embodied, verbal, pictorial, and symbolic representations, with the pictorial aspect open-ended with respect to the types of visual and diagrammatic tools utilized. For example, the Experience component encourages gesture and tacit thinking, the Language component specific use of rich spatial terminology, the Pictorial component the use of 2D and 3D manipulatives—both concrete and mental representations. The framework also goes some way to ensuring that an over emphasis on symbolic representations does not occur frequently or too early in concept formation (as described by Jirout and Newcombe).

The ELPSA framework advocates for the use of concrete manipulatives. Research outlined by Jirout and Newcombe suggested that use of concrete manipulatives can assist students to think spatially while acting as a scaffold for more abstract mathematical processing. The embodied nature of engaging with manipulatives in both the Experience and Pictorial phases, assists with language development as students discover explicit language associated with spatial concepts and undertake tasks with a focus on lived experiences. In a similar vein, Casey and Fell described methods for teachers to help students generate images, which they suggest is a critical aspect of children thinking spatially and essential for utilizing spatial strategies across mathematics content areas. This aligns with the pictorial phase of ELPSA. As students proceed through the visualize, predict, experiment, and check cycle, explicit opportunities for generating images, both mentally and concretely, are created.

The transition of the ELPSA framework from the pictorial to symbolic aligns with Casey and Fell’s conclusions since students with stronger spatial skills are better equipped to utilize both spatial and analytic strategies when solving arithmetic problems. ELPSA allows students the flexibility to move between pictorial and symbolic/analytic processing, and as they gain content knowledge and confidence with the analytic strategies, they will fold back less and less to representing their thinking spatially or pictorially. When students are faced with a novel or complex task, they should be encouraged to revert to spatial/pictorial strategies until they are more fluent with the analytic approaches (Lowrie & Kay, 2001; Martin, 2008). The iterative nature of the framework offers a solid pedagogical foundation for students’ spatial concept development, since the framework encourages three phases of representation to be considered before symbolic representations are introduced or applied.

Our colleagues in cognitive psychological tend to be more focused in their research designs, than those typically framed in mathematics education—providing opportunities for aspects of learning and concept developed to be quarantined. The ELPSA framework allows for the blending of the two approaches in ways that provide synergies between cognitive psychology and mathematics education.

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