Looking within and beyond the geometry curriculum: connecting spatial reasoning to mathematics learning
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This commentary adopts a broad perspective in considering the contributions of papers from cross- and interdisciplinary fields of mathematics education, psychology, child development and neuroscience. The discussion aims to complement the commentary by Dindyal, focused on background research on geometry and implications for pedagogy and curricula. Spatial reasoning is considered as a common underlying theme salient to most of the papers in this issue. The fundamental role of spatial ability or visual thinking skills in young children is traced through the various theoretical approaches of studies ranging from concepts such as perspective taking, symmetry, and two- and three-dimensional shape, to the role of technological tools in mapping and location. Intervention studies to promote early spatial reasoning also provide insight into effective transformative practices. New questions are raised about the crucial development of spatial reasoning in Science, Technology, Engineering and Mathematics (STEM) education. The commentary suggests the need for a more strategic research agenda that aims to coordinate the key questions and methodologies that investigate what may seem common problems.
This special issue of ZDMI is opportune in its focus on geometry and early spatial learning and reasoning. Collectively the suite of papers contributes much to a more coherent body of research beyond the domain of geometry to key aspects of spatial ability and visual thinking both within and beyond mathematics education research. This includes studies from child development and cognitive psychology perspectives, investigations of pedagogical approaches to teaching and learning geometry, and the role of dynamic technologies in geometric learning. A cross-disciplinary view provides opportunity to revisit key theoretical approaches concerning the fundamental role of spatial ability in relation to the origins of learning and teaching of geometry. More broadly, the scope and meaning of what constitutes geometry in mathematics learning and curriculum in the twenty-first century and beyond is reconsidered. This has important implications for pedagogy and curriculum. Some papers raise new questions about the crucial development of spatial reasoning as salient to Science, Technology, Engineering and Mathematics (STEM) education (Wai, Lubunski, & Benbow 2009).
The contributing papers refer essentially to the early childhood and elementary grades, focused on children aged 4–8 years of age, and in some contexts are concerned with older children. These studies contribute to growing research evidence that highlights the strong mathematical capabilities of young children not previously considered possible. Much can be gleaned from this work to inform the teaching and learning for students in the upper grades. The affordances of a challenging and more substantial geometry curriculum emanating from the early years are also exemplified. Some of the papers also address the professional development of teachers and the support necessary to promote their spatial learning experiences.
the use of drawings and diagrams, including digital tools, in developing conceptual understanding;
the critical role of transformation skills including concepts of symmetry and perspective-taking;
a more dynamic and integrated view of geometry curriculum including construction and de-construction of figures, classification, mapping and orientation, and
the importance of mental imagery in visualizing and manipulating two- and three-dimensional forms.
2 Key themes and research questions
Before introducing the key themes of the issue the following questions should be raised: In what ways are the papers connected, and are they drawing on similar research? Is this issue taking essentially a mathematics education research perspective or a broader view of spatial reasoning?
What this particular group of papers provides is an opportunity to view spatial reasoning research in a more coherent way by foregrounding commonalities between the studies in terms of conceptual foci and by broadening the range of research designs and methodologies utilized. The issue provides a balance between theoretical foundations, introducing new interdisciplinary perspectives, with empirical work that advances theory, as well as highlighting the importance of pedagogy through classroom-based studies and cross-cultural perspectives.
The introductory paper provides a reflection on seminal theories and new theoretical perspectives pertinent to research in spatial reasoning. Complementary to this paper is the commentary by Dindyal (2015), who traces the work of key research groups investigating geometry and highlights the importance of the geometry curriculum in the primary school.
Sinclair and Bruce (2015) trace some key theories pertaining to the role of geometry in mathematics learning, while outlining current research trends at the elementary level. They draw together key themes and discuss the papers according to topic domains and approaches: the role of spatial reasoning; recent conceptions of geometry; the use of diagrams and drawings; angles and symmetry; three-dimensional geometry and volume, and the affordances of digital tools. They begin by reviewing the connection between spatial abilities and mathematics achievement in mathematics and science; and then turn attention to a lack of emphasis on spatial skills in mathematics learning and the far-reaching effect of this in later years (Clements & Sarama, 2011). Moreover they raise important implications for learning in a dynamic environment that requires much broader applications of geometry through computer-based tools and models. Studies using digital tools, and the kinds of visual processes involved in problem-based situations with younger children are exemplified.
The influence of eminent theorists such as Hans Freudenthal (1971) are highlighted as well as the contribution of mathematicians focused on spatial imagery such as Tahta (1980). Sinclair and Bruce’s paper provides a bridge between leading theorists and new ideas raised by contemporaries in the 1980s and beyond (for example, Bishop, 2008; Lehrer & Chazan, 1998; Rivera, Steinbring, & Arcavi, 2014; van Hiele, 1985). What is common is attention to the role of spatial ability and visualization across mathematics. Each of the papers in this issue attends to various notions of ‘space’ and the way that children come to interact with ‘space’. In some sense Freudenthal’s initial notion of ‘grasping space’ (Freudenthal, 1971) is revisited within new spatial problems and learning environments.
A reconceptualized view of geometry is presented in the paper by Soury-Lavergne and Maschietto who distinguish three kinds of spaces interacting in the teaching and learning of geometry: physical, graphical and geometrical space. They assert that the physical space is the world of physical objects that pupils perceive and act upon; that is, the “real” world where a concrete problem is to be solved. Geometrical space contains Euclidian theory, deductions and axioms, and provides pupils with tools to solve problems. Graphical space contains diagrams, drawings, schemas and artifacts that are utilized in the problem-solving process. They pay particular attention to the graphical space, which acts as a ‘bridge’ between the other two spaces. Similarly, Thom and McGarvey (2015) adopt an embodied cognitive view of the study of children’s drawings, and how the physical process of drawing allows geometric understandings to emerge.
New conceptions of what constitutes spatial reasoning are perhaps well exemplified in the study of 7-year olds’ mapping skills. Kotsopoulos and colleagues (2015) examined how children represent motion in large-scale mapping tasks that they referred to as “motion maps”. Underlying the ability to represent and describe their maps was the critical use of transformational skills. From an embodied cognition point of view the use of gesture is highlighted as a vehicle to express motion.
The particular focus of several papers in this issue is on symmetry and related transformation skills such a perspective taking (van den Heuvel-Panhuizen, Iliade & Robitsch, 2015). In a cross-cultural study they investigate the relationship between mathematical ability and perspective taking. This study employs challenging tasks to assess concepts systematically. Their findings advance our understanding of the difficulties young children face in visualizing objects from another viewpoint. The study draws attention to perhaps this most important aspect of spatial reasoning within a tradition of research calling for prioritization of symmetry and rotation in the geometry curriculum. And in this issue we see innovative work conducted with young children not typically expected to deal with such complex concepts.
Bruce and Hawes (2015) focus on both 2D and 3D symmetry and the relationship between these aspects using a Lesson Study approach, as part of a larger team approach. Teaching strategies are foregrounded with improvement in mental rotation shown after just 4 months of work with children from a range of abilities.
Similarly, Hallowell and colleagues (2015) focus clearly on spatial reasoning where first graders compose and decompose geometric figures in order to seek congruency with a given shape. They identify some key difficulties in children’s spatial estimation pertaining to scaling and the significance of triangular vertices, and relating lines on 2D diagrams to 3D boundaries. The result that their reasoning was inconsistent across tasks points the need for more in-depth longitudinal studies to follow the development of their spatial reasoning over time. The papers by Ng and Sinclair (2015) and Kaur (2015) extend the scope of investigations to the use of technological tools in dynamic learning environments.
Moss, Hawes, Naqvi and Caswell (2015) exemplify the powerful impact of attention to spatial reasoning through a purpose-designed geometry curriculum adapted from a Lesson Study approach in the Math for Young Children (M4YC) program. The integration of play-based activities supports a range of investigations that promote spatial reasoning in tandem with geometric concepts. This paper also contributes to a growing number of studies that provide support for professionals, for example, Building Blocks (Clements & Sarama, 2011), three-dimensional geometry (Casey et al., 2008) and the Patterning and Early Algebra Program (PEAP) (Papic, Mulligan, & Mitchelmore, 2011), that each promote spatial reasoning. In this issue the paper by Tsamir and colleagues (2015) turns attention to teaching professionals geometric concepts with a view to supporting improved pedagogy.
2.1 Spatial reasoning and geometry
This special issue reflects the mutual concern that spatial capabilities of young children continue to receive inadequate attention in both research and practice, and consequently in mathematics curriculum and assessment. Despite efforts over the past three decades to draw attention to the importance of mathematics learning processes such as ‘spatial sense’, ‘spatial reasoning’, and ‘geometric thinking’, there has been limited take up in practice.
Spatial reasoning (or spatial ability, spatial intelligence, or spatiality) refers to the ability to recognize and (mentally) manipulate the spatial properties of objects and the spatial relations among objects. Examples of spatial reasoning include: locating, orienting, decomposing/recomposing, balancing, diagramming, symmetry, navigating, comparing, scaling, and visualizing (Spatial Reasoning Study Group, 20151).
Studies of spatial reasoning have been found to be strong predictors of success in mathematics and science and school subject domains beyond mathematics and science and subsequent careers related to STEM. The need for visual literacy has become fundamental to functioning in modern society—this has been fueled by information-based communication where interfaces have become less and less alphanumeric and more and more visuo-spatial.
3 Research on spatial reasoning
Despite the seminal work The Child’s Concept of Space (Piaget & Inhelder, 1956) that influenced the research of developmental psychologists and mathematics educators, and that led to the formulation of developmental theories, interest in neo-Piagetian models of spatial learning has waned since the 1980’s. Three decades later, the community has come to realize that these developmental models do not take into consideration the sophisticated mathematical thinking that children are capable of, given rich and challenging tasks, including those with contemporary technological tools.
Although there is a large and coherent body of research on individual mathematical content domains, particularly in counting and arithmetic and rational number, there have been fewer studies in mathematics education that have attempted to focus explicitly on the developmental characteristics of geometric concepts including spatial reasoning (Clements, 2004). Much of the work on spatial reasoning has arisen from the domain of psychology focused on the issue of spatial ability.
Recent developmental studies have drawn attention to the positive impact of the early development of spatial skills on mathematical development (Newcombe, 2010; Verdine et al., 2013). New research in mathematics education has also shown that spatial reasoning skills can be developed from an early age and these are malleable and can be augmented over time (Casey et al., 2008; Hawes, Tepylo, & Moss, 2015; Uttal et al., 2013). On the other hand, lack of attention to the sustained development of these critical skills can result in deterioration if not supported and challenged.
Studies in early mathematics learning have also highlighted the role of spatial ability in the development patterning skills (Clements & Sarama, 2011; Papic et al., 2011), and the relationship between patterning and spatial structuring (Mulligan & Mitchelmore, 2013; van Nes & de Lange, 2007). Similarly, Rivera (2010) adopts a visuo-spatial approach to the development of pattern and early algebraic thinking.
The implications of these studies for teaching and learning and professional practice are what make these studies peculiar to a mathematics education view, (with the emphasis on education). They differ from studies exclusive to the psychology domain because of the design and formulation of spatial reasoning tasks and instruments; the attention to children’s growth of broader mathematical conceptual ideas and their application in practice.
4 New directions in research on spatial ability and mathematical development
For many decades the mathematics curriculum and much associated research focused almost exclusively on the development of number and computation. But, with new developments in the research on spatial ability and its connection with mathematics and science learning the papers help to elevate the relative importance of geometry and spatial reasoning in the development of mathematics overall. Further, these papers raise critical questions about the need for differentiated teaching, assessment and curriculum or intervention programs for learners who display wide differences in spatial ability not exclusive to mathematics learning.
Devlin sees mathematics ability as inclusive of number sense, but includes logical, relational and spatial reasoning abilities (Devlin, 2012, p 10). In line with Devlin’s view, Grandin (2009) argues that modern mathematics does not consider adequately the spatial sense required for people who predominantly ‘think in pictures’. These thinkers may benefit from mathematics teaching approaches that are not based solely on analysis of, and generalization from, number patterns.
Studies on the development of geometric and spatial reasoning may well be supported by growing evidence from mathematics education research as well as neurocognitive studies (Butterworth, Varma, & Lauillard, 2011; Dehaene, 2009) that connect the development of number and arithmetic processes with spatial processes. Procedures in school mathematics, for example, may depend on such attributes as number sense, subitizing (the rapid and accurate perception of small numerosities), comparison of numerical magnitudes, location on a number line, axis differentiation and symmetry (e.g., Dehaene, 2009). This view of development has support from neurocognitive studies of students (and non-school age adults) who were exceptional in that they did not have, or could not use, all of these attributes (e.g., Butterworth et al., 2011; Dehaene, 2009).
The identification of cognitive aspects such as co-linearity and axis differentiation and symmetry in studies pertaining to the neuroscience field provide strong synergies with the study of spatial processes in mathematics learning. It is not surprising that recognition of structural features such as co-linearity is critical to the visualization and manipulation of mental images. The link to number concepts and quantification of spatial attributes is also evident. The notion of spatial structure has long been recognized as an important feature of constructing measurement units and geometric properties. Battista (1999) first identified and defined spatial structuring as:
the mental operation of constructing an organization or form for an object or set of objects. It determines the object’s nature, shape, or composition by identifying its spatial components, relating and combining these components, and establishing interrelationships between components and the new object (p. 418).
Battista and colleagues (1998) found that students’ spatial structuring abilities provided the necessary input and structural organization for the numerical processes that the students used to calculate the number of squares in an array. This finding explained how attempts at enumeration sometimes engender spatial structuring, which in turn provides the input and organization for enumeration. Hence, spatial structuring is “an essential mental process underlying students’ quantitative dealings with spatial situations” (Battista et al. 1998, p. 503).
In more recent studies spatial structuring has been found integral to an Awareness of Mathematical Pattern and Structure (AMPS), and in intervention studies overall mathematics performance of low-achieving students improved including their performance in number (Mulligan & Mitchelmore, 2013). AMPS incorporates elements of spatial differentiation as well as the categorisation and analysis of patterns that are critical to spatial reasoning. Studies of AMPS in young children also integrate the ability to use visual memory to record spatial representations. Such features, including co-linearity, spatial structuring and spatial memory, may underpin development of number rather than complement it, as well as other mathematical concepts (Mulligan & Woolcott, 2015).
This is where the paper by Sourcy-Lavergne and Maschietti (2015) shows strong connections with the notion of spatial structuring, and structural features inherent in using grids and nets in a dynamic environment. They report on teaching experiments that integrate physical and digital artifacts using Cabri-Elem. Similarly, a focus on structural aspects of the pathway of robotic toys turns attention to the role of technological tools in the development of spatial reasoning. The study by Bartolini-Bussi and Baccaglini-Frank (2015) utilizes a teaching experiment design as they observe children’s capability in programming simple robotic toys and how they monitor its path. Theoretically they approach the study in order to observe processes of semiotic mediation, engaging students in a multi-modal dynamic learning experience.
Another theoretical perspective focused on spatial representations is put forward in the paper by Mamolo and colleagues (2015) where they use the term ‘geometric’ reasoning in connection with spatial representations. They develop a theoretical model for restructuring mathematical tasks, with a network of spatial visual representations designed to support geometric reasoning for learners of disparate ages, stages, strengths, and preparation. From a broader perspective the work of Mulligan and Woolcott shows synergies with the approach on networking of spatial concepts (Mulligan & Woolcott, 2015).
5 Finding research connections: promoting spatial reasoning
Recent trends have seen the promotion of spatial approaches such as through the volume, Toward a Visually-Oriented School Mathematics (Rivera, 2011) supporting a curriculum exploring a unified theory of visualization in school mathematical learning. In connection with this issue is the recent issue of ZDM (Rivera, Steinbring & Arcavi, 2014) focused on the fundamental role of visualization across the curriculum and the importance of visual imagery in mathematical abstraction. More closely connected perhaps are the synergies with the recent volume, Spatial Reasoning in the Early Years: Principles, Assertions, and Speculations (Davis & the Spatial Reasoning Study Group, 2015) that has common authorship with those represented in this Issue. These connections give support to disseminating and developing research collaborations and interest groups—the Spatial Reasoning Study Group (SRSG) formed in 2011 has disseminated their work in both volumes and fuels further research focused more broadly on spatial reasoning. Second, the research contributions of the Research Forum on Spatial Reasoning for Young Learners, featured at the recent annual meeting of the International Group for the Psychology of Mathematics Education (PME) and PME NA (Sinclair & Bruce, 2014) is well represented in this Issue. This Research Forum provided a critical analysis of the theories and practices that define contemporary geometry curriculum and pedagogy across a range of countries and contexts. The issues raised at this forum largely through discussion and through utilizing poster presentations are articulated in this Issue. In particular the process of ‘spatializing’ the curriculum’ is provided through examples of classroom innovations that might help to bring a stronger spatial reasoning emphasis into school mathematics. This is an advantage of this Issue as it represents international cutting-edge research that advances the field without necessarily being exclusive to ZDMI.
6 Implications for curriculum and practice
The research presented in this issue provides a compelling and coherent argument grounded on empirical studies of the need for attention to a spatial-oriented curriculum. In essence the issue presents studies that advance children’s mathematical learning through spatial reasoning in ways we did not think was possible. We have seen how the papers focus on key features of early spatial reasoning rather than geometry per se, as well as advocacy for the kinds of spatial learning critical for the twenty-first century. Reviews of mathematics curricula now reflect to some extent stronger support for geometry, for example, in supporting new curriculum standards in the USA (Lehrer, Slovin, Dougherty, & Zbiek, 2014).
7 Looking forward
In what ways can we view this collection of papers in part as a synthesis of research on early spatial reasoning and how can it be extended beyond mathematics education to more closely connect across academic domains into fields such as learning sciences and STEM education?
The papers in this issue have contributed directly to an emergence of a cross-disciplinary perspective arguing for greater acknowledgment that spatial reasoning is critical to functioning in twenty-first century society especially in careers associated with sciences, technology, engineering, and mathematics, i.e., the STEM disciplines.
Given the developments in research on young children’s cognitive capacities and the influence of cognitive, developmental and neuro-scientific approaches, mathematics education researchers are adopting a more concerted and collaborative approach to raising the profile of spatial reasoning in early mathematical development.
It is not practicable to second guess the exact kinds of visual skills and applications of spatial reasoning that will be essential for learning generally in the future or to support a society focused on visuo-spatial functioning in a rapidly changing environment dependent on digital technologies.
What we may need is a more strategic research agenda that aims to coordinate the key questions and methodologies that investigate what may seem common problems. The collaboration of such groups as the international SRSG aims to address these important issues by recognition of the importance of spatial reasoning, an appreciation of its malleability, and the noticing of a lack of interdisciplinary dialogue. This work should ultimately inform and strengthen not only the geometry curriculum but the whole integrated curriculum.
The Spatial Reasoning Study Group (SRSG) is a group of 20 researchers across education, mathematics, psychology, cognitive science and network analysis.
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