PROLOGUE
Spatial thinking is ubiquitous; it occurs in both our subconscious and conscious lives. At times, we operate spatially when we use our subconscious to navigate from one place to another. It is not uncommon, for instance, to hear people report arriving at a destination without being aware of how they got there. Every day, people leave work, get in their cars, and drive home without a remembrance of a single landmark. A good deal of time, we are involved in completing activities without even being aware that they rely heavily on spatial thinking skills. Examples include the spatial orientation of notes on a musical score, mathematical symbolism (e.g., fraction notation), and the process of interior decorating a home.
At other times, however, we are consciously aware that we are required to use cognitive skills that rely heavily on spatial ability. Examples in this case include graphic design, interpreting an architect’s blueprint, and working with geometric figures. Seven shapes that are commonly known as tangrams are shown in Figure 0.1. While looking at these shapes, notice that there is no specific order in their arrangement. The shapes are placed on the page, each with its own individual properties and attributes. We ask readers to examine these shapes and then mentally arrange them on the page to create one unified shape in the form of a square. When completing this task, see Figure 0.2 for one possible solution.
This challenge provides an example of the difficulty in tapping critical, multilayered spatial thinking skills, and how diverse humans are in their abilities to use spatial thinking skills when solving problems. In attempting to complete this puzzle, which represents a visuo-spatial task, some people may be able to rotate and manipulate the shapes mentally, while others would find it more conducive to re-represent the situation at hand through drawing or cutting the pieces out and having a physical model for manipulation. Still, others might find it easier to
solve a problem like this one through verbal discourse and logic. Each of these methods provides some insight into the various modes of cognitive functioning that humans display.
In this book, we take a global view of spatial intelligence through its intrinsic connections to practical, theoretical, and empirical domains of inquiry.That said, care must be taken when one attempts to classify complex behaviors because there is often overlap and integration of content represented by the subskills that comprise these behaviors. Such is the case when considering how to discuss and examine a psychological construct like spatial intelligence. Psychological perspectives on spatial thinking and cognition research have focused primarily on empirical and theoretical approaches to inquiry. Less emphasis has been placed in the spatial literature on the practical aspects of how spatial thinking can be used in our everyday lives. However, in recent years, researchers have devoted more attention to the study of the use of spatial skills in terms of its applicability in careers, education, and everyday activities. Accordingly, each of the chapters in this book challenges the idea that spatial thinking is detached from our daily lives; instead, we consider the manner in which spatiality plays an intrinsic part of our everyday existence. In fact, spatial cognition is not only an area of inquiry in and of itself, but also a skill set that provides a foundation
FIGURE 0.1 Random Placement of Tangrams
Prologue
that is necessary to both understand and conduct many life activities. This can be seen from children’s block building, as discussed in Chapter 8, to an engineer’s working of an architect’s blueprint of a planned structure discussed in Chapter 7. Each of these activities demonstrates the human ability to start with contemplative, abstract concepts that eventually lead to real-life structures that may engender functional characteristics.
From a theoretical standpoint, spatial intelligence can inform the reader how to generalize about spatial constructs within the human experience. In Chapter 6, we connect the theoretical perspective of affordance with the use of visuo-spatial constructive play objects (VCPOs)—examples of which include blocks, bricks (such as Lego), and planks. Our definition of affordance with regard to spatial intelligence differs from the typical psychological view, which contends that affordance enables the individual to advance through prompts and sensations that are attributed to the object or place. In contrast, we argue that the greater the affordance of an object, the less it will allow the individual the opportunity to think creatively and to fully understand the essence of the object’s properties in relationship to form and function. In Chapters 2 and 3, we highlight the development of spatial thinking models through an examination of five theoretical positions: Piaget and Inhelder’s theory of the child’s
FIGURE 0.2 Solution to the Tangram Problem at the Beginning of this Prologue
Prologue xi
concept of space; the Vygotskyan socio-cultural perspective on spatial cognition; the nativist perspective on spatial thinking; the interactionist approach; and aposteriori universalism.
From an empirical perspective, spatial intelligence research can be quite informative in explaining differences in spatial ability, development of spatial cognition, and the effect of spatial thinking on inquiry-related behaviors. In Chapter 4, we devote our attention to the influence of biological factors such as genes and hormones and their influence, combined with environmental conditions, on the development of spatial intelligence. Our notion of g to infinity reflects our position that the conception of intelligence has expanded from a unitary construct into a vast set of factors that include spatial ability as one of the prime components of intelligence. Therefore, we consider spatial intelligence to be the “hidden intelligence” that supports the successful completion of many everyday activities that we engage in. Spatial intelligence encompasses a skill set that some recognize as practical intelligence, where outcomes from an activity are tangible. As a hidden intelligence often overlooked in formal educational settings, spatial intelligence is firmly engrained in the work of mechanics, kitchen designers, cartographers, and many other professions. To account for the importance of thinking about space in practical terms, we explore the relationships among navigation, mapping, and spatial awareness in Chapter 5. Through the examination of Seymour Papert’s theoretical framework for learning called constructionism, which grew out of the basic principles of Piagetian constructivism in which cognitive development precedes and influences what can be learned, we have sought out connections between technological advances and the development of spatial thinking. Accordingly, explorations in the use of technology and its intrinsic connections with spatial intelligence are discussed in Chapter 9. In Chapter 10, we broaden the scope of spatial intelligence by viewing it through the ecological lens, and examine how the natural landscape influences spatial development. In this chapter, we discuss spatial dominance, which refers to the human desire to manipulate and organize space. Spatial dominance is an attempt to control the environment in which we come into contact, and arrange it in a manner that is suitable to our needs.
In Chapter 1, we begin our discussion of spatial intelligence by considering the importance of spatial thinking as a means of understanding and interacting with the physical world. We then consider the views of a diverse group of specialists in space and spatial thinking. This group comprises physicists, mathematicians, psychologists, architects, engineers, geographers, philosophers, and educators.
From the combined content of all ten chapters in this volume, we posit that spatial intelligence is truly a basic intelligence, or as we are concerned, a primal intelligence that serves as a keystone to all considerations of spatial ability upon which many other activities depend. In this volume, we explore how spatial intelligence is developed, displayed, and acted upon throughout our lives. Now we return to the initial challenge at the beginning of this prologue. Were you able
to find a correct pattern and complete the tangram puzzle successfully to meet the challenge? Whether successful or not is unimportant. What is important is that one begins to realize the ubiquitous nature of spatial thinking and its close relationship to everyday activity in a theoretical, empirical, and practical manner. Because the risk of leaving out a key individual when considering the development of an “Acknowledgments” section is considerably high, we, the authors, are culpable for any oversight that may be apparent between this book’s covers. Indeed, any work such as this one is only possible by the sifting, winnowing, and labor of others who have come before us to share their insight and findings through publications and presentations to advance knowledge and scientific thinking. Thus, we atone for any omission that we have made in our acknowledgments. That said, we are grateful to several individuals who have influenced us in shaping this work into its present form. Lynn Liben, at The Pennsylvania State University in University Park, was instrumental in leading us on a successful path of inquiry. As a pioneer in the field of spatial cognition and its multidisciplinary connections, Lynn has graciously taken the time to read our manuscript and make suggestions to advance the usefulness and completeness of the text. We thank Kathy Hirsh-Pasek and Nora Newcombe, both at Temple University, for their correspondence and the work that they have added to the corpus of literature in spatial cognition. Thanks goes to Eleanor Duckworth, at Harvard University, who was influential in having us reconsider the role of affordance in creative thinking as it applies to spatial intelligence. We also wish to acknowledge the work of the following individuals whose research and publications helped to inform our conceptions of spatial cognition and intelligence: Ken Cheng, at Macquarie University in Sydney, Australia, for his research in modularity theory; Peter Kahn, University of Washington, who influenced our views about the interaction among nature, technology, and space; Rik Pinxten, at Ghent University in Ghent, Belgium, for his work on a posteriori universalism as it applies to non-Western conceptions of space, time, and spatial–temporal constructs; Barbara Tversky, at Columbia University, for her insights on spatial language and cognition; and David Uttal, Northwestern University, for his approaches to the consideration of spatial thinking as it applies to diverse disciplines. We thank Alex Masulis, Editor, and Lauren Schuhmacher, Editorial Assistant, at Routledge as well as Emma Harder, Project Manager of Out of House Publishing. Without their assistance, this book would not have been written. We are grateful for the Summer Research Grant from St. John’s University, which enabled the reconsideration of the SPAGAR coding system (discussed in Chapter 8). We are indebted to Judith Mangione and the faculty and staff of the Department of Curriculum and Instruction at St. John’s University and to Eleanor Armour-Thomas and the faculty and staff of the Department of Secondary Education and Youth Services at the City University of New York, Queens College for their ardent support. Their passion for planks as a tool for spatial thinking and development, cognitive advancement, and social, emotional, and intellectual enrichment warmed our
Prologue xiii hearts. Finally, we are grateful to Chia-ling Lin, at Nassau Community College, for lengthy philosophical debates that lasted through the early morning hours about spatial thinking and naturalistic observation methodology, and to Mark Diercks at the CERTA Learning and Research Center for the use of facilities, equipment, and an endless supply of coffee.
Daniel Ness, Stephen J. Farenga, & Salvatore G. Garofalo New York, New York
1 THE SPATIAL DEFINITION
A Time-honored Enigma
There’s the added element of adrenaline if you’re performing.You’re aware of spatial relationships and the music.
Kim Gordon
We begin our journey on spatial intelligence by introducing four figures that are shown in pairs.While examining these pairs, special attention should be placed on how they compare and contrast with each other. Moreover, consider the following questions: first, what is the significance between each pair of figures? And second, what relationship exists, if any, among all diagrams in each of the four pairs? Before answering these two questions, consider each photograph or diagram in each of the four figures (Figures 1.1, 1.2, 1.3, and 1.4).
As for the first question, what is the significance of Figure 1.1? We begin by emphasizing the need to represent objects, in this case, water molecules, as spatial representations. In Figure 1.1 a sixth grader, who did not yet learn specifically about water molecules in the science curriculum, was asked to represent her version of the behavior of water molecules in a one-liter flask containing approximately a half liter of liquid water just reaching the boiling point (100°C or 212°F). Her depiction of this account is astonishingly accurate—namely, her visual representation of the liquid water molecules show greater density and packedness in terms of proximity of location than that of the water vapor above the liquid water line. The water vapor shows more separation between molecules. It is also worthy of note to analyze our interpretation of the sixth grader’s spatial representation. In our own interpretation, we use terms like “proximity,” “above,” and “separation” to explain her visual representation. At the same time, from a cognitive perspective, if asked to represent a situation or phenomenon using a diagram, the sixth
2 The Spatial Definition: A Time-honored Enigma
FIGURE 1.1 A sixth-grade student draws a pictorial representation (a) of the density of water molecules in a one-liter flask containing approximately a half liter of liquid water at boiling point (100°C or 212°F). A tenth-grade student (b) learns how to represent the elemental composition of a water molecule in a chemistry class
FIGURE 1.2 A fourth-grade student (a) constructed two squares—one 16 square units and the other 5 square units—using a Geoboard with elastic rubber bands. An eleventh-grade student (b) is preparing for a college entrance examination with a question that asks for an area of a given square with included side lengths
grader clearly cannot escape representing the task without employing constructs of spatial relations. The same case can be made with regard to Figure 1.1 on the right, namely, the tenth grader who is learning about the atomic composition through a visual account of the water molecule. What is fascinating here is that even at the atomic or microscopic level of physical reality, it is still possible to represent something spatially. Further, spatial considerations are given to the angular positioning of the two hydrogen atoms in relation to their bonds to the single oxygen atom—a measure that approximates 104.45 degrees. In most chemistry
FIGURE 1.3 Dr. John Snow, a nineteenth-century physician from London, tracks the number of cases of cholera by constructing a map (shown) and traces the illness to a water pump in the London community of Soho in 1854 (a). Using Geographic Information Systems (GIS), epidemiologists on the border of the West African countries of Sierra Leone and Guinea identify settlements surrounded by dense tropical forests that are likely to increase human interaction with fruit bats, one of the main vectors of the Ebola virus (b)
FIGURE 1.4 The Suzhou Star Map rubbing (a) is a stellar constellation map that influenced calendar reform in late twelfth-century China.The Swan (Omega) Nebula is a Hubble Image that captured a vast ocean of gleaming hydrogen gas with smaller quantities of oxygen, sulfur, and other elements as well. This Hubble photograph (b) shows a small section within the Omega Nebula, also known as Messier 17 in the constellation Sagittarius—a breeding ground for star formation
Photo courtesy of NASA.
Spatial Definition: A Time-honored Enigma 5 curricula, it is imperative for chemistry teachers to cover the angular positioning of atomic bonds within any given molecule. Through the lens of the water molecule, angular positioning of hydrogen atoms is important to learn because these figurations indicate how a single water molecule interacts with other water molecules. The topic of angular positioning of atoms in intra-molecular bonding clearly exemplifies the need for students to think spatially, both in terms of their understanding and in terms of eventual expertise in chemistry-related fields. Not only are these depictions spatial—i.e., spatial representations—but their referents, in other words, the water molecules themselves, are spatial, too.
What about the significance of parts (a) and (b) of Figure 1.2? Figure 1.2a is a photographic representation of a Geoboard—a mathematical manipulative that is used to help students improve their conceptual formations of mathematical concepts, particularly those in geometry. In this specific Geoboard spatial representation, we can say that the rubber band that depicts the larger square consists of 16 small squares. Notice how the pegs, depicted by the dots are represented in terms of organization; they are neatly organized in rows and columns, and the distance between each peg can be considered equal to one unit. That is, a 1 unit × 1 unit square is equal to one square unit. The outer rubber band thus creates an inner square whose area is 16 square units. Moreover, if we stipulate that sides representing diagonals divide a rectangle into two congruent right triangles, it is then possible to determine the area of the interior square—five square units (we leave the explanation for the area of this square to the reader). The main idea here is that we are thinking and considering all the parameters involved in this example from a spatial perspective. Now, in terms of Figure 1.2a, b determining the area of a given square as a question on a college entrance examination, notice the similarity between this depiction of a square and the inner square represented in Figure 1.2a; without question, they are strikingly similar. In fact, the answer to the question in Figure 1.2b is the same answer that we find when determining the area of the inner square in Figure 1.2a. Again, these two visuals demonstrate spatial considerations; there is essentially no way around it. Students and other individuals who are introduced to the Geoboard or the question posed in Figure 1.2b are in essence required to think spatially. Again, notice our language when analyzing Figure 1.2a: “formation,” “geometry,” “square,” “organization,” “rows,” “columns,” “distance,” “square unit,” “inner,” “area,” “sides,” “diagonals,” “rectangle,” “congruent,” “outer,” “right triangles,” “constructed,” “Geoboard”—these are all spatial terminology. Regardless, from a cognitive perspective, anyone who comes into contact with a Geoboard or a question such as the one asked in Figure 1.2b will be required to exhibit and use spatial thinking skills.
Next, what is the significance of parts (a) and (b) of Figure 1.3? To begin with, both diagrams demonstrate how spatial thinking plays a dynamic role in the field of epidemiology. Dr. John Snow, who developed methods that many would
The Spatial Definition: A Time-honored Enigma
argue catapulted the field of epidemiology—the study of diseases and the spread of diseases by various contagions—into the limelight of medical and academic research, generated a map of Soho that listed all, or most, of the cholera cases in the Soho vicinity by indicating individual cases of cholera using hash marks. Snow produced this map in 1854—a time when contemporary technology included the nascent stages of the daguerreotype, steam engine, and processing mills. In other words, GIS, as we know it in the twenty-first century, did not exist, and all cases had to be documented by hand or etched onto paper. Early epidemiology was a dangerous way to earn a living in 1854 because the extent to which one could come into contact with tainted drinking water (in the case of cholera) or infectious disease (such as tuberculosis) was exponentially higher than it is today; the Industrial Revolution in Great Britain was more than 60 years in the making, and this resulted in part, with large numbers of families moving into urban areas, thus making the opportunity for spreading contagious disease all the more possible. Further, Snow’s proto-epidemiological search and investigation during the late summer of 1854 occurred nearly a century before antibiotics were developed and used to treat infection and deadly maladies. To be sure, Snow demonstrated something very important with respect to his map of Soho: spatial thinking skills can enable us to grapple with adverse, and oftentimes life-threatening situations of utmost criticality. By identifying cases with hash marks on a map of Soho, Snow was able to identify the cause of the upsurge of cholera cases in the Soho vicinity—a well with a water pump serving the community tainted water with potentially deadly cholera bacteria.
In comparison to Snow’s diagram, Figure 1.3b is a GIS image of the border between Sierra Leone and Guinea near the west coast of Africa. This image is a magnificent example of a twenty-first-century version of Snow’s epidemiological map of 1854. Given that fruit bats were determined to be one of the main lines of transmission of the Ebola virus, and that this species of bat depends primarily on dense tropical forest and wetlands environments for survival, it was critical for physicians and Red Cross personnel to hone in on specific locations where fruit bats congregate. While dense rain forests are key for fruit bat survival, fruit bats carrying the Ebola virus were found primarily where they can get sources of nourishment—namely, oil palm trees. The GIS snapshot in Figure 1.3b shows buildings and other human dwellings outlined on the right and oil palm cultivations outlined on the left. The need for spatial thinking with respect to this image is unequivocal—the groves where the fruit bats congregate are within yards from communities with homes and other buildings where people do business and other forms of transactions in everyday life. Clearly, distance and proximity—both spatially charged concepts—impact the extent to which Ebola infection may occur.
What about the answer to the first question with respect to Figures 1.4a? As we indicated previously (p. 4), the Suzhou Star Map rubbing is a representation of
constellations that led to changes in the Chinese calendar system during the late twelfth century. Astronomers during the twelfth century completed the star map that connects concepts of space with time. Through the innovations of this star map, the Chinese calendrical system influenced later astronomical representations, such as that of the Tenmon Bunya no Zu that was conceived and worked out by Harumi Shibukawa on the orders of the Japanese shogunate with the intention of revising the lunisolar calendar in the late seventeenth century. Figure 1.4a differs from all previous examples in that the Suzhou Star Map rubbing is a representation of space on a macro level—as it represents galaxies and related features of the universe. So, too, does the picture in Figure 1.4b. Consisting of hydrogen gas and other elements, Swan (Omega) Nebula, a Hubble image, shows a swath of space that is approximately 5000 to 6000 light years from earth and some 15 light years in diameter. As we have indicated earlier, Messier 17 is only a section of the entire Swan Nebula—a star formation hotbed (no pun intended). Both Figures 1.4a and 1.4b demonstrate space on a massive scale. Space on the macro level—“outer space”—is often the way in which many individuals refer to space. At the very least, it is the way in which most people use the term in the everyday context.
We move on to the second question posed: what relationship exists, if any, among all diagrams in each of the four pairs? In answering this question, we refer to the work of Urie Bronfenbrenner and his contributions to ecology as it relates to human development as we consider the notion of ecological space (2009). Just as understanding the earth from an ecological perspective—such as the categories of species, population, community, ecosystem, and biosphere—can shed light on the nature of environmental change, so, too, is it possible to reflect on human development and cognition in general and space or spatial thinking in specific through the lens of an ecological framework. Thus, the relationship of each of the eight depictions in Figures 1.1 through 1.4 is evident when considering spatial cognition from an ecological perspective. From an ecological standpoint, then, Figures 1.1a and 1.1b clearly exemplify what specialists in spatial cognition would refer to in ecological contexts as a micro system in spatial thought. Next come Figures 1.2a and 1.2b what we believe falls into the category of the meso-system of spatiality. This is because our encounters with Geoboards and (spatial) test questions are physically experienced in our everyday lives; they are things we can describe clearly in that they are in proportion within our visual field, and, therefore, can be utilized and discussed through sight and touch. Figures 1.3a and 1.3b, we argue, fall into the category of the exo-system of spatial thought. This has to do with the fact that these depictions—the physical map of Soho in London and the bird’s-eye view of an Ebola-affected region—cannot be touched or viewed in the same manner as a book or a Geoboard because their referents are relatively larger spaces than items in our local environments that can be compared to our personal contexts; at the same time, they are not large to the extent that we are unable to fathom where something is located within a map or in a GIS. In contrast, Figures 1.4a and 1.4b are difficult to conceptualize within our
The Spatial Definition: A Time-honored Enigma
local contexts. Again, this has to do with our contexts in comparison with those of stellar environments—those that are almost entirely bereft of personal perspective.
To explain this phenomenon, we refer to books, periodicals, or the Internet as examples. Go to nearly any news website on the Internet. You will undoubtedly encounter a story about a newly found exoplanet, a Goldilocks exoplanet (one that may be suitable for life), up-to-date research on the oldest galaxy to date, a sighting of a black hole. The list goes on. To be sure, newly discovered events, not only in our solar system, but just about anywhere in the universe, are posted as news stories at a near-exponential rate. The point is that we’re so far away from these events that it is next to impossible to identify an actual and real representation of objects in space at this level, what, in ecological terms, would be referred to as the macro system, or astronomical space. The depictions of newly discovered phenomena that are parsecs from Earth are virtually always depicted in a way that is described by astrophysicists. We often see new exoplanets, for example, with captions that read: “An artist’s depiction of Planet X” and the like. So, in reconceptualizing the meaning of spatial thinking, it is important to consider the ecological model as a starting point in research and analysis of space and spatial thinking. All systems within the ecological framework are important in a variety of ways that affect our lives—from the spatial relations at the atomic level to those of the Goldilocks exoplanet that may be many parsecs from Earth.
So, based on the answers to the two initial questions that begin this chapter, we can safely argue that spatial thinking is universal—literally. However, our analysis of these diagrams and photographs leads us to a time-honored enigma: what is spatial thinking? Restated, what is space?
The Importance of Spatial Thinking
In the second edition of Geography for Life: National Geography Standards, the National Council for Geographic Education (NCGE) revised its standards to emphasize the need to include spatial thinking as an essential topic in the geography curriculum (NCGE, 2012). In fact, Sinton, Bednarz, Gersmehl, Kolvoord, and Uttal (2013) addressed the need for spatial thinking in geography by organizing this area of inquiry around four domains: life spaces, physical spaces, social spaces, and intellectual space. Examples of life spaces include those having to do with the fundamental concepts of location, scale, and movement and the idea that we live and make plans in space. The notion of physical spaces involves the idea that humans modify their physical environments while that of social spaces involves the idea that space can be construed in terms of culturally relevant norms, such as playgrounds, school zones, and neighborhoods. The idea of intellectual space has to do with how humans construe space in terms of how everyday things or ideas can be designed, planned, or organized. As a school subject, geography was a staple of the school curriculum during most of the twentieth century. However, with the onset of content standards during the 1980s, geography seemed to have
Spatial Definition: A Time-honored Enigma 9 been downplayed in school curricula, and since that time, geography content was integrated in the subjects of social studies and physical science.This has resulted in a meager treatment of spatial thinking in the K-12 curriculum, and in cases where the subject of social studies lacks discussion of geography, spatial thinking has been absent altogether (Holcomb & Tiefenbacher, 1989; Unwin, 2013; Winter, 2009).
While NCGE has made important inroads in addressing the importance of spatial thinking skills in geography, organizations representative of mathematics and science have either done so tangentially or have not clearly articulated the topic. Given its emphasis on English language arts/literacy and mathematics, the Common Core State Standards Initiative (CCSSI) has attempted to present a comprehensive overview of these important subjects. Yet within the mathematics component of the CCSSI, it is difficult to pin down any indication of spatial thinking as a significant component of mathematical knowledge (CCSSI, 2010). The closest implication of spatial thinking in the CCSSI Mathematics Standards can be found in Standard 4.G.A.3, which emphasizes symmetry, or Standards 8.G.A.1, 8.G.A.2, 8.G.A.3, and 8.G.A.4, which address geometric transformations, namely, reflections, rotations, translations, and dilations. Likewise, the Next Generation Science Standards (NGSS) is sorely wanting with regard to spatial thinking skills (NGSS, 2013). Although NGSS has alluded to spatial constructs in two standards (MS-ESS2-2 and HS-ESS2-1, middle-school and high-school earth science, respectively), nowhere prior to these seemingly demanding experiences do students have familiarity with specific content skills involving spatial thinking. Thus, based on the literature in the fields of cognition and education, it is unrealistic to expect students to think spatially in a deliberate and routine manner when engaging in these complex activities because doing so makes the learning experience difficult and frustrating (Bransford, Brown, & Cocking, 2000).
It is important to articulate the relationship between the K-12 school curricula with spatial skills because the ability to think spatially has become an indispensable part of numerous professions and practical applications (Kell & Lubinski, 2013). It would seem quite necessary, then, to render the teaching of spatial ability a recognized and required educational objective. To do so would be natural because the conceptual narrative emphasized in the NCGE, Common Core, and NGSS standards (Reys, 2014) parallels the conceptual constructs inherent in spatial thinking.
From a developmental perspective, the study of young children’s spatial thinking can enhance our understanding of spontaneous geometric propensities that are closely related to the study of science, technology, engineering, and medicine (STEM). Spatial thinking ability is a cross-cutting skill that overlaps several content areas (STEM, language, social sciences, and arts and humanities, to name a few). Even after accounting for verbal and mathematical ability, it has been shown that higher spatial ability predicts interest and success in STEM disciplines (Newcombe, 2010; Wai, Lubinski, & Benbow, 2009; Verdine et al., 2014). Key findings from NASA’s systems engineering studies support our assertion that students must be
exposed to physical construction materials prior to moving to the digital representations. Several reasons account for the cognitive transition from physical models to more abstract understandings of space. Despite excellent marks in college and superior research backgrounds, younger engineers had more difficulty when working in ambiguous situations than did older engineers (Brown, 2009). Further, younger engineers were found to engage in less creative reasoning when compared to older engineers (Casey, Jarvis, & Amatucci, 2008; Williams, & Derro, 2008). Archer and Lloyd (1982) state “in older children that there is a clear relationship between high spatial performance and experience with three-dimensional forms—for example wood working, model making, or toys such as Lego and Meccano” (p. 236). Siann (1977) suggests that certain activities that are linked to spatial ability are also considered traditionally masculine or feminine and may only be encountered if appropriate to one’s gender. As noted by Johnson (1984), if the science curriculum is built around interest, attitudes, and experiences that are present for one sex and not the other, identical treatment of the sexes will only accentuate the inequity. The same holds true for the mathematics curriculum as well.
While the ability to think spatially is an essential skill for school as well as success in the professional world, it may often seem hard to define. There has been general agreement in the literature, particularly in the areas of cognitive psychology and geography, about what spatial thinking is. Among the community of geography scholars, spatial thinking has been defined as “an ability to visualize and interpret location, position, distance, direction, relationships, movement, and change over space” (Sinton et al., 2013, p. 44). Clearly, this ability taps requisite skills necessary to become an architect, engineer, physician, scientist, and even artist. The NRC’s Committee on Support for Thinking Spatially claimed that “the key to spatial thinking as a constructive amalgam of three elements: concepts of space, tools of representation, and processes of reasoning” (NRC, 2006, p. 12). In light of these definitions, we have defined spatial thinking as one’s ability to perceive, recognize, or conceptualize physical or intellectual constructs in terms of their position or location in both static and dynamic systems. What these three definitions have in common is the idea that spatial thinking involves one’s interpretation and representation about space and constructs within it. Examples of spatial thinking skill sets include, but are not limited to, conceptualizing space, using tools of representation, reasoning and proving, problem finding, problem solving, visualizing relationships, analyzing static and dynamic systems of objects, observing how objects behave in their environment, recognizing the relationship between two- and three-dimensional constructs, and differentiating between Euclidean space and other geometric models. It should be pointed out that the spatial skill sets mentioned are vital in learning about relationships between tension and compression, columns and beams, trusses, and arches— concepts inherent in engineering principles. Equally important to note is that the act of spatial thinking can be, on the one hand, a deliberate activity involving intent and purpose and, on the other, one that is spontaneous and unplanned. As
Sinton et al. (2013) point out, “Spatial thinking is a constant and pervasive act in which we all participate, at times automatically and intuitively and at other times very methodically and deliberately” (p. 15).
So, we know from experts in the fields of spatial cognition and neuroscience that spatial thinking skill is an important factor not only in success and accomplishment in several professions and occupations, both within and outside STEM disciplines, but also as a way to go about numerous activities of everyday life. But even with the growing corpus of research literature on spatial thinking that has shown the benefits of spatial thinking skill, for example, that experience with mental rotation and spatial change detection tasks in infancy and early childhood predicts high levels of mathematics performance of four-year-old children (Bonny & Lourenco, 2015), or the high correlation between spatial skills and STEM disciplines (Uttal et al., 2013), it is nevertheless next to impossible to pin down precise meanings of “space” or “spatial.”
The Elusiveness of Spatial Thinking
Our discussion and interpretation of the discourse regarding the elusive nature of the terms “space” and “spatial” that we present on p. 8 are not new. In her extensive examination of multiple interpretations of space and spatial representation, Lynn Liben (1981) describes our seemingly intuitive and spontaneous ability to describe space and spatial representation on the one hand, and our ambiguity to define these terms on the other. Liben begins with an account of the definition of space, and outlines in a fitting manner the parallelism between the evolutionary progression from absolute to relative space and from the Euclidean paradigm to non-Euclidean representations of space:
Just as there has been a shift from absolute to relative spatial concepts in physics, there has also been a shift from an exclusively Euclidean, three dimensional model of space, to non-Euclidean models of space with the possibility of more than three dimensions. (p. 4)
Liben continues:
The distinctions between place and space . . . foreshadow a distinction drawn . . . between “environment” and “spatial abstraction,” that is, a distinction between location or places in particular, and spatial concepts or abstraction in general. (p. 5)
This distinction between place and space is intrinsically connected with ideas regarding psychological space and physical space. Liben points out that, contrary to the views of many cognitive psychologists who argue in favor of compartmentalization of these spaces—namely, psychologists study psychological space and
physicists study physical space—the two spaces are intertwined and not mutually exclusive. An understanding of psychological space presupposes the study of spatial behavior and spatial representation. Liben (1999) has us also reconsider what it means to understand spatial representations in general and external spatial representations in particular. By introducing the seemingly simplistic view of the socalled transparency model, that is, the notion of representing an external referent in a generally direct and somewhat spontaneous process, Liben suggests a more complex model, referred to as the embedded view, one that takes into account a constructivist perspective in which the child interacts with her environment. To this end, spatial representation is not solely based on sensation, as in a manner derived by stimulus-response, but one centered on the cognitive developmental view that integrates one’s place in space.
Liben (2006) revisited the problems of terminology with respect to “space” and “spatial” by broadening the audience on the topic of spatial thinking. Similar to our rationale for writing this text, Liben set out to address three main audiences: scholars in cognition and spatial thinking; budding scholars who are beginning to recognize and appreciate the interplay between research and practice; and practitioners and caregivers. One type of symbolic artifact that has the potential to increase awareness through connections of space and spatial thinking to our everyday lives is the map. Indeed, maps have multiple purposes and functions. Perhaps the most commonly perceived purpose is the identification of direction between or among two or more locations through the process of locomotion (i.e., getting from one place to another by walking, riding, driving, boating). As Liben emphasizes, maps also help us record and store information and make sense of data either through exploration or summarization—as in the case of an informational map that includes figures regarding varying quantities of resources or social preference (e.g., political candidate inclination). In sum, the map is categorized as a specific type of spatial representation, namely, a spatial product, that can serve as a useful starting point in education and spatial thinking skills. It is with the map that education for spatial thinking needs to be made explicit (Vygotsky, 2012). Unfortunately, with marked, adverse changes in school curricula over the past three to four decades, geography has become a steadily lifeless subject, one that has been deconstructed; its components have been made diffuse in a way that smatterings of its subject matter have been reduced to bits of content in other subjects, such as history, earth science, and political science. The slow evolution of the marginalization of geography has left the skill set connecting spatial skills with everyday experience in an indeterminate state.
Liben notes the evolving nature of individuals with that of physical entities and humans’ changing spatial representations of these entities. As an example, one need only look at a photograph of a community street scene, say, 75 or 100 years ago, and a Google Map image of the same scene today in order to appreciate her point regarding evolving environments and the potential role they play in developing spatial education curricula to be embedded in current school subjects.
In sum, what is fascinating about Liben’s earlier and later accounts of the problem with definition is how numerous dichotomies are brought to debate, especially those that have, from a historical perspective, caused great confusion in the quest for understanding space and spatial thinking. Liben’s earlier studies, then, form the backdrop of a highly arguable issue—and that is the problem of definition of “space” and “spatial” [thinking].
The Polysemy of Spatial [Thinking]
The word “polysemy,” meaning literally many (“poly”) signs (“semy”), is a term not often used in everyday discourse. But it makes a great deal of sense when it applies to the terms “space” and “spatial.” The Oxford English Dictionary defines polysemy as “The coexistence of many possible meanings for a word or phrase.” This definition is particularly applicable when referring to a word’s semantic structure. Thus, from a polysemic perspective, the task of exacting a singular definition of “space,” or of “spatial,” is so difficult to accomplish that even experts in numerous fields in the social and natural sciences have found it exceedingly difficult to do so as well.
When a noun is modified by the adjective “spatial”—such as “spatial thinking,” “spatial cognition,” “spatial sense,” “spatial ability,” “spatial orientation,” and the like—we are almost always at a loss when attempting to arrive at a definitive meaning. To frustrate matters further, this lack of definition is no novelty; it has been an enigma for millennia and remains so to the present day. As a word, “space,” as well as “spatial,” is a polysemy—a term with multiple meanings and definitions. History has shown that the terms “space” and its derivatives—“spatial thinking” with respect to the topic of this book—are elusive. In this chapter, and in the book, while we may convince most readers in setting the record straight, as it were, to find the meaning behind anything spatial or spatially related, the meaning of the term clearly has been a conundrum for practitioners and scholars throughout history.
Views of a diverse group of specialists in space and spatial thinking (including physicists, mathematicians, psychologists, architects, engineers, geographers, philosophers, and educators) and analyses of these perspectives follow.
Albert Einstein
In illustrating the difficulty in defining “space” or “spatial,” we proceed in this section by providing a historical backdrop of anecdotes by scholars, philosophers, and laypersons, many of whom have had to grapple with using the terms “space” and “spatial” in their work and in their specific science. We start with a classic quote by one of the twentieth century’s—and some would argue history’s—most influential people: Albert Einstein. The following quote by Einstein essentially
The Spatial Definition: A Time-honored Enigma
sums up the difficulty in defining something spatial. Einstein wrote the following passage in his Foreword to Max Jammer’s book Concepts of Space: The History of Theories of Space in Physics (1954).
If two different authors use the words “red,” hard,” or “disappointed,” no one doubts that they mean approximately the same thing, because these words are connected with elementary experiences in a manner which is difficult to misinterpret. But in the case of words such as “place” or “space,” whose relation with psychological experiences is less direct, there exists a far-reaching uncertainty of interpretation. (p. xiv)
Like Einstein, Jammer was a world-renowned physicist. So, the term “space” that Einstein discusses in the quote broadly refers to how it might be used within the context of a construct in physics. Regardless, his point is loud and clear: the meaning of “space” is nebulous; the term conjures up different meanings to different people when uttered, even more when context is lacking or unavailable. Einstein points out that certain words are difficult to misconstrue because their usage prevents any room for misapprehension. He uses the words “red,” “hard,” and “disappointed” as examples. When one refers to a red car, red hair, or “all red in the face,” we know that the speaker is referring to color. Even if one were to use the phrase “in the red,” which is not based on the color red, the meaning is crystal clear—namely, a cliché to describe someone or some entity that owes money. In short, the use of “red” in a sentence indicates clarity of meaning. So, too, the use of “hard.” While the word “hard” has multiple meanings, its context makes the term definitive. “Hard” can mean “difficult” or “physically inflexible.”
Not so for the words “space” or “place,” words that are so elusive that their very use can lead to even more complexity. The word “space” is all the more elusive in meaning when considering the role of word choice based on contextual associations. What better word can psychometricians and test preparers use on standardized test questions when testing language concepts or meaning within reading passages? “The word ‘space’ in the passage most nearly means . . . .” This all depends on the context of the passage. So, if finding the definition of “space” is so difficult to pin down, doing so for the term “spatial thinking” is certainly no easy task either.
Elizabeth Fennema
Next, we turn to an anecdote from an eminent mathematics education researcher, Elizabeth Fennema, Professor Emeritus from the University of Wisconsin, Madison. The following excerpt comes from Fennema’s article entitled “Mathematics, Spatial Ability and the Sexes,” a paper she had presented in 1974 at the annual meeting of the American Educational Research Association in Chicago.
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that environment into which it has its outlet. My conception of my spiritual environment is not to be compared with your scientific world of pointer readings; it is an everyday world to be compared with the material world of familiar experience. I claim it as no more real and no less real than that. Primarily it is not a world to be analysed, but a world to be lived in.
Granted that this takes us outside the sphere of exact knowledge, and that it is difficult to imagine that anything corresponding to exact science will ever be applicable to this part of our environment, the mystic is unrepentant. Because we are unable to render exact account of our environment it does not follow that it would be better to pretend that we live in a vacuum.
If the defence may be considered to have held good against the first onslaught, perhaps the next stage of the attack will be an easy tolerance. “Very well. Have it your own way. It is a harmless sort of belief—not like a more dogmatic theology. You want a sort of spiritual playground for those queer tendencies in man’s nature, which sometimes take possession of him. Run away and play then; but do not bother the serious people who are making the world go round.” The challenge now comes not from the scientific materialism which professes to seek a natural explanation of spiritual power, but from the deadlier moral materialism which despises it. Few deliberately hold the philosophy that the forces of progress are related only to the material side of our environment, but few can claim that they are not more or less under its sway. We must not interrupt the “practical men”, these busy moulders of history carrying us at ever-increasing pace towards our destiny as an ant-heap of humanity infesting the earth. But is it true in history that material forces have been the most potent factors? Call it of God, of the Devil, fanaticism, unreason; but do not underrate the power of the mystic. Mysticism may be fought as error or believed as inspired, but it is no matter for easy tolerance
We are the music-makers And we are the dreamers of dreams Wandering by lone sea-breakers
And sitting by desolate streams;
World-losers and world-forsakers, On whom the pale moon gleams: Yet we are the movers and shakers Of the world for ever, it seems.
Reality and Mysticism. But a defence before the scientists may not be a defence to our own self-questionings. We are haunted by the word reality. I have already tried to deal with the questions which arise as to the meaning of reality; but it presses on us so persistently that, at the risk of repetition, I must consider it once more from the standpoint of religion. A compromise of illusion and reality may be all very well in our attitude towards physical surroundings; but to admit such a compromise into religion would seem to be a trifling with sacred things. Reality seems to concern religious beliefs much more than any others. No one bothers as to whether there is a reality behind humour. The artist who tries to bring out the soul in his picture does not really care whether and in what sense the soul can be said to exist. Even the physicist is unconcerned as to whether atoms or electrons really exist; he usually asserts that they do, but, as we have seen, existence is there used in a domestic sense and no inquiry is made as to whether it is more than a conventional term. In most subjects (perhaps not excluding philosophy) it seems sufficient to agree on the things that we shall call real, and afterwards try to discover what we mean by the word. And so it comes about that religion seems to be the one field of inquiry in which the question of reality and existence is treated as of serious and vital importance.
But it is difficult to see how such an inquiry can be profitable. When Dr. Johnson felt himself getting tied up in argument over “Bishop Berkeley’s ingenious sophistry to prove the non-existence of matter, and that everything in the universe is merely ideal”, he answered, “striking his foot with mighty force against a large stone, till he rebounded from it,—‘I refute it thus’” Just what that action assured him of is not very obvious; but apparently he found it
comforting. And to-day the matter-of-fact scientist feels the same impulse to recoil from these flights of thought back to something kickable, although he ought to be aware by this time that what Rutherford has left us of the large stone is scarcely worth kicking.
There is still the tendency to use “reality” as a word of magic comfort like the blessed word “Mesopotamia”. If I were to assert the reality of the soul or of God, I should certainly not intend a comparison with Johnson’s large stone—a patent illusion—or even with the ′s and ′s of the quantum theory—an abstract symbolism. Therefore I have no right to use the word in religion for the purpose of borrowing on its behalf that comfortable feeling which (probably wrongly) has become associated with stones and quantum coordinates.
Scientific instincts warn me that any attempt to answer the question “What is real?” in a broader sense than that adopted for domestic purposes in science, is likely to lead to a floundering among vain words and high-sounding epithets. We all know that there are regions of the human spirit untrammelled by the world of physics. In the mystic sense of the creation around us, in the expression of art, in a yearning towards God, the soul grows upward and finds the fulfilment of something implanted in its nature. The sanction for this development is within us, a striving born with our consciousness or an Inner Light proceeding from a greater power than ours. Science can scarcely question this sanction, for the pursuit of science springs from a striving which the mind is impelled to follow, a questioning that will not be suppressed. Whether in the intellectual pursuits of science or in the mystical pursuits of the spirit, the light beckons ahead and the purpose surging in our nature responds. Can we not leave it at that? Is it really necessary to drag in the comfortable word “reality” to be administered like a pat on the back?
The problem of the scientific world is part of a broader problem— the problem of all experience. Experience may be regarded as a combination of self and environment, it being part of the problem to disentangle these two interacting components. Life, religion, knowledge, truth are all involved in this problem, some relating to the
finding of ourselves, some to the finding of our environment from the experience confronting us. All of us in our lives have to make something of this problem; and it is an important condition that we who have to solve the problem are ourselves part of the problem. Looking at the very beginning, the initial fact is the feeling of purpose in ourselves which urges us to embark on the problem. We are meant to fulfil something by our lives. There are faculties with which we are endowed, or which we ought to attain, which must find a status and an outlet in the solution. It may seem arrogant that we should in this way insist on moulding truth to our own nature; but it is rather that the problem of truth can only spring from a desire for truth which is in our nature.
A rainbow described in the symbolism of physics is a band of aethereal vibrations arranged in systematic order of wave-length from about .000040 cm. to .000072 cm. From one point of view we are paltering with the truth whenever we admire the gorgeous bow of colour, and should strive to reduce our minds to such a state that we receive the same impression from the rainbow as from a table of wave-lengths. But although that is how the rainbow impresses itself on an impersonal spectroscope, we are not giving the whole truth and significance of experience—the starting-point of the problem—if we suppress the factors wherein we ourselves differ from a spectroscope. We cannot say that the rainbow, as part of the world, was meant to convey the vivid effects of colour; but we can perhaps say that the human mind as part of the world was meant to perceive it that way
Significance and Values. When we think of the sparkling waves as moved with laughter we are evidently attributing a significance to the scene which was not there. The physical elements of the water—the scurrying electric charges—were guiltless of any intention to convey the impression that they were happy. But so also were they guiltless of any intention to convey the impression of substance, of colour, or of geometrical form of the waves. If they can be held to have had any intention at all it was to satisfy certain differential equations— and that was because they are the creatures of the mathematician
who has a partiality for differential equations. The physical no less than the mystical significance of the scene is not there; it is here—in the mind.
What we make of the world must be largely dependent on the sense-organs that we happen to possess. How the world must have changed since man came to rely on his eyes rather than his nose! You are alone on the mountains wrapt in a great silence; but equip yourself with an extra artificial sense-organ and, lo! the aether is hideous with the blare of the Savoy bands. Or—
The isle is full of noises, Sounds, and sweet airs, that give delight, and hurt not. Sometimes a thousand twangling instruments Will hum about mine ears; and sometimes voices.
So far as broader characteristics are concerned we see in Nature what we look for or are equipped to look for. Of course, I do not mean that we can arrange the details of the scene; but by the light and shade of our values we can bring out things that shall have the broad characteristics we esteem. In this sense the value placed on permanence creates the world of apparent substance; in this sense, perhaps, the God within creates the God in Nature. But no complete view can be obtained so long as we separate our consciousness from the world of which it is a part. We can only speak speculatively of that which I have called the “background of the pointer readings”; but it would at least seem plausible that if the values which give the light and shade of the world are absolute they must belong to the background, unrecognised in physics because they are not in the pointer readings but recognised by consciousness which has its roots in the background. I have no wish to put that forward as a theory; it is only to emphasise that, limited as we are to a knowledge of the physical world and its points of contact with the background in isolated consciousness, we do not quite attain that thought of the unity of the whole which is essential to a complete theory. Presumably human nature has been specialised to a considerable extent by the operation of natural selection; and it might well be debated whether its valuation of permanence and other traits now
apparently fundamental are essential properties of consciousness or have been evolved through interplay with the external world. In that case the values given by mind to the external world have originally come to it from the external world-stuff. Such a tossing to and fro of values is, I think, not foreign to our view that the world-stuff behind the pointer readings is of nature continuous with the mind.
In viewing the world in a practical way values for normal human consciousness may be taken as standard. But the evident possibility of arbitrariness in this valuation sets us hankering after a standard that could be considered final and absolute. We have two alternatives. Either there are no absolute values, so that the sanctions of the inward monitor in our consciousness are the final court of appeal beyond which it is idle to inquire. Or there are absolute values; then we can only trust optimistically that our values are some pale reflection of those of the Absolute Valuer, or that we have insight into the mind of the Absolute from whence come those strivings and sanctions whose authority we usually forbear to question.
I have naturally tried to make the outlook reached in these lectures as coherent as possible, but I should not be greatly concerned if under the shafts of criticism it becomes very ragged. Coherency goes with finality; and the anxious question is whether our arguments have begun right rather than whether they have had the good fortune to end right. The leading points which have seemed to me to deserve philosophic consideration may be summarised as follows:
(1) The symbolic nature of the entities of physics is generally recognised; and the scheme of physics is now formulated in such a way as to make it almost self-evident that it is a partial aspect of something wider
(2) Strict causality is abandoned in the material world. Our ideas of the controlling laws are in process of reconstruction and it is not possible to predict what kind of form they will ultimately take; but all the indications are that strict causality has dropped out permanently. This relieves the former necessity of supposing that mind is subject
to deterministic law or alternatively that it can suspend deterministic law in the material world.
(3) Recognising that the physical world is entirely abstract and without “actuality” apart from its linkage to consciousness, we restore consciousness to the fundamental position instead of representing it as an inessential complication occasionally found in the midst of inorganic nature at a late stage of evolutionary history.
(4) The sanction for correlating a “real” physical world to certain feelings of which we are conscious does not seem to differ in any essential respect from the sanction for correlating a spiritual domain to another side of our personality.
It is not suggested that there is anything new in this philosophy. In particular the essence of the first point has been urged by many writers, and has no doubt won individual assent from many scientists before the recent revolutions of physical theory. But it places a somewhat different complexion on the matter when this is not merely a philosophic doctrine to which intellectual assent might be given, but has become part of the scientific attitude of the day, illustrated in detail in the current scheme of physics.
Conviction. Through fourteen chapters you have followed with me the scientific approach to knowledge. I have given the philosophical reflections as they have naturally arisen from the current scientific conclusions, I hope without distorting them for theological ends. In the present chapter the standpoint has no longer been predominantly scientific; I started from that part of our experience which is not within the scope of a scientific survey, or at least is such that the methods of physical science would miss the significance that we consider it essential to attribute to it. The starting-point of belief in mystical religion is a conviction of significance or, as I have called it earlier, the sanction of a striving in the consciousness. This must be emphasised because appeal to intuitive conviction of this kind has been the foundation of religion through all ages and I do not wish to give the impression that we have now found something new and more scientific to substitute. I repudiate the idea of proving the
distinctive beliefs of religion either from the data of physical science or by the methods of physical science. Presupposing a mystical religion based not on science but (rightly or wrongly) on a self-known experience accepted as fundamental, we can proceed to discuss the various criticisms which science might bring against it or the possible conflict with scientific views of the nature of experience equally originating from self-known data.
It is necessary to examine further the nature of the conviction from which religion arises; otherwise we may seem to be countenancing a blind rejection of reason as a guide to truth. There is a hiatus in reasoning, we must admit; but it is scarcely to be described as a rejection of reasoning. There is just the same hiatus in reasoning about the physical world if we go back far enough. We can only reason from data and the ultimate data must be given to us by a non-reasoning process—a self-knowledge of that which is in our consciousness. To make a start we must be aware of something. But that is not sufficient; we must be convinced of the significance of that awareness. We are bound to claim for human nature that, either of itself or as inspired by a power beyond, it is capable of making legitimate judgments of significance. Otherwise we cannot even reach a physical world.[49]
Accordingly the conviction which we postulate is that certain states of awareness in consciousness have at least equal significance with those which are called sensations. It is perhaps not irrelevant to note that time by its dual entry into our minds (p. 51) to some extent bridges the gap between sense-impressions and these other states of awareness. Amid the latter must be found the basis of experience from which a spiritual religion arises. The conviction is scarcely a matter to be argued about, it is dependent on the forcefulness of the feeling of awareness.
But, it may be said, although we may have such a department of consciousness, may we not have misunderstood altogether the nature of that which we believe we are experiencing? That seems to me to be rather beside the point. In regard to our experience of the physical world we have very much misunderstood the meaning of our sensations. It has been the task of science to discover that
things are very different from what they seem. But we do not pluck out our eyes because they persist in deluding us with fanciful colourings instead of giving us the plain truth about wave-length. It is in the midst of such misrepresentations of environment (if you must call them so) that we have to live. It is, however, a very one-sided view of truth which can find in the glorious colouring of our surroundings nothing but misrepresentation—which takes the environment to be all important and the conscious spirit to be inessential. In our scientific chapters we have seen how the mind must be regarded as dictating the course of world-building; without it there is but formless chaos. It is the aim of physical science, so far as its scope extends, to lay bare the fundamental structure underlying the world; but science has also to explain if it can, or else humbly to accept, the fact that from this world have arisen minds capable of transmuting the bare structure into the richness of our experience. It is not misrepresentation but rather achievement—the result perhaps of long ages of biological evolution—that we should have fashioned a familiar world out of the crude basis. It is a fulfilment of the purpose of man’s nature. If likewise the spiritual world has been transmuted by a religious colour beyond anything implied in its bare external qualities, it may be allowable to assert with equal conviction that this is not misrepresentation but the achievement of a divine element in man’s nature.
May I revert again to the analogy of theology with the supposed science of humour which (after consultation with a classical authority) I venture to christen “geloeology”. Analogy is not convincing argument, but it must serve here. Consider the proverbial Scotchman with strong leanings towards philosophy and incapable of seeing a joke. There is no reason why he should not take high honours in geloeology, and for example write an acute analysis of the differences between British and American humour. His comparison of our respective jokes would be particularly unbiased and judicial, seeing that he is quite incapable of seeing the point of either. But it would be useless to consider his views as to which was following the right development; for that he would need a sympathetic understanding—he would (in the phrase appropriate to the other side of my analogy) need to be converted. The kind of help
and criticism given by the geloeologist and the philosophical theologian is to secure that there is method in our madness. The former may show that our hilarious reception of a speech is the result of a satisfactory dinner and a good cigar rather than a subtle perception of wit; the latter may show that the ecstatic mysticism of the anchorite is the vagary of a fevered body and not a transcendent revelation. But I do not think we should appeal to either of them to discuss the reality of the sense with which we claim to be endowed, nor the direction of its right development. That is a matter for our inner sense of values which we all believe in to some extent, though it may be a matter of dispute just how far it goes. If we have no such sense then it would seem that not only religion, but the physical world and all faith in reasoning totter in insecurity.
I have sometimes been asked whether science cannot now furnish an argument which ought to convince any reasonable atheist. I could no more ram religious conviction into an atheist than I could ram a joke into the Scotchman. The only hope of “converting” the latter is that through contact with merry-minded companions he may begin to realise that he is missing something in life which is worth attaining. Probably in the recesses of his solemn mind there exists inhibited the seed of humour, awaiting an awakening by such an impulse. The same advice would seem to apply to the propagation of religion; it has, I believe, the merit of being entirely orthodox advice.
We cannot pretend to offer proofs. Proof is an idol before whom the pure mathematician tortures himself. In physics we are generally content to sacrifice before the lesser shrine of Plausibility. And even the pure mathematician—that stern logician—reluctantly allows himself some prejudgments; he is never quite convinced that the scheme of mathematics is flawless, and mathematical logic has undergone revolutions as profound as the revolutions of physical theory. We are all alike stumblingly pursuing an ideal beyond our reach. In science we sometimes have convictions as to the right solution of a problem which we cherish but cannot justify; we are influenced by some innate sense of the fitness of things. So too there may come to us convictions in the spiritual sphere which our nature bids us hold to. I have given an example of one such
conviction which is rarely if ever disputed—that surrender to the mystic influence of a scene of natural beauty is right and proper for a human spirit, although it would have been deemed an unpardonable eccentricity in the “observer” contemplated in earlier chapters. Religious conviction is often described in somewhat analogous terms as a surrender; it is not to be enforced by argument on those who do not feel its claim in their own nature.
I think it is inevitable that these convictions should emphasise a personal aspect of what we are trying to grasp. We have to build the spiritual world out of symbols taken from our own personality, as we build the scientific world out of the metrical symbols of the mathematician. If not, it can only be left ungraspable—an environment dimly felt in moments of exaltation but lost to us in the sordid routine of life. To turn it into more continuous channels we must be able to approach the World-Spirit in the midst of our cares and duties in that simpler relation of spirit to spirit in which all true religion finds expression.
Mystical Religion. We have seen that the cyclic scheme of physics presupposes a background outside the scope of its investigations. In this background we must find, first, our own personality, and then perhaps a greater personality. The idea of a universal Mind or Logos would be, I think, a fairly plausible inference from the present state of scientific theory; at least it is in harmony with it. But if so, all that our inquiry justifies us in asserting is a purely colourless pantheism. Science cannot tell whether the world-spirit is good or evil, and its halting argument for the existence of a God might equally well be turned into an argument for the existence of a Devil.
I think that that is an example of the limitation of physical schemes that has troubled us before—namely, that in all such schemes opposites are represented by + and -. Past and future, cause and effect, are represented in this inadequate way. One of the greatest puzzles of science is to discover why protons and electrons are not simply the opposites of one another, although our whole conception of electric charge requires that positive and negative
electricity should be related like + and -. The direction of time’s arrow could only be determined by that incongruous mixture of theology and statistics known as the second law of thermodynamics; or, to be more explicit, the direction of the arrow could be determined by statistical rules, but its significance as a governing fact “making sense of the world” could only be deduced on teleological assumptions. If physics cannot determine which way up its own world ought to be regarded, there is not much hope of guidance from it as to ethical orientation. We trust to some inward sense of fitness when we orient the physical world with the future on top, and likewise we must trust to some inner monitor when we orient the spiritual world with the good on top.
Granted that physical science has limited its scope so as to leave a background which we are at liberty to, or even invited to, fill with a reality of spiritual import, we have yet to face the most difficult criticism from science. “Here”, says science, “I have left a domain in which I shall not interfere. I grant that you have some kind of avenue to it through the self-knowledge of consciousness, so that it is not necessarily a domain of pure agnosticism. But how are you going to deal with this domain? Have you any system of inference from mystic experience comparable to the system by which science develops a knowledge of the outside world? I do not insist on your employing my method, which I acknowledge is inapplicable; but you ought to have some defensible method. The alleged basis of experience may possibly be valid; but have I any reason to regard the religious interpretation currently given to it as anything more than muddle-headed romancing?”
The question is almost beyond my scope. I can only acknowledge its pertinency. Although I have chosen the lightest task by considering only mystical religion—and I have no impulse to defend any other—I am not competent to give an answer which shall be anything like complete. It is obvious that the insight of consciousness, although the only avenue to what I have called intimate knowledge of the reality behind the symbols of science, is not to be trusted implicitly without control. In history religious mysticism has often been associated with extravagances that cannot
be approved. I suppose too that oversensitiveness to aesthetic influences may be a sign of a neurotic temperament unhealthy to the individual. We must allow something for the pathological condition of the brain in what appear to be moments of exalted insight. One begins to fear that after all our faults have been detected and removed there will not be any “us” left. But in the study of the physical world we have ultimately to rely on our sense-organs, although they are capable of betraying us by gross illusions; similarly the avenue of consciousness into the spiritual world may be beset with pitfalls, but that does not necessarily imply that no advance is possible.
A point that must be insisted on is that religion or contact with spiritual power if it has any general importance at all must be a commonplace matter of ordinary life, and it should be treated as such in any discussion. I hope that you have not interpreted my references to mysticism as referring to abnormal experiences and revelations. I am not qualified to discuss what evidential value (if any) may be attached to the stranger forms of experience and insight. But in any case to suppose that mystical religion is mainly concerned with these is like supposing that Einstein’s theory is mainly concerned with the perihelion of Mercury and a few other exceptional observations. For a matter belonging to daily affairs the tone of current discussions often seems quite inappropriately pedantic.
As scientists we realise that colour is merely a question of the wave-lengths of aethereal vibrations; but that does not seem to have dispelled the feeling that eyes which reflect light near wave-length 4800 are a subject for rhapsody whilst those which reflect wavelength 5300 are left unsung. We have not yet reached the practice of the Laputans, who, “if they would, for example, praise the beauty of a woman, or any other animal, they describe it by rhombs, circles, parallelograms, ellipses, and other geometrical terms”. The materialist who is convinced that all phenomena arise from electrons and quanta and the like controlled by mathematical formulae, must presumably hold the belief that his wife is a rather elaborate differential equation; but he is probably tactful enough not to obtrude
this opinion in domestic life. If this kind of scientific dissection is felt to be inadequate and irrelevant in ordinary personal relationships, it is surely out of place in the most personal relationship of all—that of the human soul to a divine spirit.
We are anxious for perfect truth, but it is hard to say in what form perfect truth is to be found. I cannot quite believe that it has the form typified by an inventory. Somehow as part of its perfection there should be incorporated in it that which we esteem as a “sense of proportion”. The physicist is not conscious of any disloyalty to truth on occasions when his sense of proportion tells him to regard a plank as continuous material, well knowing that it is “really” empty space containing sparsely scattered electric charges. And the deepest philosophical researches as to the nature of the Deity may give a conception equally out of proportion for daily life; so that we should rather employ a conception that was unfolded nearly two thousand years ago.
I am standing on the threshold about to enter a room. It is a complicated business. In the first place I must shove against an atmosphere pressing with a force of fourteen pounds on every square inch of my body. I must make sure of landing on a plank travelling at twenty miles a second round the sun—a fraction of a second too early or too late, the plank would be miles away. I must do this whilst hanging from a round planet head outward into space, and with a wind of aether blowing at no one knows how many miles a second through every interstice of my body. The plank has no solidity of substance. To step on it is like stepping on a swarm of flies. Shall I not slip through? No, if I make the venture one of the flies hits me and gives a boost up again; I fall again and am knocked upwards by another fly; and so on. I may hope that the net result will be that I remain about steady; but if unfortunately I should slip through the floor or be boosted too violently up to the ceiling, the occurrence would be, not a violation of the laws of Nature, but a rare coincidence. These are some of the minor difficulties. I ought really to look at the problem four-dimensionally as concerning the intersection of my world-line with that of the plank. Then again it is necessary to determine in which direction the entropy of the world is
increasing in order to make sure that my passage over the threshold is an entrance, not an exit.
Verily, it is easier for a camel to pass through the eye of a needle than for a scientific man to pass through a door. And whether the door be barn door or church door it might be wiser that he should consent to be an ordinary man and walk in rather than wait till all the difficulties involved in a really scientific ingress are resolved.
[49] We can of course solve the problem arising from certain data without being convinced of the significance of the data the “official” scientific attitude as I have previously called it But a physical world which has only the status of the solution of a problem, arbitrarily chosen to pass an idle hour, is not what is intended here.
