EMBODIED COGNITIVE SCIENCE AND MATHEMATICS Laurie D. Edwards Saint Mary's College of California The purpose this paper is to describe two theories drawn from second-generation cognitive science: the theory of embodiment and the theory of conceptual integration. The utility of these theories in understanding mathematical thinking will be illustrated by applying them to the analysis of selected results from a study of mathematical proof. The argument is made that mathematical ideas are grounded in embodied physical experiences, either directly or indirectly, through mechanisms involving conceptual mappings among mental spaces. INTRODUCTION The goal of this paper is to clarify the central concepts and potential utility of two existing theories from outside of mathematics education for understanding mathematical thinking. The "outside" theories derive from what is known as "second generation cognitive science," which includes neurophysiology, emotion, perception and the body in its models of cognition, in contrast to first generation cognitive science, which built a model of thinking based on the metaphor of mind as computer, employing rules, physical symbols and productions systems (Lakoff & Johnson, 1999). In this paper, we will look at the theory of embodiment in cognitive science (Johnson, 2007; Lakoff & Johnson, 1999), and the theory of conceptual integration (Fauconnier & Turner, 2002). The first theory takes a broad view in connecting cognition and meaning to bodily origins and existence, and the second offers a specific set of analytical tools and proposed mechanisms for explicating how new meanings are generated from existing mental structures. The paper presents a sketch of each theory, illustrated with empirical examples drawn from recent research on mathematical proof carried out with doctoral students (Edwards, 2009b), in order to illustrate their utility in, and connections to, mathematics education. EMBODIMENT THEORY Embodiment theory offers an answer to the question of how meaning arises, and of how thought is related to action, emotion and perception. Embodiment theory proposes that meaning and cognition are deeply rooted in physical, embodied existence, on at least three levels: 1. Phylogenetic: At the level of biological evolution, our particular capacities for perception, emotion, cognition and the construction of meaning are both enabled and constrained by the current evolutionary state of our bodies, including our modes of movement, organs of perception, and nervous systems. Furthermore, this state is the result of millennia of interaction with various environments, in which the capabilities that permitted survival (including pattern-noticing, inference and problem-solving) were selected for and refined over evolutionary time.
2. Ontogenetic: At the level of the individual organism, a child is born with (or soon develops) a set of basic perceptual and cognitive capabilities as a result of the evolutionary processes described above. Recent research in infant cognition has demonstrated what could be called "proto-arithmetic", in that they can detect changes in small numbers of objects as well as "impossible" changes in the number of objects displayed to them (Deheane, 1997). The development of the individual, however, is also based to some degree on his or her specific physical experiences within a particular environment. Thus, embodiment theory would predict that children who learn mathematical concepts and procedures with hands-on mathematical manipulatives would have a different conceptualization of them than those who are taught only with symbols and two-dimensional representations. Given the fact that humans are bipedal, symmetric with respect to left and right, with a distinct front and back, and live in a world with gravity, a relatively small collection of common perceptual/cognitive constructions that seem to be common across cultures has been delineated. These constructions are called image schema, and are defined as "recurrent, stable patterns of sensorimotor experience...[that] preserve the topological structure of the perceptual whole [and have] internal structures that give rise to constrained inferences" (Johnson, 2007, p. 144). A simple example of an image schema derives from the fact that we stand in a vertical relationship to the ground, and that if we stack objects one on top of another, the pile becomes taller. As a result, we develop the image schema UP IS MORE, which is used, unconsciously, in numerous situations where we want to express an increase, both linguistically and in conventions of representation. We say, for example, "The numbers go up" when we mean, "The cardinality of the numbers increases." In the Cartesian coordinate system, the y-axis displays numbers that increase "upwardly," although, in theory, this convention could have been reversed. A second example of an ubiquitous image schema is the SOURCE-PATH-GOAL schema, which is based on our basic experience of goal-directed movement, which originates at a particular physical location, proceeds along a given path (possibly encountering obstacles or detours) and arrives at the goal location. Thus, image schema are common (but generally unconscious) building blocks of cognition available to all thinking humans living on earth. This construct from embodiment theory has already been used in the analysis of mathematical ideas (e.g., Lakoff & Núñez, 2000). 3. Microgenetic: At the level of individuals in interaction with each other, or the immediate environment, embodiment is also omnipresent. Humans must use the bodies we have to do things in the world, and also to engage in social interaction and symbolic production. Although in studying human interaction and symbolic production, many have reduced these phenomena to the words exchanged or written inscriptions produced, these things do not happen without the engagement of concomitant modalities, including physical motion, bodily stance, gesture, facial expressions, prosody and rhythm. These modalities, in particular gesture, are now
being analyzed as part of a move toward a more complete understanding of human cognition and communication (Edwards, 2008; McNeill, 1992, 2005). Embodiment proposes a theory of meaning that contrasts with both traditional linguistics, in terms of the definition of meaning, and with a representationalist, information-processing view of the mind. Rather than an objectivist view of meaning as a connection between concepts in the mind and objects in the world, mediated by symbols and words that somehow "carry" meanings, the theory of embodiment sees meaning and thought as emerging from interactions between the knower and the environment. Similarly, cognition is not seen as the manipulation of internal representations of the outside world by an internal "processor" or viewer, but as the dynamic interaction of patterns of neural activation, responding to perceptions (or equivalent re-imaginings) and preparing for action. There is support from recent neuroscience for an embodied theory of cognition and mathematics, whether from the discovery of mirror neurons that are activated when one simply thinks of an action as well as when one enacts it (Gallese & Lakoff, 2005), or the fact that the area of the human brain responsible for counting is the same as that which controls the fingers (Dehaene, 1997). Johnson (2007) summarizes what he calls an "embodied, experientialist view" of meaning, based both on the work of pragmatists philosophers like Dewey and William James, as well as empirical work in contemporary cognitive science: Meaning... arises through embodied organism-environment interactions in which significant patterns are marked within the flow of experience. Meaning emerges as we engage the pervasive qualities of situations and note distinctions that make sense of our experience and carry it forward. The meaning of something is its connections to past, present, and future experiences, actual or possible (p. 273). One of the central principles of embodiment theory, as well as of pragmatism, is that of continuity. Under this principle, thinking is an activity that is fundamentally connected to other life activities, like moving, perceiving and feeling. In addition, human cognition may differ in complexity from that of other living things, but it arises under the same circumstances described above by Johnson, and shares many common features (see, for example, research on counting abilities among primates and certain birds (Dehaene, 1997)). The principle of continuity breaks down the longheld distinction between body and mind, where image schemas and other concepts are products of the interaction between the thinker and the world, not disembodied abstractions. The mind is the on-going cumulative trace, in the form of neural patterns, of experienced and imagined action. Under the principle of continuity, "concrete" thought is not ontologically different from "abstract" thought, and mathematics is not ontologically different from other realms of thought. Instead, one of the tasks of cognitive science is to delineate how it is that the mechanisms that have allowed people to survive and thrive have also supported the creation of art, language, monuments, music and mathematics.
The principle of continuity has relevance to theories of mathematical thinking. Under some theories, there are different kinds of mathematical thought, some embodied and some not (e.g., Tall, 2007). However, the theory of embodiment, as originally conceived in contemporary cognitive science, does not recognize kinds of thinking that are ontologically distinct in this way. Although there are certainly more and less complex kinds of thinking, all cognition is "built" using the same set of mechanisms and working from the same "raw materials;" all thinking is ultimately embodied. The next section considers cognitive mechanisms utilized in the construction of ideas and inferences, whether in mathematics or in other domains of thought. THE THEORY OF CONCEPTUAL INTEGRATION The theory of conceptual integration was developed in order to explain how ideas emerge from other ideas, and how the inferential structure of one domain can be imported or mapped to another, permitting logical reasoning and the construction of more complex networks of thought out of simpler ones. The theory is based on the construct of "mental spaces" (Fauconnier & Turner, 2002). Mental spaces (which can be compared to the notion of "schema" in cognitive psychology) are partial conceptual structures, made up of elements and relations among them, derived from and elicited by our experiences and interactions. Fauconnier and Turner (2002) call them "small conceptual packets constructed as we think and talk, for the purposes of local understanding and action" (p. 40). Examples of mental spaces from mathematics are legion: we are presumed to construct mental spaces corresponding to everything from whole numbers to polygons to proofs (Lakoff & Núñez, 2000). The interesting question is how these mental spaces relate to each other, and how they are constructed. It is assumed here that the construction of mental spaces is constrained and facilitated by multiple influences, including the physical body, social interactions and cultural contexts. Taking these influences as a given, we focus on a specific mechanism for creating new mental spaces, conceptual integration. As described by Fauconnier and Turner (2002), conceptual integration connects input spaces, projects selectively to a blended space, and develops emergent structure (p. 89). In other words, conceptual integration (also referred to as conceptual mapping or conceptual blending) begins with one or more mental spaces, designated as "input spaces." Selected elements, inferences and relationships within the input space(s) are mapped to a newly created mental space, referred to as a conceptual blend or blended space. An example of a conceptual blend in mathematics is the number line (Lakoff & Núñez, 2000). A number line is neither strictly an arithmetic entity nor a geometric one - it has elements drawn from both domains. It conceptually "maps" numbers to points on a line, blending properties of numbers (for example, that 2 is greater than 1) with properties of points on a line (for example, that point B is to the right of point A). The resulting conceptual blend (that is, the mental space for "number line") has a useful emergent inferential structure that is not found in either of the input spaces.
A conceptual blend that maps a single input space ("source domain") to a single output space ("target domain") is called a single-scope blend or a conceptual metaphor (Fauconnier & Turner, 2002). An example of a conceptual metaphor for mathematical proof, based on the image schema SOURCE-PATH-GOAL, is shown in Figure 1. Source Domain: Journey Target Domain: Proof Starting location of journey Destination A path that physically leads from starting location to destination Process of finding the correct path Moving along a sub path that does not lead to the destination or the correct path ("dead end") Figure 1. The "A Proof is a Journey" Metaphor Premises Conclusion Sequence of logically linked statements from premises to conclusion Process of generating the correct sequence of statements Generating a statement not relevant to the desired sequence RESEARCH ON METAPHORS FOR PROOF Although the metaphor of a proof as a journey has plausibility as an analysis of how mathematical proof is generally talked about (and although much of the analysis of conceptual blends is carried out by using only existing exemplars of language), this kind of analysis has more validity when it is supported by purposefully collected empirical data. In a study on how proof is conceptualized, the author collected data from twelve mathematics doctoral students, who were interviewed in pairs for about 90 minutes. During the interviews, they were asked about their mathematical specializations, whether they thought there were different types of proofs, and their experiences teaching undergraduate mathematics. They were also asked to work together to create a proof for an unfamiliar conjecture, and to evaluate whether a visual argument constituted a proof. Further information about the study can be found in Edwards (2010). There was evidence for the metaphor of proof as a journey in one student's speech, as well as in his gestures. In answer to a question about what kind of proof was difficult for him, the student stated: And then the question is, well, can I fill in those steps that I have // cause you start figuring out, I m starting at point a and ending up at point b. There s gonna be some road where does it go through? And can I show that I can get through there? While speaking, the student traced a "path" in the air, as shown in Figure 2a. Thus, both the student's words and his co-produced gesture support the interpretation that he was thinking about proof as a kind of journey, as diagrammed in Figure 1.
This, however, wasn't the only metaphor for proof present in this student's speech and gesture. The first line of the utterance above was, "And then the question is, well, can I fill in those steps that I have," which was accompanied by a gesture in which he held his right hand almost horizontally at chest level, then moved it down a few inches, then down again (Figure 2b). a. A Proof is a Journey b. A Proof is a Text Figure 2: Two Metaphors/Gestures for Proof Although the student used the word "steps," which taken literally might be interpreted as referring to physical steps along a path, his gesture indicated that instead he was using the term metaphorically to refer to the sequential lines in the text of a proof, as written from the top to the bottom of a page or blackboard. Thus, the theory of conceptual integration helps to surface two ways of thinking about proof illustrated within a few moments of discourse by this student. LOGICAL INFERENCE AND EMBODIMENT In this section, the theories of embodiment and conceptual integration will be used to analyze the most basic building block of proof: logical inference. In particular, we will examine the logical construction that states that if a premise (A) is true, then another statement (B) must also be true ("if A then B"). From the point of view of embodiment, such a relation is not a disembodied abstraction, nor a transcendent, eternal truth. Instead, the ultimate grounding for the ability to make and understand logical "if-then" statements can be found in common physical experiences. Note that these experiences are not claimed to be the only source required to construct logical inference; rather, the argument is that they provide a necessary embodied grounding. The primary embodied experience underlying the later construction of logical inference is proposed to be physical situations in which one event precedes and is inferred (from empirical experience) to physically cause another event. The development of the ability of human infants to empirically infer physical causality has been investigated both by developmental psychologists and by cognitive linguists (cf, Sperber, Premack & Premack, 1996). Very young infants are able to respond and act in a way that indicates that they understand that one action can cause a subsequent action (ibid.). The sequentiality of the two actions is important in constructing the notion of physical causality - the learner has to observe one event consistently occurring after another ("following it") in order to empirically infer causality. The
language of sequentiality is also found in discourse about logical proof, i.e., "B follows from A". Thus, the overall argument is that recurrent patterns in which one physical event occurs after another are used to construct a basic schema or mental space for physical causality. This schema is subsequently used, unconsciously and metaphorically, to support other kinds of reasoning, among which is logical inference. Figure 3, read from bottom to top, sketches this argument. Figure 3: Logical implication built from schema for physical causality Linguists have analyzed constructions within multiple languages that extend or map the notion of physical causality to both social situations and logical statements (Lakoff & Johnson, 1999). The theories of embodiment and conceptual integration hold that the image schemas that arise out of situations involving physical actions and reactions form the cognitive basis for the more abstract kind of reasoning involved in formal logic and proof. The implication is that rather than being an abstract, purely formal enterprise, mathematical proof is based in perceptions and experiences that are firmly embodied. As Johnson (2007) notes, "According to this view, we do not have two kinds of logic, one for spatial-bodily concepts and a wholly different one for abstract concepts. There is no disembodied logic at all. Instead, we recruit bodybased, image-schematic logic to perform abstract reasoning" (p. 181). RELATIONSHIPS AMONG THEORIES Embodied cognitive science, including the theory of conceptual integration, came onto the mathematics education scene at a time in which well-established theories were already doing useful work. These theories included radical constructivism, socio-cultural theory, various specific theories of reification (process-concept transformations), semiotics, and information processing theory. The theory of conceptual integration is based on the principles of embodiment, but offers specific mechanisms to account for how our embodied experiences become reflected in our thought and language (where language is taken broadly to include such things as gesture, written inscriptions and external imagery).
But how do these theories related to the major theories in mathematics education? At a foundational level, embodied cognition is incompatible with any theory that views meaning as an objective coupling between the external world and internal representations, or cognition as a set of rules that could be instantiated in silicon chips just as well as in the brain/body. In embodied cognitive science, cognition requires an active organism (a brain within a body) engaged in ongoing interaction and adaptation within an environment, and thinking, even logical reasoning, is ultimately rooted in physical experience. Thus, certain perspectives from information processing psychology (for example, the notion that reasoning can be modelled solely by the manipulation of propositions) would contradict embodiment theory. However, embodiment is compatible with many other theories used in mathematics education. In my view, the situation is like that of the blind men grasping the elephant, with each theory giving only part of the whole picture. Embodiment is consistent with the tenets of radical constructivism that hold that there is no "godseye," objective view of reality, but only the individual's constructions based on his or her experience. However, it proposes a grounding for these constructions in physical experience. It is likewise consistent with models of intellectual development in which more complex thinking and capabilities emerge from simpler ones. The theory and constructs of conceptual integration offer a mechanism for the construction of new ideas (mental spaces) that is compatible with schema theory. However, Piaget's discontinuous stage theory, in which strict demarcations between levels of conceptual development are proposed, would be rejected, on the principle of continuity. The theories of embodiment and conceptual integration are also fully compatible with socio-cultural theory, situated cognition, and theories that emphasize discourse. Embodiment and conceptual integration acknowledge that the environment in which cognition develops in humans includes other people, as well as the cultures and institutions they have created. It also stipulates that language and discourse are a vital part of the medium within which thought develops. Mental spaces and conceptual blends do not emerge in isolation from the surrounding culture; in fact, they fully reflect (and contribute to) that culture. Returning to proof as an example, for many secondary school students in the United States, a proof must be presented in a twocolumn format, with statements on the left and justifications (in the form of alreadyproved theorems) for the statements on the right. This convention, which is culturally specific, would form part of their mental space for proof. What the theory of embodiment insists is that although intellectual constructions, including mathematical ideas, are socially constructed, they are not unconstrained or arbitrary. Instead, they are made possibly by, grounded in, and constrained by physical realities (Nuñéz, Edwards & Matos, 1999). As noted in the introduction, these realities include the way our bodies and brains have evolved, how they develop throughout our life spans, and how we learn through multiple modes of engagement with our environment. From the perspective of embodied cognitive science, the human intellectual product, mathematics, is grounded in embodied physical
experiences, either directly or indirectly, and grows through the mechanism of conceptual integration as well as other transformations of mental spaces (Fauconnier & Turner, 2002). The recent work utilizing the theory and tools of semiotics in the analysis of mathematics shares the goals of embodiment theory and conceptual integration, to understand the construction of mathematical meaning, including attention to the important roles of shared signs and symbols. However, embodiment looks for meaning beyond relations among signs or within semiotic systems, and is careful to avoid the objectification of these human constructions. That is, signs and symbols are not characterized or investigated as formal systems, or as the "carriers" of meaning (indeed, the idea that any kind of linguistic expression can "carry" meaning is a pervasive objectivist metaphor). Instead, according to embodiment theory, the physical (and possibly even the social) world is first experienced at a non-linguistic level, and such experiences are needed in order to attach meaning to culturally created semiotic systems. Anna Sfard (2004) and others have highlighted an important construct in mathematics, the idea that mathematical activities and processes are often reconceptualized and treated as "objects" by mathematics learners and thinkers. Sfard has called this conceptual process "reification" or "objectification", and has proposed that this process is the source of a basic metaphor in mathematics, that of the mathematical "object" (loc. cit.) Although a full analysis or synthesis of the relationship between reification and embodiment theory has not been carried out, it would be fruitful to investigate whether the situation might actually be reversed. That is, does our knowledge of physical objects and actions provide input spaces for a conceptual blend that allows us to see mathematical objects as processes? This may not be the case, but if not, then another cognitive and/or neurobiological source for the capability of transforming processes to objects must be sought, and, under the principles of embodiment and continuity, this is unlikely to be a capability that applies only to mathematical thought. One of the goals of carrying out research and building theory in mathematics education is, presumably, to reach a more complete understanding of mathematical thinking, learning and teaching. Although we may be a long way from seeing the whole elephant, I would argue that to reach, eventually, an integrated and comprehensive theory of mathematical thinking we will need to incorporate the kind of knowledge gained from contemporary work in embodied cognitive science. REFERENCES Dehaene, S. (1997). The number sense: How the mind creates mathematics. Oxford, Oxford University Press. Edwards, L. D. (2008). Conceptual integration, gesture and mathematics. In O. Figueras, & A. Sepúlveda. (Eds.). Proceedings of the Joint Meeting of the 32nd
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