Principles of grounded and embodied cognition address the role of the body and body-based resources to shape cognition. Our objective is to present a theory of grounded and embodied mathematical cognition (GEMC) consistent with Stokes’ (1997) ‘use-inspired research,’ by contributing generalizable models of mathematical thinking and learning that support the application of theory to the design of effective, scalable learning experiences in science, technology, engineering, and mathematics (STEM) education. In addition to presenting the theory, we discuss how we used our theory of GEMC to guide the design of a video game that engages players’ action systems in order to promote mathematical reasoning. Our specific focus is on understanding and improving mathematical proof skills for geometry. As an example of the application of the GEMC theory, we describe early findings from a small pilot study of middle- and high-school students to understand the influences of action-based interventions on their mathematical insights and proofs. We use this occasion to discuss the inherent challenges of designing effective STEM learning environments derived from cognitive theory.

Geometric proof is a valuable content area for making strides for theories of GEMC. First, geometry is seen as the study of the properties of space and shape, and therefore should be suitable to a GEMC perspective. Second, it is an area of advanced mathematics, typically studied by students planning to attend college and other post-secondary educational programs (Pelavin & Kane, 1990). Third, geometric proof is primarily concerned with universal statements about space and objects, and therefore addresses an important area of abstract thought. Proof is an especially intriguing area of study because of the central role of conceptual understanding rather than using only ‘canned’ procedures or mathematical algorithms that might enable people to generate a valid answer with little understanding of the mathematics involved (e.g., long division). Finally, geometry plays a profound role in all of the STEM disciplines. Because of these deep connections, there is the potential for advancements in this program of research to impact mathematical reasoning and STEM education more broadly.

### Research to practice for STEM: the need to improve proof education

Justification and proof are central activities in mathematics education (National Council of Teachers of Mathematics, 2000; Yackel & Hanna, 2003). In fact, ‘proof and proving are fundamental to doing and knowing mathematics; they are the basis of mathematical understanding and essential in developing, establishing, and communicating mathematical knowledge’ (Stylianides, 2007, p. 289).

Research has long revealed that students struggle to construct viable and convincing mathematical arguments and provide valid generalizations of mathematical ideas (Dreyfus, 1999; Healy & Hoyles, 2000; Martin, McCrone, Bower, & Dindyal, 2005). Students tend to be overly reliant on examples when exploring mathematical conjectures and often conclude that a universal statement is true on the basis of only checking examples that satisfy the statement (e.g., Healy & Hoyles, 2000; Knuth, Choppin, & Bieda, 2009; Porteous, 1990). When presented with deductive proofs, students frequently find them unconvincing (Chazan, 1993), and fail to appreciate the utility of deductive reasoning for communicating generalized arguments based on logical inference (Harel & Sowder, 1998). Interviews by Coe and Ruthven (1994) showed that even advanced college mathematics students held restricted manners and attitudes toward proof. These students typically looked for standardized routines to guide their investigations, rather than seeking out methods suited to the specific conjectures at hand. Furthermore, these advanced mathematics students seldom sought out explanations that would illuminate or give them insights into the general rules and patterns, and rarely attempted to connect these patterns to broader mathematical ideas or frameworks.

In reaction, some mathematics education scholars call for more innovative approaches to proof instruction that focus on the construction and negotiation of mathematical meaning (Stylianides, 2007). Harel and Sowder (1998) define proving as ‘the process employed by an individual to remove or create doubts about the truth of an observation’ (p. 241). Thus, the process of proving encompasses a wide range of activities where students reason critically about mathematical ideas rather than focus only on an abstract, concise end product disconnected from situated reasoning.

### Grounded and embodied mathematical cognition

We view mathematical communication as a multimodal discourse practice (e.g., Edwards, 2009; Hall, Ma, & Nemirovsky, 2015; Radford, Edwards, & Arzarello, 2009; Roth, 1994; Stevens, 2012), rather than a formal, written, propositional form. When constructing valid proofs, individuals often communicate a logical and persuasive chain of reasoning using descriptive language, verbal inference, and gestures. Research on mathematicians’ proving practices has suggested that proof ‘is a richly embodied practice that involves inscribing and manipulating notations, interacting with those notations through speech and gesture, and using the body to enact the meanings of mathematical ideas’ (Marghetis, Edwards, & Núñez, 2014, p. 243). Observations show that both teachers and students use multimodal forms of talk using speech-accompanied gestures as a way to track the development of key ideas when exploring mathematical conjectures (Nathan, Walkington, Srisurichan, & Alibali, 2011). We refer to these kinds of arguments and communications as ‘informal proofs.’ While they relay the key ideas and transformations needed to explain why properties do or do not hold, they are not always organized in the propositional, deductive, and meticulous manner of formal proofs.

#### Grounded and embodied cognition

Mathematical thinking and communication, like other forms of cognitive behaviors, are of interest to the growing research program on grounded and embodied cognition (Shapiro, 2014). Grounded cognition (Barsalou, 2008, p. 619) is a broad framework that posits that intellectual behavior ‘is typically grounded in multiple ways, including simulations, situated action, and, on occasion, bodily states.’ When the focus is on the grounding role of the body, scholars typically use the more restricted term, ‘embodied cognition’. Grounded cognition is contrasted with models of cognition based on ‘AAA symbol systems’ that are abstract, amodal, and arbitrarily mapped to the concepts to which they refer (Glenberg, Gutierrez, Levin, Japuntich, & Kaschak, 2004). Yet working with symbolic notational systems and making general claims about idealized entities (such as perfect circles) through logical deduction is at the heart of mathematical proof construction. Several scholars have provided accounts of thinking about abstract entities and relationships that we never actually see or touch based on principles of grounded and embodied cognition (Casasanto & Boroditsky, 2008; Lakoff & Nunez, 2000). Thus, a central goal in this work is to explicate how a GEMC perspective accounts for seemingly abstract forms of reasoning.

#### Directed actions, gestures, and learning

One form of GEMC intervention explores the effects of directed actions on reasoning. Here we define ‘directed actions’ as physical movements that learners are instructed to formulate by some kind of pedagogical agent (Thomas & Lleras, 2009). ‘Gestures’ can be distinguished from directed actions in that they are spontaneously generated movements, often of the hand, that accompany speech and thought (Chu & Kita, 2011; Goldin-Meadow, 2005; Nathan, 2014). Our review of the literature on directed actions, gestures, and learning reveals four empirically based findings of note: (1) gesture production predicts learning and performance; (2) directed actions can influence mathematical cognition; (3) directed actions from earlier training opportunities leave a historical trace, or legacy, expressed through gestures during later performance and explanation; and (4) mathematical reasoning and learning are further enhanced when actions are coupled with task-relevant speech, leading to coordinated action-speech events that are the hallmark of contemporary gesture research. Taken together, these findings support the assertion that actions serve a valuable role in addition to language in both fostering and conveying mathematical ideas.

The first finding - that gesture production predicts learning and performance - includes content areas such as mathematics (Cook, Mitchell, & Goldin-Meadow, 2008; Valenzeno, Alibali, & Klatzky, 2003) and language (Glenberg et al., 2004; McCafferty & Stam, 2009), as well as broad influences such as general problem solving (Alibali, Spencer, Knox, & Kita, 2011; Beilock & Goldin-Meadow, 2010), inference-making (Nathan & Martinez, 2015), and cognitive development (Church & Goldin-Meadow, 1986). Conversely, when gesture production is controlled, gesture inhibition often disrupts performance and learning (Hostetter, Alibali, & Kita, 2007; Nathan & Martinez, 2015).

For example, the likelihood that students produced valid proofs for mathematical conjectures was positively associated with the presence of ‘dynamic depictive gestures’ (Donovan et al., 2014). Depictive gestures are gestures through which speakers directly represent objects or ideas with their bodies (e.g., forming an angle with their two hands; McNeill, 1992). Dynamic depictive gestures (which we will often refer to as ‘dynamic gestures’) are defined as those that show a motion-based transformation of a mathematical object through multiple states (Walkington et al., 2014). The odds of generating valid proofs were 4.14-times greater (95% confidence interval 1.57–10.92) for participants who produced dynamic gestures than those who did not (Donovan et al., 2014). Figure 1 shows a student making a dynamic depictive gesture while proving the statement that ‘The sum of the lengths of any two sides of a triangle must be greater than the length of the remaining side.’ The gesture uses motion-based transformations to show two sides of the triangle being unable to meet. Pier et al. (2014) demonstrated that the benefits of dynamic gestures for predicting performance on verbal mathematical proofs are over and above the effects of variations in speech.

The second finding comes from a growing body of empirical literature indicating that directed actions can influence learning. Superior problem-solving performance has been demonstrated when students follow directions to perform specific actions hypothesized to foster effective problem-solving strategies (Goldin-Meadow, Cook, & Mitchell, 2009). Thomas and Lleras (2007) showed that manipulating eye-gaze patterns can, unbeknownst to participants, affect the success of solving Dunker’s classic Radiation Problem. In mathematics, Abrahamson and Trninic (2015; Abrahamson, 2015) increased primary grade children’s awareness of mathematical proportions by engaging their hand and arm motions in order to achieve a particular goal state of the system (a green illuminated screen rather than a red one) once they enacted the appropriate (but tacit) proportions with their relative rates of dual hand movements. It may be said that their hand movements constituted a form of problem solving or epistemic action (Kirsh & Maglio, 1994). According to the authors (Abrahamson & Trninic, 2015), participants’ movements did not elicit proportional reasoning, tacitly or otherwise, because, in their view, such forms did not exist yet for these young children, who were engaged in an activity that could later give rise to the concept of proportion. Rather, students were engaged in manipulating objects in the spatial-dynamical problem space. Students could later reflect on their own emergent manipulation strategies, discern motion patterns, and then model these patterns mathematically. Thus, these physical experiences helped children’s subsequent performance symbolizing otherwise elusive multiplicative relationships. Fischer, Link, Cress, Nuerk, and Moeller (2015) used digital dance mats, Kinect sensors, and interactive whiteboards to promote physical experiences that children could use to understand the mental number line through embodied training. A mixed-reality environment developed by Lindgren (2015) fostered body ‘cueing’ that led to higher achievement and more positive attitudes toward learning by providing grounding for students’ understanding of physics principles. Enyedy and Danish (2015) also used a mixed-reality environment to support students’ understanding of Newtonian force using motion-tracking technology. They found that verbal and physical reflection on embodied activity and first-person embodied play allowed students to engage deeply with challenging concepts.

In the domain of geometry, Petrick and Martin (2012) describe an intervention where high-school students physically enacted (versus observed) dynamic geometric relations, and found that enactment improved learning gains. Shoval (2011) describes an intervention in which students made ‘mindful movements’ to kinesthetically model angles with their bodies, and demonstrated improved understanding of angles at post-test compared to a control group receiving traditional instruction. Smith, King, and Hoyte (2014) used the Kinect platform to engage students in making different types of angles with their bodies. They found that making conceptual connections between physical arm movements and the grounding metaphor ‘angles as space between sides’ allowed students to demonstrate greater understanding of estimating and drawing angles. However, they found that in order to benefit from the intervention, it was critical for students to connect their physical actions to the canonical geometric representation and to engage in dynamic movements in which they tested different hypotheses.

In our prior work, Nathan et al. (2014) showed that directing participants’ (*N* = 120) body actions affected the generation of appropriate mathematical intuition, insight, and informal proof for two different tasks. They looked at intuition and informal proof for a conjecture on properties of the lengths of sides of all triangles. They also looked at insight and informal proof for a task involving parity for a train of gears. Mathematical insights are defined as partial understandings of the key ideas underlying a mathematical system. Insight is related to but distinct from intuition (e.g., Zander, Öllinger, & Volz, 2016; Zhang, Lei, & Li, 2016): intuition draws on unconscious information to make a judgment (often Yes/No), without leaving a reportable trace of the decision-making process, whereas insights use conscious retrieval processes applied to both unconscious and conscious knowledge to report on one’s thoughts about a solution or to provide a partial solution. One of the challenges of insight processes is overcoming unhelpful associations (e.g., when conjectures about triangles inappropriately activate Pythagoras’ theorem).

Nathan et al. (2014) found that trials in which participants performed the directed actions were associated with significantly more accurate intuitive judgments on the Triangle conjecture, and more accurate insights on the Gear conjecture, than the trials that used control actions of comparable complexity that were less relevant to the mathematics. Participants who performed relevant directed actions were also significantly more likely to generate an accurate intuition on a transfer task for geometry (i.e., as extended to other polygons) than participants who performed irrelevant actions. However, participants were not more likely to have the key insight for a transfer task involving numerical parity of gears. Thus, directed actions can facilitate mathematical intuition and insight, though transfer appears to be highly task dependent.

While performing mathematically relevant directed actions facilitated key mathematical insights and intuitions for two tasks (Triangle and Gear), directed actions on their own did not lead to superior informal proofs compared to irrelevant actions. Rather, adding pedagogical language in the form of prompts (prospective statements) and hints (retrospective statements) that explicitly connected the directed actions to the tasks significantly enhanced proof performance on the Triangle task. The authors interpret the findings as raising questions about the reciprocal relations between action and cognition: actions on their own facilitate insight, while actions coupled with appropriate pedagogical language explicitly connecting the actions to the mathematical ideas foster informal proofs.

Our third finding is that actions from earlier training opportunities leave a historical trace that is evident when people later solve problems in new, related contexts. Body-based training on the Tower of Hanoi task led participants to integrate their motor experiences into their mental encoding of the objects and their subsequent solution processes (Beilock & Goldin-Meadow, 2010). Donovan et al. (2014) showed that directed actions influence later performance, and found that trained actions can leave an observable ‘legacy’ in learners’ subsequent gestures during proof production. Third and fourth graders learned to solve equivalent equations in one of two ways that involved different manual actions for the assigned condition. Participants using the two-handed ‘bucket’ strategy were significantly more likely to use both hands when solving post-test and transfer problems, and more likely to exhibit a relational understanding of the equal sign than control group participants who used no manipulatives.

In the fourth finding, we note that reasoning and learning with directed actions appears to be enhanced when the solver’s actions are coupled with task-relevant speech, such that action and language become coordinated. As noted in the proof research reviewed earlier, gestures and speech each make independent and significant contributions to predicting performance (Pier et al., 2014). Goldin-Meadow et al. (2009) performed a mediational analysis of students’ speech in their study of how directed actions influence performance on equivalent equations. Their results show that students come to apprehend the grouping strategy for solving the equivalence equations even though the strategy was depicted only through directed actions and never explicitly vocalized or gestured to the participants. Students who added grouping in their speech along with the directed actions had increased post-test performance. Their analysis showed that the action condition by itself predicted whether participants added the verbal grouping strategy to their repertoire, while verbalizing the grouping strategy more strongly predicted post-test performance. The authors propose that student-generated speech mediates learning from actions. Carrying out specific actions directs learner attention to solution-relevant features of the task, which helps students confer meaning to the actions.

Several summary points emerge from this literature: gesture production predicts learning and performance; in reciprocal fashion, directed actions are a malleable factor that may influence cognition; actions on their own tend to promote insight and intuition that is not well articulated verbally; reasoning and learning with actions is further enhanced when actions are coupled with task-relevant speech; and actions from earlier training opportunities leave a legacy trace shown in future problem solving. Together, these empirical findings form the basis for a set of hypotheses for ways to promote reasoning in STEM by exploring the mutual influences between action and cognition, as moderated by speech.

#### Action-cognition transduction

Evidence is mounting that sensorimotor activity can activate neural systems, which can in turn alter and induce cognitive states (Thomas, 2013). Recent work has also identified two critical modes of thinking: System 1 processes that are automatic, effortless, nonverbal, and largely unconscious (e.g., orienting to a sudden sound); and System 2 processes that are effortful mental activities involving agency, choice, and concentration (e.g., checking the validity of an argument; Kahneman, 2011). Whereas the influence of directed actions on cognition may largely be on automatic and unconscious System 1 processes, gestures, which are more intimately bound to language, may influence the verbal, deliberative processes of System 2 (Nathan, in press).

As reviewed above, an emerging literature on cognition and education shows that concepts can be learned through motoric (System 1) interventions. Specifically, ‘action-cognition transduction’ (ACT; Nathan, in press) explores the bidirectional relationship between cognition and action. ACT theory draws inspiration from reciprocal properties of electromechanical and biological systems relating input-output behavior. Physical devices, such as motors, acoustic speakers, light-emitting diodes (LEDs), and so forth, can run both ‘forward’ and ‘backward’; so input energy, often in the form of electric current, can be transduced when forcibly cranking the rotor of a motor (we call the ‘reverse’ motor a generator), shining a light on an LED (making a photoreceptor), or singing into a speaker (making a microphone).

Transduction behavior is evident in biological systems as well, with reported influences on cognitive processes. Niedenthal (2007) illustrates how affective state is surreptitiously induced through manipulations in the facial muscles to form specific facial expressions, which in turn influence the cognitive processing of emotion information when presented in writing, speaking, and images. Havas, Glenberg, Gutowski, Lucarelli, and Davidson (2010), in a similar vein, showed that Botox injections affect cognitive processing of emotion-laden sentences through paralysis of facial muscles. Niedenthal (2007) references the ‘reciprocal relationship between the bodily expression of emotion and the way in which emotional information is attended to and interpreted’ (p. 1002). As noted, interventions directing arm movements (e.g., Nathan et al., 2014; Novack, Congdon, Hemani-Lopez, & Goldin-Meadow, 2014) and eye gaze (Thomas & Lleras, 2007) have led to superior performance in mathematics and general problem solving.

ACT theory offers several testable hypotheses about thinking and learning that have implications for STEM education. One hypothesis states that directed actions*,* body movements that learners are instructed to formulate, can induce cognitive states that activate relevant knowledge. A second hypothesis is that action-based interventions by themselves are expected to induce cognitive states around ideas that are not propositionally encoded. In this way, actions can foster insights that may be nonverbal, and therefore unavailable for immediate verbalization. In this manner, action-transduced knowledge may operate outside of the awareness of the learner.

Consistent with these two hypotheses, Nathan et al. (2014) found that experimental participants who performed directed actions that were selected for their relevancy to mathematical tasks showed improved intuition and insight, even though participants were largely unaware of their mathematical relevance or influence. This work raises three important question about the epistemological basis for claims about body-based interventions influencing cognition. The first question is how actions absent any action-based goals can conjure something meaningful. The second question is whether there is evidence that thoughts induced by actions can contain entirely new ideas, or if the conjured ideas are simply due to priming effects of pre-existing knowledge. The third question is how movements performed in response to directions but without inherent meaning can contribute to specific meaning-making.

Several empirical studies speak to this first question, and show that presenting stimuli can activate motor systems even in the absence of any motoric goals to act. Skilled *kanji* writers, for example, demonstrate motor-system activation in areas associated with writing these characters, even without any intention to write (Kato et al., 1999). Isolated word presentation of action words (e.g., pick, lick, kick) can induce motor responses in the associated muscle systems (fingers, tongue, legs; Pulvermüller, 2005). Beilock and Holt (2007) showed that people reported preferences for letter dyads (FJ over FV) without cuing any action-based goals because these were less demanding to type, but this held only for skilled typists. These studies illustrate ways that people invoke action-based meaning for presented stimuli even when action-oriented goals are not explicitly cued.

On the second question, Leung et al. (2012) provide evidence across several experiments that embodied interventions can increase the generation of entirely new ideas, rather than only priming prior knowledge. Here, interventions that embodied creative and alternative viewpoints (changing hands, being outside of a box, freely wandering) led to more creative responses on a number of convergent and divergent thinking tasks.

In addressing the third issue, we note that actions performed in response to directions can generate a specific meaning by evoking many multiple meanings that undergo real-time selection. One way that actions may generate new ideas is through mental simulation (e.g., Barsalou, 2008). Mental simulation processes literally ‘run’ or ‘re-run’ multimodal enactments of external sensory and motoric signals along with internally generated introspective events. This offers one account for why we perceive similarities between enacting, observing, and recalling specific behaviors; and understand the minds and behaviors of others (Decety & Grèzes, 2006). The GAME framework proposed by Nathan and Martinez (2015) provides a computational account of how actions that are initiated without specific meaning contribute to specific meaning-making through mental simulation. The GAME framework builds off the MOSAIC architecture, which provides an account of movement regulation in uncertain environments (Haruno, Wolpert, & Kawato, 2001; Wolpert & Kawato, 1998). In this model, as a movement commences, it simultaneously initiates the parallel production of multiple, paired predictor-controller modules. Each module is intended to anticipate one of the myriad of plausible next states of the motor system. Each predictor-controller pair receives feed-forward signals of expected movement and feedback signals of the actual movement, which provides rapid access to the difference between the projected state of the motor system in the simulated mental model and its actual state as movement occurs. This coupling between actual and simulated motor activity establishes a pathway for transductive influences of actions on the cognitive state of the agent that may start out as nonspecific, and ultimately induce specific, contextually relevant cognitive states. As the movement progresses, there is continuous competition among these predictor-controller modules, each serving as a potential future state of the mental simulation. The system favors those modules that most closely track the external influences from the environment and the internal influences from the current cognitive states. Selection of the most helpful predictor-controller modules is used to update the reader’s current mental simulation, enabling idea generation, while the current action potentially alters current cognitive processes. Those specific predictor-controller pair modules that are found to most accurately predict both the state of the world and the state of mind receive greater activation for the future, thus improving body response and action-cognition alignment. Nathan and Martinez (2015) provide evidence in support of the prediction that the execution of motor control programs during movements such as gesture production can influence simulated mental model construction processes, and enable the generation of novel inferences. In this way, even nonspecific movements can induce specific mental states through ACT that can lead to novel cognitive processing, and support the generation of insights through nonverbal means.