Thank you for your comments. I was also drawn to your project as well as it too explores the role of the body for communication and learning and teaching! I feel there is a growing community using various motion capture tools to add a more human dimension to educational technology. It's great to see.

Thank you for posting! I know you work with math teachers at a state-wide level. Any thoughts you have about how this can help them, either in their own math learning of math, or in their classroom instruction? One the first point, I think teachers can use embodied ways of knowing to improve their understanding of the generalized properties of space and shape. This can extend their knowledge beyond shape recognition and memorizing definitions. For instruction, we are exploring how students can create math conjectures + movements that they then share with their classmates and observe how the classmates investigate mathematical ideas through collaborative movement. I also see the potential for teachers to get more attuned to their students' uses of gestures as a way to diagnose student understanding. Glad to hear from you!

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I think this would be a wonderful tool for students who are currently learning virtually.They can engage in active learning.  I can see that this tool can support teachers in their own understanding as well as students. The research on math conjectures and movement will be powerful to help teachers to think about how what to assess, how to assess and then conduct whole class discussions that extend students understanding of the properties of space and shape.

Thank you for these comments. These are great questions! I do believe this is a very general framework for gaining insights into mathematical phenomena so it would apply to many areas of physics (I have one student looking into this) and other areas of mathematics, such as numbers and operations (e.g., arithmetic, algebra), measurement, and probability. For me, there is something compelling about contributing to geometry and other non-numerical areas of math education, since number often has such an out-sized rule in K-12 education.

I really appreciate this comment. There is so much room to make mathematics more engaging, creative, and meaningful for students than we seem to allow in most K-12 programs. Are you interested in education research? Thank you for stopping by!

I'd love to hear more about what you did with your students!  Very cool that you are here!  

Hi Brandi and Judy.

The Interface is a little confusing and I cannot seem to REPLY to Brandi without replying to Judi. Candace and Brandi did an amazing job on curriculum development for this project so I hope they have a chance to respond. Briefly, students spent three days exploring embodied ways of expressing ideas of quadrilaterals, which also, inevitably, drew on their intuitive notions of triangles. And therein lies the power of this approach: As students explore ideas of shape and space they often make important connections through movement and the collaborative discussions that follow. The study took place this spring so Candance does not yet have a formal paper to share, but we hope to soon! Thank you for your question!

Thank you Brandi and Mitch! We helped bring the game into Brandi's classroom for her quadrilaterals unit, and one thing that was really striking was how well it complemented her general instructional approach to geometry. Brandi uses a lot of gestures and motions as she teaches, and I think math teachers tend to have great intuitions about how important they are. The game allowed the students to really get into making motions to "think with" and "think together" too!

Thanks for your replies. I can't wait to hear more about the work in Brandi's classroom, Candace.  So excited about this work.

Nancy, I really appreciate your perspective on this work as a neuroscientist. It would also be interesting to explore a cross-cultural study with students in Cuba, if you were so inclined. For example, I would be interested in how Spanish speaking students in the US and students in Cuba might do things similarly and differently. In general, mathematics can be so mysterious to students sometimes and embodied approaches seem to give them a more meaningful way to express their thinking and observe the thinking of others. Thank you for this post!

Thanks for the question! We just began data collection in February for our site's study where students create their own actions, so we don't yet have a good public-facing repository yet. However, you can see an example at this link: https://smu.box.com/s/ewgc8xr6lezx1mqyiyilbprk2gt6cvhn.

 These are motions some high school geometry students created for the conjecture "Base angles of an isosceles trapezoid are always congruent." The proof is done by making two right triangles on either side of the trapezoid using altitudes. You can see the students' motions first show the right triangle on the left side of the trapezoid, then show the trapezoid's two parallel bases, then show the right triangle on the right side of the trapezoid.

This is so interesting how gesturing transcends all the way to high school! We have seen gesturing in our K-2 work, so it's pretty amazing to see how it extends!

Thank you for making that connection. Yes we even see it among adults and experts, like math professors. Gesture seems to play an important role in conveying concepts in ways that compliment language. Whereas language is sequential and syntactic, gestures seem to be more holistic and spatial-temporal. In this project we also see that gestures help reveal nonverbal ways of knowing. Our statistical models, for example, show that just using language doesn't tell the whole story of what students know, and that the models of mathematical reasoning performance are significantly improved when we include both speech factors and gesture factors. I conclude from this that some of the knowledge students are conveying is uniquely embodied. I think this may be even stronger for younger children who are earlier in their language development and generally more physical and active. I really appreciate your post -- thank you!

Thank you, Judy.  

RE: Spatial Skills

Over the course of our current grant, we have conducted a number of studies.  Using Harel & Sowder's (2005) criteria for transformational geometric proof: (1) logical, (2) operational, and (3) generalizable.  By video recording participants' responses, we look for evidence that demonstrates these criteria of robust conceptual understanding of the conjecture.  Among our pretest measures are batteries that assess spatial skills.  Analyses thus far show that students with lower spatial skills scores who incorporate dynamic gestures into their explanations are associated with significantly better performances that students with low spatial skills scores who do not gesture.

We are in the midst of analyzing data from a randomized controlled quasi-experiment conducted in high school classrooms and hope to corroborate our earlier findings. 

Hi Judi - great question! I'm sure Candace has additional views to share on this,. The relation of gesture production and spatial skills has often cropped up in our studies in this project. Michael shares one of the important findings. In a forthcoming study of proof performance (interviews conducted without The Hidden Village, while it was in development) we looked at the behavior of math experts and non-experts (N=90) and saw that spatial skills were a reliable predictor of proof, but then it drops out when we add gesture production to the model. The effect size for dynamic gesture production on predicting proof performance is pretty big, d >1.0. See the link below to access the J. Ed Psych. paper that is currently accepted with minor revisions. We will be looking for this spatial skills–gesture trade-off with our game studies that are currently being analyzed.

Nathan, M. J., Schenck, K., Vinsonhaler, R., Michaelis, J., Swart, M., & Walkington, C. (under review). Embodied geometric reasoning: Dynamic gestures during intuition, insight, and proof. Journal of Educational Psychology.

https://drive.google.com/file/d/1y0-IeMRuE9Zz0K...

Awesome. Thank you Michael and Mitch.  I will check out the paper.  

hi. It’s still in final review. Can I send it to you when I hear back? Cheers!

Alison, I really like your question. The Gesture Studies literature is rich with examples of cultural variation of gesture production. There are so many varied norms of what gestures are acceptable to make in public and when. We now have data that will allow us to compare some cultural differences of recent arrivals to the US as compared to native born high school students, but we don't have those analyses to share yet. Interestingly, while there are differences in degree and type, co-speech gesture production seems to be a universal across cultures, just as with language.

I love your personal note. In fact, it is these types of uninvestigated anecdotes that so often offer big breakthroughs in education research! For example, the whole notion that gestures matter for learning and teaching is something that has flown under the radar for decades. Many mathematicians and physicists have shared in their autobiographies the central importance of nonverbal ways of thinking (I think of Poincare, Einstein, Feynman as examples), yet these seldom show up in curriculum and professional standards. I agree there is wisdom there that deserves scientific investigation. This is very thought provoking! Thanks!

It is not directly related to cultural differences, but one thing I have noticed is that the students' social relationships to each other can really transform the way in which they use gesture and motions. Students who know each other well can gesture about math together in very coordinated ways, engaging in what we call "joint gestures" where they form a single mathematical representation together using gesture.

Alison,

I realize now I may not have directly addressed your point about entho-mathematics, and ways that cultural practices have contributed to mathematical practices and formalisms. I imagine that how we move and manage material objects provide the likely origins of much of our notions of geometry and number, but I don't personally have  the historical or anthropological background to shed much light on this. Are there other scholars I should investigate in this regard? cheers!

Thank you Mitchell for the thoughtful reply. Here's another wacky wondering... Kids impacted by trauma often dissociate from their bodies. Given this, I am curious if and/or how trauma influences a student's embodied cognition. Any thoughts?

I learned about ethnomathematics from Swapna Mukhopadhyay, retired faculty from Portland State University. She's amazing and I'm sure she'd love to talk with you more if your interested. Let me know and I'll reach out to her. 

Wow you are offering  an amazing connection of embodiment and childhood trauma. It's not something we address directly in this project. In my grad course on embodied cognition we do read a nice review paper by Gjelsvik et al. (2018; see below) where they discuss mindfulness-based
cognitive therapy (MBCT), which you might already be aware of, and its value for helping patients with trauma. I think there are some powerful emerging ideas for clinical practice that are coming out of embodiment research. Great connection!

I do find ethnomathematics really engaging but I am a total novice. What appeals to me in particular is when I see cultural practices (e.g., weaving is an excellent example) that draw on very sophisticated math principles, which are mediated by the material artifacts and craft practices, yet I would find it baffling to represent these same patterns formally, even though I can appreciate them visually, tactilely, etc. I would very much like to know more.

Gjelsvik, B., Lovric, D., & Williams, J. M. G. (2018). Embodied cognition and emotional disorders: Embodiment and abstraction in understanding depression. Journal of experimental Psychopathology, 9(3), pr-035714

I'm also noticing the limits of the interface, replying here rather than to Candace's point on joint gestures and social relationships (above).  I observed students doing something similar in biology classes to help with understanding biological concepts.  

From Candace's reply:

"Students who know each other well can gesture about math together in very coordinated ways, engaging in what we call "joint gestures" where they form a single mathematical representation together using gesture."

Thanks also to Allison for reminding us of potential problems with trauma and embodied cognition and ethnomathemetics.  So much to think about.

oh that’s good to hear about these observations in biology education. I am always looking for examples outside of math and physics, which seem to be over represented in this kind of research. 

Elizabeth, I agree that we may find that the role of embodied mathematics may be especially helpful for those trying to navigate language differences. This is suggested by findings from wonderful work by Breckie Church and her colleagues, such as

Hi Leanne. Thank you for stopping by! I appreciate the supportive comments. At this stage, the process data we collect (their gestures, verbal explanations, and the like) require a good bit of processing and we use it primarily to inform our research questions. The product data (outcomes like their choices of proof explanation) could be shared in real time, but for now, again, driven by our focus on hypothesis testing and modeling, we try to orient our students toward getting the motions correct (here we can reflect back to them their performance in real time in the game) and producing clear explanations to us and their collaborators. It’s a good thing for us to explore for the future — thank you for this suggestion!

Kenne, Both of your points, about ELL and teacher support, are really important. We have seen a lot of ways that the students' gestures clearly support their mathematical participation, regardless of their language proficiency. Your questions about how to best support teachers is more complex, as you already seem to grasp. We have a grant proposal to explore this issue. What we propose is to work collaboratively with teachers. Teachers would first become acquainted with some of the research that shows how movement can support children's mathematical reasoning. We also want teachers to understand how student thinking can be revealed through gestures (as well as speech). One arena we have done this successfully is the Nevada Mathematics Project, where we worked with teachers  from every school in the State to integrate effective gesture teaching practices into their STEM education. We found that teachers became more positive about gestures as part of their instruction and were able to incorporate it into their teaching. We hope for some of the things we developed in that project to contribute to this one focused on geometry. In my mind, we still have a ways to go (The Hidden Village was a Goal 1/Exploration grant from IES)  yet we also have some really encouraging directions to pursue. Another element to your question is addressed somewhat in Candace's response to Brandi's question. Thank you for asking. These are very much on our mind these days now that we feel we have something concrete to take to more teachers!

Hi Sarah, Yes there does need to be considerable professional learning. As a Goal 1/Exploration grant, we have not yet addressed these issues in general, we established the proof of principle and theoretical basis for this work. We now have a Goal 2/Development grant proposal, which, if funded, would enable us to address this question. Our approach is to support collaborative co-design activities with teachers within a context of how to bring about the benefits of movement and gesture production. One factor that seems important is teachers' initial beliefs about the role of gestures in learning, for which we recently developed a reliable survey instrument that is being used in a number of studies, including our own. I welcome input on this issue of how to best provide the support for teachers, for example how to effectively use coaches. Thanks for raising this issue!

We have not done this control study against regular textbook learning. Partly this is because we are advocating a very different method of performing proofs--framing it in terms of transformational geometry as opposed to the classical two-column proof. This makes the comparisons difficult, especially early on while we are studying the theoretical basis of learning through embodied mathematics. I agree that now that we have built up a body of supportive evidence and been able to provide a theoretical rationale for when and how eliciting body based thinking helps with geometric reasoning, this makes more sense to compare them in at a curriculum level (with many materials, practices, and processes all changing at once).

We are really happy to reach out to schools who would be interested to work with this. It's not a turn-key system, since there is specialized hardware involved for the sensor array that tracks players' movement, and some limits on which laptops can currently support the game. Again, now that we can see its value and potential, we have plans to make it more portable across different platforms.

As to its role in other subjects areas: I most readily see its applicability in STEM areas, but that is my own bias coming from a background in engineering and /mathematics. I would think approaches like this for teaching music can have great potential. I would think the real test of the approach is for music theory, rather than performance, which is more obviously embodied already. There is a wealth of research on music education from an embodied perspective, including the role of gesture. Perhaps you already know of this work, but otherwise I am happy to point you to some of it.

Hi Tensia! Thank you for stopping by! This is a question I have been thinking about recently. Individual game play shouldn’t be effected (so long as the player has the proper hardware- the Kinect sensor). I also have a graduate student doing an online version in a study while using BB Collaborate live and doing the camera recording, and I think that will work out pretty well (we’ll know soon!). Losing the shared space for collaborative gestures I would expect to show a big drop since it’s just less natural, and the collaborative build gestures that Candace has documented wouldn’t seem possible at a distance. So I’m hoping this is temporary and not the permanent state of the world—in which case we’d have bigger issues to tackle. The other work I’ve done with Martha and Voicu developing the instructional avatar I think is a viable method for online instruction. I’d value your thoughts on this, about this project and your own work, which also involves embodied engagement! Thanks for the great question!

Thank you for this comment. So much is nonverbal and this is surprisingly absent from teaching and curriculum standards. I appreciate you stopping by!

I love this idea! In the past students playing the game had mainly made motions with their partners, but not outside those groups. In our final study this Spring, we for the first time had students sharing the motions they created with the whole class, and proposing mathematical ideas to the larger group through motion and gesture.We have a lot of analyses left to do, but it was such an interesting classroom dynamic!

Thank you for your encouraging comment. Given these new social developments we are starting to consider a version of the game environment that could work with players remotely located, but it would have to operate without the Kinect 2 sensor array that we currently use. There are workarounds I can imagine and I appreciate your support for pursuing this line of innovation. Thanks for stopping by!

In the game the player encounters different villagers - like a village scientist or dancer - who can give the player energy strings to fix their spaceship. The player can perform tasks for these villagers that involve movement (e.g. the scientist has never seen a human before, and asks the player to make motions within his scanner while he takes some readings).

Players have landed on an alien planet and their ship needs refueling.  In their search for the energy they need, they stumble upon The Hidden Village, where their encounters with different villagers have players performing movements that, unbeknownst to them, are enactments of geometric transformations!  Then, having been primed by these movements, players are asked to assess the validity of the geometric conjectures. Once they've met all the villagers, the village leader assists them in refueling the ship to return home!! 

Jeremy, thank you for your supportive comments on the project. It's great to have this encouragement as you know, its a loooong path from idea to design to data to principles that can really inform classroom learning experiences. We are getting there and it helps to have this community of scholars, designers and practitioners to help us along the way! Thanks for stopping by!

A great question and certainly something we've contemplated.  There are a couple of answers:

1) In some of our latest work, many students decided to sit during play since the game focuses mainly on upper body movements.  The Kinect did not experience any issues with picking up students arm movements when they were seated.  This inclines us to think that minimally, we could accommodate students with lower-extremity mobility issues.

2) Also, in the latest version of the game, in which students can design their own movements, there is a setting for "tolerance" that students can set.  This setting allows students to vary the accuracy (i.e., how discrepant a students' movements and arm positions are from being 100% aligned).  Since the range of this tolerance ranges from 0 to 100%, this means that students with mobility issues could play a version where the tolerances are very high, which would allow them to successfully complete the movements given any mobility issues.

As a final note, we did work with a student who was partially-sighted.  As an accommodation, we had a researcher read the text to her and help her in discerning the movements.  Consequently, this experience is informing features that we hope to add in subsequent redesign of the project.  

Excellent.  Variability in the settings would certainly cater to more varied student needs. 

These are really helpful comments. It also makes me wonder about the educational implications that follow from embodiment: Are there concepts in math and science  that are more difficult for us to grasp (even those who are "fully" abled) because of the limits we all have to our mobility? I am inclined to think the answer is a strong YES. We also know physical variations lead to different cognitive processes -- left handed people are more included to think of Good as being to the left; whereas righties do the opposite; and so on. So it would follow that we might see corresponding differences in cognition with physical variability. E.g., Might stroke patients have difficulty with symmetry? Of course, we also have language systems and symbol systems that can help us transcend these limits (i.e., we can think about infinites and phenomena that exist past the visible spectrum by using abstractions), but they may impair the early learning and how these abstractions are grounded for people. Great ideas to consider!

I really appreciate your comments.  Gestures are such a powerful part of communication outside the classroom. We are hoping to better understand their role in learning and assessment and harness this for future STEM education. Thank you for stopping by!

Anouschka, thank you so much for stopping by and leaving these comments. It has been a multiyear effort to develop a theoretically guided design that allows us to conduct a variety of experimental investigations where we can control certain important factors hypothesized to be important for supporting mathematical thinking, while also providing an engaging learning experience for students and teachers. There is indeed a great deal of collaboration that goes on when students work in groups, and we even find students will cross groups to entertain their ideas. One important finding of this research is Candace Walkington's framework for describing collaborative gestures, where players employ the bodies of their collaborators to make and explore general properties of shapes and space. We have a recent paper (see the reference, below) documenting this with teachers who played The Hidden Village in groups, and  we have others in the works showing it among students, and also showing how collaborative gestures among one's group members affects one's mathematical reasoning and individual gesture production. It is great to read your comments!

Walkington, C., Chelule, G., Woods, D., & Nathan, M.J. (2019). Collaborative gesture as a case of extended mathematical cognition. Journal of Mathematical Behavior. Doi: 10.1016/j.jmathb.2018.12.002

The collaborative work between students is a really key part of effective learning from the embodied game. We have run studies where kids play individually and studies where they play collaboratively, and the quality of their reasoning and their ways of moving and embodying math concepts fundamentally changes when they work together, in an awesome way.

Mina, These are really helpful comments. You might imagine we had some challneging constraints placed on us for making the video when we our campus closed down in March. Still, the point is well taken that a slower-paced video that went into more depth of one example problem would add a great deal to the story line. Thank you for stopping by!

Hi Mina! I mocked up a single task - you should be able to view it here: https://smu.box.com/s/rsr8flfakg2ub1hqgu5kgp518.... This was one created by a group of kids. Thanks so much for visiting!

Mina, we also have a poster you can download as a resource that summarizes and goes over a task.

Olympia, Thank you for these supportive comments! I also really appreciate the questions. For one, we do indeed see what we refer to as the "legacy" of the directed actions that are elicited during game play show up in some of the players' co-speech gestures when they explain their mathematical thinking. This is actually hypothesized as one of the mechanisms through which this embodied form of learning fosters enhanced mathematical reasoning. The idea is called Action-Cognition Transduction, and theorizes that some of the movements put players into certain cognitive states that facilitate their reasoning. It's a little like when we use gestures to help us come up with a word during that tip-of-the-tongue phenomenon. In the newest work, we then expect to see students use those helpful gestures when they create new math+movement content that they try out on their classmates. We are just starting to explore this really exciting development! I am providing references to a couple of papers that elaborate on this, in case you are interested in exploring this further.

As to your second question, we are exploring exactly these questions--how mathematical performance is affected when players participate in the activity alone vs. with a peer vs. with additional teacher input! We have different experiments designed to test each of these. We are currently analyzing data investigating if students learn as much if they observe someone acting out the movements for a math concept vs. enacting the movements themselves. We also are in the process of investigating the question of how pedagogical support -- in the form of hints -- helps students by making the connection between the movements and the math principles tied to those movements more explicit. We have explored this previously in a low-tech form of this intervention in the last paper references below.

Thanks for the wonderful questions, and for stopping by.

Nathan, M. J. & Walkington, C. (2017). Grounded and embodied mathematical cognition: promoting mathematical insight and proof using action and language. Cognitive Research: Principles and Implications. DOI: 10.1186/s41235-016-0040-5.

Nathan, M. J. (2017). One function of gesture is to make new ideas: Evidence for reciprocity between action and cognition. In R. B. Church, M. W. Alibali & S. D. Kelly, (Eds.) Why gesture? How the hands function in speaking, thinking and communicating. (Chapt. 8, pp. 175-196). Philadelphia, PA: John Benjamins Publishing Company. doi 10.1075/gs.7.04der

Nathan, M. J., Walkington, C., Boncoddo, R., Pier, E. L., Williams, C. C., & Alibali, M. W. (2014). Actions speak louder with words: The roles of action and pedagogical language for grounding mathematical proof. Learning and Instruction, 33, 182-193. DOI: 10.1016/j.learninstruc.2014.07.001

John, I agree that there is great potential for understanding our embodied experiences as a way to understand how and when we learn. Mindfulness if an important area in this regard and it is just entering educators' awareness. Thanks for stopping by. 

Thank you for these supportive comments! Glad you could stop by.

Sarah, This is a really good article for looking across the range of research as it relates to classroom learning and instruction. I personally include movement and embodied learning under a common framework. The important aspect to me is how our bodies provide resources for supporting and enabling thinking, even when it is about things other than movement. Performing movement -- and even imagining movement -- has that role. WE  an also see ways uses of metaphor about movement and space enable this. As a classroom intervention, movement seems to be particularly important early on in learning, but I also think that overt movements can be internalized and we may engage our motoric (and perceptual) resources without visible movement (as the Pulvermüller works in the article you reference so vividly illustrate).

I am curious if you draw a different distinction and how it may help you during classroom instruction, planning, or assessment.

Thank you for the thoughtful question and sharing this reference! I appreciate you stopping by.

Hi Sarah and Mitch, 

We have a recent article aimed at K-12 teachers (https://www.researchgate.net/publication/338779...) that unpacks the differences between simply 
moving while learning, and performing movements that are meaningful embodiments of the concepts being learned. This might get at the distinction you are looking at too - thanks for your question!

Thank you! That was the distinction we made as well--motion for the sake of incorporating physical movement while learning vs. motion that embodied the concept being learned. In your words, "Not every physical activity in a math class meets this definition. For example, if students were to do jumping jacks in between problems, this may have some side benefits for learning [e.g., increased circulation leading to more focused attention (Jenson, 2000)], but the students’ movements would not be directly related to the concepts they were learning." I would assign activities like Balance Points to a moving while learning category and activities in your paper as more embodied. Each kind of activity has its merits, but, overall, I find embodied activities to be more helpful for internalizing math concepts.