Monthly Archives: March 2017

Situating Mathematics and Science in the Classroom

T. N. Carraher, D. W. Carraher, & A. D. Schliemann (1985) bring forth an interesting insight in “Mathematics in the streets and in schools”. Through their study, these researchers found children involved in the street markets making complex mental math calculations daily and successfully; however, when these same children were brought into a setting with similar pencil and paper mathematical tasks, many of them underperformed. Their results showed that “context-embedded problems were much more easily solved than ones without a context” (p. 24). In their context (the market) when solving successfully, “actual items in question were physically present” (p. 25). Too often in our math classrooms we are asking students to deal with operations and mathematical problems that are “in a very real sense divorced from reality” (p. 28).

As I have mentioned in previous posts, when considering this research and others we have encountered earlier in this course during our exploration of the Jasper series, I set out to revamp some of the problem solving I was using in my grade 3 math class. After exploring many problems that I tried to base in my students’ lives and make more real to them, our next project was to have students create their own problems. I had students use Google slides, accessed by an easy bit.ly address, to compile a class set of problems. Next, we took pictures to add to the slides that showcased the problems using as many props and settings relevant to the problem as possible. This collection was then put on our class blog for students to access from home over spring break to work on. If you would like to see how the project turned out visit: http://mrskostiuksclass.edublogs.org/2017/03/17/solve-me/

Using programs such as GLOBE and virtual field trips are ways to utilize the accessible affordances offered by technology in this day and age. GLOBE not only connects classes with real life scientists and experts in their fields, but it also provides a platform for students to contribute meaningful data to ongoing studies. Showing students future careers in different fields and ways they can contribute in the present day is impactful. Additionally, GLOBE “encourages students to understand the context of their own environment” (p. 12) by immersing them in conducting research around them. As evidenced in the Carraher, T. N., Carraher, D. W., & Schliemann, A. D. (1985) study, showing students how to solve problems in context is more likely to later be recalled in context when needed.

Similarly, Adedokun, O. A., Hetzel, K., Parker, L. C., Loizzo, J., Burgess, W. D., & Paul Robinson, J. (2012) find that virtual field trips can be “viable alternatives for providing students with learning opportunities and experiences that would have otherwise been unavailable to them” (p. 608) while exposing students to scientists and their real, authentic work.

In summary, I believe that providing students with as many experiences as possible that are situated in context and engaging in problem solving not only for problems they may encounter in the work force but also for problems they currently encounter in their everyday lives as children and students, we can better prepare them with skills necessary to succeed in the math and sciences.

Adedokun, O. A., Hetzel, K., Parker, L. C., Loizzo, J., Burgess, W. D., & Paul Robinson, J. (2012). Using Virtual Field Trips to Connect Students with University Scientists: Core Elements and Evaluation of zipTrips™. Journal of Science Education and Technology, 21(5), 1-12.

Butler, D.M., & MacGregor, I.D. (2003). GLOBE: Science and education. Journal of Geoscience Education, 51(1), 9-20.

Carraher, T. N., Carraher, D. W., & Schliemann, A. D. (1985). Mathematics in the streets and in schools. British journal of developmental psychology, 3(1), 21-29.

GLOBE – Anchored Instruction

Question

Globe researchers have suggested that Globe is an example of anchored instruction. Do you agree or disagree with this statement and why?

Response

After analysis of the GLOBE program, I agree it is an example of anchored instruction. First, anchored instruction is summarized followed by the reasoning for how GLOBE fits this description.

Cognition and Technology Group at Vanderbilt (CGTV) (1992a) explored The Jasper Series and described it as an example of anchored instruction. The group defined anchored instruction as an “…approach to instructional design, whereby instruction is situated in realistic, problem-rich setting (p. 78). Prado and Gravoso (2011) also explain that “…this approach situates learning in realistic or authentic problems, which allows students to experience the kinds of complex, challenging problems that experts encounter…” (p. 62). To summarize, anchored instruction is authentic, realistic and meaningful instruction that exposes students to challenging problems that experts face in the field of math or science.

GLOBE has two attributes that fit this description. These attribute are detailed further.

I) Realistic Setting

Penuel and Means (2004) explain “GLOBE is an international environmental science and science education program focused on improving student understanding of science by involving young people in the collection of data for real scientific investigations” (p. 295). The collection of data that pertains to real scientific investigations qualifies GLOBE to be situated in a realistic setting. When students contribute to the program with data, they “…are not just collecting data as part of an isolated laboratory experience but as contributors to actual scientific studies” (Penuel and Means, 2004).

II) Experiencing Problems as Experts

Penuel and Means (2004) further explain that GLOBE is an example of a “…so-called network science [program]…[that draws]…on networked technologies such as the Internet to create virtual communities that engage students not just as learners but as scientists themselves, collecting and analyzing data that are part of larger scientific investigations” (p. 297). GLOBE provides students with access to and influence scientific research by contributing data in their local environments. Moreover, it provides scientists with an enormous amount of data gathered by students to study from. It is a two way access between research and the classroom.

Hence, GLOBE is truly anchored instruction as it provides realistic research experiences to students in their own classrooms by collecting and submitting data that can be harnessed by scientists and experts in the respective fields of research.

Question for feedback from peers:

Penuel and Means (2004) describe barriers in data reporting as a result of surveying teachers that use the GLOBE program. The biggest barrier described is “…difficulty teachers face in integrating GLOBE with the curriculum (p. 307). I personally found this to be both a problem and equally surprising. With a push for more authentic teaching and learning experiences in math and science, I imagined it would be easier to implement the scientific process in the classroom using programs like GLOBE. A second barrier to reporting data was “difficulty teachers face in finding time to report data” (p. 307).

In your opinion, what would be the necessary steps needed to reduce the barriers of curriculum integration and lack of time to report data in today’s math or science classroom?

References

Cognition and Technology Group at Vanderbilt (1992a). The Jasper experiment: An exploration of issues in learning and instructional design. Educational Technology, Research and Development, 40(1), 65-80.

Penuel, W.R., & Means, B. (2004). Implementation variation and fidelity in an inquiry science program: Analysis of GLOBE data reporting patterns. Journal of Research in Science Teaching, 41(3), 294-315.

Prado, M. M., & Gravoso, R. S. (2011). Improving high school students’ statistical reasoning skills: A case of applying anchored instruction. Asia-Pacific Education Researcher (De La Salle University Manila), 20(1).

Knowledge Construction, Collaboration and Virtual Reality in Science Class

Speculate on how such networked communities could be embedded in the design of authentic learning experiences in a math or science classroom setting or at home. Elaborate with an illustrative example of an activity, taking care to consider the off-line activities as well

The use of virtual field trips and interactive virtual expeditions (IVE) are valuable tools that an educator can use to make science come alive in the classroom. While there is inherent value in students actually going on a field trip the logistics are often daunting. In my school district students arrive at 8:50 am and dismiss at 3: 30 pm. (asking for the bus to arrive early or return late is a logistical nightmare as 95% of our students are bused home and many come from homes with commuting parents or working farms. If we miss the home busses parents need to pick up their child at the school which often means we end up waiting up to 2 hours for parents who are late for the pick up).

We are located in a rural community in the Niagara region of Ontario. Often field trips become social outings rather than educational experiences. Students spend at least two hours on the bus in each direction, which leaves approximately two hours for exploration and lunch. The cost of bussing has become so high the average field trip costs in excess of 40 dollars per child an amount many of our families cannot afford. So, we must weigh the costs and benefits. Often time the costs outweigh the benefits.

Virtual field trips and IVE are life savers for schools like mine. Students enjoy them, learn from them and often continue to explore them on their own time. Niemitz et al (2008) report that “the use of interactive virtual expeditions in classroom learning environments can theoretically be an effective means of engaging learners in understanding science as an inquiry process, infusing current research and relevant science into the classroom, and positively affecting learner attitudes towards science as a process and a career (p. 562).”

The researchers report that studies have shown that virtual field trips can enhance learning (Cox & Su, 2004; Tuthill & Klemm, 2002; Woerner, 1999), achieve the same gains in student achievement as physical field trips (Garner & Gallo, 2005), and provide an effective supplement to physical field trips (Spicer & Stratford, 2001). As such, we can apply many of the best practices of effective virtual field trips (Klemm & Tuthill, 2003; Woerner, 1999) – purposeful trip planning, learner-centered experiences, active student learning, cooperative learning activities, teachers as guides who scaffold learning experiences, differentiated instruction, and multiple opportunities for learner success – to the field of IVE. (Niemitz et al, 2008 p. 566).”

Collaboration amongst students is possible on the virtual reality field trip as much as on a traditional field trip. Often collaboration in both settings provide students with the opportunity to question and test their hypothesis, discuss findings and eliminate misconceptions. According to Driver et al (1994) “Scientific knowledge is socially constructed, validated and communicated (p. 11).” While Lamon, Laferriere & Breuleux, (in press) reported that research shows that knowledge construction is rarely done in isolation but rather by creating and forming a knowledge building community and the goal for learning communities is that a group of students with focused common issues complete tasks better than any single person.

Working collaboratively in math and science requires three important personal characteristics:

INTELLECTUAL COURAGE: we should be ready to revise any one of our beliefs.

INTELLECTUAL HONESTY: we should change a belief when there is a good reason to

change it…

WISE RESTRAINT: we should not change a belief wantonly, without some good reason, without serious examination (Lampert, 1990 pp. 7-8).

Collaboration among students and access to virtual learning environments need to become integral parts of our daily classrooms. After exploring several of the websites this week GLOBE, Exploratorium and virtual field trips I was reminded of a project Trish Roffey and I created last term in ETEC 565A. This project required us to create a google classroom module for the subject and age group of our choice. We chose Engineering for grade three students ( this module could easily be used with almost any grade level). What I was reminded of was that given today’s technology we can create our own virtual reality digital stories and field trips.

Trish used a Ricoh Theta 5 camera that films in 3D to create a virtual tour of the amusement park at the West Edmonton Mall. Her module was based on an end project where students had to design a ride or “car” that would accommodate a special needs classmate. The classmate wanted to enjoy the amusement park as well. All kids could relate to that.

The video can be accessed via https://www.thinglink.com/video/850811682614673410

It is best watched using google cardboard or on a tablet (many laptops will not display it properly).

I created a video that took students to the plains in Africa where a young student had made his own wind turbine from found materials. This turbine solved many issues for his family including refrigeration and crop irrigation. In the video, studetns saw the geography, weather patterns and crop growth for the area. They like the boy in the story had to create a device from found materials that would solve a social justice issue in any area of the world.

All that being said what Trish and I found to be the best “gotcha” with the students is that they were not just expected to learn new information but they had to work together to solve a problem. This made the learning real and valuable and students saw the connections to real life.

Here are some screen shots of our Google classroom:

If you would like to look at it more in-depth or look at the entire module contact me and I will provide a user name and password.

Catherine

References:

Driver, R., Asoko, H., Leach, J., Scott, P., & Mortimer, E. (1994). Constructing scientific knowledge in the classroom. Educational researcher, 23(7), 5-12.

Falk, J. & Storksdieck, M. (2010). Science learning in a leisure setting. Journal of Research in Science Teaching, 47(2), 194-212.

Lamon, M., & Laferrière, T., & Breuleux, A. (in press). Networked communities. In P. Resta, Ed., Teacher development in an e-learning age: A policy and planning guide, UNESCO.

Lampert, M. (1990). When the problem is not the question and the solution is not the answer: Mathematical knowing and teaching. American educational research journal, 27(1), 29-63.

Niemitz, M., Slough, S., Peart, L., Klaus, A., Leckie, R. M., & St John, K. (2008). Interactive virtual expeditions as a learning tool: The School of Rock Expedition case study. Journal of Educational Multimedia and Hypermedia, 17(4), 561-580.

Knowledge construction in networked communities

How is knowledge relevant to math or science constructed? How is it possibly generated in these networked communities? Provide examples to illustrate your points.

In the past few weeks, much of the discussion regarding knowledge and STEM education has focused on the construction of knowledge in practical, hands-on environments. In these cases, the relevant knowledge that the learner constructs is knowledge that helps bridge the user’s understanding of the world and their ability to function and interact with it. Carraher, Carraher, & Schliemann’s (1985) study of working youngsters in Brazil seemed to corroborate this, with findings that point to the street youth excelling in math problems that had real-life context and using strategies different from the traditional ones they would have learned had they stayed in school.

While this certainly bodes well for the ideas of constructivism, I do feel that this type of or learning is limited to practical knowledge and not to more abstract, higher level concepts. In Carraher et al.’s study, the youth were able to calculate using strategies they had developed selling fruit on the streets of Brazil including repeated addition (in place of multiplication). However, further examination showed an inability for the children to solve problems using more tradition school-taught strategies. This certainly supports the idea that knowledge construction that occurs in the real-world can provide a stronger functional ability with the necessary concepts, despite not building a stronger theoretical ability that lays the groundwork for more abstract and higher level concepts.

Networked communities provide a method of bridging this gap by connecting students with people and places that allow them to ground their knowledge in practical, real-world contexts. Spicer & Stratford (2001) found that the link between students, experts, and real-world context is something that students saw as necessary for their own learning. In their study, they set up a “virtual field trip” by using a program called Tidepools that simulated intertidal marine life and their responses to low oxygen environments. While a survey of the undergraduate students that took part showed a general amicability to the simulation, all the students acknowledged that it could not nor should not replace real-life field trips, not can it replace interactions between students and experts. Instead, the simulations would be ideal in a supporting role to either pre- or post-field trip as a way to introduce or review the topics.

However, despite knowledge coming from a variety of sources and the learning being spread across all participants, the construction of knowledge needs to be carefully monitored and guided by the teacher. Moss (2003) noted concerns regarding the difference in knowledge and ability level between students and scientists in global communities. This difference manifests in the activities that the students participate in which often resembles that which would normally be given to technicians, such as data collecting, and do not experience the full spectrum of scientific research. His study into students using the JASON project supported this concern, showing that while students did benefit in the short term, their knowledge gains were not maintained at the end of the year.

From one perspective, the results from all three studies suggests that the constructed knowledge is that of functional, working knowledge required for students to be proficient at the tasks required in that environment. However, care must be taken to ensure that the constructed knowledge is not constricting due to the limited foundational knowledge the students bring to the community. Ideally, the activities and the community should foster the development of knowledge while still allowing students to take part as peers; a balance is easier said than done.

References

Carraher, T. N., Carraher, D. W., & Schliemann, A. D. (1985). Mathematics in the streets and in schools. British journal of developmental psychology, 3(1), 21-29

Moss, D.M. (2003). A window on science: Exploring the JASON Project and student conceptions of science. Journal of Science Education and Technology, 12(1), 21-30

Spicer, J., & Stratford, J. (2001). Student perceptions of a virtual field trip to replace a real field trip. Journal of Computer Assisted Learning, 17, 345-354

Making Sense of the Chaos – Thoughts on Role Play in Mathematics and Sciences

I have been the facilitator of gathering students together to represent the unseen phenomena of molecular movement in states of matter. Students who are “solid” stand very close together and jiggle on the spot, while the “liquid” students stand further apart and move a bit more freely. The students who represent gas find their own space and move around in comparable bliss. I have had students dramatize the story of Archimedes and the king’s golden crown, and have seen a line of students model each part of the ear as sound moves through it. These students are taking on the roles of scientific phenomena, but their role play, as Resnick and Wilensky (1998) would suggest, is merely representing the results rather than “the processes and interactions that give rise to the results” (p.168).

Traditionally role play has found itself in the arts and humanities, helping students view themselves and society through varied lenses, making connections and altering perspectives. Winn (2003) quotes Reyes and Zarama suggesting that in the sciences, too, perspectives of self can be changed. The learned distinctions can often “tell us more about ourselves than about the world we are describing” (Winn, 2003, p.19). As well, Resnick and Wilensky (1998) have found that “role-playing activities provide a framework in which learners can start to make … distinctions – learning to project only the specific parts of their own experiences that are useful for understanding other creatures and objects” (pp.168-9). Can role play in the sciences and mathematics classroom aid in growing these distinctions? In subject areas where traditionally there is one correct answer, can seemingly random and indeterminate role play help bring order and understanding to complex ideas?

Resnick and Wilensky (1998) would affirmatively attest that role play is not intended for simply representing a result, but for “developing new relations with the knowledge underlying the phenomena” (p.167). In fact, they assert that for complex and system sciences, role play is ideal for providing “a natural path for helping learners develop an understanding of the causal mechanisms at work in complex systems. By acting out the role of an individual within a system…, participants can gain an appreciation for the perspective of the individual while also gaining insights into how interactions among individuals give rise to larger patterns of behavior” (p.167). Gaining insights into how localized patterns influence larger-scale, or globalized activity, is essential in understanding the intricacies of a complex system.

The enactivism theory of cognition supports Resnick and Wilensky’s affinity for role play within the sciences and mathematics. As described by Proulx (2013): “[e]nactivism is an encompassing term given to a theory of cognition that views human knowledge and meaning-making as processes understood and theorized from a biological and evolutionary standpoint. By adopting a biological point of view on knowing, enactivism considers the organism as interacting with/in an environment” (p.313). As the organism and environment interact, both change and adapt in response to the interaction, making them even more compatible. This evolution of structure is referred to as coupling (Proulx, 2013). Learning through enactivism is neither simple nor linear, but rather complex and undetermined. Using role play to understand mathematics and complex and science systems takes the student through an evolutionary process of change. The student takes on a role, interacting with the problems (environment) presented, and through this interaction poses new problems and pathways of solution. Along the way, the student finds their initial role is changing too, in order to adapt to the changing environment.

Interestingly, the chaos theory of instructional design also recognizes the value of instruction and learning that is evolutionary in nature (You, 1993). Similarly to Resnick and Wilensky, the chaos theory allows for patterns and order to emerge from seemingly randomness and chaos. You (1993) states that central to the chaos theory is “[t]he discovery that hidden within the unpredictability of disorderly phenomena are deep structures of order” (p.18). Quoting from Hayles (1990, 1991), the characteristics of the chaos theory are described with such phrases as a pattern of order within disorder; chaos is the precursor and partner to order rather than the opposite; and chaos is paradoxically locally random, but stable within a global pattern (You, 1993).

To bring this back to role play in mathematics and sciences, there is a need to recognize that complex ideas can be defined and understood through role play scenarios and interactions whether technology-based or non-technology-based. Through role play, localized complexities can be more clearly defined through continual problem solving and problem posing that allow the learner to begin to see and interpret patterns that emerge. As Proulx (2013) states, “The problems that we encounter and the questions that we undertake are thus as much a part of us as they are part of the environment; they emerge from our interaction with it” (p.315). Perhaps by opening the world of role play to mathematics and science students, we will see more students acting like Barbara McClintock, a Nobel-winning biologist who attributes “her greatest discoveries to the fact that she had a “feeling for the organism” and was able to imagine herself as one of the genes within the corn (Keller, 1983)” (Resnick & Wilensky, 1998, p.168). Perhaps McClintock’s experience is a call for educators to consider further the possibilities for when students are handed permission to relate and interact through imagination, and hence are given opportunity to experience phenomena.

The possible’s slow fuse is lit by the imagination. ~ Emily Dickinson

Resnick, M. & Wilensky, U. (1998) Diving into complexity: Developing probabilistic decentralized thinking through role-playing activities, Journal of the Learning Sciences, 7(2), 153-172. DOI: 10.1207/s15327809jls0702_1

Proulx, J. (2013). Mental mathematics, emergence of strategies, and the enactivist theory of cognition. Educational Studies in Mathematics, 84, 309-328.

Winn, W. (2003). Learning in artificial environments: Embodiment, embeddedness, and dynamic adaptation. Technology, Instruction, Cognition and Learning, 1(1), 87-114.

You, Y. (1993). What can we learn from the chaos theory: An alternative approach to instructional system design. Educational Technology Research and Development 41(3), 17-32. Retrieved from http://www.jstor.org/stable/30218385

Let’s Teach Geometry in the Gym

As a kinesiologist, I always look for ways to connect learning to movement. This week’s readings were right up my alley. Especially considering I had just tried Leap Motion technology and a 3D geometry activity with my class.

The question:

How could you use what is developed in these studies to design learning experiences for younger learners that incorporates perception/motion activity and digital technologies? What would younger children learn through this TELE (technology-enhanced learning experience)? Fit perfectly with what I was working on.

In the readings for week 11 a good foundation for using motion and perception with teaching in math was laid out. For example, Winn (2003) stated “once we start to think of cognition as the interaction between a person and their environment, it is necessary to consider how that interaction occurs. This, in turn, requires the consideration of how our physical bodies serve to externalize the activities of our physical brains in order to connect cognitive activity to the environment (p. 93).” Students need to use their bodies to interact with the environment and the concept they are trying to understand. Geometry, including 3D geometry is an excellent example of this. If students do not get a chance to understand how objects move in 3-dimensional space how can they be expected to learn this on their own? Looking at a rectangular prism on paper is much different than holding one in your hand and manipulating it by sliding, flipping it or rotating in in real space.

As modules continue students will scaffold their learning from concrete to abstract and then consolidate their actions into gestures so that deeper learning, as well as easier transfer and recall will occur. Novak et al. (2014) stated that “gesture promotes transfer of knowledge better than action, and suggest that the beneficial effects gesture has on learning may reside in the features that differentiate it from action (p. 445).” Lindgren et al (2013) reported “there is increasing evidence that body movement, such as gesture, can serve as a “cross-modal prime” to facilitate the retrieval of mental or lexical items (p. 447). Finally, Pouw et al (2014) found that:

Under certain conditions, perceptual and interactive richness can alleviate cognitive load imposed on working memory by effectively embedding the learner’s cognitive activity in the environment (Embedded Cognition claim).
Transfer of learning from manipulatives does not necessarily involve a change in representation from concrete to symbolic. Rather, learning from manipulatives often involves internalizing sensorimotor routines that draw on the perceptual and interactive richness of manipulatives (Embodied Cognition claim) (p. 53)

As a kinesiologist I have always benefitted from doing rather than imagining. My body is the instrument I use to understand my world and my place in it. As an educator, I want my students to rely on their bodies as a learning tool. If the use of manipulatives enriched learning, imagine the leaps and bounds that could be made if at an early age students kinesthetically understood what these terms implied? For example, with students as early as kindergarten and continued through primary education what if we take geometry on a cross curricular journey into the gym.

Using the technology of a white board or projector the teacher could introduce the idea of translations (sliding across a surface), rotations (spinning their bodies right, left, forward, backward) and flipping an object in the same manner. Students could use their bodies to change their shape, curl up into a ball, spread out into a star fish. Once the idea of the movement in three D space has been introduced large scale objects could be moved, such as yoga balls, large cardboard boxes, large cylinders. As students become more comfortable with the movement of the object the size of the object can be diminished until it fits in their hand. A final step would be to use technology (with programs such as the Leap motion 3d geometry app) to have students manipulate virtual objects. Following these steps would build and reinforce neural pathways and eventually students (as they mature) would be able to use this information to try and do the manipulation mentally.

Kim et al (2011) noticed that students in their study often naturally gravitated to using their bodies to mimic actions. They state “her thinking develops in and through her gestures, and her gestures further develop her thinking. Her gestures constitute the thinking with and about shapes and motion (p. 230).”

They found firstly “that children’s bodies (bodily orientations and gestures) constitute an integral part of knowing, thinking, and learning supporting the appropriate of geometrical concepts before the age thought possible (p. 233).” Secondly, the children’s co-emerging gestures allowed new concepts to be enacted for themselves and others and, thereby, for new concepts to become reflexive objects available to individual and collective inspection (p. 233).”

If students benefitted from using their bodies while seated in a classroom imagine the possibilities of using their bodies in the 3D space of a gymnasium.

Kim et al (2011) conclude that “as children think, develop, and express knowledge through their bodies, their bodily engagement needs to be realized as integral to student learning. Their bodies are necessarily engaged in coping with the abstractness of knowledge. Their bodies embody the knowledge of science and mathematics and become part of knowing itself (p. 235).”

As I noticed when my students tried the Leap Motion 3D geometry technology and app in the classroom (in groups) they were moving their bodies through space, helping each other visualize the end result of a manipulation in space. The collaboration seemed to help them solve problems and persevere more than when students worked alone. This coincides with the research by Hwang et al. (2013) that students were more successful when collaborating using new technology (p 318).

Finally, by observing students at each of the various steps outlined above there would ample opportunity for the teacher to evaluate or assess the students’ knowledge in a new way. Bodily movement and the manipulation of objects using technology can be assessed over a paper pencil test. This unit would also provide an excellent way for students to document their growth in the math using a digital portfolio, digital story or digital movie or interview. Can’t wait to give this a try.

Catherine

References:

Hwang, W. Y., & Hu, S. S. (2013). Analysis of peer learning behaviors using multiple representations in virtual reality and their impacts on geometry problem solving. Computers & Education, 62, 308-319.

Kim, M., Roth, W. M., & Thom, J. (2011). Children’s gestures and the embodied knowledge of geometry. International Journal of Science and Mathematics Education, 9(1), 207-238.

Lindgren, R., & Johnson-Glenberg, M. (2013). Emboldened by embodiment: Six precepts for research on embodied learning and mixed reality. Educational Researcher, 42(8), 445-452.

Novack, M. A., Congdon, E. L., Hemani-Lopez, N., & Goldin-Meadow, S. (2014). From action to abstraction: Using the hands to learn math. Psychological Science, 25(4), 903-910.

Pouw, W. T., Van Gog, T., & Paas, F. (2014). An embedded and embodied cognition review of instructional manipulatives. Educational Psychology Review, 26(1), 51-72.

Winn, W. (2003). Learning in artificial environments: Embodiment, embeddedness, and dynamic adaptation. Technology, Instruction, Cognition and Learning, 1(1), 87-114