Though the bulk of these technologies had been developed decades ago, it is only more recently that they have entered the mainstream, becoming accessible to students and educators to integrate into educational experiences. There is no guarantee, however, that emerging technologies will cue the kinds of body movements that have shown promise for effectively teaching mathematics.

This leaves it up to researchers, designers, and educators to make deliberate efforts to implement novel technologies in ways that trigger movements that support, rather than hinder, targeted learning outcomes. We hope to further advance this conversation and provide insights for the research and development of embodied learning experiences. In the following sections we review the available literature about embodied cognition and mathematics in the domains of manipulatives, hand gestures, and whole-body movements.

Additionally, we discuss how technology can be leveraged to enhance mathematical learning experience through embodied cognition and discuss related design opportunities. Finally, we will discuss future directions in the areas of design, implementation, and assessment of embodied learning of mathematics. Several complementary theoretical assumptions provide insight as to how manipulatives impact learning.

First is the thought that manipulatives support the development of abstract reasoning for younger children who have greater dependency on physically interacting with their environment to extract meaning Montessori, ; Piaget, Supporting this notion, a recent meta-analysis showed that children ages 7—11 years benefitted most from concrete manipulatives whereas ages 3—6 years found little benefits from using manipulatives.

Concrete manipulatives have also been found to be less beneficial for older students, a finding that can be partly explained by their increased ability to reason abstractly Carbonneau et al. A second theory is that manipulatives provide the learner with an opportunity to enact the concept for improved encoding. When later asked to do similar addition problems, students have access to multiple codes and the retrieval of one could activate the other, resulting in improved learning outcomes.

This building, strengthening, and connecting of various representations of mathematical ideas enhances mathematical understanding. Finally, manipulatives work by affording opportunities for learners to discover mathematical concepts through their own exploration. These effects are thought to occur because the processes during generation of knowledge, compared with being given the knowledge, are in greater alignment with those used to produce answers during testing for details on the related transfer-appropriate processing framework, refer to Morris, Bransford, and Franks It is also cognitively more effortful to generate a solution by yourself, which in turn might induce more active processing and strengthen the knowledge in memory Bertsch et al.

This supports the possibility that students who generated their own solution had an easier time accessing their memory of a solution method and needed to engage working memory processes at test to a relatively lower degree than those who did not generate their own solutions. The question of whether or not manipulatives are effective though does not have a clear-cut answer. A recent meta-analysis of 55 studies Carbonneau et al. An analysis of moderators showed that a host of factors mattered such as age and the associated developmental status of the child, perceptual richness of the material, level of instructional guidance, and mathematical topic.

Lower perceptual richness e. Martin and Schwartz , for example, taught children about fractions using either pie wedges or tiles. Findings showed that those who used tiles were better able to transfer their fraction addition skills to other manipulatives than those who used pie wedges. Therefore, while pie wedges can initially help students with problem solving in that specific context, the added structure added perceptual richness that prevented them from being able to transfer that knowledge to other types of problems.

In another study that looked at perceptual richness, fourth- and sixth-grade students were either given perceptually rich bills and coins i.

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There is a trade-off between the two types of manipulatives because younger students who have not learned the relevant school-based algorithms may need the help of bills and coins to solve the problems whereas older students who have more domain knowledge may not gain any benefit from perceptually rich bills and coins McNeil et al.

The level of instructional guidance also has a significant impact on the benefits of the use of manipulatives Carbonneau et al.

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In particular, findings suggest that unassisted discovery does not benefit learners, whereas feedback, worked examples, scaffolding, and elicited explanations do Alfieri et al. Digital manipulatives can allow students to make the connection between concrete objects and more abstract objects to gain a better symbolic, conceptual understanding of the mathematical concept.

A challenge of the concreteness and perceptual richness of some physical manipulatives, however, is that it makes it difficult to transfer knowledge and generalize to other contexts. Students may not recognize, for example, that a circle with one fifth shaded, the decimal representation of 0. It is possible that physical, concrete manipulatives are better to learn with at the beginning, but that digital manipulatives are better for transferability.

So, if learners can transition from concrete to digital, they may get the benefit of both. As a scaffolding technique, younger students can start with concrete manipulatives before using digital ones. Our later case illustration in this manipulatives section illustrates one way this could be approached. The ability to control which elements learners can and cannot move can be leveraged to design more effective embodied learning experiences.

As an example of this, we turn to the possibility of constraining options of what can be moved in ways that encourage learners to use different strategies. The children were asked to provide all the ways that a certain amount can be combined e. Researchers found that children provided significantly more unique solutions in the manipulative condition with small plastic blocks as opposed to the condition with paper and a writing instrument.

These differences could be explained by the affordance of the block condition which allowed the children to move multiple blocks at a time which was not possible in the paper condition. This has implications for strategy use and mathematical understanding because, for example, reversing combinations e. As a digital application of this, students could be first constrained to moving one block at a time on a touchscreen and then be given the opportunity to move two blocks at a time to develop their strategies and mathematical understanding. The use of technology for educational purposes often induces a mental image of students interacting directly with digital devices.

Viewing technology as a general tool, however, allows us to put it to work in different ways. The power of 3D printing technology, for instance, can be leveraged to create a customized and personalized manipulative that is not available or feasible to get otherwise. Manipulatives are generally associated with early learning to help students to formalize some of the early school mathematics ideas. But, with the emergence of 3D printing and their digital design platforms, one could also use technology to create nonstandard manipulatives to convey more complex mathematical concepts beyond the early learning that the vast majority of manipulatives typically target Carbonneau et al.

The manipulative shown in Fig. Panel a shows a parabola with a positive b -value; Panel b shows a parabola with a negative b -value. Originally designed specifically for commercially available consumer 3D printers, this manipulative is intended for learning to graph quadratic functions.

However, it can be challenging to understand how the linear coefficient b -value influences the graph of a parabola. Because of this, students are not exposed to how this coefficient behaves in a graphing context. This manipulative allows students to change the b -value and see how it affects the graph of a parabola.

Digital manipulatives allow for understanding very small or very large scales such as the mathematical concept of exponential growth. For instance, in a game called Circle Exponents Fig. The software can zoom out to allow students to keep track of an exponential growing pattern, something unfeasible in a tactile environment. Additionally, students are able to see the circles organized into patterns that reinforce the mathematical concepts. Panel a shows the screen resulting from a student selecting which card to apply to the circle in the large rectangle.

Panel b shows the effect of applying the card to start generating the pattern on the left. Panel c shows the multiplicative effect of repeated use of the card. Students interact with the game by selecting a card with the correct number of spokes to replicate the yellow circle with the goal of creating the shape on the left Fig. The game moves from visual to symbolic as students describe the exponential pattern using visual cards, then repeated multiplication, then exponential notation Fig.

By determining which card will replicate the circles, students are to create the graphic on the left. Students are asked to connect exponential notation with the graphic by generating the pattern and not just the total number of circles. Students connect the pattern with exponential notation by writing an expression that models the pattern on the left. Though such a task can theoretically be accomplished with physical manipulatives, it would be logistically expensive and taxing, requiring the purchasing and space for spreading out thousands of objects in the classroom, ample time, and a great degree of patience from students.

For further enhancing the embodied experience of exponential growth, one could conceive of virtual reality tools allowing students to feel exponential growth of objects all around them. As a case illustration, we exemplify a way that technology can be used to connect the concrete with the abstract through better scaffolding of knowledge. One of the issues with learning with manipulatives is that if the manipulative is too concrete and too rich perceptually, students likely struggle to generalize that knowledge to other cases. However, especially for younger children, starting with an abstract example may be too challenging to grasp and learn.

Technology has the potential to scaffold and link the concrete and abstract in ways that help students build those connections. As an example, we highlight a spatial thinking game that we enhanced through technology to provide a deeper embodied learning experience in reference to the embodiment taxonomy outlined earlier Fig. The game called Upright JiJi Fig. The aim is to turn the penguin to an upright position by thinking ahead, selecting a series of rotations on a touch device, and then watching those answers unfold as the penguin rotates on the screen Fig. Panel b depicts a solution that was entered before the penguin begins the selected rotations.

Rather than selecting the desired rotation and watching the animation, the learner is able to create the desired movement with a bluetooth-enabled gyroscopic device Fig. A student playing Gyro JiJi by rotating a gyroscope that is wirelessly connected to a computer that generates feedback on the screen. As learners rotate the gyroscope in different ways, they interact with a combined physical and digital manipulative that translates their rotational hand movements into the rotations needed to rotate the penguin into an upright position.

Representative gestures include pointing gestures to indicate objects or locations e. Gestural congruency refers to the alignment of the type of gestures used, such as on a touchscreen e. Hand movements made with intention are often taken as evidence that the body is involved in thinking. Further, actions can also guide perceptional encoding. One mechanism by which this occurs, as demonstrated by a study that looked at memory and gesturing, is that gesturing while explaining math problems enables speakers to maintain more unrelated information in memory than they can when they do not gesture Cook et al.

This representational format is thought to free cognitive resources that can then be used to encode new information in a more lasting format. Gestures have been shown to improve learning by helping learners process existing ideas with less cognitive load. Since hands are already commonly used to manipulate objects, gestures provide additional feedback and visual cues by simulating how an object would move if the hand were holding it.

In a series of experiments with allowed-to-gesture and prohibited-to-gesture groups of young adults, Chu and Kita found that people who have difficulty in solving spatial visualization problems spontaneously produce gestures to help them, and that the use of gestures is related to improved performance.

As participants solved more problems, frequency in gestures decreased. It is thought that the spatial computation supported by gestures becomes internalized, and the gesture frequency decreases. This in part provides an explanation as to why the benefit of gestures persisted even in subsequent spatial visualization problems in which gesture was prohibited. Gestures are also involved in creating and shaping new ideas by introducing new ways of thinking through movement. In one line of related work with elementary school students, one group was told to gesture while solving algebraic equivalence problems e.

Findings showed that the gesture group solved more equivalence problems correctly in the post-instruction test compared to peers who were instructed not to gesture. These studies point towards an untapped potential for prompting cued body movement to improve learning. Gestures have been shown to be effective in helping learners to retain knowledge long term. Research demonstrates that learners should be more likely to grasp a concept if told to produce gestures instantiating that concept during learning than if told to verbally articulate the concept without using gestures.

Although all groups improved comparably, from pretest to immediate posttest, what is particularly interesting is that in a follow-up a month later, those in one of the conditions with gesture retained more knowledge than those in the speech-only condition. In explaining these results, the research team presented three hypotheses: 1 gesture offers a representational format that requires relatively little effort to produce, thereby freeing resources that can then be used to encode new information in a more lasting format, 2 gesturing directly facilitates encoding in long-term memory through producing stronger and more robust memory traces than expressing information in speech because of the larger motor movements involved or because of the potential for action-based, bodily encoding, and 3 gestures that indicate objects and locations in reality may make it easier for learners to link developing mental representations to relevant parts of the external environment to reduce processing demands.

The use of congruent gestures helps to construct better mental representations and mental operations to solve mathematical problems, which has implications for the ways that digital interfaces can be designed. This suggests that asking learners to perform a specific type of gesture could mentally prime them to solve the problem in a particular way. In particular arithmetic is a discrete task and should be supported by discrete rather than continuous actions whereas estimation is a continuous task and should be supported by continuous rather than discrete actions.

If action supports cognition, performance should be better with a gestural interface designed such that the actions map conceptually to the desired cognition. As such, tapping with a finger on a virtual block or clicking with a mouse on a virtual block to count and add are gestures that are congruent with the discrete representation of counting.

In contrast, sliding the finger vertically over a series of blocks or dragging a mouse across a series of blocks to count them are continuous movements that are not congruent with the discrete procedure of counting. Also as expected, students who did numerical estimation with a sliding gesture performed better than those who did the task with a tapping gesture.

Later in this section we provide an example of applying this recommendation of cueing gestures that are congruent with the mental representations of the mathematical concept in the context of the mathematical understanding of slope. Children were taught to spread out two fingers to make V-point gesture with the two fingertips each pointing to the first two numbers the 4 and the 3 in this instance and then have a finger on the other hand pointing at the blank on the other side of the equation.

The purpose of these movements was to help the children see that the problems can be solved by grouping and adding the two numbers on the left side of the equation that do not appear on the right side and then putting the sum in the blank. Children who were asked to produce these hand movements during a math lesson were able to extract the grouping strategy despite the fact that they were never explicitly told what the movements represented.

Future design and development work could explore ways to leverage technology to encourage gestures that contribute to effective strategy use. In this case illustration, we provide an example of leveraging technology to provide gestural congruency to provide students with another level of understanding of the concept of slope. Typical approaches teach slope as a formula to memorize and be applied to a static graph. Students look at two points and simply remember which to subtract and where to put the answers.

Often, students learn the formula and practice calculating with it to the point that the formula abstracts away the understanding and hides what slope really means. To address this issue, a game called Tap Tempo was designed to experience the difference in slope as a tapping tempo, providing the embodied experience of slope as a literal rate over time.

Tapping influences the vertical movement such that faster tapping creates more steepness and, by having embodied control of this mathematical process, students can internalize that slope is about a vertical change over a specific horizontal distance Fig. A puzzle of Tap Tempo that can be solved by tapping the button to move vertically over time.

This allows students to experience the embodiment that a linear path has a constant slope and connect that with the understanding that a line is the only type of function for which a constant cadence along the x-axis corresponds to a constant cadence along the y-axis. As a form of gestural congruency in this game, lines with different slopes can be embodied with a faster or slower tapping tempo. In a touch environment, the game can be designed such that the student controls both horizontal and vertical movement simultaneously. This affordance made possible by multitouch screens is not possible to implement with a computer mouse as input device.

Another domain that has been repeatedly linked to embodied cognition in mathematics is full-body movement. Critically, by whole-body movement, we do not refer to exercise or workouts that strengthen muscles and the cardiovascular system, but instead focus on bodily activities that are closely related to mathematical content with a negligible fitness component. This theory proposes, for example, that stored and strongly interlinked information about concrete objects in the world include not only perceptual features, such as color, shape, size, and smell, but also action-related properties such as chewing and swallowing.

Critically, if an object is processed, all of its features — perceptual and action-related in nature — are activated. During later retrieval, the activation of one feature i. Moreover, the activated features i. But this may also work the other way around, that is, the activation of features of a planned action facilitates the perception of objects and carrying out other actions that share features with this planned action. Although TEC serves reasonably well as a theoretical starting point to explain embodied cognition, its original conception did not have embodied cognition in mind.

However, that extension is straightforward, not only in regard to body movements, but in regard to embodied cognition in general Hommel, Most of the currently available intervention studies on the impact of whole-body movements on mathematical learning focused on magnitude representation involving a number line. To our knowledge, there is currently only a limited set of intervention studies available that incorporated whole-body movement into the training part.

Since this corpus of literature is so small, we are going to review the individual studies in more detail than we did with the literature in the previous sections where the available published research is considerably more substantial. Fischer et al. They analyzed data of 5—6-year-old children who formed the experimental group and at the same time also served as a control group in a cross-over design.

In the experimental condition, children had to decide whether a presented number was smaller or bigger than a presented target number using a digital dance mat. If the number was bigger than the target, they had to move their whole body by stepping to the right; otherwise they had to step to the left. With the goal to further increase the effectiveness of the intervention, a spatial number line was also presented in the experimental condition.

In the control condition, the same task was performed on a computer with a touch interface without presenting a spatial number line. Identical number sets were used in both conditions. The authors reported that after the digital dance mat condition, children performed better in a traditional spatial number line estimation task and also in an untrained verbal counting task. No effects were found in object counting, knowledge of digits and number words, and simple addition and subtraction problems. We note that the design of this study does not allow for disentangling the potential effects of the dance mat component and the potential effects of the presentation of the number line, both of which were only present in the experimental but not the control condition.

The participants in the above described intervention study Fischer et al. Since the number line is assumed to be continuous, however, it makes sense to implement a continuous oriented intervention, which is what Link, Moeller, Huber, Fischer, and Nuerk did. In their study, they trained first-grade students on a number line task that required participants to walk to a position on the line that corresponded to a given number. The response of the participants, that is, the position where they stood on the number line, was tracked with a Microsoft Xbox Kinect device.

The control condition also involved a walking component that was not related to mathematical content such that participants were instructed to walk to a tablet computer and estimate the position of a given number on a virtual number line using a touch interface. This procedure ensured that neither the walking component nor the presentation of a number line represented confounding variables. The children were trained over three sessions in both conditions. The authors reported that both groups reliably improved in a number line task but that the improvements in the experimental and control conditions were not significantly different from each other.

Neither were there effects found in symbolic and nonsymbolic number comparison tasks nor in tasks that measured place-value understanding. However, the authors found differential training effects in favor of the experimental group in two out of three measures of mental addition. An issue of the two intervention studies reported so far is that they are not very scalable because the training sessions had to be conducted in a one-on-one setting.

This inherently limits the potential reach of these embodied training approaches in the classroom. In order to overcome this problem, Fischer, Moeller, Huber, Cress, and Nuerk conducted a pilot study in which they used an interactive whiteboard as a means to train number magnitude in an embodied way. The authors argue that whiteboards are readily available in classrooms and are also big enough in order to implement embodied training of number magnitude with them. In their intervention, second-grade students had to mark given numbers on a number line drawn on a whiteboard.

Critically, the whiteboard was big enough so that children had to walk a few steps in order to reach the position at which they marked the number on the number line.

## Processing Mathematics Through Digital Technologies: The Primary Years by Nigel Calder - yvoropijajif.gq

The whole intervention including pretest and posttest assessments was conducted in just one session. Two control conditions were implemented, one in which children had to discriminate colors on the whiteboard media-matched control group and another control condition in which participants performed a number line estimation task on a computer tablet task-matched control group. The authors reported that the experimental group performed better in a number line estimation task than the two control groups but there were no other differential group effects in a mental addition task, a number comparison task, and a task in which children had to select the closest number to a given reference number.

In the study by Ruiter, Loyens, and Paas , the authors compared two-digit number building training in an embodied condition with a control condition that did not involve mathematics-related whole-body movement. In the embodied condition, the participants were instructed to build the numbers by taking steps along a ruler on the floor.

Large steps corresponded to the 10s, medium steps to the 5s, and small steps to the 1s. In the nonembodied condition, participants had to build the numbers verbally, without taking steps, followed by indicating on a ruler where the number that they just built was located. After a brief intervention of constructing 10 numbers, results revealed that the embodied condition outperformed the nonembodied condition on two tasks that required building numbers using Lego blocks, in a similar way as the numbers were constructed during the intervention.

It should be noted that this study implemented a posttest-only design, that is, participants were not tested on the criterion task before the intervention and, therefore, it is uncertain whether the groups were comparable at baseline. Finally, in a recently published study, Dackermann, Fischer, Huber, Nuerk, and Moeller trained children to segment spatial distances into equal intervals with the hypothesis that a better understanding of the concept of equidistant spacing would lead to a better understanding of number magnitude.

In the embodied condition, children started to walk at the beginning of a number line that was taped to the floor and were instructed to choose their stride length so that it segmented the number line into a requested number of pieces. Every step defined one segment. Their steps were tracked by a Microsoft Xbox Kinect device which was necessary for performance measurement and generating performance feedback that was presented via video after each trial. In the control condition, children had to segment a number line shown on a tablet computer into equidistant pieces using a touch interface.

It should be noted that the control condition also included a full-body movement component that was unrelated to any math activity, i. The authors report that in the embodied condition children were better in segmenting a number line compared to the control condition. However, no condition differences were found for number line estimates and arithmetic performance. The reviewed intervention literature that focuses on whole-body movement as an embodied way to improve numerosity is small and more work is needed to evaluate the potential of an embodied training approach in this domain.

The current evidence of efficacy is based on relatively small sample sizes per study — with the exception of Ruiter et al. It is conceivable that longer interventions would result in more pronounced effects, assuming that more training leads to better learning. The reviewed studies demonstrate that participants got better in the condition of interest but this improvement was not always superior than the improvement in the control condition.

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Additionally, there is inconsistent support that participants also improve in tasks that were not part of the intervention such as verbal counting and mental arithmetic. However, there are large procedural differences between the linear number line board game studies and the whole-body movement studies such as group versus one-on-one administration and number of training sessions to name only a few, that do not allow a straightforward comparison of the two intervention approaches.

In sum, the available whole-body movement studies that aim to enhance mathematics learning are inconsistent and a clear advantage of the embodied approach is not yet fully established. The basic idea of whole-body movement as an embodied activity is that it has a positive impact on cognition by allowing a student to learn a certain mathematical concept better than without movement. Here, we were only interested in movement that is closely related to mathematical content and not in movement for the sake of fitness or motivation.

In the following we want to explore in what way we can use emerging technology as a superior tool to increase the efficiency of mathematical learning. It is worth mentioning at this point, that this area of research is relatively young and with the exception of one study Ruiter et al. However, as we also later discuss in more detail, there are other technologies that to the best of our knowledge have not yet been implemented in mathematical intervention studies, such as pedometers, Global Positioning System GPS trackers, and floor-projected imagery that changes as a function of where participants stand.

Body movements can be tracked through technology. If such movements are executed to embody a mathematical principle then such tracking allows us to study the mathematical thinking processes that an individual is going through. As a consequence, we can present the learner with immediate, real-time feedback e. The tracking of movements as an expression of mathematical thought processes is possible without interrupting the thought processes of a learner. In contrast, tracking thought processes in a more traditional setting often requires learners to deliberately put their own thinking into words by writing them down or saying them out loud as they are problem solving.

The problems of the latter approach are that it is more intrusive and that a learner might fail to mention an important step in the train of thoughts or the description of thoughts is lacking detail. Further, real-time feedback on thought processes is very challenging to implement in traditional settings. On the other hand, one has to keep in mind that body movements are not a direct expression of thoughts; however, if implemented carefully, they might provide insights into ongoing thoughts that are not possible to access otherwise.

Finally, recorded body movements can serve as an excellent source to reflect on the understanding of the mathematical problem at hand and allows for the discussion and comparison of mathematical solving approaches of different individuals. An explicit goal of the following case illustration was to create an embodied activity that made efficient use of technology that would enhance learning without putting the technology itself in the focus of attention.

In this activity, students first measured their average stride length and were then asked how far away they thought a certain object in the environment was, for example, a wall. Students were then instructed to give an estimate before pacing out the distance. Next, students walked the distance to the target and counted the number of steps. In order to improve their own measurement accuracy, they were asked to walk back to the point from where they initially started and calculate the average of their two measurements. While the students did that, a pedometer that was given to them beforehand, also measured their steps and allowed a comparison with their own measurement.

In a next step, students could then calculate the distance by multiplying the counted number of steps with their average stride length. In order to get a more objective measurement of the real distance to the target, they were provided with a laser-based distance measurement tool which again allowed a comparison with their calculated distance. This activity was then repeated with different targets that were located in varying distances from the starting point. In this environment the main goal was to train number sense through practicing estimation skills, but other mathematical aspects, such as calculating averages and comparing expected numbers with real measurements, were also part of the exercise.

Here, technology takes a more subtle role, and movement becomes a vehicle of mathematical intent. In this way, the power of technology is elevated due to its ability to facilitate without it being distracting. In a somewhat related activity that is also on display in different variants in mathematics museums, such as the The National Museum of Mathematics MoMath in New York City USA , students are asked to interpret functions of graphs through movement. A common approach is to define the x-axis as time. Often, graphs are perceived to be static, but there is mathematical foundation for them to be active.

For example, one way to interpret the mathematical idea of parameterizing a curve, may it be linear or nonlinear, is acting it out over time. This creates an opportunity for students to act out functions while technological devices are visualizing the movement on a monitor; for example, through data acquisition of a pressure-sensitive floor mat.

To interact with the technology, students stand on a mat several yards long. As they move forward or backwards on the mat, their position is interpreted in the context of the graph on the monitor. Walking forward on the platform, towards the monitor, is interpreted as movement upwards on the graph. Taking steps back on the platform models a vertical decrease. Visitors start at a fixed middle point and must step forward or backwards to stay as accurate to a given graph as possible. In such an environment, the slope of the graph becomes synonymous with speed along the y-axis. Obviously, environments such as this evoke a high degree of physical movement, and the movement is intentionally suggested by the graph.

We reviewed the influence of manipulatives, hand gestures, and whole-body movements on mathematical learning in the context of emerging technologies, and we discussed the potential to increase the impact that embodied interventions have in these domains. We find that the embodiment across all three reviewed domains can benefit mathematical learning, likely by providing an additional representation of the mathematical conception to strengthen encoding, by reducing cognitive load to provide more processing power to deeply think and problem solve, and by inspiring the use of strategies and modes of thinking other than what nonembodied approaches evoke.

At the same time, however, it also becomes clear that the research on embodied cognition and mathematics is still quite young, in some domains more so than others. The available body of research on manipulatives as well as hand gestures is more comprehensive than the work on whole-body movement. Across the domains of manipulatives and whole body movements, most of the work in mathematics understanding has been conducted with younger children and, therefore, it is unclear what effects can be expected with an older population, such as college students, who are learning more sophisticated mathematics.

Focusing on the research on the effectiveness of manipulatives, it seems that manipulatives are generally beneficial; however, results are strongly moderated by the contexts in which they are used and, in particular, the amount of instructional guidance and the perceptual richness of the material. Finally, it is an open question as to whether the different levels of embodiment, in the sense of the taxonomy suggested by Johnson-Glenberg et al. In providing guidance for how we can leverage technology to design embodied experiences for mathematical understanding, the examples that we provided are intended to illustrate the possibilities and the constraints involved, but we acknowledge that there are other design approaches and technologies that can be used.

Future research and reviews certainly have much ground to cover in those areas, but there is still quite a bit that we can take away from and build on with what is currently known and what our reflections across three areas of embodied cognition and mathematics have revealed. The effectiveness of the case illustrations that we provided also has not been formally assessed which constitutes another necessary area for future exploration. In guiding the application of embodied cognition to the design of mathematical tools, we discuss thematic considerations for the design, implementation, and assessment in the space of technology, embodied cognition, and mathematics education.

With the link between movement and thought, one of the opportunities that we have for design is that we can guide movements to influence thoughts in ways that improve mathematical cognition. This can be done on touch interfaces through cuing appropriate gestures, such as discussed earlier see also Richland, , tapping for discrete arithmetic operations, sliding for estimation problems, and tapping at different rates to experience different steepness levels of slope.

A challenge to using body movements as a foundation or even as a metaphor for activating new learning opportunities is that individuals may not remember the details of the movements that they engaged in. For learners to reflect on their movements and the mathematical concepts that those movements enact, there needs to be a way for them to observe and reflect on these movements. To address this challenge, technology may allow a playback feature or provide real-time metrics that allow learners to observe what their body is doing with respect to the important features of the target learning domain.

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Technology is a tool. If treated as the focus, technology actually adds more distraction to the environment and may not result in better learning. Using technology does not have to be flashy and can be incorporated in subtle ways. This is what our case example does in which students are instructed to estimate distances and compare their estimates with their own real-world measurements using pedometers and laser-based measuring tools. If the technology is too perceptually rich, it will distract from the learning content, as found in the case of certain manipulatives.

Ideally, the technology blends perfectly into the activity, so that you are not blinded by it but you are able to fully focus on the content. Technology can help to make some things more scalable such as the use of manipulatives. When it is digital, one does not have to buy several copies of objects, set them up, and clean them up.

The falling price of technology also helps with scaling technologically enhanced experiences. However, there are also aspects of embodied learning that are not as easily scalable. With whole-body movements, for instance, it is hardly feasible to have 20 or more digital dance mats or Microsoft Xbox Kinects in the classroom given the cost, required space, and supervision needed to implement this. Further, 3D printing allows for the creating and sharing manipulatives using free software and open-source devices. As 3D printers become increasingly affordable and available in schools, custom manipulatives will become more scalable.

Consequently, schools with 3D printers could receive digital files and print the manipulatives as needed.

They could print in any size that suits their needs or even with the colors or materials that best fit their desires. This puts customized and personalized manipulatives into the classroom quickly and at low cost. Learners may manipulate physical objects seemingly meaningfully, but that does not necessarily mean that the abstract concepts that these objects aim to convey are understood. In an example illustrating that, working with Cuisenaire rods, Holt , pp. We therefore assumed that children, looking at the rods and doing things with them, could see how the world of numbers and numerical operations worked.

The trouble with this theory is that [my colleague] and I already knew how the numbers worked. Qualitative and quantitative video data is used by the authors who identify 95 potential helping and hindering affordances among the 18 apps they trialled. As was the case in previous research on iPads Larkin ; Moyer-Packenham et al.

The remaining articles in the special issue shift the focus to examinations of possible appropriate pedagogies to best utilise the affordances of the mobile technologies available to classroom and university teachers. A variety of theoretical frameworks are proposed and the educational contexts include very young children, upper primary, junior secondary and senior secondary students, and undergraduate education students.

As was the case with the first four articles, these pedagogically oriented articles encompass a range of national and international contexts. Accompanying this demographical shift are increased community expectations that such teachers can embed information communication technologies ICTs effectively into the teaching of mathematics. In this article, Attard and Orlando investigate how four early career primary school teachers use ICT in their teaching of mathematics.

Two important factors, developed from the research presented in this paper, suggest that ECTs uses of technology to teach mathematics may be more complicated than first envisaged. Ingram, Williamson-Leadley and Offen report on a qualitative investigation into the use of Show and Tell tablet technology in mathematics classrooms. A Show and Tell app allows the user to capture voice and writing or text in real time.

The article starts from the premise that teaching secondary mathematics has a number of challenges, including the expectations that teachers cover the prescribed curriculum, help students learn difficult concepts and prepare students for future studies and, increasingly, that they do so incorporating digital technologies. Their analysis indicates that the teacher and students were positive about their experiences with a flipped classroom approach, that utilised mobile technologies and that students were motivated to engage with the teacher-created online mathematics resources.

In so doing, the study adds to the limited research literature related to student and teacher perceptions of the affordances of the flipped classroom approach in secondary school mathematics. This article, by Bray and Tangney, explores how a combination of a transformative, mobile technology-mediated approach facilitated the development of mathematics learning activities in 54 Irish, secondary school students. This article proposes clear, logical connections between aspects of the activity design and their impact on student attitudes and behaviours in learning mathematics.

Although situated in an Irish context, there are clear implications in this research for mathematics teaching in many other countries. The final article in this issue, from Handal, Campbell, Cavanagh and Petocz, reports on a survey of Australian, pre-service students studying primary school education. The technological pedagogical content knowledge TPACK model is used as the conceptual framework for the analysis of these students engagement with iPad mathematics apps. The respondents examined three different apps using a purposely designed instrument in regard to their explorative, productive or instructive instructional role.

All articles in this special issue have undertaken the normal blind, peer-review MERJ process, and we deliberately sought national and international colleagues who are both knowledgeable in the field and also not contributing authors for this special issue. Nigel and I would very much like to acknowledge the wonderful reviewing work of the following colleagues, many of whom did several reviews of an article in this special issue. Clearly, without their contributions this special issue would not have been possible.

We look forward to further editorial work with both Robyn and Peter. Finally, we would like to express our support for Ms. Belle Mojado Springer Press , who answered many, many emails regarding the Springer Editorial system—thank you very much Belle. The special issue identifies some themes and challenges as the mathematics education community continues to examine the usefulness of mobile technologies to enhance the learning of mathematics, either from the particular affordances of the media that allow teachers to reshape the learning experience, or from the particular pedagogy associated with the technologies.

Questions arise when technology is considered in the teaching process: what is the relationship between features of the tablets, the architecture of the apps, and the learning that a reshaped environment might afford? In terms of pedagogy, what approaches might best optimise student engagement and mathematical thinking? As well, how might the notion of scaffolding be re-envisaged to include feedback from digital sources and a greater element of self-assessment?

Equity issues concerned with access both within domestic school communities and globally still remain, and require ongoing examination. Some planning and management aspects also need to be considered. What comes first when planning — the mathematics, the app or the pedagogy? What are the best ways to manage the introduction of new apps? Researchers also need to continue to critically examine the influence of mobile technologies on learning across a range of contexts and the ways that they interact with other pedagogical media.

The articles in this special issue reveal considerable potential for mobile technologies to enhance student engagement and mathematical thinking, but the full scope of opportunity and the relationships with other learning approaches are still to be fully unravelled. Nevertheless, they make a considerable and very worthy contribution to further understanding in this critical endeavour.

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### Curriculum structure K–12

Mathematics education and mobile technologies. Article First Online: 23 December While there are a range of technologies and associated pedagogical approaches that can be incorporated into mathematics education, it is important to consider the ways that they might reshape the learning experience and influence engagement and understanding. In this issue, the focus clearly concerns either mobile technologies themselves iPads, iPhones, Androids or pedagogies associated with the use of mobile technologies flipped classrooms, twenty-first century learning.

In our call for articles, we sought contributions with the following features. Calder, N. Processing mathematics through digital technologies: The primary years.