In mathematics education, simple passive manipulatives provide valuable “hands-on” approaches to teach students abstract concepts, especially when the students start to learn a novel unit of math, e. g., arithmetic or algebra. Based on the assumption that activity supports learning, these approaches are in-line with models from didactics like Bruner’s concrete-representational-abstract approach  or the constructionist objects-to-think-with approach  that suggest to use physical objects for abstract concepts, especially for beginners. While a considerable body of research on using tangible user interfaces (TUIs) for learning has been conducted (c.f. , ), more research efforts are needed to address how TUIs can support the learning of abstract concepts such as algebra, and how the tangibles need to be designed in order to meet the needs of teachers and students.
In our research, we investigate the potentials of smart objects for algebra learning. The objects are based on traditional algebra tiles, which are passive manipulatives (see Figure 1) used in many schools in Northern America to support algebra learning. We are extending these tiles to smart tiles by integrating them into an interactive multi-touch surface and by adding multimodal input and output capabilities as well as adaptivity and feedback.
In educational research tactile models are the object of interest for a few decades already, as publications by Bruner , Montessori , McNeil and Jarvin , Carbonneau et al.  or Kieran  show. For example, Bruner’s and Montessori’s learning concepts encompass using physical objects to teach small children abstract concepts. Kieran mentions, that “algebraic meaning” can be achieved by modelling the situations with physical objects. McNeil and Jarvin  summarize the research regarding manipulatives and draw the conclusions, that it seems to be easier for children to learn these abstract concepts when “the problems draw on children’s practical understandings”. Common digital learning platforms, such as Dragonbox,1 an App for algebra learning, lack the benefits that come with tangibility. By transforming the algebra tiles into multimodal TUIs, we combine the feedback provided by digital platforms with the benefits of tactile interaction. With this approach, we aim at supporting the transition from the enactive stage via the iconic stage to the symbolic stage – as classified by Bruner . Although these stages have first been established with regard to young children, they have in variations proven helpful also where older learners with different abilities are to be addressed in one learning setting, allowing some children to work at their appropriate stage longer than others.
In the following sections of this article, we will provide an overview on related work, introduce the approach of algebra learning with tangible tiles, and then present insights from two comprehensive field studies we conducted with students, teachers and textbook authors in order to derive design implications. Finally, we present and discuss the resulting system design.
2 Related Work
Tangible user interfaces emerged in the 1990s, when Ishii and Ullmer ,  described their vision for “tangible bits”, physical objects that are connected to digital data or functions and can be directly manipulated for interaction. Examples for using tangibles for math learning have been provided by Falcaō et al. , Girouard et al. , Manches and O’Malley , and Marichal et al. , amongst many others. Others, like Rick , have incorporated touch to be able to directly manipulate math objects presented on a screen. Research about how tangibles can support learning or how learning theories can inform tangible development has for example been presented by the “Tangible Interaction Framework” by Hornecker and Buur  or the “Tangible Learning Design Framework” by Antle and Wise . They propose design principles for tangibles and a taxonomy about the relationship between TUIs, interactions and learning. Furthermore, Marichal  and Marshall, Price and Rogers  have critically discussed how tangibles can support learning. Results from neuroscience research suggest that concept learning is improved when using someone’s motor skills , . A study by Kiefer et al. for example implies that action information helps conceptual processing.
With this research, we add to this body of work by exploring tangible interaction for algebra learning based on the concept of algebra tiles. Inspired by earlier tangible systems such as the sifteo cubes  or Actibles , we are especially focusing on the design of multimodal feedback based on smart objects for interaction (e. g., through integrating haptic elements with embedded displays and LEDs).
3 Algebra Learning with Tangible Tiles
Algebra tiles as shown in Figure 1 consist of three types of tiles: single units used as “ones”, variable-tiles and variable2-tiles. For the ease of reading, variables are denoted as x-tiles/x2-tiles, even though any other symbol could be used as the variable placeholder. Each tile has two sides being differently colored, red represents the negative and the other color the positive sign, the latter color also stands for the type of tiles, yellow for single “one” units, green for x-tiles, blue for x2-tile. To represent a linear equation, the tiles are placed on a 2×2 grid (see Figure 2), where the two areas on the left represent the left side of an equation and the two areas on the right represent the right side of the equation. The top areas on both sides are the “addition zones” (tiles are connected by addition), while the lower areas are the “subtraction zones” (tiles there are subtracted from the top ones). Figure 2 (top) shows how an equation, in this case , is set up with algebra tiles. After initialization, the goal is to apply a sequence of legal actions in order to transform the equation to finally isolate one remaining x-tile. Therefore, it is allowed to add or subtract by removing the same tiles in value from the same zone on both sides or the same tiles in value from opposite zones on one side of the equation. Division is represented by setting up the same number of equal groups on each side and then continuing to work with only one of them. Additionally, it is allowed to put a negative and a positive tile of the same kind into the same zone (a so called “zero pair” – their values add up to 0) to transform the equation (c.f. step 1 and 2 in Figure 2).
With the traditional tiles, a student can transform the equation but does not get any feedback about the correctness of the actions. Our approach combines tangible algebra tiles with visual and haptic feedback with an interactive surface to provide an interactive learning experience. To better understand the challenges of and requirements for our system, we conducted two studies. Based on the results of these, we derived design implications for a tangible algebra system and present our design approach.
4 Insights from a Field Study with Paper Tiles in the Classroom
A paper version of the tiles was tested in a realistic setting in school. Over the course of 9 sessions (à 75 to 90 minutes) a group of 12 students of grade 9 (age 14–16) with persisting problems in elementary algebra worked with paper representations of the algebra tiles. The use of self-made paper tiles allowed to easily create a set of tiles and a mat that already addressed some issues that had been identified by the team. Unlike the commercially available algebra tiles, the paper versions were white (ones) and blue (x-tiles), with the negative side printed over in grey. The subtraction zone (as described in the previous paragraph) was an extra sheet to be placed parallelly to the vertical axis inside the addition zone to stress the importance of the left-right-divide and to overcome the obstacle of not being able to divide with tiles in the negative area. To illustrate division, straws were given to the students. The straws function as divider to highlight the equal groups that are necessary for a valid division.
During the field study, the students were instructed to work collaboratively in pairs. All sessions were videotaped with three cameras (each camera filming one pair of students) and observed. The observers took notes during the sessions. The video material was coded regarding (1) usability and user experience problems, (2) value adding elements for the planned system and (3) possible solutions .
The video analysis revealed that a digital system could benefit from automatically providing the next task and therefore reducing idle times between the tasks. Furthermore, a common problem was that tasks were either not completed or not performed as intended. The students had to shift their foci between the tasks on paper, the documentation of their steps and the paper tile system, which was a necessary workaround. This clearly led to errors, omitting tasks, and problems in focus. The analysis showed that many of the students would have benefited from feedback regarding tile meaning and legal moves as they took away an x-tile on the left side and a unit tile on the right side or divided unequal groups. Even with prior knowledge regarding negative numbers, they struggled with the negative sides of the tiles and the subtraction zone. The division was confused with the steps of a subtraction. The collaborative task design revealed that in unequal groups one student typically performed all actions and rarely explained his or her actions to the second person.
5 Insights from an Expert Study with Textbook Authors
The second study was an expert study with authors of mathematics textbooks, who are experienced teachers. The authors formed fixed teams based on the textbook they contribute to. The study had three phases that built on each other (see Figure 3): first focus groups and then two consecutive questionnaires. Within the 80 minutes of the focus groups (size: 5 and 12) the authors learned the concept of algebra tiles, discussed models as used in textbooks, online tools like Dragonbox, and the concept idea of a tangible algebra learning system. The questionnaires built upon the results of the respective prior round – questionnaire one on the focus group, questionnaire two on questionnaire one. Both questionnaires asked for agreement on statements from previous rounds on a 5-point-Likert scale. The first one additionally contained open questions regarding algebra learning models and the advantages or disadvantages of them, 22 and 23 authors from three different textbooks participated.
The experts highlighted the importance of the enactive and iconic aspects of algebra teaching models. Some noted they prefer simpler systems with constraints over complex ones since they need less explanations. While organizational issues like storage space and availability speak against physically available models, they exceed purely digital models in their enactive aspects. A digital one convinced the experts regarding easy availability, flexibility and adaptability, which was emphasized in the context of teaching students with different needs and learning paces. The experts would welcome a system that reduces their workload and allows the students to work on their own and in their individual pace. As 83 % underline the importance of the transition from the enactive/iconic to the symbolic stage, they expect a system supporting this and giving clear feedback. But as costs will likely restrict the availability, teachers do not want to depend on the system. When exposed to the algebra tiles concept, the teachers were slightly confused with the different colors for each of the tiles. Moreover, some disliked that the length of the rectangle shape of the x-tile appeared to be three or four times the unit tiles, as the value of x should not be considered to be three or four due to the length. Problems with equivalent transformations also occurred and the x² actions caused some trouble in the beginning as well. The authors liked that negative numbers are represented and began to think about utilizing the inherent riddle aspect for the better students as a challenge. Exposed to the Dragonbox game, some teachers commented that for the target group of students this application might be too “childish” regarding design.
6 Design Implications for a Tangible Algebra Tiles System
From the results of the field and the expert study we derive the following design implications (also inspired by Nielsen’s 10 heuristics for user interface design  and Antle’s and Wise’s and “tangible learning design framework” ):
- DI 1.
The system should provide clear guidance through the tasks and thereby aid understanding of the system.
- DI 2.
As both students and teachers had problems with the colors and new ways of interacting with the model, a tutorial introducing each functionality step by step is highly recommended.
- DI 3.
Furthermore, hints should be available to provide solutions to the current situation, e. g. when the user is stuck or an action cannot be performed as the user tried.
- DI 4.
This includes going back to a former situation as in keeping track of previous steps.
- DI 5.
The user needs the opportunity to get information on what each tile means and what the possible interactions are.
- DI 6.
As collaboration is a great opportunity when working with tangible tiles, the system should support this use case, e.g. by distributed controls as proposed by Antle and Wise .
- DI 7.
A tangible system is a tool to support the transition from the enactive/iconic stage to the symbolic stage. Thus, all ways of representations should be available and it should be possible to continuously increase difficulty.
- DI 8.
The continuous increase in difficulty should be also available regarding the tasks, as one goal is to provide benefit for students with different learning capabilities and should therefore be adjustable to each individual student.
- DI 9.
A clear and intuitive color scheme is important to convey the necessary information without confusing the user.
7 Design Approach: A Scalable System for Algebra Learning
We combine the concepts from literature, the requirements and the design implications from the studies to create a system for algebra learning. As having a large interactive surface for every student is not feasible, among the project goals was also to create a scalable system ranging from a tablet-based multitouch version mostly focusing on the iconic and symbolic stage to an interactive table version with interactive tangibles and a strong focus on the enactive stage. Here we focus on the tangible system.
In our scalable algebra learning system, we reduced the color set to two colors, one for the positive sign (blue) and one for negative sign (red) (see design implication (DI) no. 9). We incorporated a preview for the tiles – so called “ghost tiles” – to support during equation set up especially during the tangible scenario. Additionally, the ghost tiles are used as hints to guide the user when help is required (one approach to provide help as advised in DI 2 & 3). To support the transfer to the symbolic stage, we also show the symbolic representation of each equation updating in real time while moving tiles on the canvas (DI 7). As maintaining the equivalence is an important aspect, we can give either instant or delayed feedback. In our current design, the feedback is always slightly delayed to allow the user to finish an interaction, for example when removing tiles from both sides. To cope with the different screen sizes of each version, task descriptions are currently placed in a sidebar menu. The current version of our system can be seen in Figure 4.
8 Discussion and Conclusion
Developing a scalable system for touch and tangible use on different screen sizes poses challenges such as the available screen space or that the tangibles occlude text displayed on the screen. There are differences in terms of the possible interaction on pure touch devices compared with a combined touch and tangible approach. One example is the division. On the tablet with pure touch there is a nice solution with a dividing gesture and fading tiles, as it can be seen in Figure 5. As tangibles cannot just disappear, we have to carefully design the interaction affordances to lead to the same result – just the result of the division remains on the canvas while all other tangibles are removed. When focusing on specific concepts like for example zero pairs, tangibles can enrich the interaction equipped with dynamic constraints.
The presented work has the following limitations: The students participating in the field study had prior knowledge in algebra from previous instruction. Therefore we do not know how novice students would have interacted when exposed to the concepts of algebra mediated by the algebra tiles. Additionally, it was a group of students who were selected by their teachers by their need of extra lessons in algebra. Therefore the group likely excluded students who might have been ready to work with symbols alone at an earlier point in instruction. The expert study with textbook authors was a Delphi study  designed to lead to different types of consensus. As the textbook authors were representing the existing variety of school types, we only can expect biased results because of the different group sizes. However, due to the teachers’ responsibility as textbook authors they might have been in favor for supporting books over technology.
In summary, a multimodal system for algebra learning with tangibles requires a clear strategy, starting with a clear and intuitive color set, an easy to follow introduction into the system and its functionalities and comprehensive didactic concept. Additionally, when developing a scalable system, a consistent and comparable set of interactions is needed in order to ease switching between the systems. As a next step, we aim at testing the interactive prototypes with students to evaluate the benefit of the digital and tangible system. Based on the results we will refine the guidelines and improve our system for future testing.
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About the article
Anke Reinschluessel is a research assistant/PhD student at the Digital Media Lab led by Professor Rainer Malaka at the University of Bremen, Germany since 2016. She has Bachelor Degree in Cognitive Science and Master Degree in Human-Computer Interaction. Her research focuses on innovative multimodal user interfaces.
Dmitry Alexandrovsky is a Ph.D student in computer science at the Digital Media Lab at the University of Bremen. He is lead developer in the multimodal algebra learning project. His work focuses specifically on reality based interaction in learning environments and game user research.
Tanja Döring is a research associate (Post Doc) in the Digital Media Group at the University of Bremen, where she currently leads an interdisciplinary research project on tangible user interfaces for math learning. Her research focuses on human-computer interaction and materiality, digital fabrication, and novel interaction techniques, including interactive surfaces, tangible, gestural, and mobile interaction. Tanja Döring graduated in Computer Science and Art History at the University of Hamburg (Germany) and holds a PhD in Computer Science (Dr. Ing.) from the University of Bremen (Germany). She serves as program chair, PC member and reviewer for major conferences and journals and as member of the steering committee of the German Tangible Interaction Group (GI Fachgruppe Be-Greifbare Interaktion).
Angelie Kraft just finished her bachelor degree in psychology and computer science. Her interest is in multimodal and tangible interaction.
Maike Braukmüller studied Mathematics (MSc) and is currently working as a PhD student at the textbook publisher Westermann Gruppe (Braunschweig). In cooperation with the department of Mathematics Education at the University of Bremen, her research is located in the field of combining textbooks and digital tools in mathematics education.
Thomas Janßen is part of the mathematics education working group at the University of Bremen. In his dissertation, he has investigated how students develop a sense for algebraic structures, in particular for linear equations. Finding that the teacher’s guidance was crucial in the observed learning activities, he is now investigating how much of this guidance can and should come from a machine and how it can be optimized.
David A. Reid has been researching proof and proving in mathematics education for two decades. His other research interests include enactivism as a theory of learning and a methodology, the use of technology in mathematics teaching, and comparative perspectives on mathematics pedagogy.
Estela A. Vallejo Vargas has been working on research about proof and proving for about six years. Her current research interests are developing Proof-based teaching sequences to engage school students in reasoning-and-proving and identifying what factors might support teachers and teachers’ trainers with the implementation of Proof-based teaching classrooms.
Angelika Bikner-Ahsbahs is professor of mathematics education at the University of Bremen. One of her main research interests is design based research, specifically addressing the question of how students build new mathematical knolwedge and how conditions, such as tools, tasks and additional resource, can be shaped to foster these learning processes. In this respect, algebra learning plays a key role in school.
Rainer Malaka is Professor for Digital Media at the University of Bremen since 2006. The focus of Dr. Malaka’s work is Digital Media, Interaction, and Entertainment Computing. At the University of Bremen, he directs a Graduate College “Empowering Digital Media” funded by the Klaus Tschira Foundation. He is also director of the TZI, the center for computing technologies at the University of Bremen. Before joining the University of Bremen, he lead a research group at the European Media Lab in Heidelberg. He did his PhD at the University of Karlsruhe on modeling brain structures using neural networks. He is also steering board member of the Interdisciplinary College for multiple years, a multidisciplinary international spring school on cognitive science, neurosciences and AI. Rainer Malaka is the German representative in IFIPs technical committee TC14 for Entertainment Computing and he will act as chair of the TC starting January 2018.
Published Online: 2018-11-14
Published in Print: 2018-12-19
This work was funded by the German Federal Ministry of Education and Research (BMBF) in the grant program “Erfahrbares Lernen” (experienceable learning).