Project proposal

INFO 262, Fall 2024.

TUI in ed-tech: making mathematics tangible

Team Members: Julien Putz, Sebastian Pereyra, Frank Feng, Eric Tsai, Elyssa Tong

Figure 1. Brainstorming sketches from our first group meeting.

Traditional mathematics instruction often emphasizes symbolic procedures over embodied sensemaking. While the advent of touchscreen devices and dynamic geometry environments has increasingly allowed students to engage in dynamic manipulations of mathematical figures, the focus remains on visual registers. In this project, we wish to develop TUIs that bring multimodality and physicality into mathematical activities. We present four potential design ideas, all of which combine the affordances of digital technology and material objects.

Idea 1: Mobile Mathematics

Problem:

The transition from basic arithmetic to algebra is difficult for many young learners. According to Gunawardena Egodawatte (2009), a Doctorate in Mathematics and former professor with the Department of Curriculum, Teaching and Learning the University of Toronto, reasons children struggle are multifactorial, ranging from the psychological, to intuition, to syntactic/semantic and more. Algebraic thinking requires children to shift their reasoning from concrete concepts to the abstract and symbolic, and is important, not only for understanding higher levels of mathematics, but is essential for developing skills in logic and critical thinking (Peker and Acar, 2022).

Objective:

What if we could make the abstract concrete? Mobiles are an excellent example of balance and equilibrium and are often used as a representation of a system of equations. However, traditional mobiles and their components are fixed, and utilizing them for teaching remains in the digital realm. The goal of this project is to combine the innate quality of mobiles with physical computation and sensors for customized “equations”, allowing pre-algebra learners the opportunity for a hands-on, tactile experience of physically manipulating symbolic shapes and representational values to experiment and understand the basics of algebraic thinking.

Figure 2. Examples of mobiles (without digital augmentation). Images taken from: PBS, “ArtSEA: Going mobile with Alexander Calder at Seattle Art Museum”.

Idea 2: Linear Shadows

Shadow projections under parallel light provide a natural phenomenon that instantiates important concepts from linear algebra. In particular, the shadows from a movable Cartesian grid can produce linear transformations, including shear, rotation, and reflection (see Figure 2). By experiencing linear algebra concepts as natural phenomena, students may gain a deeper conceptual understanding in this area of mathematics, rather than focusing only on symbolic procedures. While simple shadow projections provide a concrete context for illustrating linear transformations, they do not immediately allow for precise quantitative reasoning. By augmenting the basic principles shown in Figure 2 with digital components — giving the receiving surface a “memory” or making the grid movements programmable — students could (1) precisely measure different linear transformations and their effects, (2) investigate the eigenvectors of a given transformation, (3) experience the effect of transformations on a variety of geometric objects, and (4) investigate/demonstrate key properties of linear transformations. The design could also foster collaborative sensemaking, for instance by dividing basic ways of moving the grid (rotating, shearing…) among different students; if the students wish to create more complex transformations out of these basic ones, they would need to carefully coordinate their movements.

Figure 3. Linear transformations as shadows of a movable grid.

Idea 3: Pinchable equations

Figure 4. Experiencing mathematical symbols as dynamic objects.

Core Concept

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Allow learners to interact with individual elements of an equation. By pinching, dragging, or tapping variables, constants, or operators, students can move them to different parts of the equation to see how the equation changes.

Features

• Visual feedback, such as the equation balancing itself automatically when terms are moved, so users see the mathematical implications (e.g., flipping signs when moving across the equals sign).
• A physical balance using some features to indicate different numbers and operators. Then detect the equation and show the results intuitively.

Digital prototypes: https://www.nctm.org/Classroom-Resources/Illuminations/Interactives/Pan-Balance----Numbers/

Idea 4: Tangible diagrams

Motivation:

Consider a circle with an inscribed square and a circumscribed square (see Figure 5, top). What is the ratio of the areas of the two squares? There are at least two ways of answering this question (see Figure 5, bottom). The first way relies on the observation that the circle diameter is common to both squares: it is the side of the larger square and the diameter of the smaller square. A brief calculation involving the Pythagorean theorem shows that the sought ratio is 1:2. The second way starts by rotating the inner square 45°. We now observe that the corners of the outer square can be folded exactly onto the inner square, showing that the outer square is twice as large as the inner square. Both solutions rely on distinct resources. The first relies on traditional pen-and-paper mathematics, while the second relies on physical manipulations.

Figure 5. A mathematical problem with two solution pathways. The first solution (bottom left) is static and relies on calculations. The second solution (bottom right) is dynamic and relies on physical manipulations (rotating, folding).

Design:

The design genre of TeleTangibles (Lambert et al, 2022) aims to pair manipulatives with digital twins to create inclusive learning activities. Drawing on this design genre, we seek to augment a particular digital geometry environment, Numworx, with material resources such as foldable paper, hinges, (elastic) strings, (electro-) magnets. The goal is to create a diagrammatic reasoning environment that merges the affordances of digital and material objects. Such an environment could readily accommodate both solutions to the problem in Figure 5. While the design ideas mentioned above focus on particular concepts and skills — balancing equations or enacting linear transformations — the goal of “tangible diagrams” is to create a more general-purpose reasoning tool.

References

Egodawatte , G. (2009). Is algebra really difficult for all students?. Is Algebra Difficult for All Students? https://files.eric.ed.gov/fulltext/EJ1052268.pdf

Lambert, S. G., Tancredi, S., Fiedler, B. L., Moore, E. B., Gorlewicz, J. L., Abrahamson, D., & Gomez Paloma, F. (2022). Getting a grip on geometry: Developing a tangible manipulative for inclusive quadrilateral learning. Italian Journal of Health Education, Sports and Inclusive Didactics, 6(1), 1–21. https://doi.org/10.32043/gsd.v6i1.604

Peker, B. & Acar, S. (2022). Algebra and algebraic thinking. In O. Tunaboylu & Ö. Akman, Current studies in social sciences 2022, (pp. 191–205). ISRES Publishing