Multi-Modal Mathematical Investigations with 4^th GRADERS

Overview

      We are investigating the impact of integrating a popular dynamic geometry environment with haptic devices, which allow users to not only see and manipulate geometric figures on a screen but also feel, through force-feedback, and touch via direct contact with a screen, the result of interacting with such objects. We report preliminary findings of how young children can yield expressive language that is intimately connected to the mathematical concepts being presented. This is linked to the NCTM's priority area of instructional interventions.

Educational Significance & Conceptual Perspective

Haptic literally means "ability to touch" or "ability to lay hold of" (Revesz, 1950) and has evolved in a technological era to be an interface for users to virtually touch, push, or manipulate objects created and/or displayed in a digital visual environment. Haptic devices can provide force-feedback or tactile feedback.

The activities we have created allow users to interact and explore geometric figures and shapes and their properties. We exploit the technological affordances of force-feedback devices to touch and feel attributes of shapes as well as allow young students to lay their hand or fingers on the iPad to directly manipulate complex mathematical constructions in simple and successively iterative ways. Our study is assessing and evaluating what new or enhanced learning experiences can be created by the synergistic integration of dynamic geometry with new haptic hardware as a multi-modal environment. Multi-modality reaches into education in various ways, intersecting deeply with a multi-media approach. Multi-modal approaches have also focused on the role of gesture and mathematical expressivity.

The dragging of well-constructed objects to establish whether the mathematical constructs that underlie its engineering can be preserved upon manipulation offers another dynamic perspective on geometric diagrams and is referred to as a "drag test." Such embodied actions of pointing, clicking, grabbing and dragging parts of the geometric construction allows a semiotic mediation (Falcade et al, 2007) between the object and the user who is trying to make sense, or induce some particular attribute of the diagram or prove some theorem.  We explore the additional conceptual benefits of adding direct touch and feedback to these dynamic figures and explore the meditational effects through discourse.

Research Question & Design

We ask: What are the perceived benefits of haptic technology in addition to visual technologies and how can we investigate and evaluate these benefits? In order to investigate and evaluate such benefits we analyzed the discourse of students working on activities within a multi-modal (visual and haptic) environment. We investigated two environments: 1. a development version of The Geometer's Sketchpad¨ on the iPad tablet and 2. prototypes with a PHANTOM Omni^ñ haptic device. It is our hypothesis that activities within these multi-modal environments mediate multiple meaning-making tracks, as well as elicit discourse regarding conjectures and refutations in early learners' perceptions of complex geometric shapes.

These activities are not designed to instruct students, and instead they afford students the opportunity to utilize and enrich their prior experiences, scholastic and non-scholastic, while making sense of a mathematical task within a visual-haptic environment. The iPad activities focused on 2D investigations of geometric transformations, measurement and estimation. During an activity students worked in groups, and also engaged in whole class discussions. In one activity, called "Etchasketch", one student controls the lateral-moving "Point 1" and another student controls the vertical-moving "Point 2" in concert, in an effort to produce an output, a colored blob that traces a circle.

The PHANTOM Omni^ñ activities involved student investigations of 3D shapes, such as cubes, spheres, and pyramids. Students worked in groups of four, and engaged in discussions with their peers and the interviewers. In one activity, students were asked what types of shapes are formed when two objects intersect. A visual bug locked onto the intersection for students to guide as they received force-feedback when they traversed the shape.

Data Collection & Analysis

      The data in this presentation is part of a larger project that collected data from 150 fourth-grade students from an elementary school in the Southcoast region of Massachusetts in 2011, while they worked on activities in both of our multi-modal environments. The research team conducted semi-structured interviews guided by the design intentions of the activity space. These were recorded and transcribed for discourse analysis purposes. We identified episodes where students gave indication that the multi-modality of an activity's environment gave them some insight into the mathematical properties of the task at hand, i.e. the visual and physical supported or contradicted their investigation.

Summary of Findings

      Our initial analysis of student discourse with our activities has shown that multi-modal, visual-haptic environments produce experiences where students experience a conflict between what they see (visuals) and what they feel (haptics), leading to rich descriptions of early learners' perceptions of complex geometric shapes. In the "Etchasketch" activity, as students explored the effects the lateral- and horizontal-moving points had on their trace, they began to realize the need for coordinating the movements of the points in order to create one output; the trace of the circle. Mathematically, this finding is significant because it shows fourth-grade students enacting the implicit function of a circle, which is a concept not discussed until later grades.

      The "Intersections of a Cube" activity provides another exemplar of cross-modality conflict. In the episode, one student situates the plane such that the resulting intersection is a trapezoid, whereas the visual makes the whole group think it is a pentagon. Tracing the sides enables them to feel only 4 sides, which results in a rich discussion, explicit drawings and debate on how such a shape could exist. One student sums up the cross-modality conflict by stating, "you see a pentagon, but you feel a trapezoid."

      During our presentation we will engage the audience in an interactive discussion about these findings, as well as additional visual-haptic activities.

References

Revesz, G. (1950). The psychology and art of the blind. London: Longmans Green.

Falcade, R., Laborde, C., & Mariotti, M. A. (2007). Approaching functions: Cabri tools as instruments of semiotic mediation. Educational Studies in Mathematics, 66, 317–333.