Multi-Modal Mathematical Investigations with 4th 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
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 Omniñ 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 Omniñ 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.