Gerofsky, Susan. Seeing the graph vs. being the graph. (2011)
Summary:
The reading reports the findings from the first two years of
a multi-year research study that examined secondary students’ gestures when
asked to describe mathematical graphs. The research does not examine the use of
gestures and other embodied methods in effective teaching; rather, it seeks to
discern whether there is a language underlying gestures and their variations
when students are asked to describe mathematical graphs. The research appears
to be built on the premise that gestures can provide a window into the
conscious and unconscious mind (p. 245). The gestures were elicited by showing pictures
of graphs of functions. Participants later reported that they were not
consciously aware of the details and choices they made while gesturing the
given graphs. Overall, the participants’ gestures varied widely in terms of
“placement of the axes (especially the x-axis), symmetry, acceleration,
direction of movement, handedness, and large vs. small kinesthetic engagement”
(p. 249). Some students did not merely gesture the graph but embodied it, as if
they were “in” the graph (p. 251). Several factors were identified that could
account for the variations, including bodily, cognitive, and neurological
differences. The researcher linked these observations to two concerns: (a)
whether these gestures offer insights into how students notice patterns, and
(b) whether students could be helped to identify patterns more effectively (p.
254).
A few questions came to mind:
Stop 1:
Where and how do we distinguish between culturally learned ways of expressing oneself in social situations and gesturing mathematical ideas? For instance, the report states that “many
participants treated the x-axis as a representation of time” (p. 249). I
wondered whether this might have been influenced by how students in secondary
classes are exposed to historical timelines.
Stop 2:
I also wondered about the cultural, educational, and
socio-economic backgrounds of those involved in the study. Some cultures tend
to be more visually oriented in how they perceive and relate to the world, as
reflected in their languages and child-nurturing practices, while others tend
to be more analytical. Some families place more emphasis on self-expression and
active engagement in the learning process from an early age, while others may
emphasize memorization, and so on.
Stop 3:
I also wondered about the role of imagination in how
the mind (not reduced to brain) relates to the body, especially considering
that participants reported they were not consciously aware of the details and
choices of their gestures.
Thank you for sharing this summary of the very interesting research that Gerofsky has been conducting. I am a math and business education educator with five years of experience in grades 8 to 10, and the message of this article certainly resonated with me.
ReplyDeleteI observe students using gestures in their explanation of graphs, particularly in explanations of concepts like the slope of a line or a parabola’s path in a graph. The result regarding students embodying the graph is spot on in different instances in my class where a student has actually leaned or turned their arm in a certain way to “become” a graph they were talking about.
Your "stops" are excellent questions which I struggle with every day:
The identification of the x-axis with the time axis is a near-universal experience of mine. In business education classes that plot supply, demand, or revenue, the typical graph almost always has the x-axis marking time. I have begun to use examples that are not temporal in nature (e.g., price vs. quantity) to fight this instinct, but it is a hard habit to break.
The presence and role of background are so important. In the same classroom, I have children who will comfortably embody a graph with their entire body and others who feel embarrassed by the mere thought. This is not a personality difference; it is a difference in learning and expression norms. My role, of course, is to accept and encourage each one.
"The role of the unconscious and imagination is the most fascinating one." When I ask college students "show me the graph with your hands" before attempting to graph it on paper, they will often demonstrate an unconscious sense of direction, curvature, and rate of change that their conscious, calculation-oriented minds have not yet put into words. This experience has led me to include more body-oriented warm-up exercises in my teaching.
Gerofsky’s work confirms the intuitions of many educators about mathematics, namely, mathematics is much more than simply an intellectual exercise, unconnected with the body. For many students, their body is the primary means of making mathematics meaningful.
Thank you for the thoughtful summary and questions.