The Role of Gestures in Mathematical Discourse: Remembering

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The Role of Gestures in Mathematical Discourse
Remembering Problem Solving

• Laurie D. Edwards
• St. Marys College of California
• ledwards_at_stmarys-ca.edu

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Theoretical context

• Cognition is embodied, at multiple levels
• In the moment (microgenetically), through gaze,
  gesture, speech, imagery
• Developmentally (ontogenetically), through
  individual experiences, for example, with our own
  bodies, physical objects, electronic artifacts,
  inscriptions
• Biologically (phylogenetically) through
  constraints and capabilities developed through
  evolutionary time.
• The only mathematics we have access to is the
  mathematics we are physiologically capable
• of understanding (Lakoff Núñez)

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Gesture and multimodality
Pervasive Multimodality (Nathaniel Smith)
Gesture is integrated with speech, writing and
drawing in the construction and communication of
mathematical ideas. General research goal To
investigate the form and functions of gestures in
the context of talking about and doing
mathematics.
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McNeills Typology/Dimensions

• Iconic gestures close formal relationship to
  the semantic content of speech
• Metaphoric gestures the pictorial content
  presents an abstract idea
• Beat indexes the word or phrase ... as being
  significant
• Deictic pointing movement that selects a part
  of the gesture space

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Adequacy of McNeills Framework

• McNeills typology was produced using data
  gathered in a specific context subjects
  producing a narrative describing a cartoon they
  have just seen.
• Additional research goal To determine whether
  McNeills categories/dimensions are adequate for
  describing gesture produced in mathematical
  situations.

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Methodology

• 14 women, college sophomores, enrolled in math
  content course for prospective elementary school
  teachers
• Interviewed in pairs about fractions
• (note not a learning situation, not involving
  motion or graphing)
• Solved five problems involving operations with
  fractions, with questions fractions before and
  after the problem-solving portion
• Interviews videotaped, transcribed, gestures coded

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Interview Questions

• How were you first introduced to the idea of
  fractions?
• Do you remember anything difficult about learning
  fractions?
• Have you ever used fractions in everyday life, or
  in other classes?
• How would you introduce fractions to children?
• How would you define a fraction to children?

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Fraction Problems

• Which is larger 3/4 or 4/5?
• 2/3 1/2
• 1 1/4 3/8
• 7/10 x 5/9
• 3/5 1/10

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Data

• 6 half-hour interviews, 3 hours total of
  videotape
• A total of 86 gestures
• Ranging from 3 - 21 gestures per student in
  half-hour sessions (mean7.7)
• During problem solving portion, all gestures
  except one were deictic, pointing at written
  algorithm. These deictics were not included in
  the current analysis.

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Gestures when remembering

• Four salient kinds of gestures when asked to
  describe how they learned about fractions
• Iconic gestures referring to physical
  manipulatives or actions
• Iconic gestures referring to inscribed
  representations of physical manipulatives
• Iconic gestures referring to specific written
  algorithms
• Metaphoric gestures referring to generic or
  abstract mathematical actions or objects
  (formulas, comparisons, etc.)

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Iconic gestures referring to physical
manipulatives or actions
KP I think we did, like, just a stick or a
rod (went on to talk about dividing it again
and again, using a repeated chopping motion)
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SS one portion colored in
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Iconic gestures referring to inscribed
representations of physical manipulatives
JB I know we did the pie chart
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Iconic gestures referring to specific written
algorithms (iconic-symbolic)

• Algorithms in the air
• Gestures that index a remembered image of a
  written mathematical algorithm or procedure
• The motions and pointing refer to a remembered
  physical inscription (kinesthetically and via
  visual imagery)
• Yet the inscription refers to an abstract
  mathematical procedure (chain of signification)

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R Remember anything about multiplying
and dividing with fractions? KG Isnt
that GG Dont you like cross them? KG Yeah you
cross like GG The bottom. KG The bottom and
the top of the opposite.
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MB I remember learning that you put one under
the other, the fractions and adding and
subtracting and then finding the least common
denominator, or multiple, and then put it under
there and then having to multiply it out to
figure it out, but mostly we would have to figure
it out and then just adding it together.
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Metaphoric gestures referring to generic or
abstract mathematical actions or objects
SS like the different formulas
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KG And according to their statistics and
depending on like whether or not your team was
going to win
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KG If it was more than what the bottom
was then it would become, like, one and...
(one other example of more to the right, and
one example of less to the left)
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Gestures in communicationg about problem solving

• Only one student used a gesture sequence in
  explaining her thinking when she solved one of
  the fraction problems
• Problem was
• Which is larger, 3/4 or 4/5?

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AT Well, I mean its like Im thinking if I had
a pie and I had 5 people versus 4 people then,
R Ah. you know, were each kinda getting less
of a piece R Ah. because theres a fifth piece
we have to like, put out to the other four
people.
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• Except for initial emblem (Im thinking),
  ATs gestures highlighted important aspects of
  the task and her reasoning about it.

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Highlights relative size of denominators (fourth
vs fifth)

• were each kinda getting less of a piece

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Highlights number of pieces, i.e., the numerators

• because theres a fifth piece

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Highlights sharing operation

• we have to like, put out to the other four
  people.

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Discussion/Conclusion

• Gestures are not epiphenomenal to thinking, but
  rather an integral part of embodied
  understanding.
• Gestures may refer to remembered physical
  objects, actions with these objects, written
  inscriptions, as well as more abstract
  mathematical relationships, procedures and
  objects.

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Manipulatives

• The physical objects we call mathematical
  manipulatives do not embody or represent
  mathematical ideas.
• Instead, students actions with these objects,
  linked to speech/labels provided by the teacher,
  are the source of construction of mathematical
  ideas (through the creation of semiotic nodes).
• Gesture can re-evoke these actions and assist in
  the reconstruction of these ideas

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Remembering Algorithms

• Students kinesthetic actions and visual memory
  of written symbolic algorithms are an essential
  part of their knowledge of these algorithms or
  procedures.
• The way students describe functions shows deep
  traces of their actions and interactions with
  instruments and representations. Such traces are
  not complementary to the concept but are an
  essential component of its meaning.
• Robutti, O. (2003) Approaching Algebra through
• Motion Experiences PME 27 Proceedings, p. 1-113

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A New Framework for Gestures in Mathematics?

• In everyday discourse, concrete objects do not
  typically refer to anything beyond themselves
• By contrast, in mathematics, concrete objects
  have been designed to refer to mathematical
  ideas
• And symbols compress many layers of previous
  sign/symbol construction (chains of signification)

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• Furthermore, outside of mathematics, written
  symbols are not usually manipulated as if they
  were objects
• Thus, there is a need for a more sensitive
  framework for describing the form and function of
  gesture in mathematical contexts.
• Initially, created new category
• Iconic-Symbolic

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Other referents in fraction data
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• We need to develop a framework that attends (at a
  minimum) to
• Form
• Referent
• Speech genre (narrative, explanation,
• instruction, proof)
• Setting (classroom, interview,
• group homework session)
• Involvement of artifacts

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• And, most importantly, a framework that can help
  us understand the
• Function
• of gesture within mathematical discourse,
  learning and teaching
• Gestures may be
• rhetorical/conversational/pragmatic
• assist in developing the mathematical content
• assist at a meta-content level (structuring an
  
• argument, guiding reasoning, etc.)