Manipulatives, Limit Objects, and Mathematics Learning

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Manipulatives, Limit Objects, and Mathematics
Learning

• Ricardo Nemirovsky

Work supported in part by the Math in Motion
grant from the National Science Foundation
(REC-0087573).
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Mathematical Abstractions

• If mathematics education aims at familiarizing
  students with abstractions. What does this mean?
• Abstractions as entities that cannot be directly
  touched, seen, heard, etc.
• Why would bodily activity be relevant to learning
  about mathematical abstractions?

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Abstractions and Tool Use (1)

• How and why would the use of tools which engage
  eyes and hands in drawing, writing, manipulating,
  or touching would be relevant to learning about
  abstractions?

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Abstractions and Tool Use (2)

• The use of tools in mathematics learning is often
  seen as a process of internalizing them
  developing mental versions of the tool.
• From this perspective abstractions are mental and
  tool-uses are physical. Tool use (in the
  physical domain) stimulates abstractions (in the
  mental domain)

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Abstractions and Tool Use (3)

• In this presentation we want to advance an
  alternative view.
• Instead of splitting mental vs. physical we want
  to describe doing and thinking as woven in body
  activity.
• Body activity in its infinite forms eye motion,
  drawing, writing, grasping, gesturing, talking,
  and so forth.

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Abstractions and Tool Use (4)

• Body activity can be largely covert or imaginary.
  We might imagine ourselves drawing, for
  instance, without actually drawing. But this act
  of imagination has much in common with actual
  drawing.
• Examples from neuroscience research (e.g. imagine
  seeing something far away)

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Abstractions and Tool Use (5)

• The episodes selected for this presentation
  include one from a high school classroom where
  the students are using the drawing machine to
  draw a circle, as well as several episodes from
  interviews with Greta, a high school student

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Our Focus Fluency with Mathematical Objects
that Can Be Approached

• Lets call those mathematical objects that can be
  approximated in space Limit Objects.
• Limit Object is a kind of abstraction.
• Examples circle, point, number, line, and so
  forth.

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Our Focus Fluency with Limit Objects

• x2 y2 r2

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Our Focus Fluency with Limit Objects
Limit objects cannot be seen, touched, etc.
They are unperceivable as such
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Working with Limit Objects

• Drawing closer to the limit object
• Reflective Turns

Noticing their own ways of acting with the
drawing machine
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Working with Limit Objects
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Limit Objects and Their Footprints

• Making visible its footprints
• Making present its absence
• Tangible emptiness

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Limit Objects and Their Footprints

• Limit objects cannot be seen, touched, etc.,
  unlike their footprints
• We conceive of limit objects by tracing,
  imagining, and talking about their footprints.

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Reflective Turns
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Abstractions and Tool Use (6)

• Instead of splitting mental vs. physical we want
  to describe doing and thinking as woven in body
  activity with tools and with others.
• How does the use of tools participate in the
  development of fluency with limit objects?
• We will explore these questions by examining
  interview episodes with Greta. A high school
  student interviewed by David Carraher.

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Greta The problem of projecting a circle
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Greta The problem of projecting a circle

• Greta predicting 1.42
• Marking Circle .28
• Tracing Circle .58
• Tracing with the monitor .36

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Greta The problem of projecting a circle

• Line of vision 4.29

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Greta The problem of projecting a circle

• Approaching limit objects

Reflective turns
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Greta The problem of projecting a circle

• Approaching limit objects

Reflective turns
Gretas line of vision and the mediation of
language and symbol use
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Reflective Turns (1)
Seeing Ourselves Seeing
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Reflective Turns (2)
Seeing Ourselves Seeing from passive
contemplation Involves positioning ourselves
and shifting perspectives
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Abstractions and Tool Use (7)

• Instead of splitting mental vs. physical we want
  to describe doing and thinking as woven in body
  activity with tools and with others.
• How does the use of tools participate in the
  development of fluency with limit objects?
• We will illustrate with Gretas episodes two
  critical roles of tool use (1) Developing
  sensitivities and (2) Readiness for action

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How does the use of tools participate in the
development of fluency with limit objects?

• 1. Developing Sensitivities
• Greta I have trouble, like, visualizing it two
  dimensional David two dimensional like, I can
  when I look through. I see it the rope on the
  floor I don't see it on the screen.

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How does the use of tools participate in the
development of fluency with limit objects?

• 2. Readiness for action
• Greta like if I personally if I just stand here
  without the camera I'd see the same thing as the
  camera, like I'd see it that way