Measurement Units and Logical Relations in a Dynamic Geometry Environment

1
Measurement Units and Logical Relations in a
Dynamic Geometry Environment

• Jeffrey E. Barrett
• Illinois State University
• Jbarrett_at_ilstu.edu
• February 8, 2003, Chicago, IL

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Activities I have worked on for measure and
geometry

• Distance from point to a line precision, GRADE 3
• Triangle Perimeter units, and segment as
  shortest path, GR 4, 7
• Playground defining objects, GR 3
• Letter World tracing and segment definition, GR 3

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Grade 3/4 kids on sketchpad

• They did not naturally move to using the cursor
  tool, it had to be dragged out of them! It
  helped to say the cursor tool was a way of
  talking to Nick to show him what they wanted to
  work on next.
• They tended to want to create multiple cases of
  the triangles with perimeter 8 cm by drawing more
  segments.This helped point out the dynamic nature
  of the triangles, and the case-based
  argumentation.

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Theories of Learning in Dynamic Environment

• Van Hiele roughly in a hierarchy of abstraction,
  effected by verbal and active interactions with
  object
• Arzarello et.al. (1998) various modalities of
  dragging wandering, testing, lieu muet
  (isolating an invariant amidst variation).
• Yu Barrett (2002) various prototypes via
  concrete carriers as images, related
  contemporaneously, not hierarchically

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Trajectory for young learners on measurement
(grades 2/3)

• Younger kids will need to associate sliding a
  unit object along a longer object
• Hopping motion associates movement with a record
  of the movement (rabbit)
• A coordinated trace of front and back of a unit
  object shows potential for coordinating the
  leading and trailing ends of shifting unit

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Precision Distance Sketch, Day 1

• How far is it from a point in the plane to a line
  (or segment).
• Kids had not thought of this question before.
  They approached it intuitively, but had little
  language for the angle they needed to describe.
• The precision at units of cms was not an issue
  to them, even though it changed little.
• Needed a story to situate the activity
  BasketBall free throw line extended and the
  distances to the basket
• They use the straight path, or find the
  shortest!

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Distance and a Fair Circle Game, Day 2

• On the second day, we began by asking about a
  childrens game Mother May I, and how to make
  the game fair for several children (based on
  previous days work)
• Kids predicted a fair arrangement (equidistant
  spacing of players)
• Measured several locations and recorded
• Used rulers to check they were wrong scale!
• Sketched and measured their own ideas
• Lined up front to back, Side to side, In an arc,
  In a circle and then measured to check their
  ideas.
• The circle shape emerged from this experiment

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Precision for Triangle Perimeter

• Grade 2, finding triangles with P20
• Able to construct and measure with three
  reporters, teacher wanted the rubberband
  geoboards!
• Grade 4, finding triangles with P8
• Broke triangle inequality with cm units, but not
  with hundredths of cm units
• See sketches msmath
• Grade 7, finding triangles, P24
• Precision whole units
• Suspected rounding problems (like calculators)

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Building Triangles with a specific perimeter

• Some issues their were many cases with
  hundredths of cm units, but it also avoided
  breaking the Tri Inequality Thm.
• When we begin with cm units (grade 7) the
  students noticed the inconsistency between a sum
  of 24 and parts like 1,12 and 12. But, this did
  not connect to spatial arguments.
• The children in grades 4, and grade 7 classes
  were inclined to say they wanted to use parts of
  units, and suspected rounding problems.
• They benefited from seeing the arcs of circles
  with side lengths that met at the base (boundary
  case)

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Raises questions

• One Have students ever been required to decide
  on units for sake of precision
• Example, comparing two bent paths with several
  different unit options (or a dynamic unit)
• Two Do kids even know that precision is a an
  aspect of measurement (Do pre-service teachers
  know this?) What most have is an idea of number
  rounding
• Three the measure reporters are not seen by the
  kids as immediate length values.

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Problems for unit precision in sketchpad

• Kids have not developed the meaning of zero
  units (like 0 centimeters).
• Currently sketchpad reports 0 centimeters as a
  possible length when we are at unit precision
• Currently, sketchpad Edit/Preferences menu
  describes options as unit, tenths, hundredths
  etc. This should be addressed as whole units,
  tenths of units, hundredths of units, etc.
• I suggest activity with a sketch like the
  swinging doors that allows the child to see
  boundary case.

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On the interpretation of a measure tool
environment

• Ideal teach kids to interpret the tools
  according to their abstractions for space and for
  number
• (since we cannot avoid using tools and depending
  on them without recourse to visual checks, say
  in nanometers, or in light years in astronomy)
• Rounding is related closely to measuring to the
  nearest unit, which should be encouraged.
  Otherwise, precision cannot be taught!
• Can kids actually see what their unit is (what is
  the unit of practice?) Say when they are using a
  ruler marked to 16 parts per inch, is the unit
  1/16 of an inch, or one inch.