Measurement Units and Logical
Relations in a Dynamic Geometry
Environment
Jeffrey E. Barrett
Illinois State University
Jbarrett@ilstu.edu
February 8, 2003, Chicago, IL
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
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.
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
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
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 cm’s 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!”
Distance and a Fair Circle Game,
Day 2
On the second day, we began by asking about a
children’s 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
Precision for Triangle Perimeter
Grade 2, finding triangles with P=20
Able to construct and measure with three reporters,
teacher: wanted the rubberband geoboards!
Grade 4, finding triangles with P=8
Broke triangle inequality with cm units, but not with
hundredths of cm units
See sketches: “msmath”
Grade 7, finding triangles, P=24
Precision: whole units
Suspected rounding problems (like calculators)
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)
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.
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.
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