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 of an inch, or one inch.