Java MathWorlds Activities
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| Here are some photos and descriptions of our activites. |
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| Staggered Start, Staggered Finish: The simplest case is to produce families of functions defined by a single parameter,
such as the "b" in Y=2X+b, where b varies according to, say, group number. Then each student in a group produces the same function, a group's linear graphs all overlap
(same Y-intercept), and the different groups' graphs are parallel. When animated, the screen objects "representing" the members of a given group move alongside each other "as a group," and the different groups move at the same velocity but are offset by their initial positions . |
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| Staggered Start, Simultaneous Finish: A third basic example of student-indexed parametric variation points to other affordances. Here we ask students to "start at 3 times your count-off number and finish in a tie with Y=2X after 6 seconds" [Link 9]. The aggregated result is a
family of linear functions whose slope depends on starting point, and by deliberately including a group whose size exceeds 4, any student with count-off number larger than 4 starts on the opposite side of the finish line (at 12) and moves "backwards." We have a family of lines converging on (6, 12) whose
slopes and formulas vary systematically, and where a dramatic set of motions converges to the finish line, with some objects moving "backwards," and one (representing the student with Count-off number 4) that does not move at all since it starts at 3*4=12. Importantly, the family of formulas, lines and motions
contextualizes each student's case across representations, particularly the special cases. This is but an introductory example of a highly generative parametrically structured activity where (a) the given conditions of a problem situation vary parametrically, (b) the parameters are indexed by the students themselves,
(c) individual attention can be sharply focused on a particular solution ("Where are you?" or "Who's that?"), (d) the set of solutions contextualizes each in ways that preserve the identity of the individuals, (e) attention is systematically elevated to the family of solutions to reveal patterns, structures and
higher-order objects not apparent at the individual level and (f) generalization is built as a habit of mind from daily embedded experience of such. |
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