Across the curriculum, students learning mathematics confront dichotomies of discrete and continuous phenomena, of constancy and change. Yet the tools we give them for thinking about these opposing ideas rarely bridge the span between them. We show them a drawing on the blackboard; but this offers only a single example--a "case study"--of a mathematical idea. In it, one might see that some condition is true, but rarely how or why it came to be so, or when--perhaps--it might no longer obtain. We then deliver a symbolic expression that generalizes all possible related examples. But where in this fixed symbolism can one find the rich mathematical diversity it encodes?
Dynamic manipulation software bridges this gap. As students vary a parameter directly, they see--and more, they generate--a near-infinite number of continuously-related case examples. Their figure is no longer merely illustrative; through dynamic manipulation, it approaches the general case.
Example: The Orthocenter of a Triangle
Given the figure at left, a student might observe that the altitudes of a triangle concur in a point, and that this point, the orthocenter, is located inside the triangle. Other examples will demolish this conjecture, and show that sometimes the orthocenter must fall outside the triangle. But will these examples reveal when the orthocenter falls inside its triangle, and when outside? Or why?
Experimenting in a dynamic manipulation environment, a student observes (by dragging) that each of the three vertices contribute equally to the location of the orthocenter. Dragging one vertex, she finds it possible to "push" the orthocenter outside the triangle, and that when it leaves the triangle, the orthocenter always exits through a vertex. Investigating each vertex at the moment the orthocenter passes through it, she realizes that each exit or re-entry of the orthocenter occurs as the vertex angle passes through 90 degrees: once the angle is greater, the orthocenter must fall outside the triangle. Thus, only acute triangles have interior orthocenters; and obtuse triangles must have exterior orthocenters. Moreover, the case in which the orthocenter coincides with a vertex--the right triangle--emerges not as some third and separate mythical entity of the geometry curriculum, but instead as the natural "border" between obtuse and acute, where opposing tensions are held in equilibrium.
Proposed Standard: Starting in about grade 6, students should experience problems and situations in which continuity between one state and another allows them to reason about intermediate states. Since dynamic manipulation software helps students to create and work with such problems, students should have some of these experiences using such software at each grade level.
As you drag one object on the screen, the objects that are linked to it change as well. Sometimes you think of these linkages as dependencies: "The size of this residual depends on the location of this point." Sometimes you see causality: "Increasing the exponent causes the curve to go up more sharply." Or you describe an implication: "As this vertex angle becomes 90 degrees, the side opposite has to get closer to being a diameter of the circumcircle." These insights characterize the heart of mathematics as the study of relationships; and dynamic manipulation provides learners with tools for experiencing and investigating such relationships.
Example: Least Squares Regression
An early prototype of Fathom, a computer learning environment for data analysis and statistics, illustrates how dynamic manipulation can reveal the workings of the algorithm for computing a least squares regression line.
Each small square is constructed from a residual, the difference between the value predicted by the fitted line and the actual data value. The large square's area is the sum of the area of the small squares. As you drag the line you see the squares change size and you can adjust the line for a minimum sum of squares. We are convinced, even without a controlled experiment, that playing with this model demystifies how this algorithm works, suggests other algorithms for fitting a line, and provides insight into how an outlier can have a great deal of influence over the slope of the fitted line. These discoveries contextualize mathematical knowledge, helping us understand how an analysis works, and when and why we might wish to apply it.
Proposed Standard: Given a dynamic mathematical model, students should be able to discover and describe in mathematical language the relationships that exist between the model's parts.