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 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 |

**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.

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