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Instructional
programs from prekindergarten through grade 12 should enable all students
to—
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When students can see the connections across
different mathematical content areas, they develop a view of mathematics
as an integrated whole. As they build on their previous mathematical understandings
while learning new concepts, students become increasingly aware of the
connections among various mathematical topics. As students' knowledge
of mathematics, their ability to use a wide range of mathematical representations,
and their access to sophisticated technology and software increase, the
connections they make with other academic disciplines, especially the
sciences and social sciences, give them greater mathematical power.
Students in grades 9–12 should develop an increased capacity to link mathematical ideas and a deeper understanding of how more than one approach to the same problem can lead to equivalent results, even though the approaches might look quite different. (See, e.g., the "counting rectangles" problem in the "Problem Solving" section in this chapter.) Students can use insights gained in one context to prove or disprove conjectures generated in another, and by linking mathematical ideas, they can develop robust understandings of problems.
The following hypothetical example highlights the connections among what would appear to be very different representations of, and approaches to, a mathematical problem.
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The students in Mr. Robinson's tenth-grade mathematics class suspect
they are in for some interesting problem solving when he starts
class with this story: "I have a dilemma. As you may know, I have
a faithful dog and a yard shaped like a right triangle. When I go
away for short periods of time, I want Fido to guard the yard. Because
I don't want him to get loose, I want to put him on a leash and
secure the leash somewhere on the lot. I want to use the shortest
leash possible, but wherever I secure the leash, I need to make
sure the dog can reach every corner of the lot. Where should I secure
the leash?" |
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After Mr. Robinson responds to the usual array of questions and comments (such as "Do you really have a dog?" "Only a math teacher would have a triangle-shaped lot—or notice that the lot was » triangular!" "What type of dog is it?"), he asks the students to work in groups of three. All their usual tools, including compass, straightedge, calculator, and computer with geometry software, are available. They are to come up with a plan to solve the problem.
Jennifer dives into the problem right away, saying, "Let's make a sketch using the computer." With her group's agreement, she produces the sketch in figure 7.36.
Before moving
on to work with other groups, Mr. Robinson works with the members
of Jennifer's group on clarifying their ideas, using more-standard
mathematical language, and checking with one another for shared
understanding. Jennifer clarifies her idea, and the group decides
that it seems reasonable. They set a goal of finding the position
for D that results in the line segments DA, DB, and
DC all being the same length. When Mr. Robinson returns,
the group has concluded that point D has to be the midpoint
of the hypotenuse, otherwise, they say, it could not be equidistant
from B and C. (Mr. Robinson notes to himself
that the group's conclusion is not adequately justified, but he
decides not to intervene at this point; the work they will do later
in creating a proof will ensure that they examine this reasoning.)
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Small-group conversations continue until several groups have made observations and conjectures similar to those made in Jennifer's group. Mr. Robinson pulls the class back together to discuss the problem. When the students converge on a conjecture, he writes it on the board as follows:
He then asks the students to return to their groups and work toward providing either a proof or a counterexample. The groups continue to work on the problem, settling on proofs and selecting group members to present them on the overhead projector. As always, Mr. Robinson emphasizes the fact that there might be a number of different ways to prove the conjecture.
Remembering Mr. Robinson's
mantra about placing the coordinate system to "make things eeeasy,"
one group places the coordinates as shown in figure 7.37a, yielding
a common distance of |
Jennifer's group returns to her earlier comment
about the three points A, B, and C being on a
circle. After lengthy conversations with, and questions from, Mr.
Robinson, that group produces a second proof based on the properties
of inscribed angles (fig. 7.37b). » Pedro
presents his group's solution showing how they constructed a rectangle
that includes the three vertices of the right triangle (fig. 7.37c)
and reasoned about the properties of the diagonals of a rectangle.
Anna presents a solution using transformational geometry (figure 7.37d).
Since M and M' are the midpoints of  and  , respectively, the triangle MAM' is similar to
the triangle BAB', with each of the sides of the smaller triangle
half the length of the corresponding side of the larger triangle.
The same relationship holds for triangles BMC and BAB'. Using this
fact and the fact that BAB' is isosceles (since  reflects onto  ), Anna shows that triangle MAM' is congruent
to triangle CMB, from which it follows that CM and MA are the same
length.
Mr. Robinson congratulates the students on the quality of their work and on the variety of approaches they used. He points out that some basic mathematical ideas such as congruence were actually part of the mathematics in a number of their solutions and that some of their thinking, such as Alfonse's comment about the Pythagorean theorem, highlighted connections to other mathematical ideas. Taking a step backward to reflect, the students begin to see how different approaches—using coordinate geometry, Euclidean geometry, and transformational geometry—are all connected. Mr. Robinson notes that it is good to have all these ways of thinking in their mathematical "tool kit." Any one of them might be the key to solving the next problem they encounter. |
Although the students learned a great deal from working on the problem,
the class was not yet finished with it. Mr. Robinson had selected this
problem for the class to work on because it supports a number of interesting
explorations and because the students would be exploring the properties
of triangles and circles as they worked on it. And, indeed, as the students
worked on the problem, they remarked that they were "seeing circles everywhere."
(The following discussion is inspired by Goldenberg, Lewis, and O'Keefe
[1992].) »
| One
group decides to look at the set of all the right triangles they can
find, given a fixed hypotenuse. A group member starts by constructing
a right triangle with the given hypotenuse and then dragging the right
angle (fig. 7.38a). Another group decides to fix the position of the
right angle and look at the set of right triangles whose hypotenuses
are the given length (fig. 7.38b). They observe that the plot of the
midpoints of the hypotenuses of the right triangles appears to trace
out the arc of a circle. At first the students are ready to dismiss
the circular pattern as a coincidence. But Mr. Robinson, seeing the
potential for making a connection, asks questions such as, "Why do
you think you get that pattern?" and "Does the circle in your pattern
have anything to do with the circle in Jennifer's group's solution?"
As the groups begin to understand Mr. Robinson's questions, they begin
to see the connections among the circles in their new drawings, the
definition of a circle, and the fact that their problem deals with
points that are equally distant from a third point.
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| Mr. Robinson adds a final challenge for homework: can the students connect this problem (or problems related to it) to real-world situations or to other mathematics? The students create posters illustrating the mathematical connections they see. Most of the posters depict situations similar to the original problem in which something, for some reason, needs to be positioned the same distance from the vertices of a right triangle. One group, however, creates an experiment that they demonstrate for the class in one of the dark, windowless rooms in the building. They put on the floor a large sheet of white chart paper with a right triangle drawn on it, place candles (all of the same height) at each vertex, and stand an object shorter than the candles inside the triangle. The class watches the shadows of the object change as one of the group members moves it around inside the triangle. The three shadows are of equal length only when the object is placed at the midpoint of the hypotenuse—a phenomenon that delights both Mr. Robinson and his students. This activity concludes the discussion of right triangles, but it is far from the end of the class's work. Mr. Robinson reminds the students of the problem that started their discussion and asks them how the problem might be extended. "After all," he says, "not all backyards have right angles or are triangular in shape." This comment sets the stage for abstracting and generalizing some of their work—and for making more connections. » |
The story of Mr. Robinson's classroom indicates many of the ways in which teachers can help students seek and make use of mathematical connections. Problem selection is especially important because students are unlikely to learn to make connections unless they are working on problems or situations that have the potential for suggesting such linkages. Teachers need to take special initiatives to find such integrative problems when instructional materials focus largely on content areas and when curricular arrangements separate the study of content areas such as geometry, algebra, and statistics. Even when curricula offer problems that cut across traditional content boundaries, teachers will need to develop expertise in making mathematical connections and in helping students develop their own capacity for doing so.
One essential aspect of helping students make connections is establishing a classroom climate that encourages students to pursue mathematical ideas in addition to solving the problem at hand. Mr. Robinson started with a problem that allowed for multiple approaches and solutions. While the students worked the problem, they were encouraged to pursue various leads. Incorrect statements weren't simply judged wrong and dismissed; Mr. Robinson helped the students find the kernels of correct ideas in what they had said, and those ideas sometimes led to new solutions and connections. The students were encouraged to reflect on and compare their solutions as a means of making connections. When they had done just about everything they were able to do with the given problem, they were encouraged to generalize what they had done. Rich problems, a climate that supports mathematical thinking, and access to a wide variety of mathematical tools all contribute to students' ability to see mathematics as a connected whole.
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Alfonse, who is explaining this solution, proudly remarks that it
reminds him of the Pythagorean theorem. Mr. Robinson builds on that
observation, noting to the class that if the students drop a perpendicular
from M to AC, each of the two right triangles
that result has legs of length a and b; thus
the length of the hypotenuses, MC and MA, are
indeed
