Technology in the Mathematics Classroom: Helping Students Make
Connections
Terry Wyberg, Department of Curriculum & Instruction, University
of Minnesota
The Curriculum and Evaluation Standards for School Mathematics
(NCTM, 1989) recommend that high school students should be able
to do the following: "represent and analyze relationships using
tables, verbal rules, equations, and graphs; translate among tabular,
symbolic, and graphical representations of functions; recognize
that a variety of problem situations can be modeled by the same
type of function; and analyze the effects of parameter changes on
the graphs of functions" (p. 154). The teaching of functions by
emphasizing the tabular, symbolic and graphical representations
and the connections between them became known as "The Rule of Three."
Functions can also be represented by real-world situations themselves
so "The Rule of Three" later was called by some as "The Rule of
Four". These representations should not be learned in isolation
and that true learning of the concept of function occurs when a
person can easily make connections between the various representations
and see how changes in one representation effects the other three.
One of the most widely used tools in the mathematics classroom
is the graphing calculator. Most graphing calculators currently
available have the capability to represent mathematical situations
graphically, symbolically, and as a table of values. The real power
of the graphing calculator is that the user can enter one representation
and easily switch to the other two. An example of an activity that
emphasizes the translations between representations is given below.
This activity is also used to demonstrate how communication in the
classroom can be enhanced when students have access to graphing
calculators. Conversations during this activity should be focused
on making sense of the translations between representations and
it will be the students that will be doing the explaining of their
reasoning rather than the teacher explaining some rules to be followed.
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The Problem (Real-World Representation)
According to standardized growth
and development charts used by many American pediatricians,
the median weight for girls during their first six months
of life increases at an almost constant rate. Starting
at 7.0 lbs. at birth, for each additional month of life
the female median weight increases by 1.5 lbs. (Kime,
1997)
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The problem given above is a real-world representation of a linear
function. Students are next asked to generate a table of values
that show the relationship between age and median weight for the
first six months of age and enter this information in the graphing
calculator. Figure A is the display of a Texas Instruments TI-83
graphing calculator after a student has entered the data. The L1
column is the ages in months and the L2 column is the median weight
for girls in pounds.
Note that students studying linear functions can and should be
able to do this first part of this activity without the use of a
graphing calculator. The next part of the activity is where the
use of a graphing calculator is essential.
Students are asked to enter a function (symbolic representation)
of the form y = mx + b, for example y = 5x + 2, into their calculator
as shown in Figure C. The corresponding graphical representation
is shown in Figure D. The goal for the students is to change the
values of the m and b so that the line passes through all the points
on the scatter-plot. The focus of this part of the activity is on
the connection between symbolic and graphical representations and
on the effects of parameter changes on the graphical representation.
Students usually find the equation that fits, y = 1.5x + 7, in a
short amount of time. The discussion that takes place during this
part of the activity usually connects the symbolic representation
y = 1.5x + 7 and the real-world situation. Students notice that
the 1.5 is the same as the 1.5 lbs. increase per month and this
creates an opportunity to discuss slope and average rate of change.
The 7 corresponds to the initial weight of 7 lbs. and there becomes
and opportunity to discuss the y-intercept. One of the purposes
of this article is to emphasize how the communication between the
students and the teacher is changing in a classroom. First, the
content of the communication is on making connections between representations
rather than just on the representations themselves. Second, it is
the students who are explaining rather than the teacher. The teacher's
role is that of a facilitator of knowledge rather than an information
dispenser.
References
Kime, L. A., Clark, J (1997). Explorations
in College Algebra. John Wiley & Sons.
National Council of Teachers of Mathematics
(NCTM) (1989). Curriculum and Evaluation Standards for School
Mathematics. Reston, Va.: NCTM.
National Council of Teachers of Mathematics
(NCTM) (1998). Principles and Standards for School Mathematics:
Discussion Draft. Reston, Va.: NCTM.
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