EXCELerating Mathematics
Valerie Watts* (watts@math.arizona.edu)
Pallavi Jayawant (jayawant@math.arizona.edu)
Maria Robinson (mkr@math.arizona.edu)
Technology, and in particular simulations, can be used as an
effective tool to demonstrate abstract mathematical concepts
that students find difficult to understand. Spreadsheets can
handle large simulations easily and efficiently. We will use
simulations in Excel to illustrate the following three concepts:
the convergence of a power series at a point, the probability
distribution of a large sample, and the expected value of a
random variable.
In calculus we are interested in the interval of convergence of
a power series, but students do not always understand what this
means. In practice students use the ratio test to determine the
interval of convergence. However students have trouble making
the connection between the result from the ratio test and what
that actually means. We have designed an Excel worksheet where
the partial sums of a power series for a particular function are
computed. To demonstrate the idea of the interval of convergence
we can input different values of x and see if the sequence of
partial sums (at x) converges. This shows the students that for
different values of x we have a different series. So, for every
real number the convergence of the series needs to be analyzed.
In statistics students are taught that the probability distribution
of a large sample will approximate the theoretical probability
distribution. This concept is time-consuming to demonstrate through
a manual simulation in class. In our Excel worksheet we simulate a
large sample of a known probability distribution and then compare
the sample probability distribution to the theoretical distribution.
Of course, we cannot do this type of demonstration for every
experiment since the theoretical distribution is not always known.
However, the Excel worksheet validates the law of large numbers.
Another concept in statistics that students find difficult is the
expected value of a random variable. Students have trouble
interpreting the expected value as a long-term average. They
often think that the expected value is one of the values of the
random variable. In our Excel worksheet we simulate a large
number of payoffs from a slot machine. We take the average of
these payoffs and compare it with the theoretical expected value.
With a large simulation these two numbers are similar. Therefore,
the students can see that the expected value is the average over a
large number of trials and is different from the values of the
random variable.
Technology, when appropriately used, can enhance the learning
environment in the classroom. We have used these simulations,
and others, in the classroom, and we have found them to be useful
in explaining these mathematical concepts.