Learning by Doing:
       The Role of Active Learning in the Reform Movement

                       Elizabeth Mayfield

                          Julie Ayres
                          Aba Blankson
                         Elizabeth Kyle

             Dept. of Mathematics and Computer Science
                          Hood College
                   Frederick, Maryland  21701
                    mayfield@nimue.hood.edu

Abstract:

One of the hallmarks of reform mathematics classes is that
students are actively involved in the learning process.  They
perform experiments, they collect data, they work on projects
with other students, they conduct and write up labs.  This panel,
consisting of one faculty member and three students, will discuss
some of the ways in which students have become active
participants in their mathematics classes here at Hood College.

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In teaching mathematics, I have rarely stood at the front of the
room and lectured, while my students sat passively in their seats
and took notes -- and I'll bet you haven't either.  For most of
us, teaching is a kind of dialogue: we write something on the
board and ask the students what they make of it, they respond, we
continue, they interrupt with questions... We are engaged in a
discussion with our students, and I would have said that they
were active participants in the classroom, and actively involved
in the learning process.

But my notion of what constitutes "active learning" was cranked
up a notch when I attended a Project CALC workshop several
summers ago at Duke University.  And that's what I'd like to talk
about today.  Here are some of the things we do in mathematics
classes at Hood now that I think of as "learning by doing":

Classroom demonstrations and experiments

     We bought some Vernier hardware and software which allows us
     to use a motion detector to track a student's distance and
     velocity as she walks across the room, and graph them on a
     computer screen in real time, so we can look at the graphs
     and talk about possible relationships between them.  We can
     take a temperature probe and plunge it into a hot cup of
     coffee, or a glass of ice water, and look at a graph of
     temperature over time, then talk about its shape, and try to
     fit a curve to it.

     My students still try to find that open-top box with maximum
     volume by cutting the corners from a piece of cardboard, but
     we actually make those boxes -- I hand out pieces of paper,
     and students cut out squares of different sizes from the
     corners, and tape the sides up to make boxes, so that the
     classroom is soon littered with boxes of different sizes.
     We talk about the different shapes of the boxes, and try to
     predict which will have the largest volume.  Then, of
     course, we can compute the volumes, pick the winner, and go
     on to talk about how to generalize the problem and solve it
     using graphs and derivatives of functions.

Computer labs

     We have a two-hour computer lab scheduled every week in each
     semester of the calculus sequence.  The students investigate
     all sorts of problems -- population growth, the path of a
     mortar shell, the spread of an epidemic -- using real, messy
     data.  They work in pairs, and a lot of arguing and
     discussion goes on in the labs as the students discover
     results and write them up in lab reports.

Group projects

     Often in class, we will stop and break up into small groups
     and work problems or investigate a new concept, with the
     students "in charge" instead of the instructor.  And several
     times during each semester, students do large-scale projects
     in groups of three or four.  They may do research in the
     library, perform experiments, and use the computer.  Some
     examples are: learning about radiocarbon dating; using a
     topographical map of Yosemite National Park to find the path
     of steepest ascent from one point to another; discovering
     the similarities among electrical circuits, cooling bodies,
     and falling raindrops.

Reading and talking (and writing) about mathematics

     I think the hardest part, for me, of teaching mathematics in
     this "new way" has been to encourage my students to read the
     textbook before coming to class -- to allow them to steal my
     thunder, to anticipate results before I have revealed them
     in class.  Students are taught to "read actively" -- to keep
     a pencil with them when they read, to make notes in the
     margin, to stop and work problems along the way to make sure
     they understand, to read and re-read.  And because students
     always work in lab with a partner, and work so often in
     groups, they must talk about mathematics.  They are not
     allowed to be passive learners in a class like this.


Student presentations

Three of my students are here with me today to tell you about
some of their favorite activities in math classes.

Aba Blankson has just completed her freshman year at Hood, and
took Calculus I and II this year.  She is going to talk about a
couple of class experiments we did on the very first day of class
in Calculus I, two experiments straight out of the Project CALC
instructor's guide.

Superball experiment:  students bounce a ball, measuring its
initial height and bounce height.  We collect lots of data and
make a scatterplot on the classroom computer.  As a class, we
look for patterns and fit a straight line to the data.

Pendulum experiment:  students make a pendulum out of a
doorknob and a piece of string.  For different lengths of string,
they time how long it takes for the pendulum to swing through ten
full swings.  Then, again, we collect the data and make a
scatterplot of period vs. length of string.  This time the
relationship is not linear; we fit a more complicated model to
the data.

Julie Ayres just graduated from Hood as a mathematics major.  She
will talk about two group projects she participated in when she
took calculus.

Centers of mass project (Calculus II):  Students discovered
properties of centers of mass in one and two dimensions.  For the
one-dimensional part, students balanced a wooden dowel on a
bracket and then balanced metal washers on each side (3 at a
distance of 2 on one side, and 2 at a distance of 3 on the other
side, for instance).  For the two-dimensional part, they took
cardboard cutouts of planar regions (like the region   between
the curve y=3Dx2 and the x-axis, for x between 0 and 1) and
balanced each one on a pencil eraser.  They wrote down the
coordinates of this balance point and then worked on some
discovery exercises, drawing regions and filling them with
rectangles and summing the centers of mass of each rectangle, to
come up with a theoretical formula for the center of mass which
they could compare with their collected data.

Spirograph project (Calculus III).  Students drew pictures with
a Spirograph and then found parametric equations to represent the
curves they had drawn.  To check their work, they used the
computer to graph the parametric curves.

Izzy Kyle has just completed her junior year at Hood; perhaps you
recognize her from her photo in the last issue of Math Horizons.
This past spring, she was a student in my differential equations
class, which I taught using the new textbook by Blanchard,
Devaney, and Hall.  This is a textbook which will appeal
especially to students who have taken calculus from a reform
text, and who are accustomed to doing projects and using a
computer, and who expect to solve problems in lots of different
ways -- students who have been NAGged.  She is going to talk
about two projects from that course.

Rate of memorization project.  In groups of two, students timed
how long it took them to memorize a list of numbers.  They
collected data and then fit a curve to the points, using a
learning curve model we had discussed in class.

Moldy bread project.  Each student took a piece of bread and
grew mold on it.  Using a transparent grid, she counted how many
squares were covered each day until the entire piece of bread was
covered.  She then fit a mathematical curve to the data,
comparing exponential and logistic models to predict the growth.



Conclusions

Using "active learning," letting students discover things on
their own, is certainly a lot more work than just telling them
everything myself.  But I have to believe that it's better for
them in lots of ways. 

It keeps them awake in class!  It involves them in ways that
they may not have been involved in a mathematics classroom
before. 

It places the responsibility for learning squarely on their
shoulders.  They must read the textbook and be prepared to ask
questions if they don't understand; they must come to class and
to lab, ready to participate; they must think about how to design
experiments and write them up.

If we believe the literature on the subject, we have to believe
that students learn better, and remember things longer, when they
are actively involved in the process.

And, most important of all, this way of teaching, and of
learning, is so much fun!  I hope we have given you a glimpse of
how exciting this kind of learning is.  Once you have tried it, I
don't think you'll ever want to go back.