Monday, May 2 is my last day of teaching. Not that I will never step into a classroom again,
but it marks the end of my full-time employment by Macalester College. My responsibilities
for the future will not require any teaching. I am looking forward to the freedom this phased
retirement will bring to focus on my writing and other educational activities, but I also
approach this date with some sense of loss. For forty years, I have defined myself as a
teacher first. I am taking advantage of this column to reflect on a few of the lessons I have
learned over these four decades in the classroom.
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| Class on Revolutions, taught at Penn State, Spring 1993, with John Harwood and Phil Jenkins (whose daughter Catherine is also in the picture). Taken on a field trip to the Joseph Priestley House, Northumberland, PA. |
Students absorb far less than we think they do. As teachers, we must identify what is
truly important and do all in our power to ensure that students are internalizing this
knowledge.
The first time this reality hit home for me was a differential equations class I taught at the
University of Wisconsin in 1980–81, before the days of computers in the classroom. As
taught back then, it was all about finding exact solutions to certain limited classes of
differential equations. I thought I did a pretty good job of explaining the motivation and
usefulness of these techniques, but homework and midterm exams were focused on finding
these exact solutions. For the final exam, I wanted to make it interesting by presenting
situations that my students would have to model using one of the types of differential
equations we had studied, and then they would solve it. Not only did the students totally
bomb this exam, they were angry that I was asking them to do something—modeling real
world situations with these differential equations—that they had never been asked to do
during the semester.
They were absolutely right. Students learn by doing, not by listening. Knowing how to solve
a differential equation is very different from knowing how to use a differential equation. It is
far too easy to focus our instruction on the easily tested mechanical skills and assume that
understanding and an appreciation for context will come as fortuitous byproducts.
My second major experience of this truth occurred in a minicourse run by Kathy Heid and
Joan Ferrini-Mundy at the 1994 Joint Math Meetings (in frigid Cincinnati). The theme was
the use of student interviews as a tool for deep assessment of student learning. I practiced
on a volunteer, a student from a local university who had completed four semesters of
calculus and was now a junior. Eight months after the last of these classes, I wanted to find
out what he had retained. I was horrified. As much as I pressed, differentiation for him
carried no connotations beyond a method for turning functions into simpler functions (in
the sense that 3x
2 is “simpler” than x
3), and integration reverses that process. Nothing about
tangents or rates or areas or accumulations had stuck. It drove home to me how little it is
possible to learn while still passing a mathematics class.
These experiences explain why I am such a fan of Inquiry Based Learning (IBL). It uses class
time to focus on what we really want students to learn and provides a means of constantly
probing student understanding. In my columns from February and March of this year,
What We Say/What They Hear and
What We Say/What They Hear II, I described work in
mathematics education that has validated my observation that what we think we are
communicating is not what most students hear. I appreciate Stan Yoshinobu’s recent blog,
A Practical Solution to “What We Say/What They Hear,” that illustrates how an IBL approach
can address this problem of conveying meaning.
Very little of what is learned at the post-secondary level happens inside the
classroom. As teachers, we have a responsibility to structure how students interact
with the mathematics beyond the classroom walls.
This point really struck home in 1990–91, the year I taught AP Calculus at the State College
Area High School. The leisurely pace through the AB curriculum meant that I could actually
watch light bulbs coming on in my class, something that I had never witnessed in the large
lecture calculus classes at Penn State. This has always been for me the greatest distinction
between high school and college mathematics. For the latter, students have to know how to
learn outside of class from notes and textbook and in exchanges with other students.
My last year at Penn State, 1993–94, I surveyed students in one of these large lecture
sections to try to understand their experience. I found that most of them were very
conscientious about studying, usually spending about two hours each evening that followed
class going over that day’s lesson. But I also found that most of them had no idea how to
study. They would read through their notes, paying particular attention to the problems
that had been worked out in class, and then they would tackle problems from the end of
that section, practicing the techniques that they had seen demonstrated. Anything that
strayed too far from what had been worked by the instructor was considered irrelevant.
Uri Treisman has shown the importance of students working together to clarify
understandings (see [1]). I have found that if I want students to think more deeply about the
mathematics introduced in class and to share those understandings with others, I have to
structure out-of-class assignments designed to accomplish this.
Final exams carry far too much weight. As teachers, we need to administer frequent
and varied assessments that truly measure what our students are learning and that
provide opportunities for students to learn.
I hate final exams. I have had colleagues who promise students that if they do really well on
the final, any poor test results earlier in the term will be forgiven. I have even seen students
who manage to pull off a superior performance on the final despite a record leading up to it
that would not have predicted this. Over forty years, I can count them on one hand. On the
other side, I have seen many of my students who were steady and successful during term
completely fall apart at the final. It is a stressful time, not just in my class but in almost
every class a student is taking. It is a time of late nights and cramming and incredible
pressure. The final straw for me was about fifteen years ago when there was a major
incident of cheating on my final exam. This was by good students who had been doing well
and who I knew did not need to cheat, but they were overwhelmed by the fear of doing
poorly on this major component of their final grade for which there would be no
opportunity to overcome a poor result.
My policy now is—with rare exceptions—to never count a final exam for more than 15% of
the total grade. As a complement to this, I test early, trying to get the first major assessment
in by the end of the fourth week of class; I test often; and I assess student performance using
a wide variety of measures: in class tests and quizzes that focus on procedural knowledge,
take-home tests with problems that challenge students to apply their knowledge in
unfamiliar situations requiring multiple steps, in multi-week projects that will be critiqued
and returned for revision, in Reading Reflections—short answers to questions about the
material read
before class that help inform me before class begins of what students do and
do not yet understand, and in written questions collected at the end of each class. Many of
these have been inspired by Angelo and Cross’s
Classroom Assessment Techniques, but there
is nothing I have picked up from that book that I have not reshaped to fit my own style and
needs.
I also believe in looking for ways to enable students to learn from the assessments I use.
Every major project is turned in for feedback before the final submission. For each exam
during the semester, students are allowed to earn back some of the lost points by explaining
where they went wrong and how to do the problem correctly. By giving myself some
flexibility on how much of the grade students can earn back, I find I that I can give very
challenging exams without needing to grade on a curve. Grading on a curve is a practice I
consider to be particularly pernicious because it communicates to students that they are
competing against each other, that what matters is less how much you have learned than
how much better you can perform than your neighbor.
I especially value group projects as a tool for teaching as well as assessing student learning.
Most of my classes include several major projects. It was my last year at Penn State when I
had the great good fortune to be able to teach the early
Project CALC materials with David
Smith present on campus (on sabbatical from Duke). I was able to meet with him weekly
over lunch to talk about how the course was going. I have been able to watch how effective
students are at teaching each other. It still never ceases to amaze me that I can say
something in class without it registering with some student until the person next to them
restates it as his or her own insight, though often verbatim, in a private conversation
around solving a problem they are working on. I have found that group grades are
problematic, and have experimented with a variety of techniques over the years to make
them fairer. Now, as much as possible and for at least one project per class, everyone in the
group is required to write up his or her own report of what the group has found. Clarity of
exposition is every bit as important as the correctness of the results, a requirement that
usually catches out those who simply attempt to reproduce the work done by other
members of the group.
With many writing assignments, projects, and exams, my students always complain about
how much work they have to do for my classes. But no single assessment counts for very
much, and my students have many opportunities to learn and recover from a bad
performance. I am proud of the fact that it has been many years since a student complained
that my grading was not fair. I am also proud that, despite complaining about the amount of
work, my students also note how much they have learned.
Most students get too few opportunities to appreciate the culture of mathematics. As
teachers, our instruction should communicate the true nature of mathematics.
I find it deeply discouraging that so many students graduate from college without any
appreciation for mathematics as a rich venue for discovery and innovation. Burger and
Starbird have done an excellent job of communicating this side of mathematics in
The Heart
of Mathematics. I have tried to do it through the history of the subject. This is reflected in all
of the textbooks that I have written, and my students will attest that I am constantly
interjecting the history of the subject into my classes.
This spring, as a swan song, I am teaching a 100-level class on the history of mathematics
for the first and last time. Fifty students are enrolled, most of whom are taking it to satisfy
Macalester’s quantitative reasoning requirement. I am using Berlinghoff and Gouvêa’s
Math
through the Ages, a perfect book for my vision of this course. Their text consists of a stripped
down history of mathematics, complemented by thirty short vignettes that survey topics
from the development of negative numbers to non-Euclidean geometries to the rise of the
computer. I use class time to tell the stories I love and introduce my students to the people
who have been instrumental in the development of this vast subject. The real learning takes
place in the writing students are required to do: short questions that must be submitted at
the end of each class and from which I pick a few to answer at the start of the next, Reading
Reflections in which students must tie what they have read to their own experiences, and
many short papers in which they must pursue some of the many references provided by
Berlinghoff and Gouvêa and explain to a younger version of themselves something about
mathematics that they wish they had known earlier in their mathematical career. These
have been especially insightful. I have seen so many of my students who were interested in,
even excited about mathematical ideas early in their schooling, but had had that interest
pounded out of them. It is emotional for me to see them, jaded as they now are about
mathematics, reaching back to that younger self, trying to blow that ember back to life.
None of the insights I have presented here are particularly original. Others have described
them much more eloquently. But for me they are hard-won truths achieved through years of
constantly striving to be a better teacher. After all, that is how we learn, not by listening to
someone else expound or by reading a book or column, but by observantly striving to
master our chosen profession. What we hear or read can suggest fruitful directions in which
to explore and grow. Ultimately, this is our challenge, to constantly seek to improve how we
teach.
[1] Eric Hsu, Teri J. Murphy, Uri Treisman. 2008. Supporting high achievement in
introductory mathematics courses: What we have learned from 30 years of the Emerging
Scholars Program. Pages 205–220 in M.P. Carlson & C. Rasmussen (eds.),
Making the
Connection: Research and Teaching in Undergraduate Mathematics Education. MAA Notes
#73. Washington, DC: Mathematical Association of America.
