Slides from a talk given by Tom Judson and me at the MAA Session on Research in Undergraduate Mathematics Education. Abstract: "Every teacher of calculus encounters various degrees of student understanding. To be a successful teacher, it is essential to understand student misconceptions and to make clear explanations to one’s students. Our project is concerned with how new teachers develop their ability to understand student thinking. We conducted individual interviews with graduate students teaching calculus for the first time. We interviewed each graduate student before and after their first teaching assignment. The interviews were transcribed and coded for analysis. We will present the results of our findings in this talk. Our hope is to provide information to that will be useful in developing more effective teaching training programs for graduate students who will teach undergraduate mathematics."

1. How beginning teachers understand
student thinking in calculus
JMM San Francisco 2010
Thomas W. Judson, Stephen F. Austin University
Matthew Leingang, New York University
January 16, 2010
Thursday, January 21, 2010

2. Calculus and Linear Algebra Classes
Instruction for calculus and linear algebra is done
in sections of 25–30 students by teaching fellows
(TFs).
TFs are graduate students, postdocs, and regular
faculty.
A faculty member acts as the course coordinator
for all sections and writes a common syllabus.
Students have common homework assignments
and common exams.
Thursday, January 21, 2010

3. Preservice Training for Graduate Students
Graduate students are supported for their
first year and have no teaching duties.
Graduate students attend a one-semester
teaching seminar where they learn
speaking skills, pedagogical mechanics,
and have some opportunities to work with
actual calculus students.
Thursday, January 21, 2010

4. The Apprenticeship
Each graduate student is required to apprentice
under an experienced coach.
The apprentice attends the coach’s class for several
weeks and holds office hours.
The apprentice teaches the coach’s class three times.
At the end of the apprenticeship, the graduate
student will be put in the teaching lineup with the
coach’s approval or the coach will recommend
additional training for the graduate student.
Thursday, January 21, 2010

5. Mathematical Knowledge for Teaching
Common Content Knowledge (CCK)—Formal
mathematical knowledge that mathematicians have
developed through study and/or research.
Pedagogical Content Knowledge (PCK)—
Knowledge used to follow student thinking and
problem solving strategies in the classroom.
Specialized Content Knowledge (SCK)—
Mathematical knowledge that is used in the
classroom but has not been developed in formal
courses.
Thursday, January 21, 2010

6. l’Hôpital’s Rule is a consequence of the
Cauchy Mean Value Theorem or Taylor’s
Theorem (CCK).
Students armed with the sledgehammer
of l’Hôpital’s Rule will use it on limits
which are not in indeterminate form and
arrive at wrong answers (PCK).
Thursday, January 21, 2010

7. ples in [18]. Ma examined the complex mathematical k
elementary school teachers. For example, Ma posed t
Specialized Content Knowledge
to both American and Chinese teachers.
Students performed the following multiplication
123
Liping Ma gives the
following example: × 645
Suppose that a student 615
performs the following 492
multiplication. What
would you say to the 738
student? 1845
What would you say to these students?3
Thursday, January 21, 2010

8. Participants
We interviewed seven graduate students
before and after their first teaching
assignments.
The graduate students were from Asia,
eastern Europe, and the U.S.
Both men and women were represented.
Thursday, January 21, 2010

9. Pre-Teaching Interview
“Can you talk a little bit about your
background, and how you got here?”
“Can you tell us about your career plans
and how you see teaching as part of those
plans?”
Each participant was given four questions
involving different calculus scenarios.
Thursday, January 21, 2010

10. All of the TFs planned a research career or
saw research as a strong component of their
future career.
All thought teaching was important. Those
planning an academic career thought that
teaching would be an important duty.
Several looked forward to the teaching.
All had some idea of the need for PCK in
the classroom.
Thursday, January 21, 2010

11. graphics.nb
3. The graph of f (x), given below, is made up of straight lines and a semicircle.
f HxL
4
2
x
-5 -3 -1 1 3 5
-2
-4
We define the function F (x) by
x
F (x) = f (t) dt
0
One of your students understands that F (2) = 4 but believes that F (−2) is undefined.
What would you say to the student?
Thursday, January 21,often
4. Students 2010 have difficulty working in three dimensions. One of your students comes
to you and asks how to match each of the following equations with the appropriate

12. Several participants gave an explanation
by appealing to signed area.
“You could say, why do we have this rule
in the first place? One reason for it is that
we want the Fundamental Theorem of
Calculus to hold.”
No one gave an explanation using the
integral as net change without some
prompting.
Thursday, January 21, 2010

13. Post-Teaching Interview
“Now that you've had a chance to work
with students, has your view of teaching
changed at all?”
“What surprised you about teaching?
What happened that you didn’t think would
happen?”
We asked four more questions involving
different calculus scenarios.
Thursday, January 21, 2010

14. View of Teaching
“I’ve always thought that the professor
doesn’t like to have all that many
questions. And it just sounds silly
sometimes. And then when I taught, I
realized that even the serious questions,
I really wanted those questions. ... It was
a very different perspective that I got.”
Thursday, January 21, 2010

15. “It went great. I really loved it. I mean, I thought I’d
like teaching, but it went better than I expected. I was
nervous, but only for the first couple of classes. Then I
really became comfortable with them. ... They asked a
lot of questions. They are pretty demanding. They
really want to know things. And you can’t just get
away with stuff with them. There will definitely be at
least one person who has something to say, you know.
So I thought that was great. But I realized how much I
love questions. I mean, whenever they were a little
tired and they weren’t asking so many questions, I felt
sad, you know? It feels great when they have
questions and you feel that they understand
everything.”
Thursday, January 21, 2010

16. What Surprised Them
“I was surprised at how heterogeneous
the students were that I had in terms of
mathematical ability. Some of them had
trouble understanding that x/2 and (1/2)
x were equal to one another, and others
were well over prepared for the class.
They’d taken calculus in high school.”
Thursday, January 21, 2010

17. “When I was teaching, students would really
ask me sometimes some questions that I would
never expect. I saw at first, for example, for
log x times a constant. Everyone knows the
derivative is 1/x times the constant. Then, I
put some kind of extra constant, then people
are very confused ... I think this is should be
kind of easy and obvious to me, but it’s really
not obvious to the students. It’s a little bit
surprising to me, so I really have to know
what students are really thinking about.”
Thursday, January 21, 2010

18. A: So I felt that I could assume that this is well-known to
students, so I can just move faster when deriving or finding
[something on the] blackboard. But then—Well, since students
always ask the question, but why the equation is true or ... how
could I get second line from first line like that? So after that I
found I need to be more careful and I needed to be prepared.
Q: Do you think it’s that these basic facts about algebra and
trigonometry is that they don’t know them, or that they just lack
the necessary fluency?
A: Oh, it’s just lack.
Q: Lack fluency?
A: Yeah. They’re just slow, yeah. ... if I just do it line by line
slowly
Thursday, January 21, 2010

19. “There are some things I guess
everybody could use help with. They
have trouble doing derivatives that
involve recursing more than twice. If
they need to use the product rule alone,
that’s fine. If they need to use the
product rule on the chain rule, that’s
fine. But if you need to use the product
rule, the chain rule, and something
else...”
Thursday, January 21, 2010

20. ∞
∞ ∞
ak ≤ an+1 + a(x) dx ≤ a(x) dx.
k=n+1 n+1 n
What would you say to the student?
3. Consider the following problem. Let
x 1/x
1 1
F (x) = 2
dt + 2
dt,
0 1+t 0 1+t
where x = 0.
(a) Show that F (x) is constant on (−∞, 0) and constant on (0, ∞).
(b) Evaluate the constant value(s) of F (x).
What sort of difficulties would would a student encounter when trying
to solve this problem? What would you say to the student?
4. Students often have difficulty working in three dimensions. One of
your students comes to you and asks contour plots. If the contour plot
of f (x, y) is given below, at which of the labelled points is |∇f | the
Thursday, January 21, 2010 smallest? What would you say to this student? What
greatest? The

21. Two TFs found at least three different solutions
to the problem.
Students will integrate 1/(1 + t2) and then get
stuck.
Students will be able to differentiate the first
term using the Fundamental Theorem of
Calculus but will have difficulty differentiating
the second term.
No one mentioned that students will have
difficulty with locally constant functions.
Thursday, January 21, 2010

22. Conclusions
Pedagogical content knowledge comes with
teaching experience. It is difficult to “teach”
PCK.
Pre and inservice training should train TFs to
look for PCK and provide in depth examples.
TFs should have opportunities to work with
real students BEFORE they enter the
classroom as the primary instructor.
Thursday, January 21, 2010

23. Acknowledgements
Thanks to our participants and
colleagues.
Thanks to the generous support of the
Educational Advancement Foundation.
Thursday, January 21, 2010