how beginning teachers understand student thinking in calculus

How beginning teachers understand student thinking in calculus

JMM San Francisco 2010Thomas W. Judson, Stephen F. Austin University

Matthew Leingang, New York UniversityJanuary 16, 2010

Thursday, January 21, 2010

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.


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.


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.


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.


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).


Specialized Content Knowledge

Liping Ma gives the following example: Suppose that a student performs the following multiplication. What would you say to the student?

We classify statements like these as PCK.

In order to avoid these pitfalls, experienced educators may construct

lessons on L’Hopital’s Rule that

• illustrate why limits in indeterminate form have no “law” associated to

them, in contrast to the Quotient Rule for limits when the limit of the

denominator is not zero.

• give examples of manipulating other indeterminate forms (for instance,

∞−∞) into indeterminate quotients.

• show how the conclusion of L’Hopital’s Rule can fail if the hypotheses

are met.

Thus, PCK also governs teaching practice.

A third categorization of the mathematical knowledge needed is mathe-

matical knowledge that one uses in teaching but that is not ordinarily de-

veloped in formal courses. Liping Ma calls this type of knowledge profoundunderstanding of fundamental mathematics (PUFM) and gives several exam-

ples in [18]. Ma examined the complex mathematical knowledge required of

elementary school teachers. For example, Ma posed the following problem

to both American and Chinese teachers.

Students performed the following multiplication

123

× 645

615

492

738

1845

What would you say to these students?3

Ma found that the American teachers made procedural corrections, but

the Chinese teachers had a much deeper understanding of the relationship

between the distributive law and the standard algorithm used for multipli-

cation. This relationship is not always taught in standard courses, although

3Check Ma’s book to get the exact quote.

4


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.


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.


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.


3. The graph of f(x), given below, is made up of straight lines and a semicircle.

-5 -3 -1 1 3 5x

-4

-2

2

4

f HxL

graphics.nb 1

We define the function F (x) by

F (x) =

� x

0

f(t) dt

One of your students understands that F (2) = 4 but believes that F (−2) is undefined.

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 how to match each of the following equations with the appropriate

graph below. What would you say to this student?

(a) x2 + 4y2 + 9z2 = 1

(b) x2 − y2 + z2 = 1

(c) y = 2x2 + z2

(d) x2 + 2z2 = 1

(e) 9x2 + 4y2 + z2 = 1

(f) −x2 + y2 − z2 = 1

(g) y2 = x2 + 2z2

(h) y = x2 − z2

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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.


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.


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.”


“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.”


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.”


“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.”


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


“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...”


x ≥ 1 and that ak = a(k) for all integers k ≥ 1. A student asks you to

explain why

∞�

k=n+1

ak ≤ an+1 +

� ∞

n+1

a(x) dx ≤� ∞

n

a(x) dx.

What would you say to the student?

3. Consider the following problem. Let

F (x) =

� x

0

1

1 + t2dt +

� 1/x

0

1

1 + t2dt,

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

greatest? The smallest? What would you say to this student? What

sort of difficulties do you think students might encounter when learning

about contour plots?

44


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.


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.

Conclusions


Acknowledgements

Thanks to our participants and colleagues.Thanks to the generous support of the Educational Advancement Foundation.
