How beginning teachers understand
              student thinking in calculus
                JMM San Francisco 2010
      ...
Calculus and Linear Algebra Classes

                 Instruction for calculus and linear algebra is done
                ...
Preservice Training for Graduate Students


                 Graduate students are supported for their
                 fir...
The Apprenticeship

                 Each graduate student is required to apprentice
                 under an experienced...
Mathematical Knowledge for Teaching

                 Common Content Knowledge (CCK)—Formal
                 mathematical ...
l’Hôpital’s Rule is a consequence of the
                 Cauchy Mean Value Theorem or Taylor’s
                 Theorem (...
ples in [18]. Ma examined the complex mathematical k
elementary school teachers. For example, Ma posed t
      Specialized...
Participants

                 We interviewed seven graduate students
                 before and after their first teachin...
Pre-Teaching Interview

                 “Can you talk a little bit about your
                 background, and how you go...
All of the TFs planned a research career or
                 saw research as a strong component of their
                 ...
graphics.nb
    3. The graph of f (x), given below, is made up of straight lines and a semicircle.

                      ...
Several participants gave an explanation
                 by appealing to signed area.

                 “You could say, w...
Post-Teaching Interview

                 “Now that you've had a chance to work
                 with students, has your v...
View of Teaching

                 “I’ve always thought that the professor
                 doesn’t like to have all that ...
“It went great. I really loved it. I mean, I thought I’d
                 like teaching, but it went better than I expecte...
What Surprised Them

                  “I was surprised at how heterogeneous
                 the students were that I had...
“When I was teaching, students would really
                 ask me sometimes some questions that I would
                ...
A: So I felt that I could assume that this is well-known to
                 students, so I can just move faster when deri...
“There are some things I guess
                 everybody could use help with. They
                 have trouble doing de...
∞
                                            ∞                      ∞
                        ak ≤ an+1 +               a...
Two TFs found at least three different solutions
                 to the problem.

                 Students will integrat...
Conclusions

                  Pedagogical content knowledge comes with
                 teaching experience. It is difficu...
Acknowledgements



                 Thanks to our participants and
                 colleagues.
                 Thanks t...
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How beginning teachers understand student thinking in calculus

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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."

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How beginning teachers understand student thinking in calculus

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

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