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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
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
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
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
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
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
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
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
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
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
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
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
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
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
“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
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
“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
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
“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
∞
                                            ∞                      ∞
                        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
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
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
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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How beginning teachers understand student thinking in calculus

  • 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