Of the following timeless tales, all but #8 are true stories. Tales #7 and #11 are very close to me.
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1. Analogy: French in high school and college. I knew that the course French Elements (340.101-102) covers about the same material as the first two years of high school French. This is typical of first-year college language courses. Also, the semesters are shorter here, and one can calculate that the material is covered approximately three times as rapidly here as in high school.
After looking at the catalog description of French Elements, I called the instructor. I felt sure that there w
--------------- as more to it than just the triple speed. "Yes," she replied. "In our course, we aspire for fluency."
I admit that I had four years of French in high school, but no one ever talked of fluency. Anyway, it should be obvious that most of the work must occur outside of class. You can expect something like the tripling of high school pace, a lot of work outside of class, plus aiming for the mathematical analogue of fluency (perhaps command is the correct word), in a calculus course here.
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2. From the so-called Jekyll-Hyde compilation. In a departmental course evaluation questionnaire, the following contrasting comments were made about me by two students from my Fall '94 110.109 course:
"[He] is a giant. The man is humorous and very intelligent. He actually makes math interesting."
"I don't go to lecture anymore because I never learn anything from the instructor. He seems to be talking about irrelevant topics and phrases questions in ways which are hard to understand."
(Pairs of contrasting comments could be found in every instructor's evaluations.) What is the most likely explanation of how such opposite views can be held by students in the same course? I think it is more likely a difference of expectations than of talent.
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3. Story of J.M. (course grade B+) One of the students in the Fall '94 Calculus II course came up to me at the end of the semester to thank me for the way the course was conducted. A bit surprised, I asked him why. He explained that though he was unhappy with the lectures at first, he came to appreciate them later. I asked him to put his thoughts in writing. Of the lectures he wrote, "Dr. Zucker made the material very easy to understand, if and only if you were doing the work necessary to keep up with the class." I think he was trying to convey the message that a student who was not keeping up would be unable to see that the material was being explained in a clear and helpful manner.
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4. Story of A.D. (course grade A-) I remember vividly when, during the second or third week of the Fall '93 Calculus II course, a student came to see me in my office, feeling he was hopelessly lost. I probed with a few questions, after which both of us could see that he was very close to understanding the material. He, like so many other high school graduates, had been trained to absorb mathematics in tiny controlled doses, which are to be memorized and later regurgitated. It is no wonder that the presentation of conceptual material in the lecture often gets perceived by students as irrelevant theoretical digression, rather than the means to better comprehension that it is.
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5. Story of J.S. (course grade A-) This student from my Spring '98 Calculus II course and I would occasionally cross paths on campus in later semesters. Every time he saw me, his face would light up with a big smile. One time, I decided to ask him why. His reply still astounds me: "Until I took your course, I didn't know that I was good in mathematics."
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6. Story of S.Y. In Fall 1994, a student was in my office, taking a make-up exam at the blackboard. I started her with a fairly simple problem from the heart of the material. In working it out, she committed a standard blunder, and I pretty much told her that. She thought a moment, pulled an example out of her head, carried out the analogous calculation, and saw what her mistake was. I was actually quite impressed.
However, her performance on the two midterms was so-so. Her work declined at the end of semester. On the Final, worth 200 points (40% of the grade), she got a 21. That's right, about 10%, when the median score was over 120. Some instructors would just give her an F, for they saw what she was coming out of the course with. Instead, I just put her scores into the formula, and it came out . . . F.
Thus, I wrote F on the grade roster. In those days, we could post grades by student number, and this I did. As expected, that student came by. She wasn't happy, and she asked to see her Final Exam. Routinely, I let her look at it. She pointed out a mistake in the arithmetic: the addition was wrong. Upon changing her score to 31 (15%), I looked at her and said what I already knew, "Good, now you have a D in the course."
The action picked up. "I'm not a D-student!" "I know that, but you're still getting a D in this course. ... What was going on at the end of the semester?" After a brief pause, she said, "You know, I don't know where my time went." "I think you should figure that out, for you don't want this to happen again."
Whenever she saw me on campus in subsequent semesters, she would wave vigorously and smile.
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7. Story of S.Z. In 1966, Brown University still had Saturday morning classes---and standard curriculum. This freshman decided to get together at Yale with a few high school classmates, one of whom was a student there, on a weekend early in the fall semester for the Brown-Yale football game. This necessitated skipping the Saturday class in Expository Prose.
There was a writing assignment due Tuesday, and the instructor passed out a few pages the Thursday before concerning what constitutes good writing. I didn't really look at it before leaving town. When I got back, I spent much of an evening doing the assignment. After some time, I felt I had a good article. I would have been happy to call that "it", but it occurred to me to check the handout. I was shocked! I found it to be a bit cryptic, and what I had written didn't conform to its contents. In a panic, I desperately tried to rewrite the article, ending with something notably mediocre, which I submitted.
When the article was returned to me, I saw the poor grade it had earned me. The instructor said, "It looks as though you dashed this off Tuesday morning before class." I explained what had actually happened. He said, "If you miss a class, you are responsible for everything that transpired in that class. Had you inquired, you would have learned that I told the class on Saturday to disregard the handout."
I really couldn't argue with that. It only now occurred to me that he may have given the handout with the intention of retracting it. (I'm getting devious in my old age.)
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8. Analogy: Martial arts. An 18-year-old enters a tae kwon do studio, walks up to the instructor, and states proudly, "I want to learn how to put my hand through a stack of bricks!"
The instructor thinks a moment, then replies. "Well, that's very difficult, and will take time. First, you must develop self-control and mental discipline. Then ..."
The youth interrupts, "Don't give me that discipline crap! Just teach me how to put my hand through the bricks!"
The instructor walks away shaking his head, as does the would-be student. One of the regulars of the studio, who teaches math in a local high school, steps up to the instructor. "You know, the young man has a point. All you have to do is make the bricks out of softer material, and crack them a little in advance."
(To be clear about the analogy, I'll offer the following. The young man wants to learn to solve difficult math problems. The instructor advises him that it will take time to develop the conceptual base for achieving that goal. The impatient youth says he just wants to solve the problems! The math teacher points out that the young man can achieve his goal if you give easier problems and offer lots of hints.)
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9. Story of J. In Fall '06, this freshman started coming to my Linear Algebra class, I believe after missing the first lecture or two. He then started talking about switching to the Honors Linear Algebra course, of which I was also the instructor. Because he never attended the latter, it was clear he decided not to make the switch. I don't recall seeing him so often in lecture in Linear Algebra, either, but that was not a problem of itself.
His performance on the first midterm was unsatisfactory. When I asked students with such scores to see me in my office, he didn't show up. On the second midterm, he did a little better. His Final Exam score was consistent with the above, and his course grade was a C. He came to my office for his grade and to look at his Final.
At some early point, I told him truthfully, "You know, I was hoping that you'd fail the course." (I never contrive to give a poor grade.) It meant that I could discern a poor attitude on the part of the student, and that an unsatisfactory grade might wake him up. He looked at me, dumbfounded.
He said, "I was always good in math; it was always easy for me." I replied, "I don't doubt that for a minute!" The point had gotten through. Exactly what happened next is hard to recall, but I let him stay in my office for about half an hour. He seemed to feel the shame, (appropriately) primarily to himself, in his underachievement. At the end, he showed his gratitute, thanking me profusely. I'm hoping the message will stay with him.
Oh, yes. He mentioned being involved in a relationship ....
Epilogue: I saw J in my office in Spring 2008. He has a healthy GPA and the relationship is still on.
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10. Story of K. In my Spring '07 Calculus II course (for engineering and physical science) this freshman was struggling, and she did poorly on both midterms. She seemed to be putting in a sincere effort, on the other hand. In going through her performance on the second midterm with her, it struck me that she was choosing her methods badly, trying to "put square pegs in round holes." I pointed several instances of that out to her, comparing erroneous approaches to correct ones. Could she really have insufficient talent to sort such things out? I wondered.
After the Final was given, she e-mailed me about her grade. I asked her to come to my office, as many students do, if she had time. So she did, fearing that she had failed. In actuality, she scored around the class median on the Final, and her course grade was C. She was very happy for that.
I asked her what it was that made the difference. The answer was predictable: "I put in more time." I asked her to quantify that if she could. By her estimate, she started putting in seven hours a week outside of class instead of five. After that, I infer, she could better distinguish a square peg from a round one. She also decided to take my advice not to skip lectures just because she couldn't follow them (see Tale #11 below).
This was another instance of students' misjudging their effort until the end of the course. I told her, as if she didn't realize it already, that if she had put in commensurate effort for the beginning and middle of the course, she might have had a B as course grade. (One wonders why she hadn't adjusted in Calculus I in the Fall Semester, ....)
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11. Analogy: Receiving information in an unfamiliar language. It was near the end of December 1993. I had met up with a Vietnamese colleague in Italy, and we were to do some travelling. We could both communicate in English and French, but not Italian.
In a cafe, an old woman sensed that she could impart some useful information to us and caught our attention. I asked my prepared initial question in Italian: "Do you speak English or French?" No such luck. It had to be Italian. My facility in Italian was rather meager, coming primarily from music and Italian restaurants. The woman and I looked straight at each other, as she spoke a sentence slowly. I didn't understand; would we ever know what she had said? She repeated the same sentence, the same way. I got it! and turned to my colleague, saying in English, "She said ...." The woman had another sentence to communicate to us. It went exactly the same way: I turned to my colleague upon understanding it the second time around, and again told him what she had said.
Where is the analogy? Did the woman explain it better the second time? No. Would I have understood if I wasn't paying attention the first time? No . . . Mathematics may seem to be an unfamiliar language, something you won't get the first time around. The lectures can be the first or the second time you see the material in the course. Furthermore, you may need more than two shots at the material. You should understand it best during the last shot. If you decide to stop coming to lectures because you can't follow [some of] them, you are discarding input to your goal of eventual understanding. (See also Tale #12)
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12. Story of A.W. (course grade A-) In my office, after the semester was over, this student from my Fall '96 Calculus II course reported peer pressure to let his friends copy homework solutions from him. It was really too late for me to do anything about it to enforce the ethics code.
He then wished to confirm an impression he had picked up. It seems that your style of lecturing is to show us where we're supposed to be, then tell us to get there. Is that correct? The statement is a bit jarring, and I thought a moment and replied, "Yes, that's about right." In thinking about it further, I linked his assertion to item #6 of the 1996 document Academic Orientation.
(Ironically, few students place themselves in the majority in option a of #6.) What A.W. said might well be the impression of a student who elects option b. The week's lectures (ideally) should then be viewed as a clear sketch of the path towards understanding that week's material; the student shouldn't expect too much more than that. On the other hand, when the student takes instead the recommended option a, then what he or she has picked up from the preliminary reading is confirmed or corrected, and gets expanded during the lecture.
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13. Story of S.M. (course grade B-). This student from my Fall 2004 Abstract Algebra course (110.401) was in the bottom quarter of the class. He eventually realized that he had to start asking questions in lecture. His questions were not so profound, but I answered every one of them, having decided on the spot that they were reasonable. I do retain the option of not using class time to answer questions if they ask for discussion of past topics or of matters too elementary (those should be addressed outside of class), or if the questions become disruptive. I'll add that I post replies to students' e-mails on the course website when I feel they are relevant and of sufficient general interest.
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14. Story of G. (course grade A). This tale is about the student's responsibility of taking initiative in trying to learn the material. This Fall 2008 Linear Algebra student e-mailed me before the first midterm, concerning the problems on the posted exams from previous years. He displayed evident confusion about the subject matter, so I asked him to make an appointment to see me soon, which he did. After our meeting, which I recall vaguely having lasted about half and hour, he was largely straightened out. You saw his final course grade.
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15. Story of D.K. (course grade C-). The student from my Fall 2008 Linear Algebra course did poorly on the two midterms. The first one was a clear failure, and the second was marginally passing. After the second exam, he decided, wisely I must say, to see me during office hours. In going over the second exam, I selected a problem to start with, asking, "Do you know how to do this one now?" He didn't. I then said, "This time, I'll show you how." In this situation, it's usually better to try to get the student to progress with guidance from the instructor. Afterall, showing the methods earlier (in class or by the textbook) wasn't enough. Besides, the student will have to rely on himself during the exams.
He was able to monopolize my office hours. He apologized for that, but there was really no need to, for the rest of the students didn't seem to want my help (sic). He came to office hours faithfully up to the Final. I could sense that he would do decently enough to pass the course, and he didn't disappoint. He came by to thank me. That was also not so important either, for we both already knew how much I had helped him.
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