AUGUST 1996 NOTICES OF THE AMS 863

Teaching at theUniversity Level

Steven Zucker

One of the ironies of being a college educator inthe United States is that one is often rewardedfor not doing ones job. That sounds like a strangething to say, but I know that it is true.

I received my Ph.D. in 1974. I taught at stateuniversities (Rutgers and Indiana) before mov-ing to Johns Hopkins in 1983. When I arrived atRutgers as a fresh Ph.D., I had little teaching ex-perience, and the first few years were trying.Once, I was criticized for covering the entiretyof the math departments syllabus for a calcu-lus course when the students were not gettingsome of the topics along the way (I was expectedto put the heart before the course). I eventu-ally learned to accept the pretenure save-your-own-hide advice that I give to assistant profes-sors: teach so as to keep your ratings up. Fromthen on, I had good student evaluations. Butwas I a good university educator? Were the stu-dents learning better? Not really. Did anyonecare? Not really. At universities where the stan-dard for reappointment and promotion was qual-ity research and acceptable teaching (or evenentirely a research standard), it is obvious whyfew people wanted to rock the boat over educa-tional matters. The goal was de facto to con-centrate ones energies on research and doenough in teaching to keep the students fromcomplaining. That it kept many of them ignorantwas not at issue.

When I moved to Hopkins, I got as a bonusan improved environment for teaching. Herewas a body of students with a rather high meanSAT math score (now around 700). Believing thatthey would be a better audience (despite thelarge class size), I felt that I could comfortablyblend into the style that had emerged from my

years as assistant professor some of my idealsabout teaching calculus to science-oriented stu-dents. However, as years went by I started to feelincreasing resistancebalkingon the part of alarge portion of my class. Since I also felt thatmy presentation was getting clearer, I becamecorrespondingly irritated over their apparent re-fusal to take the course seriously. And when Itook part in the student-run course evaluationsurvey a few years ago, I discovered that theclass as a whole rated me only satisfactory.What was going on? It took a poke from my de-partment chair and a couple of years of exertionon my part to arrive at the conclusion that I nowhold. The answer is so obvious that it is embar-rassing.

The fundamental problem is that most of ourcurrent high school graduates dont know howto learn or even what it means to learn (a for-tiori to understand) something. In effect, theygraduate high school feeling that learning mustcome down to them from their teachers. Thatmay be suitable for the goals of high school, butit is unacceptable at the university level. That thestudents must also learn on their own, outside theclassroom, is the main feature that distinguishescollege from high school.

My contention is that it is possible to get col-lege freshman to learn calculus fairly well, with-out resorting to utopian tricks such as enforcedgroup projects. All we have to do is get the stu-dent to accept that learning is something that willtake place mostly outside of class; that is, justinsist that they grasp the underlying premise ofcollege education. You may wish to ask yourselfwhere, when, and how freshmen at your uni-versity get this message.

It seems to me that the right way to do thingsis to put a little effort into explicitly and imme-diately bringing the students expectations up touniversity level. I have tried this in the first week

Steven Zucker is professor of mathematics at JohnsHopkins University. His e-mail address issz@math.jhu.edu.

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of my own course (Calculus II for Physical Sci-ence and Engineering, fall semester 95I taughtthe same course in fall 93 and fall 94) withmuch success, despite the inevitable dilution ofeffort that results from our first-semester fresh-man pass-fail. It helped that my message was re-inforced by my departments new SurvivalGuide, which was passed out in all basic coursesat the beginning of the semester. The class didrather well on exams that were deemed diffi-cultI prefer thoroughby my colleagues. Weshould be putting our effort into reforming thestudents, not the calculus! My personal attemptat orientation was based on a collection of hand-outs. At the end of the semester, I saw that Icould write up their effective content on onepage; it is appended to this article.

Im told by students that during that firstweek, most of the class was thinking Strange.Why was our professor going on like that? Iteventually dawned on me that this sort of aca-demic orientation should have been getting car-ried out by the organizers of Orientation Week,who now put all of their efforts only into mak-ing students socially at ease the week beforeclasses begin. After all, isnt how to study in col-lege something important that most freshmendont know when they arrive? Still, they will beexpected to have figured it out by the end of theacademic year (not necessarily without pain).Why is this important piece of wisdom concealedfrom our entering students? And why do seri-ous educators have to endure the consequencesof receiving a batch of students with expectationsat the high school level?

When college students act as though they arestill in high school, they rate their professors ac-cordingly. If it is even suspected that the ratingswill be taken nearly at face value, as I fear theytoo often are these days, a prudent instructor willthen be tempted to make his or her coursesmore like high school. A serious instructor withambition will probably end up absorbing theconsequences of mediocre ratings. Thus, theteaching of basic university mathematics at thecollege level becomes the thankless task of theidealist. We must hold the system responsiblefor encouraging the behavior that it rewardsand ourselves for allowing such a system to per-sist.

We let them stay in high school for several rea-sons. One is the fear of negative teaching eval-uations! Its a vicious circle. To get good evalu-ations, one can simply give the students whatmost of them (think they) want: a course wherethe material moves slowly and can be picked uplargely in the classroom, exams that reflect a pre-determined list of problem types. Careful prepa-ration and a few drops of avowed concern com-pletes this recipe for an A rating with students.

And Ive never heard more than a handful of stu-dents complain that a course was too easy!

Another reason for letting them stay in highschool is that we may opt to take the path of leastresistance. It takes a lot of time and energy onthe instructors part to prepare and deliver lec-tures and to make up and grade suitable examsthat will help the students attain the goal ofcommand of the material. And it requires emo-tional strength to hold ones stance when someof the students show signs of strain while at thesame time giving help to the students during of-fice hours and beyond. In short, it is pragmaticto teach so as to keep ones ratings up and toleave the standard near the high school level.

I used to be hoodwinked by the notion thatit was unfair to test the students on anything Ididnt go over in class, even when problemswere assigned in the homework. I feel now thatit was an unwarranted concession to their intentto stay in high school. Why are we rewarding theirresistance? At Hopkins one of our finest gradu-ate student teaching assistants taught calculusin the summer session. One of her students hadthe gall to assert in the course evaluation sur-vey that he or she could not recommend her: theTA had had the effrontery to ask the class to pickup one of the last topics on their own.

One of my basic tenets is that the studentshave no right to know what an upcoming examis going to look like. (However, exams from pre-vious years are on file at the library.) I aim to pre-pare them for any reasonable exam I might comeup with. That is, Im asking them to aspire forcommand of the material of the course and notany particular subset of it. Some students thinkthis is unfairit wasnt like that in highschoolwhen in actuality asking for a sneakpreview of the exam is nothing but attemptedcheating. When the instructor helps them cheat,the students reward him or her with highermarks on the evaluation survey for giving fairexams and relevant lectures, and the commu-nity ends up with the impression that the in-structor is a good teacher. I think Ive made agood case that such people should be repri-manded, not lauded, for they contribute to theundermining of education at the college level.(The reader may be able to anticipate the corol-lary that one cannot measure the level of a coursejust by looking at the exams: an exam that lookshard may cease to be so if the students hadbeen told by the instructor to expect those prob-lems or ones just like them.)

Naturally, an instructor who plans the lec-tures carefully and delivers them well will ratebetter than one who does not; that is appropri-ate. Im only trying to enunciate a point thatevery math professor surely knows, at least sub-consciously: other things being equal, an easy

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course will rate higher than a demanding one.This factor is not treated in most course evalu-ations. (Indeed, I doubt that a reliable measure-ment of learning can be achieved by simplypolling the students at the end of the semester.)Students, especially freshmen, can and do declarethat an instructor is hard to follow just becausethe material is not presented and reinforcedcompletely in the lectures, even when the stu-dents who keep up with their share of the workassert that the material is being explained veryclearly.

In conclusion, I think I have illustrated howthe issues of academic orientation, the seriousevaluation of teaching, and the calculus crisisare linked, at least at selective universities likeJohns Hopkins. Of course, the way in whichmathematics education in any particular collegeor university can be improved depends on thecomposition of its student body. The overalltheme should be the same though. Studentsmust be told immediately that they are about toface a big jump in level from high school. Mosthigh school teaching is justifiably set to theneeds of the least talented students in the class;the better students often become convinced byhabit that this level is right for them too. Theyshould be helped to recognize that the changeis both appropriate and manageable. It is not nec-essary for their teachers to program them, forthey are quite capable of monitoring their ownlearning. It is not necessary for them to graspthings at once from the classroom presentationalone, for many things require time and effortfor attainment of the level of understanding wewould like them to achieve. And most of that cantake place only outside of the classroom.

Academic Orientation for Fall SemesterFreshman Lecture CoursesWhat follows is what an entering freshmanshould hear about the academic side of univer-sity life. It is distilled from what Ive learnedand written concerning the need for academicorientation as a result of having been the in-structor of 110.109 (Calculus II: Physical Sci-ences) in the fall semester for three consecutiveyears.

The underlying premise, whose truth is veryeasy to demonstrate, is that most students whoare admitted to a university like JHU were beingtaught in high school well below their level. Theintent here is to reduce the time it takes for thestudent to appreciate this and to help him or heradjust to the demands of working up to level.1. You are no longer in high school. The great

majority of you, not having done so already,will have to discard high school notions ofteaching and learning and replace them byuniversity-level notions. This may be diffi-

cult, but it must happen sooner or later, sosooner is better. Our goal is more than justgetting you to reproduce what was told toyou in the classroom.

2. Expect to have material covered at two tothree times the pace of high school. Abovethat, we aim for greater command of the ma-terial, especially the ability to apply what youhave learned to new situations (when rele-vant).

3. Lecture time is at a premium, so it must beused efficiently. You cannot be taughteverything in the classroom. It is your re-sponsibility to learn the material. Most ofthis learning must take place outside theclassroom. You should be willing to put intwo hours outside the classroom for eachhour of class.

4. The instructors job is primarily to providea framework, with some of the particulars,to guide you in doing your learning of theconcepts and methods that comprise thematerial of the course. It is not to programyou with isolated facts and problem typesnor to monitor your progress.

5. You are expected to read the textbook forcomprehension. It gives the detailed accountof the material of the course. It also containsmany examples of problems worked out,and these should be used to supplementthose you see in the lecture. The textbookis not a novel, so the reading must often beslow-going and careful. However, there is theclear advantage that you can read it at yourown pace. Use pencil and paper to workthrough the material and to fill in omittedsteps.

6. As for when you engage the textbook, youhave the following dichotomy:

a. [recommended for most students]Read for the first time the appropri-ate section(s) of the book before thematerial is presented in lecture. Thatis, come prepared for class. Then thefaster-paced college-style lecture willmake more sense.

b. If you havent looked at the bookbeforehand, try to pick up what youcan from the lecture (absorb the gen-eral idea and/or take thorough notes)and count on sorting it out later whilestudying from the book outside ofclass.

7. Exams will consist largely of fresh problemsthat fall within the material that is beingtested.

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