So, what did we find by looking at our students? What did studying math mean for the Black and Chinese students? For the Black students it meant this: You wake up in the morning. You go to class. You take notes. You get your homework assignment. You go home. What do you do? You do your homework religiously. You come to class. You hand it in. A day or two before the test, you study. You put in about eight hours a week for a calculus course, just what the teacher says, and what happens to you? You fail. An important point here is that the Black students typically worked alone. Indeed, eighteen of the twenty students never studied with their classmates. There was the same pattern for many blue-collar whites and rural students.
What about the Chinese students? They studied calculus for about fourteen hours a week. They would put in eight to ten hours working alone. In the evenings, they would get together. They might make a meal, usually jook, which is a rice soup. Then, they would sit and eat and go over the homework assignment. They would check each other's answers and each other's English. If one student got an answer of "pi," and all the others got an answer of "eighty-two," the first student knew that he or she was probably wrong but could pick it up quickly from the others. If there was a wide variation among the answers, or if no one could do the problem, they knew it was one of the instructor's "killers."
It was interesting to see how the Chinese students learned from each other. They would edit each other's solutions. A cousin or an older brother would come in and test them. They would regularly work problems from old exams, which are kept in a public file in the library. They would ask each other questions like, "How many hours did you stay up last night?" They knew exactly where they stood in the class. They had constructed something like a truly academic fraternity, not the more typical fraternity: Sigma Phi Nothing.
The best evidence for the power of the Asian students' group work was the contagion factor. If one student had a misconception, they would all leave with it. There is a beautiful story which illustrates this: It is 1:00 a.m., and the kids are listening to loud ethno-pop music (something like "Singing in the Rain," sung in Cantonese, to a disco beat), and they are also watching Perry Mason on television. There is this courtroom scene in which Hamilton Burger, the District Attorney in Perry Mason gives the bailiff a gun, and the bailiff says for the court reporter's record, "Let it be known that this revolver has been entered into evidence." The students picked up the expression, "Let it be known that" and began to use it in their math homework. Instead of writing, "...let X equal..." on their homework, they started writing, "...let it be known that X is..." This usage spread among their friends, and a few days later, when ten or fifteen people already were using this construction, one of them had the error corrected. The usage disappeared almost overnight.
The Black students, on the other hand, didn't have a clue what other students in the class were doing. They didn't have any idea, for example, what grades they were going to get. The exams were like a lottery: "I got a B," or, "I got a C." They had no clue where they stood relative to their classmates. Moreover, these same students were getting A's in "Study Skills," and F's in the calculus class. What they were taught in "Study Skills" wasn't worth a damn in calculus.
At this point it is useful to look at how universities attempt to deal with the problem of minority student failure. In the sixties, the university administration hired people to deal with this problem which was then seen as essentially a political one. This is not to say that the administrators didn't care about these students, but it was necessary to have visible institutional effort aimed at helping these students to survive. Because of the political character of affirmative action, the administration took primary responsibility for minority student programs, even those that addressed academic issues. The political pressure to create these programs was felt on virtually every American college and university campus. If we look at these programs, even now, we see first that they are isomorphic. They have little to do with the special mission or history of the institutions in which they exist, which is remarkable given the diversity of American higher education. They are as similar as personnel offices. Second, they have very little if any connection to the faculty. They are staffed by very caring people, many of whom are minority, and who are devoting their professional lives to helping minority students avoid failure. But, unfortunately, they see massive failure and this has led to corresponding burnout and anger. In the large, their tutorial programs are disastrous. The tutors see the students the day before the exam; the counselors see them the day after the exam. On many campuses you don't count the Black and Hispanic people who graduate in math and science . There are so few you can just name them. You can carry a photograph of each of them in your wallet.
The individuals who work in the student affirmative action operations, seeing the overwhelming failure of the students they care about, can easily develop a "bunker mentality." The faculty are the enemy. They advise their students to stay away from mathematics and science. This is a scary and depressing phenomenon, very depressing.
An equally disturbing phenomenon is the creation of remedial courses that lead nowhere, and preparatory courses that do not prepare students for subsequent courses. On many campuses, these courses have high minority enrollments and have become associated with minority students. At Berkeley, for example, we teach a course called pre-calculus. One year, 422 students enrolled in the course, only one of whom, Danny Lescano, went on to receive a grade of B- or higher in second semester calculus. It makes you want to name the course after him. The evidence is overwhelming that few students who take remedial courses ever complete science degrees. Yet the students who start these courses are not aware of this. Truth in advertising should apply to college catalogues. The course descriptions might include the proportion of students who take the course that successfully complete the courses for which it is a prerequisite.
So, at the end of our inquiry, what had we learned?
- Many Black and Hispanic students wanted to major in math and science but very few completed the prerequisite entry-level courses.
- Our ideas about why minority students failed calculus clearly were wrong.
- Affirmative action programs were not producing math and science majors. It was clear that they were helping some kids stay in school, but they weren't helping students in our field.
- Many Black students did not use the services that were designed to help them.
This last point is of special importance because many Black students are suspicious of appeals made to them based on ethnicity. These students also dislike the idea of remediation. They see themselves as the tutors, not the tutees. They do not choose to come to a Berkeley because they want to learn about being Black. They choose it because they believe in the institution's ideals and its elitism.
When the university sends a letter as ours did, "Dear Minority Student: Congratulations on your admission to Berkeley. Berkeley is a difficult institution. You are going to need a lot of help and we are here to help you," the students disregard it. Their response is, "Oh, that's for those air-heads over there." They associate "help" with the kids they had known in high school who were in the bottom of their class and in the compensatory programs. They do not relate to such appeals.
In 1978 we began to experiment with solutions. Our idea was to construct an anti-remedial program for students who saw themselves as well-prepared. In response to the debilitating patterns of isolation that we had observed among the Black students we studied, we emphasized group learning and a community life focused on a shared interest in mathematics. The program was essentially a cluster program and was an adjunct to the regular courses. In contrast to the traditional remedial programs that offered reactive tutoring and time management and study skills courses which have a questionable scholarly base, we provided our students with a warm, emotionally supportive academic environment. Most visitors to the program thought that the heart of our project was group learning. They were impressed by the enthusiasm of the students and the intensity of their interactions as they collectively attacked challenging problems. But the real core was the problem sets which drove the group interaction. One of the greatest challenges that we faced (and still face today) was figuring out suitable mathematical tasks for the students that not only would help them to crystallize their emerging understanding of the calculus, but that also would show them the beauty of the subject. Our goal was then (and continues to be now) not merely helping students pass calculus or even to excel at it but, rather, producing mathematicians (or at least students who could pursue graduate work in the field if they chose to do so.) We knew that the program goals had to be congruent with the goals of the institution, i.e.: focused on excellence, on the production of Rhodes Scholars, and the like.
It took a little work to teach the students how to work together. We were able to convince them in our orientation that success in college would require them to work with their peers, to create for themselves a community based on shared intellectual interests and common professional aims. After that, it was really rather elementary pedagogy.
In a sense, the greatest break with the past was to take a genuinely empirical stance. We did not question that minority students could excel. We just wanted to know what kind of setting we would need to provide so that they could. We also recognized early on that we only would be successful if we depoliticized the issue of minority access. We had to link our program with other issues that the faculty cared about, such as producing quality majors, and de-emphasizing the purely political characteristics of the program so that it could take hold in academic departments. From the beginning, therefore, we served students of all ethnicities, although the minorities were, in fact, a clear majority in all the sections. The effect was that many middle-class Black and Latino students found it comfortable to participate because it was a way for them to establish quickly the multi-ethnic social environment in which they were most comfortable. For the urban Black and Latino students the workshops, as we called them, were an environment in which they were the majority and the white students the minority, making it easier for cross-ethnic friendships to form. In effect, the workshops provided a buffer easing minority students' transition into the Academy.
The results of the program were quite dramatic. Black and Latino participants, typically more than half of all such students enrolled in calculus, substantially out-performed not only their minority peers, but their white and Asian classmates as well. Black students with Math SATs in the low 600s were performing comparably to white and Asian students whose Math SAT scores were in the mid-700s. Many of the students from these early workshops have gone on to become physicians, scientists, and engineers. One Black woman became a Rhodes Scholar, and many others have won distinguished graduate fellowships.
By 1982, more than 200 ethnic minority students were being served in the workshops which were then run cooperatively by a faculty committee, the College of Engineering, and the Student Learning Center. In 1983, however, when our FIPSE grant ended, there was open warfare. The faculty and administration were fighting over control of the program. Unfortunately, the faculty lost and a period of balkanization followed, with small, separate programs proliferating on campus.
But there was a more fundamental change taking place in the mid-eighties that would in any case, have forced the reorganization of these programs. Today, on the Berkeley campus, there is no longer any dominant ethnic group. Fewer than fifty percent of all undergraduates are white and roughly one-third of the incoming freshmen are Black, Latino, or Native American. The time had come when "adjunct" programs were no longer feasible or desirable. It was time to address the efficacy of the introductory courses.
We realized that in the past we had, in effect, been running a program aimed at saving our own victims. We were not trying to address the central reality that our courses were not serving their intended populations. Ultimately, one must realize that the Black and Latino students who make it into higher education are national treasures and must be treated as such. They are rare individuals and their success will have important ramifications not only for the academic disciplines and professions they pursue, but for the very fabric of American society.
I find it hysterically funny to look at the CSU and UC projections of the California students who will be served in our universities in the years 2005 and 2010. The CSU Chancellor, and the UC President, each say the State needs several new university campuses. What are their enrollment projections based on? They appear to be based on increasing the admission and retention rates of Black and Hispanic students who by then will represent a majority of the California population. No one points out that all the actual minority enrollment, retention, and graduation trends are very far from what they will need to be, if the projected figures are to be realized. Moreover, no one seems to observe that the few minority students who actually do get through the system graduate in only a handful of majors.
Now, let me look at these projections from another view. Last year, I believe that there were at most twelve Black male students who had been students in L.A. Unified who received B.S. degrees in mathematics, chemistry or physics in California public colleges or universities. For each one of those students, using FBI and LAPD data, I estimate that 100 of their peers died by violence. Yes, a Black male elementary student in L.A. is 100 times more likely to die by violence than to get a degree in math, chemistry, or physics. Such data raises many fundamental questions that we are obligated to address. They also speak to the importance of making sure that every one of our Black and Latino students excels.
Now, let me look at the issues we faced in reconstructing our workshop program. The first is the abysmally low quality of freshman instruction in large public universities and, for that matter, in many expensive private ones, as well.
Six hundred thousand freshmen a year take calculus; 250,000 of them fail. Something is fundamentally wrong. What I find even more amazing than this high failure rate is that calculus -- now here comes my prejudice as a mathematician -- is by just about any standard -- liberal arts people, trust me -- one of the greatest intellectual achievements of western civilization. The stuff drips with power and beauty. It rendered thousand-year-old questions immediately transparent. Calculus is truly amazing. But, how many students who take the course as freshmen look up and say, "Holy shit, that's amazing!" How often, math faculty members, have your students had that experience? I mean, the stuff just sort of goes by. No passion, no soul.
Why do so many students fail these courses? Our initial idea once again was to blame the students, albeit in a more sophisticated way than previously. The students, we thought, did not have "higher-order" thinking or problem-solving skills -- they just did not know how to think--they did not know how to pull the problem out from the words and find the relevant principles. However, when we tested this idea, we found once again that we were basically wrong. When we looked at students enrolled in first term physics, for example, we found there were some students who couldn't extract the problem from among the words and find the relevant principles. But they were relatively easy to help if they had enough prior exposure to physics. The majority of the students having difficulty fell into two groups. Group one were the kids who had intellectual integrity, but no meaningful prior exposure to physics. They came upon a hard idea -- inertia, angular momentum -- these are hard ideas. If you are not a physicist and you take physics, you'll know that. If you are a physicist, it is a little hard to realize these are hard, but they are hard. The students would spend four or five days trying to figure out the concepts. They had learned not to let anything pass. The result was they would get buried in an avalanche of formulas.
Other kids had really strong math backgrounds. Their response to the massive amount of material that was dumped on them was to treat everything as an isolated, abstract mathematical problem-solving task. They never had a chance to develop the underlying physical intuition (and it was sure as hell true that the standard physics labs didn't help any). They were treating physics as if it were mathematics or logic. The courses contained so much material that students had no time to develop an understanding of the physical concepts, the connections among these topics, or the relationship between the physics they were studying and the mathematics they knew.
Once again, we were forced to deal with the fact that the problem wasn't really the students. Students, in fact, respond to whatever you give them. What they were being given here were courses that had become so compressed, so devoid of life and spirit, that there was no way to really master the ideas at the level necessary to succeed, let alone become a major.
These introductory math and science courses took form at a time when there was a surplus of students interested in science. These courses came to be thought of as service courses. Everybody teaches their freshman courses for somebody else's major. Now, times have changed. Very few students are interested in math and science. CIRP data from UCLA collected and analyzed by Sandy Astin and Ken Greene indicate that in 1966, 4.6 percent of high school seniors who took the SAT were interested in mathematics as a major. Today, it is about 0.6 percent. We are teaching courses created at a time when filtering was a necessity. Now, freshman courses need to inspire students and invite them into the major.
We recognized that if we were to try at this point to improve instruction for everybody equally we could only make a slight difference. Our resources were relatively limited and we didn't want to lose the minority students. We decided to eliminate the adjunct workshops and instead to strengthen and intensify certain sections of the regular freshman calculus course. Our idea was to construct a hybrid of the regular discussion sections and the "math workshop."
In freshman calculus, when we looked at the issue of problem-solving, we faced difficulties. The first was the absence of genuine problems to solve. What passes for problems in calculus is a set of ritualized exercises that can be addressed by mastering a limited set of algorithms together with a few special cases. The exceptions -- indefinite integration, convergence of series of constants, which are fun to teach and really are excellent domains for teaching problem solving -- have unfortunately no place in a contemporary calculus course, as so many in the calculus reform movement have pointed out.
The second difficulty is that we don't have a due how to teach problem solving in a way that promotes the development of generalized skill. Using state-of-the-art materials, such as those developed by Alan Schoenfeld, for teaching a topic like indefinite integration, we can help students to get very good at one particular task. Unfortunately, experience has shown that such instruction gives students no advantage in mastering subsequent topics.
The final issue, which we are only now beginning to address, is how to make it possible for faculty members who are interested in working on course reconstruction or on the development of minority mathematicians to do so as part of their professional work. In the past the individuals who worked on these, what were then seen as quasi-professional issues, did so as personal work, almost as hobbies: "You play golf," "I work with the Black kids." The scale of the problem now is such that many mathematicians will need to engage in activities that are necessary for the future life of the profession. If this is to happen, such work must become a regular and rewarded part of departmental life. This in turn will require that faculty and administration redefine responsibilities of departments and support these redefinitions by new review and department budgeting procedures.
Let me restate this. It means that the administration has to re-think what the collective responsibilities of departments are. Are departments only responsible for research and for body-count teaching? Or are they responsible, in some way, for the future of the institution and the future of their own disciplines? If the latter is so, one has to think about ways of rewarding departments for playing their proper role. Not that every faculty member should do this, but each department has to be responsible for contributing to solutions to these problems. The rewards for the departments have to be real -- space, faculty appointments, support for more graduate assistantships, and so on.
For the most part, at least in the beginning, faculty members who do this work will have to be senior, partly because junior people need to establish research careers, but also because the changes that have to be made are structural in character. It takes ten or fifteen years to know what kind of changes will be sustainable in an institution. Junior people have little insight into such work.
If senior people are to engage in this work for the department it is important that they be able to do so in ways that do not lead to stigmatization. A hint: Don't confuse this work with better teaching. If you focus on teaching alone, you lose. It has to be about assuring the future of the profession and the future of our institutions. But it is not only the academic departments that need to change.
When the university works it does so because the faculty plays its proper role and the administration plays its. In student affirmative action, however, the administration does it all, including tasks that are clearly academic in nature. Fixing this is not a trivial matter, because there is a long history of administrative dominance in this area with which we have to contend. But that is ultimately what we will have to do.
An especially important issue will be learning to work with the EOP and minority affairs bureaucracies. They are easily and unfairly blamed by faculty members and administrators for the failures of minority students. It was ludicrous to think that their administratively constructed offices could have even addressed the problem of minority student failure in math and science.
Whereas some of the work is really academic, other parts of it are administrative. These include such issues as housing, financial aid, student organizations, and the like. We have to re-examine the ways faculty and administration work together to help students advance.
What's the bottom line here? The country is fundamentally changing. There are a growing number of schools, where all of a sudden, extraordinary results are produced. Garfield High School produces 27% of all the Hispanics in the United States and Puerto Rico who pass the AP calculus exam: One high school. I have visited ten high schools that produce relatively large number of Black and Latino students who passed the AP calculus exam. Not one is a magnet school. When you go out and look at these schools and you teach in them, what do you find? A group of senior teachers who chucked the remedial courses. They cared about mathematics and wanted to teach real courses. They built a peer group that supported kids' involvement in school work and in mathematics.
Now, instead of talking only about Berkeley, I want to talk about some of the other sites with which we've been working. At the University of Texas at Austin, a group of faculty members and an extraordinary administrator said: "Let's figure out what we have to do in calculus to produce lots of Hispanic and Black mathematics majors." They took our Berkeley idea: They intensified some sections of freshman calculus. They built group work into the course and made it clear to the students that it takes fifteen hours of work, not eight, to excel. They made it possible for the kids to take slightly fewer courses at a much greater depth and level of intensity. They unabashedly advocated for these students to become mathematicians. They set up a system where the kids in the intensive courses could be graded against the curve established by the regular sections -- same exams. What happened? Minorities: 3.53 average GPA; Others: 1.66.
At CCNY, the faculty believed that their students would never become mathematics majors. The students were all working forty hours a week and had little interest in being challenged. Inspired by the Berkeley program, they decided to test their assumptions. A team led by Laura Shapiro interviewed students, and what did they find? Lots of these students had saved up tuition money so they could go to school. The strongest students found the courses uniformly unstimulating and unchallenging. In response, the department set up more challenging "intensive" sections and the result of the first semester was a 3.2 average grade for the minorities against about 1.8 for class average.
Now let's look at another kind of institution and an administrative solution that didn't work. At Oklahoma they have a serious retention problem, the worst in the "Big Ten", or the "Big Eight", or whatever. They changed their admissions requirements and dropped the bottom quarter of test scores from admissions. What happened? Absolutely no change in retention. Retention in large measure is a function of the way students interact with each other and with the institution. It is a measure of connectivity, of the quality of life on campus.
On this campus, you did studies which looked at students' undergraduate experiences. One interesting result: Students study roughly the same amount of time, whether or not they are employed -- about ten hours a week. It seems to have nothing to do with how much time they spend at their jobs. I believe that, in fact, the amount of energy students put into their studies is more directly related to their perceptions of their chances of succeeding at the university. If students see clear evidence that they can make it, they will use whatever flexibility they have to reduce work hours and increase hours devoted to study.
In conclusion, the time has come to reexamine undergraduate instruction and to make it more responsive to the needs of today's students. We can no longer offer courses that half of our students fail, nor can we lower our standards. The challenge is to reconfigure undergraduate science and mathematics education in ways that will inspire students to make the choices we have made. This can only happen if we change the boundaries of faculty responsibility. It is the faculty that must take the lead.