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This is a description of a set of interwoven themes which may be of help to people who begin a *Math Excel*, or Emerging Scholars Program. These reflections may also be of use to other teachers of mathematics. The work we describe is centered in a freshman calculus intervention program. We indicate the success of our efforts in the section on **Progress** and describe the main features which characterize our *Math Excel* Program there. In the **Process** section we elaborate how the three themes of mathematics content, mathematical processing and group dynamics are intertwined in our workshop sessions.

These themes are well known in different areas of knowledge, but it seems useful to pull them together and document their influences in the setting of a freshman mathematics course. The state of our knowledge, and the organization of our work, are still changing. We thank Professor Uri Treisman [2] for his basic work and insights in putting together these programs. We also thank the Provost of Northeastern University, Michael Baer, for inviting Uri Treisman to this campus, stimulating the beginning of this program, and supporting it thereafter. The work has been supported by generous grants from State Street Bank, the Northrup Company, the Cabot Corporation, Polaroid Corporation, and from unrestricted funds directed to us by the President of Northeastern University, President John Curry.

The goal of the *Math Excel* Program at Northeastern University is to increase the number of people in engineering, mathematics and scientific careers. It specifically focuses on groups who come to the University well prepared but who leave these careers for non-academic reasons. It is not a remedial program. Rather it admits well-qualified people and pushes them to achieve at a high level through the formation of a community of scholars. Students are nominated for this program based on their performance on the mathematics S.A.T. exam and on their high school record.

The *Math Excel* Program is grounded in this community of academically motivated engineering, mathematics and science students. The heart of the Program is a calculus workshop in which students learn to do mathematics at an honors level by working together on challenging enrichment materials related to their courses. The workshops form a separate course, with credit, which is graded on a passûfail basis in order to discourage too much competition amongst the students. The Program is based in the ideas and experiences of Professor Uri Treisman at the University of California at Berkeley. A majority of the students in our workshops have been engineering students.

The workshops are run by graduate teaching assistants who have been trained by Professor Treisman's group, now at the University of Texas at Austin. Undergraduate engineering student mentors with the same training also help in the workshops. These mentors are chosen from the older *Math Excel* students. The point of the training is to assure proper group dynamics and work by all students. The training focuses on leading the students to discover answers while keeping groups active and at work. The workshop leaders are trained to mix students of varying abilities in each group. They seek to maintain order while allowing healthy conversation and personal characteristics to be present as well as respect for others' strengths. The academic empowerment of the students, and of the group, is the result.

The *Math Excel* Program has been a success at Northeastern University. The students have usually averaged a letter grade, or approximately 10 points, better than all non-Excel engineering majors on the common final examination in the engineering calculus course sometime during their freshman year. The program is now in its fifth year. The Program has retained many more students who began it than is the case in the other sections of the same course. The attrition in all other sections of the freshman engineering calculus course from the fall to the winter quarter has been about 28%. The attrition in the *Math Excel* section has averaged about 13% per year. At the end of their first year of studies, several of the *Math Excel* students have been invited to join the Honors Program at Northeastern University. The focus of the Program is upon African-Americans, Native Americans, Latin-Americans and women but others are invited to be part of the program. It is in no way exclusive of any groups, and students who ask to join are always welcome.

The students learn calculus in a new, or reformed, format including the use of computer and calculator technologies and a new state-of-the-art Excel Computer Lab. The Excel Computer Lab in the Mathematics Department is organized to foster group work on mathematical problems. The student mentors who manage these Labs are chosen, in part, from the successful graduates of the Excel Program. Support is provided in other engineering and science subjects for the students in this Program by NUPRIME, Northeastern University Progress in Minority Engineering.

The Excel students remain in this Program for their entire freshman year. Since the students will go into Cooperative Educational jobs related to their majors during their sophomore year, we are able to emphasize the value of learning to work in groups, in the subject area of their major, from the beginning. This will be vital to their success as coop students, as it will in their adult careers. Each spring the *Math Excel* students organize themselves and their closest friends into the same Cooperative Education Division. Thus their periods of employment in work, which is related to their major, are coordinated. They are in school during the same periods of time throughout their stay at Northeastern University. It is then possible for them to keep the community of study groups that they have formed during the freshman year. The best of the previous Excel students are undergraduate mentors for the present Excel freshman students.

These students are also of great value in the growing relationship between Northeastern University and the Boston Public Schools by providing role models for the students in Boston. The Mathematics Department at Northeastern University is quite involved in the reform of mathematical education in the Boston Public Schools. We are encouraging Excel students, as well as other mathematics majors, to help these younger students in various mentoring and advising capacities.

As the number of successful *Math Excel* classes grows, the influence of upper class graduates of this freshman program is being felt in all engineering, mathematics and science curricula as well as in freshman Excel classes. The presence of these students is raising the levels of the courses in which they enroll, as well as the attitudes and expectations of some faculty towards the populations of students in the Excel classes.

We now have a few years of experience and can give longitudinal data. Our attrition was about 4% for the first year and additional 15% in the second year for our first group of students, those who entered in the fall of 1992. By comparison, the ordinary Engineering College class lost 29% during that first year and losses were near 15% more during the second year. At the end of two years the *Math Excel* students retained 81% of their group, while ordinary engineering sections have retained about 55%. In our next class, which entered in 1993, we retained 85% at the end of that year, versus 70% in the ordinary engineering classes. These statistics continue in the same vein during the following years.It must be emphasized that our populations are lost in far greater percentages on a national scale, than the ordinary engineering student.

Maurice Gilmore, Alfred Noel and Thomas Stephens are willing to travel and speak about our *Math Excel* Program. We have described our experiences at the American Mathematical Society meeting in Lexington, Kentucky on March 19, 1994 and at the University of Massachusetts at Lowell on May 4, 1994. Professor James Graham-Eagle has now begun a similar program at UMass. There is a program to expand *Math Excel* Programs into the milieu of community colleges. This program is run by Professor Michael Freeman of the University of Kentucky at Lexington.

In what follows it is useful to keep three categories in mind. We are consciously working on three different, but intermingled, arenas of activity among the *Math Excel* students. The first and foremost is the activity of learning the content of an engineering calculus course. The second activity consists in learning how to work on mathematics which might be called mathematical processing. This amounts to acquiring personal strategies for successfully solving unusual, as well as routine, mathematical problems. Part of this activity includes the ability to be self-reflective as one works on a problem. The third area consists of the social group dynamics among the students and between the students and the teachers and managers of the *Math Excel* Program. We choose to write our descriptions in the temporal units of the three quarters of freshman year study at Northeastern University. This is the result of each of the three arenas of activity changing into different forms during each quarter.

Students in any academic support program must feel comfortable if the program is to be a success. Icebreakers and social programs for students and staff are important in the initial formation of the group. Students in new settings can be intimidated by their surroundings as well as by new people. At Northeastern University, some students are the first generation in their family to attend college. They feel a great deal of pressure to do well. There are students who have never been in an urban setting and who feel very displaced. Excel students, consciously or subconsciously, have a desire to get to know the people with whom they will be spending as much as six extra hours each week. We provide support in these matters, in particular through our academic administrator. Of concern to us as *Math Excel* Program directors is the need for students to identify in relation to their gender and ethnic identity. During freshman year, students seek people who look like themselves. It is their attempt to see themselves at some future time. Many students view the person whom they select as a role model, and the person chosen may be unaware that he/she has been chosen for such a role. In the event that there is no person connected with the program who is suitable for a student's identification, then the choice will come from elsewhere. A choice from another part of the university or an identification along lines other than gender and ethnicity may occur. Speakers who are good role models and who possibly could appeal to the Excel Students can be important persons in this process. Some of these issues of comfort continue throughout the year. They have more urgency in the fall quarter.

In analogy with a theorem from the freshman year course, we have the following useful observation by the third author:

The Comparison theorem:

If the following conditions hold true :

1. Two or more students know each other socially.

2. These same students have at least one class in common and have studied together at least once.

3. These students have attended two or more social events together.

4. These students share experiences together; they are friends.

Then at least one of these students will compare themselves to the other(s) by supposing that:

1. "I am at least as smart as the other(s)."

2. Therefore, **"I can duplicate the study habits of the other(s)."**

Freshman students, left to their own devices, need a few semesters of reality testing to learn that this theorem is false. It is important to be aware of the operation of this theorem as one advises groups of freshman students.

In the fall of 1994 we enlisted the help of Professor Rae Andre of the Human Resources Group in the College of Business Administration to work with the social dynamics, the interaction of *Math Excel* students among themselves and with our staff. At her suggestion we passed out questionnaires regarding student interactions in the group work sessions. This is now done at the end of each week during most of the fall quarter. The purpose is to get the students to reflect on their own behavior in relation to others in the groups. The students are making friends, learning to cope in the new environment of a large urban university, and assessing the difficulties of college courses. The help with mathematical and other course content, emotional support and friendships afforded in the *Math Excel* Program are judged important by the students. The matters of attendance, attention to the work, learning to be flexible and tolerant of others in group work and of mutual respect for everyone in the workshop setting are being attended to by most of the students with increasing zeal.

In this quarter the students are initiated into various majors, each involving a mode of scientific culture. The matter of mathematical process, referred to by Schoenfeld [1] as metacognition and acculturation, is also an integral part of what the teaching assistant is promoting in the workshop sessions. Often students have been unwittingly taught that mathematics problems are either solved in five minutes, or "you don't know what you're doing, so give up." We teach the opposite. Everyone can do calculus among our chosen group of students, but many need to learn to persist, to attempt different strategies on the same problem, and to develop an awareness of how to deal with their own frustration and impatience. Working with others can help enormously in matters of patience, flexibility and persistence. The refusal of the teaching assistant to give away answers usually gets some student attention early in this quarter. Students are correct in their belief that the teaching assistant could save them a lot of frustration.

The course content is that of differential calculus, with attention to the early awareness of differential equations and the modeling of scientific situations. Newton's method, and the tangent line approximation as a precursor to Euler's method in differential equations, are used in laboratory assignments as well as forming part of the course content.

**2. The Winter Quarter.**

During the winter quarter a different tone often develops in the *Math Excel* student body. The students have been measured against the commonly graded, common final exam in the fall quarter calculus course and have proven themselves. Other courses, notably physics, may be troublesome. They may call into question the commitment of so much time to mathematics. Many have friends in other majors that seem to require far less effort to attain good grades. The group support that they now give to each other across the academic board is taken for granted by some students, but in fact keeps them on task. Significantly, when they pursue a goal they do not act alone. The relatively empowered group of young adults may now look for better, and perhaps easier, ways to accomplish the same academic ends. The details vary from year to year, but in the winter quarter a larger number of problems arise. The teaching assistant usually tries to get students to think more deeply and flexibly about problem-solving, and assigns more problems that cannot be finished quickly. The result may be that a group of students will perceive the teaching assistant as the source of their problems. Other students may come to the support of the Teaching assistant. This reflects the degree to which subgroups and a pecking order now form among the entire *Math Excel* cadre. It is quite useful for those in charge of this sort of program to be aware of the flow of these sorts of processes.

On the other hand, staff members become fond of these students but must keep some degree of distance in order to stay focused on the mathematical content and process. Being a caring adult to the students may lead some of the students to believe that academic standards will be relaxed in favor of friendship. The bonding between staff and students increases throughout the year, and friendship with standards must be the rule. It is important to remain aware of the difference between academic standards and the retention of control in areas which are peripheral to the learning process.

During this time we must keep the mathematical content at a high level and improve the mathematical processes by which the students learn to work more deeply on problems. This is a central focus of the workshop. Now the more social processes are easily overlooked. When trouble arises, it may then be perceived as a merely personal threat. To know of, and to expect, the flow of the above dynamics may help the managers to adapt. It is important to face the challenges and problems that arise as real problems, worthy of open discussion with the *Math Excel* students. It is in these discussions that flexibility in areas that are peripheral to the learning process can be an asset.

The matter of self-esteem comes up in the fall quarter and continues into this quarter. By now the students know that the level of work required in a university setting is greater than that which they had to face in high school. The first few hour exams during the fall quarter are usually a serious reality check. Some students have been stung by those exams and need support to continue. It needs to be pointed out that there is a choice being made. The intensity of work done must increase in order to remain in scientific majors. This is different from not having the mental equipment to go on. The personal questions, such as the change of work habits, are hard questions with which to deal.

The mathematical content in the winter quarter is that of integral calculus. There is an emphasis on modeling, and some discussion of integrals as a method of solving differential equations including those describing motion under the influence of gravity alone. Numerical methods of integration are taught using a spreadsheet on a computer. The students will be in the workplace within six months and the spreadsheet has a broad usefulness to them.

3. The Spring Quarter.

Responses to the problems that arise in the winter quarters usually make each spring quarter of *Math Excel* at Northeastern University slightly different from other spring quarters. It may be in the number of outside speakers, the scheduling and the amount of time spent in the workshops, or the number of workshops that are devoted to reviewing physics problems to prepare for mid-term exams. It may also happen that the teacher of the calculus course or the teaching assistant who runs the workshop is asked to change how business is done. The success of the students in mathematical tasks must remain the central focus for the organizers.

The mathematical content of the spring quarter consists of differential equations with separable variables together with the method of phase-plane analysis on more difficult equations including predator-prey equations and systems. In addition, Taylor's series are studied as approximations to functions. Infinite series are used to solve differential equations. Euler's numerical method is implemented by the students using spreadsheets. The students must complete a modeling project, using differential equations. This project requires them to study a process involving change. The topic should be something close to their experience, or one that is of interest to them. They are encouraged to work in groups on this project. There are more word problems in the spring quarter. The emphasis on modeling is greater. The discussion of problems is more necessary and the possibility of different correct answers is more widespread.

The spring quarters have been quarters of hard work, students' exhaustion with school, and success in the *Math Excel* groups. There is a more serious approach by some students to the solving of problems. The struggles of the winter quarter usually pay off here. The probable arrival of a productive spring quarter may be an important ingredient in sustaining the morale of teachers who must give attention to the problems of the winter as real problems,. This future is important in order to help the managers stay focused on the mathematical tasks at hand. They are the real goals of the program, and they are the salvation of those who struggle in midstream.

One point is useful to note separately. The addition of new students to a program such as this, after a quarter of cohesiveness has gone on, should be done with care. Such students should receive a careful description of both the global goals of the program and the local conditions of this particular group. The already-formed groups should also be surveyed in order to discover which of them is most open to receiving a new member. It is our experience that weak students will not make it. Students must be strong enough to come into such a group with some degree of power and may have to prove themselves to the group.

- Schoenfeld, Alan H., "Learning to Think Mathematically: Problem Solving, Metacognition, and Sense Making in Mathematics",
*Handbook of Research on Mathematics Teaching and Learning: a Project of The National Council of Teachers of Mathematics*, ed. D. Grouws, New York: MacMillan, 1992: 334-370. - Treisman, Uri, "Studying Students Studying Calculus: A Look at the Lives of Minority Mathematics Students in College",
*The College Mathematics Journal*, Vol. 23, No. 5, Nov. 1992: 362-372.

Last modified September 27, 1996

For more information, contact: gilmore@neu.edu

Comments or corrections to: alexsuciu@neu.edu