Symbiosis of Mathematician and Math Educator in Teaching Math Teaching
Heather Calahan, Ted Gamelin, and Bruce Rothschild
Santa Monica High School, UCLA Department of Mathematics
Recently a great deal of attention has been paid to the role of mathematics departments in the preservice training of high school and middle school mathematics teachers. The Conference Board of Mathematical Sciences published in 2001 a landmark study with recommendations for the undergraduate mathematical education of teachers. A key recommendation of the CBMS report is for an undergraduate capstone course for future teachers that connects undergraduate coursework and school mathematics.
We would like to report on an experimental math teaching course that plays the role of a capstone course for undergraduate mathematics majors at UCLA who are going into secondary mathematics teaching. The main innovative feature of the course is the manner in which a mathematician and a math educator coteach in a carefully coordinated team effort. Central to our report is a series of vignettes indicating various ways this symbiosis between the mathematician and the math educator can function.
We also point out several problems we have encountered, problems that may be shared by other mathematics departments in research universities. One problem is the tightness of resources for an enriched teacher preparation program in an environment where undergraduate programs take a back seat to research and graduate programs. Another problem is the lack of mathematics ladder faculty with expertise or willingness to teach in a teacher preparation program.
We begin with a thumbnail description of the math teaching course and a brief history of how it evolved. The main part of the report is a description of the coteaching, including the vignettes. This is followed by a description of other aspects of the course, including philosophy, content, and grading. We conclude by describing several problems we have faced, and we also describe some bonus benefits.
OVERVIEW OF THE MATH TEACHING COURSE
The math teaching course offered by the Mathematics Department (Math 105AB, Teaching of Mathematics) is a twoquarter course restricted to senior math majors. The course meets once a week, and there are nine class meetings each quarter. Each class meeting extends for four hours in the late afternoon and early evening, with a short food break. Students in the course also spend some time observing in local schools. The course is currently offered fall and winter quarters, to avoid the spring exam season in the local schools.
The course serves 20 to 25 undergraduates each year. Students take the course to satisfy the requirements of the Mathematics Department’s Subject Matter Preparation Program (Math SMPP). Completion of the Math SMPP is one of two routes for satisfying the subject matter preparation requirement of the California singlesubject mathematics teaching credential. The alternative route requires that students pass a battery of exams (the CSET exams).
Most of the students enrolled in the math teaching course are in the MathematicsEducation Intern Program. This is a program sponsored jointly by the Mathematics Department and the Teacher Education Program (TEP) in the Graduate School of Education and Information Studies (GSEIS). Students in the program begin taking education courses in their senior year, and after summer school they teach for a year in Los Angeles schools while returning to UCLA weekly for seminars. At the end of the teaching year they receive a singlesubject mathematics credential and an M.Ed. degree. The financial aspects of the program are critical in attracting and enabling the students. In their senior year, participants receive stipends funded by the California Mathematics and Science Teacher Program (CMST), which is administered by the University of California, and in their teaching year they receive full teacher pay. The MathEd Intern Program has been very successful in attracting high quality UCLA mathematics majors to teaching. Retention rates are high, and more than 90% of the students who graduated from the program during the past ten years are still teaching.
UCLA has a successful multitrack undergraduate mathematics major program. With more than 900 undergraduate mathematics majors, it is the largest such program in the country. In recent academic years, more than 200 math majors have graduated each year. It is not surprising then that more students seek to enroll in the math teaching course than can be accommodated. One way we have adapted to the problem of excess demand is to adopt a posture of not simply preparing teachers but of preparing teacher leaders. Only senior math majors in the Math SMPP are permitted to enroll in the course, and a formal application process has been set up for admission to the Math SMPP. Preference is given to students who have strong academic records in mathematics, who show a serious interest in a teaching career, and who demonstrate potential to become teacher leaders.
Until recently, the math teaching course also served a handful of graduate students in the mathematics track of the TEP, seeking the mathematics singlesubject teaching credential. These students entered the class by enrolling in a Mathematics Department course in the professional course series (Math 370AB).
EVOLUTION OF THE COURSE FORMAT
The math teaching course was originally a onequarter course conducted by a math educator with a masters degree in mathematics and substantial high school teaching experience. In 1996 the course was extended to two quarters and the course format was revised. Under the new scheme, a senior Mathematics Department professor with experience in the math education scene (Phil Curtis) conducted the first two hours of the course, focusing on topics from school mathematics presented from a historical point of view. The text used was Burton’s History of Mathematics. Students solved problems chosen from the text and other sources, both individually and in groups, and they presented solutions to the class. A math educator conducted the portion of the course focusing on teaching methods and arranged for student observations. Though the instructors of the two parts of the course communicated with each other, they did not attend each other’s sessions, and the two portions of the course were largely independent.
In the 200102 academic year, responsibility for the course was assumed by a different Mathematics Department professor, Ted Gamelin, who began to experiment with course content and format. Since the 200203 academic year, Heather Calahan has assumed the role of the math educator, and since the 200304 academic year, Bruce Rothschild replaced Ted Gamelin in the role of mathematician. These four years have seen substantial changes in the math teaching course, guided by the recent CBMS report and current research on what teachers should know to teach mathematics effectively. The groundbreaking textbook of Usiskin and collaborators, treating secondary mathematics from an advanced standpoint, replaced the math history book as the principal resource for math content. However, the main change in the course was to adopt a format in which the mathematician and the math educator coteach the course in a closely coordinated team effort. It is this aspect of the course that we believe is innovative and wish to focus on.
SYMBIOSIS OF MATHEMATICIAN AND MATH EDUCATOR
In the current course format, both the mathematician and the math educator are (usually) present through the entire class. Their parts in conducting each class are carefully planned beforehand, and they alternate roles several times during the course of each class. Through our experience in the course, we have found a number of advantages to this format.

Mathematical ideas under discussion by the mathematician can be immediately connected directly to the school curriculum by the math educator. Further, mathematical topics that might otherwise seem irrelevant to teaching can be validated by the math educator, who can indicate concretely how they might appear in the school classroom.

In the other direction, mathematical topics arising in discussions of school mathematics can be immediately connected in various ways to the undergraduate math courses in which the ideas appear, courses in which the students often have been recently engaged.

The mathematician provides accuracy and depth to discussions about secondary mathematics topics, some of which are not as simple as they seem, many of which are not directly addressed in the undergraduate curriculum, but all of which must be dealt with accurately in the secondary classroom.

It is important that students have the opportunity to learn new mathematics the way that they will expect their school students to learn math. This opportunity occurs when the mathematician teaches new mathematics using pedagogical methods learned from the math educator (as opposed to the lecture style format most common in the traditional university classroom). The mathematician brings to the table advanced collegelevel mathematics that is new for the students, while the math educator brings to the table a selection of sound pedagogical techniques.

The mathematician can connect secondary mathematics topics to more advanced mathematics or to current developments in research mathematics. These connections can be presented so as to enable the future teachers to make use of them in the secondary classroom. The math educator can give tips for using the ideas to enrich the classroom and to pique the mathematical interest of secondary students.

A presentation by one of the instructors modeling a school lesson can be immediately followed by an analysis and class discussion of the presentation led by the other instructor.
In order to illustrate how these features might play out in practice, we describe a series of vignettes in which the presence of both the mathematician and the math educator strengthened the conduction of the class. Most of these vignettes have been repeated in some form each year as a regular part of the course, and each of these vignettes has transpired at least once over the past four years.
Axiom systems and proof in geometry: In this sequence of activities, instructors use
Book 1 of Euclid’s Elements to develop the idea of a formal geometry proof and at the same time to understand the parallel axiom. The mathematician guides a preliminary discussion of Euclid’s definitions, postulates, and common notions, and students warm up by finding proofs of Euclid’s Proposition 1. The instructor selects various proof formats (such as twocolumn, paragraph, and flowchart) for presentation by students and discussion. This is followed by a model high school geometry lesson, in which the math educator leads students through a series of proofs of four statements equivalent to Euclid’s Proposition 29 (the first point in the Elements where Euclid makes use of the parallel postulate). The conclusion is that the four statements can be proved to be equivalent, without the use of the parallel axiom, so that any three of the statements can be proven if the fourth is adopted as an axiom. This particular presentation impresses upon students the need for axioms, and it illustrates the need for something like the parallel axiom. The math educator is able to model this presentation just as it has been used in a high school geometry class for some time. Questions that arise in this sequence of activities provide the mathematician with various opportunities to enrich the discussion and to make connections. The mathematician can underscore the idea of an equivalence relation (appearing implicitly in Euclid’s common notions), the implicit assumptions needed to prove Euclid’s Proposition 1 (particularly the completeness axiom, which can be illustrated in the rational plane), and the increase in level of difficulty in dealing with areas and volumes (with a description of Hilbert’s third problem).
The parallel postulate and spherical geometry: In order to develop an appreciation of the parallel postulate, experience with geometry on the sphere is indispensable for both secondary students and future teachers. To provide this experience, we lead from the activities in the preceding vignette to the following sequence of activities. First, students are asked to prove the sum of the angles in a triangle is 180 degrees. The students present proofs and note the dependence of each proof on the parallel postulate. The math educator then models a school activity in which students use Lenart spheres (physical models upon which one can draw great circles and measure angles) to investigate the sum of the angles of a triangle on a sphere. Students discover that the angle sum has a lower and an upper bound, and they consider secondary level explanations of the angle sum range. Students may observe that the angle sum depends on the size of the triangle, and this provides an opening for the mathematician to connect the range of angle sums with the GaussBonet formula. The mathematician might also provide rationale for having secondary students restrict their attention to spherical triangles in a hemisphere by referring to models for spherical geometry based on projective space, where three noncollinear points determine a unique triangle.
Real numbers and decimals: The basic problem is to answer the question, “What really are real numbers?”. Many high school textbooks contain descriptions that provide good illustrations of circular reasoning. (Such as: An irrational number is a number that is not rational, and the real numbers are the union of the rational and the irrational numbers. Or: A model for Euclidean plane geometry is obtained by taking the product of the reals with the reals, and the real numbers are constructed by introducing arithmetic operations into a line provided by Euclidean geometry.) Students investigate current algebra textbook presentations of the real numbers and their properties. The mathematician is in a position to say something about the structure of the real numbers (arithmetic, ordering, and completeness), to outline the three main methods for constructing the real numbers (associated with Weierstrass, Cantor, and Dedekind), and to connect the ideas with undergraduate math courses. The math educator focuses on the Weierstrassian approach, which is to define real numbers to be decimal expansions, and to relate the ideas to the middle school classroom through activities featuring repeating decimals and rational numbers at a level appropriate for the school classroom.
Arithmetic: The mathematician conducts an activity designed to review arithmetic properties of the real numbers and various subsystems. Students connect with their undergraduate work by determining whether the subsets are groups, rings, or fields. The math educator conducts activities in which manipulative models are used to provide rationale for the commonly taught operations on integers and rational numbers. The mathematician can provide mathematically acceptable explanations of why negative times negative is positive, why one “flips and multiplies” when dividing fractions, and why one cannot divide by zero, both for the secondary level and for the undergraduate level.
Symmetry groups: This topic area includes a spectrum of activities, some appropriate to the middle school and others appropriate to the high school classroom. Activities include finding the symmetries of various geometric objects, such as an equilateral triangle and a rectangular box, and finding a “multiplication table” for the operation of composition of symmetries. The math educator identifies specific activities with specific grade levels. The mathematician connects the ideas to the math courses students have taken, particularly the algebra course in which groups are introduced and studied. The math educator models a school activity in which students construct nets for various solids as well as one in which students investigate tesselation. The mathematician introduces the regular solids and the Archimedean solids and uses a soccer ball (buckyball) to investigate properties of the icosahedral group.
A probability experiment: The mathematician begins by modeling a sampling activity appropriate for a middle or high school classroom. Students are divided into groups, and each group conducts a probability experiment in which balls are drawn from the proverbial urn (actually a paper bag) containing black and white balls. Each group records the results of a specified number of draws (with replacement) and conjectures the number of white balls in the urn. Groups report their results. The experiment is arranged so that the probability of a group predicting correctly is approximately 50%. (This provides a crude check on rampant peeking.) Then the groups are informed that in fact each urn has the same number of white and of black balls, the groups pool their results, and with better than 95% probability of success the class is able to conjecture the correct number of white balls in each urn. The mathematician discusses random sampling and says something about computing the various probabilities, connecting the ideas with the basic undergraduate probability course. (A binomial distribution is relevant, and the probabilities can be estimated by those for the approximating normal distribution.) The math educator follows up by conducting a discussion of the mathematician’s presentation of the basic probability lesson, from the point of view of a school classroom. The math educator goes on to present model probability lessons appropriate for supplementing various secondary mathematics courses. These include, for example, a probability investigation in which students determine the theoretical probability that a rod of a given length broken randomly into three pieces can form the sides of a Euclidean triangle. Students produce experimental results through the use of a graphing calculator, and they compare these results with the theoretical probability. Other activities have included the Buffon needle problem and simple geometric probability problems enhanced by programs written for the graphing calculator. The math educator discusses the role of probability in the secondary math curriculum and the fortification of the curriculum with activities that introduce probability into other topic areas.
Functions and equations: This investigation begins with an analysis of the concepts of relation, function, variable, expression, and equation. The mathematician leads an activity in which students develop and discuss various definitions for these concepts. (There are two definitions of “function” that arise at the secondary level, one as a relation, the other as an inputoutput machine. The definitions of “variable” are even more variable.) Students are then asked to solve a few equations, chosen carefully so that solution sets are at risk for being altered in the solution process, due for instance to squaring of both sides, elimination of a denominator by multiplication, or application of an inverse trigonometric function. This is treacherous ground for high school precalculus students, and also for teachers, as it is often difficult to give good explanations to students. The math educator provides many examples of student problems and student questions related to notation and to operations that can be applied to both sides of an equation without altering the solution set. Students in the course are asked to explain at the blackboard how they would respond to these student questions, and both the mathematician and the math educator may participate in the ensuing discussion.
Roots of polynomials: The math educator leads an activity beginning with the diamond method for factoring quadratic polynomials with integral coefficients and including the use of physical models for factoring and for completing the square. The activity culminates in a derivation of the quadratic formula for solving quadratic equations. The mathematician shows why the diamond method works, and then connects the diamond method to the solution of cubic equations by radicals. This leads to a discussion of the history of solving polynomial equations by radicals, featuring Cardano, Tartaglia, and Abel.
COURSE PHILOSOPHY AND COVERAGE
Throughout the math teaching course, there is a continued emphasis on four recurrent themes.
A. Problem solving: Students solve problems in groups during class time, students work regularly assigned homework exercises, and students solve problems of the week. In connection with the problems of the week, students write up both the solution and a paragraph describing how they arrived at their solution.
B. Student presentation of solutions: Most mathematics majors at UCLA have not had adequate opportunity to give an oral presentation of mathematics to a class of students. A portion of each class meeting is devoted to student presentation of solutions to the class. In some cases, this is the first time that the student makes a presentation to an entire class.
C. Multiple solutions: The diverse learners in the secondary mathematics classroom need mathematics teachers who can explain mathematical ideas in multiple ways. Mathematics teachers must be able to recognize the mathematical capital in the diverse solution approaches their students generate. With this in mind, course instructors encourage students who finish early to seek multiple solutions, and course instructors favor problems for presentation that admit of multiple solution methods. With respect to solution methods and multiple representation of solutions, the mantra of the instructors is the fourfold way: numeric, graphical, algebraic, and verbal.
D. The language of mathematics: This includes the analysis of mathematical concepts and the translation of mathematical ideas into the language of mathematics. The role of definition in mathematics is stressed. For instance, students learn to recognize the stipulative nature of math definitions, as opposed to the descriptive nature of definitions arising in normal discourse. They learn about expanding definitions, as for “fraction.” They learn that definitions can be flexible, they can vary depending upon field and circumstance, and they can be generalized often in many ways. Finally, they learn about the respective roles of undefined notions and definitions in axiom systems.
In addition to exploring mathematics topics that are important for school mathematics, the math teaching course addresses a series of other issues that are important for teaching.
1. Diverse styles of instruction: Students are exposed to a variety of instruction styles, including group, pair, and individual investigation, whole class discussion, and interactive lecture, both through their observations in local schools and through the presentations of the math educator and of various guest speakers in class. One annual guest presenter, a retired secondary teacher with a masters degree in mathematics and extensive experience teaching the high school AP statistics course, models a high school lesson on data analysis. Another annual guest presenter, a secondary school teacher with extensive teaching and coaching experience, models a lesson related to the geometry topics.
2. Appropriate use of classroom manipulatives and models: Students learn to work with tile spacers for integer operations, fraction bars for operations with fractions, cups and tile spacers for solving linear equations, and algebra tiles for computation with quadratics and for completing the square. These manipulatives are important elements in the teacher’s arsenal for teaching middle school topics. For the high school geometry classroom, students learn to use Lenart spheres in connection with a spherical geometry activity. Other physical models are used in various activities, including the soccer ball that models a truncated icosahedron. Naturally, dice are used for probability investigations.
3. Appropriate use of technology: Students learn to use various features of graphing calculators for teaching, and they learn to use compatible probes such as CBL and CBR sensors for data collection. This past year, class time was allocated to Geometer’s Sketchpad, which is a computer software program for teaching geometry.
4. Basic issues and research in mathematics education: Students become acquainted with both state and national standards through assigned readings from the Mathematics Framework for California Public Schools and the NCTM’s Principles and Standards for Secondary Mathematics along with its Research Companion. Students also read, summarize, and discuss pieces on alternative assessment, constructivism, English language learners, equity and access, literacy, and group collaboration in the mathematics classroom, to name a few. These issues are framed through assigned readings from such sources as Stigler and Hiebert’s The Teaching Gap, E.D. Hirsch’s The Schools We Need and Why We Don’t Have Them, Jeannie Oakes’ Teaching to Change the World, the NRC volume How People Learn, and NCTM’s Journal for Research in Mathematics Education.
5. Building of a portfolio: Students assemble a portfolio following careful specifications. The portfolio includes such items as a writeup of one of the problems of the week with discussion of solution process and development of several solution paths, a summary of one of the assigned readings with discussion of how it influenced the student’s conception of good mathematics instruction, and two lesson plans, each centered around a model lesson or pedagogical approach from class. Students also summarize in the portfolio their growth in understanding during the twoquarter course.
6. Professional development and support opportunities: Students learn about the NCTM and its California affiliate, the California Mathematics Council, and their associated publications and conferences. They learn about various locally based operations, such as the Mathematics Content Program for Teachers housed in the UCLA Mathematics Department, and the UCLA Mathematics Project housed in Education. They learn about the diagnostic tests and services available for school classes from the Mathematics Diagnostic Testing Project. They learn about National Board Certification as a route toward expanding professional competence.
The choice of mathematics topics covered in the math teaching course varies somewhat from year to year, though the selection has always been consistent with the CBMS recommendations. The complete log for the math teaching course for Fall Quarter 2004 and Winter Quarter 2005 will be posted on the UCLA Mathematics Department web site (see www.math.ucla.edu/mathed/). Roughly speaking, the flow of topics in the course was as follows.
The course opened with a look at concept analysis, problem analysis, and the language of mathematics, particularly definitions. There followed a look at real and complex numbers and their number subsystems, starting with the arithmetic of the integers and rational numbers, and including such topics as equivalence relations and classes, infinity, onetoone correspondences, the division algorithm, decimal representation of real numbers, repeating and terminating decimals, complex numbers, rectangular and polar representations, and stereographic projection. Abstract algebraic systems such as groups, rings, and fields were discussed in connection with concrete number subsystems of the reals and complexes. Next came a look at the function concept and at the menagerie of functions appearing in the secondary curriculum, including polynomial, rational, and exponential functions. This was followed by a look at equations, solutions of equations, transformation of equations, and polynomial equations. Finally came a long look at geometry, including Euclidean geometry, synthetic and analytic proofs, spherical geometry, areas, volumes, transformations, and symmetries. Interspersed were guest presentations on data analysis, on hyperbolic geometry, and on Geometer’s Sketchpad.
The two quarters allocated to the math teaching course are not nearly enough to treat all topics that merit coverage. The extended focus on geometry was done at the expense of probability, which had been covered in earlier years. One topic that was not covered for lack of time was the use of EXCEL for instruction in mathematics.
BASIS FOR GRADING
We have opted for a highly structured course environment, in which the expectations are carefully laid out for the students, and there are frequent checks informing both instructors and students of progress toward meeting the course objectives. This responds in part to our sense of responsibility that the students be held accountable for all that was laid out in the proposal for accreditation of the Math SMPP. The course environment models one approach that secondary teachers might find useful for handling classroom organization.
Students expecting a methodsonly experience with trivial math content learn up front that the course includes a substantial content component, though taught from a different perspective than in their other mathematic courses. The Math SMPP combined with the math teaching track of the UCLA mathematics major is actually tougher than other tracks of the mathematics major, and this challenge to the students is reflected in the high standards we expect to be maintained in the math teaching course. This philosophy echoes the views expressed by Stephen Willoughby in his autobiography The Other End of the Log. According to Patrick Callahan (personal communication), beefing up the teaching track of the mathematics major at the University of Texas in the 1990’s actually led to a large increase in student enrollment in the teaching track.
The grades in the math teaching course this past year were determined according to the following scheme.

Quizzes, 12.5%: A brief “gimme” quiz on the math content reading for the week is given during the first 510 minutes of class. The quizzes ask for two or three items or statements, designed to be answered easily by anyone who has given thought to the reading assignment.

Quickwrites, 12.5%: A brief summary and analysis of the math/ed reading is required via email each week.

Problems of the week, 15%: A novel highschool level problem is assigned each week. Students submit a thorough solution and also a description of their problem solving process.

Exercises, 15%: A homework assignment of about 6 problems from the math content textbook is due each week. Occasionally a reader has been hired to grade the math exercise solutions.

Observations, 10%: Five hours of instructorguided observation in local secondary schools are required each quarter.

Participation, 10%: Attendance, promptness, and ontask participation are required in the course.

Final paper or portfolio, 12.5%: At the end of the first quarter, students submit a paper analyzing their student observations in the light of the California Framework and the NCTM Principles and Standards. At the end of the second quarter, students submit a portfolio covering various aspects of the entire course, including exemplar lessons and a list of other items as noted earlier.

Final exam, 12.5%: The final exam focuses on math content. Typical problems are to represent a specified repeating decimal as a quotient of integers, to define carefully “function” and “inverse of a function,” and to prove that 5 is irrational. The exam also includes such problems as to show with diagrams how fraction bars can be used to demonstrate that 2/3 divided by 1/4 is 8/3, and to write a word problem appropriate for an 8^{th} grader that models this calculation.
The median grade in the math teaching course this past year was an A. This median grade is higher than Mathematics Department undergraduate courses, with the exception of certain honors courses. On account of the selective nature of the student body, the math teaching course is in some ways akin to an honors course.
COLLABORATION WITH THE TEACHER EDUCATION PROGRAM
The rewards of collaboration between mathematics departments and teacher education programs can be great. At the same time, collaboration presents some interesting challenges. There is a deep chasm between the sociologies of mathematics departments and teacher education programs. The mathematics environment is a competitive environment, rife with exams and rankings of students. Mathematics has its Cambridge wranglers, its Putnam winners, and its Fields medalists. Teacher education programs foster a nurturing environment. Exams are suspect, rankings are anathema, and the personal qualities of prospective teachers play an important role alongside content knowledge. The challenge for mathematics departments and teacher education programs is then to bridge the sociology gap and to collaborate on effective programs for producing quality mathematics teachers.
The math teaching course at UCLA is the main point of engagement between the Mathematics Department and the Teacher Education Program (TEP). As such, it has had mixed results. On the plus side, the math teaching course provides undergraduate students in the Joint Mathematics Education Program (JMEP) with an excellent preparation for entering the TEP, and the TEP facilitates the accelerated progress of these students toward a teaching credential. On the minus side, the math teaching course was not successful in melding the goals of the undergraduates with the goals of the handful of TEP graduate students in the class. The TEP graduate students in the class had no need to have their mathematics subject matter preparation certified, they were under heavy time pressure, their expectations were quite different from those of the undergraduates, and their mathematical backgrounds and levels of mathematical sophistication were more diverse.
TEACHER PREPARATION IN A RESEARCH UNIVERSITY
Can a modern research university support an enriched program for the preparation of math teachers? Not without some difficulty. We make some general observations about teacher preparation in a mathematics department in a research university, and we describe the situation at UCLA.
A math research department can obtain substantial benefits by involving itself in the preparation of future math teachers. A teacher preparation program provides a mathematics department with a mechanism for directly improving the mathematical and professional level of school math teacher leaders. It provides an entrée to the school math scene, which can be advantageous for other department outreach and professional development programs. It provides a platform for influencing public policy on the training of math teachers. A longterm effect of these various teacher preparation and outreach activities is an increased level of mathematics preparation of students entering from high school and taking courses in the department. (Several students in our math teaching course in recent years had high school math teachers who went through the course some years ago.) In the long run, these activities contribute to raising the level of math sophistication of the citizenry at large.
The training of future science and math teachers is a goal that has been recognized at the highest levels of the University of California. Production of quality school math and science teachers serves to promote the reputation of UC and to create public and legislative support. Nevertheless, when the priorities of UC are ordered, the research mission of the university must be placed at the top, along with graduate and professional degree programs. Undergraduate programs lag, and outreach programs, however laudable, fall to the bottom of the list of priorities.
At the local level, university administrators are scrambling to mount research and teaching programs with dwindling per capita resources from the state. The modern research university is becoming increasingly dependent on extramural funding for research and on gifts for endowment. Intramural allocation of resources seems increasingly to favor departments and programs that are best positioned to generate support from extramural sources, while undergraduate programmatic responsibilities seem to play a more minor role in the allocation of resources. A look at the UCLA Mathematics Department over the past fifteen years illustrates the shift taking place in the allocation of resources.
Under a strong dean and with administration support and encouragement, the UCLA Division of Physical Sciences underwent a planned restructuring over a recent decade. There was a significant reallocation of resources within the departments of the Physical Sciences, as reflected in the following table of ladder positions.

Department

1991 Ladder FTE

2004 Ladder FTE

Mathematics (Pure and Applied)

61.50

50.00

Statistics

4.25

10.00

Physics and Astronomy

57.71

62.17

Atmospheric Sciences

11.50

12.50

Chemistry and Biochemistry

43.50

42.25

Earth and Space Sciences

23.00

24.60

Total

201.46

201.52

The reallocation has favored most strongly the Department of Physics and Astronomy, which has been quite successful in generating extramural funding, and the fledgling Department of Statistics, which is very much oriented toward applications and well positioned to attract extramural support. The two departments in the Physical Sciences under heaviest pressure from undergraduate enrollments are Mathematics and Chemistry, which are precisely the departments that lost ladder positions over the time period.
In the downsized Mathematics Department, the research and graduate programs are actually thriving, due in part to extramural funding from the NSF through a research institute (IPAM) and a VIGRE grant. However the undergraduate program is under some stress due to the increased number of math majors and concurrent cutbacks in courses. This reflects the reality that undergraduates are closer to the bottom than the top of the scale of university priorities, and returns us to our original question. In the modern research university, from whose pocket does the support for an enriched program for the preparation of math teachers come?
TWO CHALLENGES
A common challenge faced by many departments at public universities is to adapt to the increasing scarcity of resources for public education. In California, the master plan for higher education is no longer considered sacrosanct, as the state struggles to deal with a massive fiscal deficit. The challenge of coping with the current budgetary crisis is particularly acute for the UCLA Mathematics Department, since downsizing has left it with a diminished base. In the past year, mathematics counseling and programming staffs have been reduced. The number of ladder mathematics faculty has dropped to its lowest point in almost half a century. Under ideal conditions, it would be desirable to expand the math teaching course to a threequarter offering, which would allow for more time to cover the full gamut of desired topics. This would be in line with the recommendations of the CBMS report, which calls for a fullyear capstone course. However, talk in the undergraduate office is not of what new courses and programs to design and mount, but rather of which courses and programs to cut, and how to minimize the number of undergraduates turned away from overenrolled courses.
Administrators expect mathematics departments to consume less resources per student than other science departments through economies of scale, such as large calculus lecture classes. However, quality teacher preparation programs are by nature economically inefficient in that they require more individual student attention from instructors than typical math programs. Further, conducting the math teaching course described above requires more energy and effort than a normal mathematics course on account of the extensive preparation time for both the mathematician and the math educator. Crediting the mathematician and math educator with a halfcourse teaching credit or reimbursement respectively for coteaching the course does not reflect the time and energy required to mount the course, yet the scarcity of resources makes it difficult to do otherwise.
A second challenge faced by the research mathematics department in mounting and maintaining a quality teacher preparation program is that research mathematics departments often do not have faculty members with sufficient background or willingness to undertake teacher training courses. Often participation in teacher preparation activities is restricted to one or two faculty members who volunteer for personal reasons stemming from observing their own children navigate local school systems. There has always been several ladder mathematics faculty at UCLA who have involved themselves in the K12 education scene. These faculty members have retired or are approaching retirement, and it is not clear where the next generation is coming from. One important resource for the department has been the willingness of retired faculty members to participate in conducting organizational and extracurricular activities connected with departmental teacher education programs.
BONUS BENEFITS
We have listed a number of advantages of the coteaching format. There are yet other benefits, which can be regarded as bonuses. Often the person learning the most from a mathematics course is the instructor. It is our experience that the coinstructors of the math teaching course have learned an immense amount from each other about “the other end of the business,” through the close collaboration. The mathematician benefits by getting insight into classroom practices and how higher mathematics plays out in the secondary classroom. Further, the mathematician learns about pedagogically sound approaches to teaching mathematics at the secondary level, which can occasionally be adapted to teaching at the university level. The mathematics department benefits in that the experience coteaching the math teaching course provides the department with a reservoir of experience, which may be useful for making informed departmental decisions on curriculum and other policy issues. The math educator benefits through a deepened understanding of the math taught in the secondary classroom. Further, the math educator emerges from the experience with new ideas for importing important mathematics into the high school classroom. Each of us has found our participation in the team to be immensely beneficial to our own knowledge base and understanding.
REFERENCES
J.L. Burton, The History of Mathematics, 4^{th} edition, McGrawHill, 1999.
T.L. Heath, The Thirteen Books of Euclid’s Elements, 3 volumes, Dover Publications, 1956.
E.D. Hirsch, The Schools We Need, Anchor Books, 1999.
J. Kilpatrick et al. (editors), Adding It Up, National Academies Press, 2001.
Jeannie Oakes, Teaching to Change the World, Lipton McGraw Hill Higher Education, 1999.
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Heather Calahan
Santa Monica High School
calahan@math.ucla.edu
Ted Gamelin
twg@math.ucla.edu
Bruce Rothschild
blr@math.ucla.edu
March 11, 2005
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