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Making Upper Level Courses More Accessible

The first section of this document is adapted from "Do we need prerequisites?" by Don O'Shea and Harriet Pollatsek, which appears in the May 1997 Notices of the American Mathematical Society (v.44, n.5, pp. 564-570). The second includes more detailed information on the seven courses.

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1. Accessible Advanced Courses

Building on our experience with the Lab, and with FIPSE's generous support, we reworked seven of our advanced courses in order to reduce the prerequisites to at most two semesters of college level mathematics. Our FIPSE grant supported sabbatical visitors whose teaching provided release time for the faculty member developing an advanced course. In addition, a visitor attended and critically commented on each new course under development. (This collegial help proved so valuable that now it is something we try to do for each other.)

By reducing the prerequisites, we hoped to attract students who were not mathematics majors, but who would enjoy and be able to use some of the ideas encountered in traditional junior-senior level courses. (Of course, we also hoped that appealing electives available early in a student's college years might attract more majors.)

We began with our own and others' experience that systematic and thoughtful use of computers often allows the introduction of relatively advanced, but exceedingly useful, ideas at an early stage in an undergraduate's career. As a result, most of our revised courses use computers in an essential way. Happily, the increased computer use also accords well with the way in which mathematics is increasingly practiced outside the classroom.

Listed in order of their development, the seven courses (and their original developers) are:

- Differential Equations with Modelling (D. O'Shea and L. Senechal),
- Analytic Number Theory (G. Davidoff),
- Mathematical Statistics (M. Davis, now at Eastern Connecticut State University),
- Lie Groups (H. Pollatsek),
- Polyhedral Differential Geometry (A. Durfee),
- Symmetry, Groups and Geometry---and applications to physics (L. Michel, I.H.E.S., and D. O'Shea),
- Theory of Equations (G. Davidoff, based on the text by Cox, Little and O'Shea).

We had substantial help from FIPSE-supported sabbatical visitors: J. W. Bruce (University of Liverpool), E. Connors (University of Massachusetts), P. Fitzpatrick (University of Cork), K. Halvorsen (Smith College), L. Michel (IHES, Paris), R. Nelsen and H. Schmidt (Lewis and Clark College), and K. Rogers (University of Hawaii).

Normally we offer each course at least once every three years, so that students have the option of a more accessible advanced course each semester. Details of the courses appear below.

Besides accessibility, other goals guided our development efforts. We sought topics that exhibit some of the range of mathematical ideas currently being investigated and applied. In particular, we wanted students to experience mathematics as a subject created and used by people. We also wanted to give our students the opportunity to use the tools, technological and conceptual, that enable them to work with these ideas.

We also considered pedagogical issues in designing these courses. From the Lab and our calculus courses, we learned that working in groups strengthens students' learning of mathematics. Similarly, we found that writing is a useful tool; like group work, it forces a student to express his or her ideas clearly and to organize them coherently. Thus, all the new courses require careful writing. Finally, we wanted our new courses to be exploratory to some degree, and some of the writing we ask of students is a description and analysis of their explorations. Beyond the justifications mentioned earlier in the description of the Lab, we note that successful exploration is not possible without taking risks. We work to create an atmosphere in which students feel free to be adventurous in their thinking. For women, especially, encouragement to let go of safe, cautious strategies and try bold ones seems important.

We have also been mindful of the pitfalls in reducing prerequisites. There is a very real danger of ending up with shallow courses that neither stretch students' minds nor prepare them well for further study or for using the ideas in other contexts. We have tried to avoid these pitfalls by building courses around substantial, explicit examples. Learning a few significant examples thoroughly and exploring their implications not only prepares a student well for more general study (and enriches the student who has already had a more general course) but also leaves the student in command of some important ideas. Very often, the computer is the tool that enables us to bring these examples to the desired level.

For instance, the analytic number theory course treats one theorem in theoretical detail (Dirichlet's theorem on primes in progressions) but introduces the statements of others, e.g., the prime number theorem, the prime number theorem for progressions, and Littlewood's theorem) through numerical experimentation. Students are asked to be quite independent as they work with the computers to find their own paths toward the correct statement of a theorem. This is a substantial adjustment for students accustomed to having results given them in lectures. However, after some initial discomfort (much less for students who have been through the Lab), they rise to the challenge.

For another example, the polyhedral differential geometry course asks students to construct models of polyhedra and to investigate lines, angles, polygons, and areas on them. The students work toward the Gauss-Bonnet theorem for polyhedra and polyhedral analogues of other differential geometric results. In addition, students investigate polygonal knots, equivalence, and Rademacher moves, and examine the invariance of knot groups and knot polynomials.

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2. The FIPSE Courses

In the following sections we briefly describe each of the seven courses, mentioning prerequisites, the phenomena under study, computer tools, the subtext and the main mathematical results. By phenomena we mean the kinds of examples we ask students to explore. By subtext we mean the broad mathematical concepts that we are seeking to convey and strengthen. In each case, we list the materials which are available for sharing, and we welcome inquiries from those who are interested.

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2.1 Differential Equations with Modelling

**Prerequisites**. The prerequisite for this course is a standard first year calculus course. In fact, most of the traditional calculus II topics (specialized techniques of integration, series) are not required.

**Phenomena**.

The phenomena under study include:

- Numerous models, including predator-prey, competing species, symbiosis, spread of disease, bounded growth, two-body problems in mechanics, celestial motion, and price dynamics.
- Behavior of solutions.
- Minima and maxima of functions such as x + cosh x, 1/(x+y-2)^2, 1/(x-y-1)^2.
- Singular points and behavior of solutions of the linearization.
- First integrals.
- Calculus of variations, with applications including the hanging cable and minimal surfaces of revolution.
- Euler, Runge-Kutta and other numerical methods, including analysis of speeds of convergence.

**Computer tools**.

Students use two programs (EULER and SLINKY) developed by James Henle of Smith College for use in the first semester of Calculus in Context, one (ODE) developed by Mark Peterson of Mount Holyoke College, and Mathematica. Henle's programs are available on his web page.

**Subtext**.

There are three broad themes: the idea that a model is a system of ordinary differential equations and defines a vector field; the dominance of equilibria; the contrast between conservative/non-conservative fields; and the view of functions as vectors in Banach space which gives a setting to compare steepest descent methods with respect to different norms.

**Mathematics**.

The course includes the Poincare-Bendixson theorem, Lyapunov theory, Euler's theorem in the calculus of variations, and the analysis of numerical methods of integration of dynamical systems.

**Materials**.

Notes and exercises are available, and a monograph is available.

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2.2 Analytic Number Theory

**Prerequisites.** The current prerequisites are a semester of calculus and our Laboratory in Mathematical Experimentation. Eventually our introductory Number Theory course will be a possible alternative to the calculus prerequisite. At institutions other than Mount Holyoke, the second requirement might be any course that gives a student some experience with computer exploration and careful argument.

**Phenomena**.

Students investigate the following phenomena:

- Primes less than 10^3, 10^4, 10^5 ....
- Primes among 2^p-1, for p < 35, and values of 2^{p-1}(2^p -1).
- Average values of the arithmetic functions phi(n), mu(n) and d(n) for n< 10^3, 10^4, 10^5, ....
- Primitive elements of (Z/nZ)^* for n < 10^3, including the question of when 2 is a primitive element.
- pi(x), pi(x)/x, pi(x)/x^{1-h} for h = 1/2, 1/3, 1/4 as well as pi(x)/(x/log x) for x=10^k, 2 <= k <= 6.
- pi_a^{(p)}(x) for p=7, a=1, ... 7; p=12, a = 1, 5, 7, 11 etc.
- The behavior, especially sign changes, of D^4(x) = pi_3^{(4)}(x) - \pi_1^{(4)} (x) as x gets large.
- Twin primes.
- Primes in {an+b} and in {n^2+1}.

**Computer tools**.

Students use a program called NEWNUM developed by Mark Peterson. In addition, with the help of a programming assistant, they write programs of their own in Pascal or C.

**Subtext**.

We want students to develop a feeling for the distribution of primes, for the meaning of big "oh" and little "oh," and for rates of convergence.

**Mathematics**.

The course includes the statement and proof of Dirichlet's theorem on primes and progressions, in addition to experimental work on the prime number theorem, and Littlewood's theorem. (The proof of the prime number theorem could be substituted for the proof of Dirichlet's theorem.)

**Materials**.

A course description, notes for the instructor and exercises are available. A text is in preparation.

###
2.3 Mathematical Statistics

Note: Currently we are offering the two-course sequence of probability and mathematical statistics in alternation with Smith College, so this version of mathematical statistics is not now being taught at Mount Holyoke.

**Prerequisites**.

Students need a first year course in calculus.

- Phenomena. The phenomena students examine include
- Behavior of dice, coins and other randomizers.
- Behavior of samples: multiple samples and samples of different size.
- The effect on different statistics (for example, the mean or variance) of increasing sample size.
- Simulation studies and the search for a best estimator.
- Comparison of sample distributions to theoretical distributions.
- Hypothesis testing via simulation and the construction of confidence intervals.
- Bootstrap and jacknife techniques.

**Computer tools**.

Students used SYSTAT the first time the course was offered, but better alternatives exist now.

**Subtext**.

The goal is for students to develop a feeling for randomness and the behavior of chance phenomena, for when theoretical results become reliable, and for distributions, estimators, and the like.

**Mathematics**.

The course includes the central limit theorem, classical theoretical results on different distributions, and the behavior of maximum likelihood estimates. (The first offering did not include bootstrap methods.)

**Materials**.

Course notes and laboratory assignments are available.

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2.4 Lie Groups

**Prerequisites**.

A first course in linear algebra is required. At Mount Holyoke, students can take linear algebra after a single semester of calculus (although few take it so soon).

**Phenomena**.

Students investigate the following phenomena.

- Isometries of the Euclidean plane and of spacetime (Lorentz transformations).
- Matrix groups over the real numbers, the complex numbers, the quaternions, and finite fields.
- Tangent spaces and differentials.
- The exponential and logarithm maps for matrices.
- One-parameter subgroups.
- Systems of differential equations.
- Lie algebras over the real numbers, the complex numbers, the quaternions, and finite fields.

**Computer tools**.

Students use Mathematica or Maple V to carry out matrix calculations with both numerical and parameter entries, in particular for finding the exponential or logarithm of a matrix by calculating partial sums.

**Subtext**.

We want students to develop a feeling for the ideas of symmetry, for morphisms of different structures, and for linearization.

**Mathematics**.

The course includes the classification of isometries of metric vector spaces, a proof that a continuous group homomorphism is differentiable, the local homeomorphism between a Lie group and its Lie algebra, the relationship between one-parameter subgroups and the exponential map, and theorems about adjoint maps, Lie subgroups and Lie subalgebras. In addition, it includes some discussion of Chevalley groups.

**Materials**.

Notes and student assignments are available, including a paper assignment and suggested bibliography.

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2.5 Polyhedral Differential Geometry

**Prerequisites**.

Students must have taken two of Calculus I, Calculus II, Geometry, Linear Algebra or the Laboratory in Mathematical Experimentation.

**Phenomena**.

Students are asked to

- construct models of polyhedra and their slices (with attention to constraints).
- measure dihedral angles, solid angles, angles in slices and their relationships.

In addition they investigate the following phenomena.

- Surface area and volume.
- Nets on the sphere.
- Knots, equivalence, and Reidemeister moves.
- Alexander and Jones polynomials.
- Knot group presentations.
- Links.

**Computer tools**.

Students can use Mathematica, Surface Evolver and Linktool.

**Subtext**.

The goal is for students to gain geometric intuition, to get a feeling for geometry on the sphere, and to appreciate the role of algebra in geometry.

**Mathematics**.

The course includes the Gauss-Bonnet theorem for polyhedra and polyhedral analogues of other differential geometric results, as well as the invariance of knot groups and various knot polynomials. It also treats the groups Pi_1 and H_1, abelianization and Lefschetz duality.

**Materials**.

Course notes are available.

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2.6 Symmetry, Groups and Geometry

**Prerequisites**.

Students are required to have taken both Calculus II and linear algebra.

**Phenomena**.

Students investigate the following phenomena.

- Groups, especially symmetry groups, including subgroups and matrix representations.
- Order and conjugacy of group elements, isomorphisms.
- The number of reflections required to express a rotation in n-dimensional euclidean space as a product of reflections.
- Spinors, Hamiltonians and Clifford algebras.
- The map from SU(2) to SO(3) and its generalizations, spin in physics.
- Classification of patterns.
- 2-manifolds, 3-manifolds and orbifolds.
- Symmetries in physics and symmetry breaking.

**Computer tools**.

Students use Geissinger's *Exploring Small Groups* (now available through the MAA), Mathematica, and 2D-Worlds (by Robert Weaver of Mount Holyoke).

**Subtext**.

The goal is for students to get a sense of mathematical structure and symmetry, to gain geometric insight into group theory, to appreciate the interplay between algebra and geometry, and to get a feeling for the idea of a manifold and a group action.

**Mathematics**.

The course includes the usual theorems of elementary group theory, together with a classification of geometric objects according to the strata of a group action. Also included is some work with associative algebras.

**Materials**.

Notes and problem sets are available.

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2.7 Theory of Equations

**Prerequisites**.

A first course in linear algebra is required (which can be taken after a single semester of calculus, although few take it so soon).

**Phenomena**.

Students investigate the following.

- Polynomials and solutions; the correspondence between an ideal and a variety.
- Bases for an ideal, ideal membership, and elimination.
- The geometry of elimination.
- Operations on ideals and geometric content.
- The computation of invariants.
- The computation of dimension.
- Properties of coordinate rings.

**Computer tools.** Students primarily use Maple V, although they also make some use of REDUCE and Mathematica.

**Subtext.** The goal is for students to get a feeling for the notion of effective computation and for the interplay of algebra and geometry.

**Mathematics.** The course includes the Hilbert basis theorem, the Nullstellensatz, Hilbert polynomials, and Groebner bases ---and their use to compute "almost anything.

**Materials.** The course uses the text Ideals, Varieties, and Algorithms, by Cox, Little and O'Shea (Springer-Verlag, 1993).

To contact us, email: Harriet Pollatsek or Don Oshea.

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