CUPM Curriculum Guide 2004:A report by the Committee on the Undergraduate Program in Mathematics

CUPM is the MAA's Committee on the Undergraduate Program in Mathematics. Its mandate is to make recommendations to guide mathematics departments in designing undergraduate curricula. It issued its first report in 1953, and updated it at roughly 10-year intervals. In this latest report, which is also available online, it has broadened its focus to include the entire college-level mathematics curriculum, for all students, even those who take just one course. This is an excellent move. Courses taken by such students can play a huge role in telling others about the power and beauty of mathematics. Furthermore many clever imaginative students take such courses while still being somewhat open to their intended area of study. They should find mathematics inviting.

The report is divided into two parts. Part I consists of a set of six general recommendations for all departments and Part II provides specific recommendations for particular groups of students.

Part I General Recommendations

Part I gives us six general recommendations. Number 1 is to know your students, pay close and caring attention to them — who they are, where they are and where they want to be going. And part of that is knowing about the big world in which they will soon be striving to find a place and in which mathematics has so much to offer. That's a big one and a good place to start though I might myself have started one big step back — know mathematics. Of course that's taken for granted, though I guess there have been curriculum design projects that have lost sight of that. But just as every waking day should start with a brief meditation on the wonder of life, so every new course we teach and even every class we enter should begin with a reflection on the power and beauty of mathematics — a big part of our own lives but something that's alarmingly easy to park just outside the classroom door.

Number 2 is to develop mathematical thinking and communication skills. That's of course what almost any version of the workplace needs and keeps asking for. They might not use the phrase "mathematical thinking" but when you look closely at what they want — the capacity to take a problem apart and put it back together again in a way that allows you to see what's really happening — that's what mathematics does. Numbers 3 and 4 concern the breadth and interconnections of the mathematical sciences and indeed of the world of knowledge. As faculty we should talk to one another about teaching and curriculum, both within and outside our discipline, fearlessly exploring different ways to draw our students into that wonderful mathematical game which sits at the centre of our own professional lives (that's number 6). And finally (number 5) we should use technology in our courses, not necessarily as a big deal, but in the same way we use it as a natural component of our work as mathematicians.

Most of us know the "truth" behind these directives, but they deserve to be always before us. Keeping to their path requires a constant realignment, so we are happy to revisit them, especially as they come to us with the support of a rich online collection of illustrative resources, which give us some fine examples of "what's happening where." I was not aware of these until I began studying the Guide — they provide lots of ideas for trying out some of the recommendations of the report, for example, the management of student reading before class.

In the third recommendation we are asked to employ a broad range of examples to illustrate and motivate the material.

When students look back on a course, it is often the examples and illustrations that are most memorable. Authentic and interesting (and sometimes surprising) applications can be powerful hooks drawing a student's interest into the mathematics under study.

This is "example" as motivator but in the discussion the report observes that the right examples will draw the students into engaging the mathematics in the right way, making connections, formulating conjectures, etc. Amen to that. In squash, the key ingredient of the stroke is the top of the back-swing. If the racquet head starts at the right place, then with the right suite of technique and balance, the stroke will unwind in a natural and fluid manner and the ball will find its true trajectory. So, for me, the key to a right class is to go in with the right example. It might be a specific problem, or a rich situation to explore, but if you have it, and you can support it well as it unfolds itself in the class, many of the exhortations of the report will simply happen as if by magic. But with lacklustre material, the stroke seems forced.

It must be emphasized that there are two aspects of "rightness" for an example, one is about content, the other is about pedagogy. Content has to do with the capacity of the example to strike at the heart of the result, pedagogy has to do with its ability to engage the class in a lively process of inquiry and discovery. That second one is a tall order in any class especially in a large introductory "service" course. But it can be done. The report provides a few examples of potential topics.

• Dynamical systems, chaos and fractals. Dynamical systems, in both their discrete and continuous manifestations, can provide a wonderful central idea for both of our mainline first-year courses, calculus and algebra. Such a centering would suit many of our partner (user) disciplines very well. And chaos and fractals. These make wonderful "show and tell" units. But to fully incorporate this topic the students must do things with it. We need to have non-trivial tasks that can engage the student in the mathematical activities we are trying to foster in the course. Too often the "neat" applications that appear in textbooks are not fully supported in this way.  
• Wavelets. Here they point out that one can do a lot without the formalism of Fourier series. The point here is that sophisticated examples can be included in the course without the traditional full slate of prerequisites. Another nice way to give a class a Fourier series experience is to get them to construct in real time a square wave out of sine waves. They have to decide which frequencies (of the allowed palette) to use and how much of each. I come armed with my laptop and a MAPLE template which allows me to quickly display any combination of frequencies proposed by the class. This unit can be supported with tone generators and other fine web resources.
• Error-correcting codes. There are wonderful ideas here, especially for the cell-phone generation. But make no mistake, the reason this unit captivated my class was because of the 3-hat problem. That's no surprise to those who know the history of this problem and how quickly it spread through the mathematical community. Mathematicians like things that are beautiful and powerful and fun. Our students are no different. Playing the seven-hat game, with 7 brave volunteers from my class of 150, and vector bars as the prize, was a fun experience for the class.

Finally one should consider the affective domain. In the "math and poetry" course (IDIS 303) I often envy my co-teacher Maggie (who teaches the poetry) because she can strike the students "where they live" more easily than I can. Often the examples which my students recall when I encounter them years later are those to which they have found an emotional connection. An example of this is the farmer, builder, tailor problem I found in a lovely little book by T. J. Fletcher (Linear Algebra, Van Nostrand, Reinhold 1972). Versions of this problem appear in standard text-books set in something like the steel industry, but that's definitely not where the kids "live." What my students argue vehemently about, is the question of why, in a closed economy with everyone working equally hard employing hard-won skills, it should not be the case that everyone gets paid at the same hourly rate. In formal terms they are asserting that, to be fair(!), the left eigenvector of a certain matrix should be (1, 1, 1)^T. But it's not, and my students can get quite worked up about that.

Part II. Specific Recommendations

My own current interest here is in part B: Students majoring in partner disciplines so I will say a bit about that. The courses we typically offer these students are designed from lists of topics and techniques that we have determined (apparently) that the user disciplines want their students to have. (Another recent MAA report addresses what exactly our partner disciplines do want.)

Such a design can weigh the course down and drain its life blood. The CUPM report suggests that we might be surprised to learn exactly what these partner disciplines want of their students; it might be different from (much less and much more) than we believe. [I believe we wouldn't be surprised, but with all we have to do and with the textbooks we have to work with we often simply continue in the old way.] And anyway, the average amount of effective learning done in the typical service course in calculus and algebra is (un?)surprisingly small, so we actually have little to lose by trying to make a few cuts to carve out some space for some of the good stuff — discussion, inquiry, exploration. The report suggests collaboration and classroom observation with colleagues in other disciplines to obtain agreements about the "essential mathematical material." That's a good idea but don't make the mistake of organizing this quest by starting with a full list of possible topics and then asking what can be left out. I have engaged in too many exercises of this type, with the inevitable conclusion that everything is important, or at least is needed as a prerequisite to something else that is. Rather I suggest the following step by step recipe.

1. Let go of all the traditional content. At this stage nothing is sacred.  
2. Get a firm hold on the big ideas of the course, at least in your own mind.
3. Find (design or beg or borrow or steal) a collection of "good" exploratory problems which embrace and do justice to this collection of big ideas.
4. Include (and insist on mastery of) the skills that support these problems. These now define the technical "content" of the course.

There's an important extra payoff in "getting it right" in these services courses. Many students enter university with good mathematical ability but with no firm commitment to a subject of study. Most of these take one or two "service" courses from us at the beginning of their studies and here we have a real chance to attract them to study more math and even to consider a math concentration.

An important sub-section (B4) deals with preservice elementary and middle school teachers. There are two huge issues. First, there is a serious looming shortage of mathematically trained teachers at all grade levels. Second, too many current teachers have weak mathematical preparation. As the report says, "the stakes could not be higher." My experience is that most mathematics departments now realize this, and at least in many Canadian universities, the math department offers a suitable (even excellent) course. The problem many of us are concerned about is that those who are interested in teaching such a course are facing retirement in some 5 years. Those following along behind are mostly much younger (demography at work) and were hired in a new age in which research was the effective sine qua non. No doubt as these younger faculty age (and talk to their children!), some of them will find their interest in educational issues growing and will make substantial investments there. But perhaps also the time has come to spread our hiring net a bit more widely.

I declared my current interest in part B, but I am also thinking a lot these days about part C: students majoring in the mathematical sciences. These students get offered "higher level" courses in calculus and linear algebra than other science majors — more theoretical and conceptual. Now there's a notable thing about those courses at least at Queen's — less than half of all the students taking them wind up with any kind of a math or stats concentration. Some of these have come in with a firm idea of what they want to study but have taken the higher level courses because of their ability, and that's great. But others are undecided and are open to an invitation, and my current view is that we should consider "lightening" these courses to make them more attractive. That word "lightening" needs explanation. It's about light in both senses of the word: opposite to heavy and opposite to dark. And it's about having fun. I believe that we can do this without compromising our standards. In fact we must. 

Peter Taylor is professor and chair of the mathematics department at Queen's University in Kingston, Ontario.