Advanced Calculus, an Informal Discussion

Written by Bill Abikoff.

Starting in the late 1930’s and even earlier, calculus was taught in two different ways. One taught skill in calculation. It was typically taught from a text written by Thomas, later with collaborators; Stewart’s text is more popular nowadays. Another track stressed theory, in addition to more sophisticated calculation. Typical texts were those of Courant and, from the 1960’s on, the books of Apostol and Spivak. These books are all still used.

The Advanced Placement exams in Calculus are called AB and BC. These evolved from the skill/theory dichotomy in the 1960’s as mentioned above. Calculus A was calculational, C was theoretical and B was a mix. Serge Lang’s Calculus was a typical B text. At that time, calculus was not a relatively standard high school course. It was very common that calculus was a required course for premeds and pre-law students in order to teach them the organized thinking necessary for their professional training — we don’t see any other reason for such requirements at that time.

Times changed in the 1980s. By then, the introduction of programmable calculators, which had started in the mid-1970’s, had a profound effect on mathematics education — to my mind, not necessarily a positive effect. By punching the right keys on a calculator, a calculus student in a calculation-based course could easily get a B on a typical exam. So we were faced with a problem: in the era of calculators, what is a viable approach to the teaching of and evaluation of performance — that means grading — calculus courses? One Harvard mathematician, during Congressional testimony, said “now we don’t need to teach people how to multiply, rather we have to teach them when to multiply”. Some people forbid the use of fancy calculators and require that memory be cleared before exams. That’s not my approach; I think we shouldn’t ignore technology, we have to adapt to new tools.

These tools present one of the new challenges. The other, also dating mostly from the 1980s, is that most of the mathematically strong students have taken calculus in high school. (I didn’t even though I attended one of the best science-oriented high schools in the country.) The emphasis and quality of instruction in high school calculus varies quite a bit and often is quite different from courses given in universities.

Our first approach at a honors sequence was called enhanced calculus. At least when I taught it, it was basically the B to C level course together with projects which were often computer-based (using Mathematica). It was called “enhanced” rather than honors reflected our recognition that it was not appropriate for all honors students taking calculus.

Around 2004, we decided that there was a need for two honors sequences in calculus. One would be a modern skill-based sequence but skill has a different meaning now. It is an honors version of the 1131Q-1132Q-2110Q sequence and is 1151Q-1152Q-2130Q. A student taking this sequence and requiring more advanced mathematics would generally have to take 2420Q (Honors Differential Equations), 2210Q (Computational Linear Algebra) and 2710 (Transition to Advanced Mathematics).

The other sequence is this one; it is called Advanced Calculus. It is four semesters long and carries four credits per semester. In effect, the transition to higher mathematics occurs on day one. The style of discussion is that of theoretical science and advanced mathematics. It is assumed that basic computational skills, at the AB level, were learnt in high school. Starting in 2004, I taught a preliminary version of this sequence as a test of this approach. Recently, I asked one of the students, an engineering major, in that sequence the following question:

What is it about the three semesters we shared that has helped you in terms of YOUR interests and goals? It’s a tough question.

Here’s her response:

Looking back on my first day of [the first course], I would have considered myself an exceptional math student … or at least, what I thought a math student really was. It seems that during high school in math classes, even in AP courses, you are given theorems, rules, etc. and you are to take them in as fact without doubt or critical thinking about where they came from. [This sequence] was a huge eye-opener for me about the world of academia in general. Each theorem has a proof and at one time, someone slaved away to develop what they have proven to be true. You are supposed to question what you learn which I think is a principle of life that really developed for me through the enhanced math courses. I remember analyzing Newton’s Method and when I was able to find the faults with it, I was honestly astonished! You mean, this simple principle of finding roots isn’t actually the without-limit method that I had learned two years previous? It’s these kind of epiphanies that make me curious and make me want to learn more and apply that same doubt to other subjects whether biology, chemistry, engineering, etc.

As far as my interests and goals are concerned, it’s a pretty simple statement: I want to know everything! Or, I guess, keep an open-mind and learn as much as I can along the way. I’m currently doing research now that reminds me a lot of my first experiences with my math projects and proofs. I’m given protocols and procedures on a sheet of paper and in my high school mind frame, I would take them as the best way possible to do them and be on my way. But, when results don’t show, you become the expert and tweak the infallible procedure because you’re taught in research to question everything; to look at the broad scale and, if necessary, repeat until a conclusion can be drawn or else change the way you’re doing it.

I feel like I’m rambling a bit, but the math courses in general opened up my mind to the possibilities while also giving me the appreciation for the foundations on our knowledge, whether math or science. Also, being a very, um how should I say this, “grade-orientated person” it was nice to be in a class where I could focus on the learning while having a professor who was observant of not just what we could regurgitate on a test, but to how much we had grown and learned throughout the course.

Math 2141 and 2142 were first given in the academic year 2006-7; I’ve often been the instructor of the first-year course and direct the sequence. It has been of interest to sufficiently many talented and highly motivated students that we now have two sections of 2141. Mostly, I think the course works for the intended audience. I do take students criticisms seriously.

Here are some comments made more recently by students late in the second semester: One wrote:

This letter is for the girl who got a 110 on her Calculus exam because there was a curve and extra credit. For the boy in the back of math class who falls asleep but somehow manages to pull out an A for the year. Last but not least, this letter is for those few students who excel at math and want math to be more than just number crunching. If any of these descriptions fit you then I recommend you take the new advanced calculus sequence. This sequence is probably the hardest set of math courses that I have ever taken but also the most rewarding. When I was in high school math came easy to me. There was always a method that would apply to the situation. However, in this class math is more of an art than knowledge. Of course, you should have a strong background in math but I’ve seen half the class filled with honors students who didn’t know what they were getting themselves into. This class explains a lot of the things that were just assumed to be true earlier. For example, how do you really know that pi is irrational? The fact that no one has found a pattern or an ending is a very good proof. Speaking of proofs this course is full of them. The first course in this sequence is dedicated to different methods of proving theorems. This part seems very easy but you must pay attention to every detail in order to understand it. Asking questions is also very important but don’t ask a question that was covered two months ago because it is expected of you to know that material. This class isn’t a class of direct answers. Numbers are only tools in this class. What really matters is logic. Every theorem that you saw last year in calculus will be derived. There are no assumptions made in this class. In fact, the first week of class will be dedicated to counting. It sounds easy but it’s not. I have helped kids in the regular calculus courses and it seems that they just get drilled on the same stuff with different numbers. In this course, every day presents a new topic so it is important to do the homework to keep up even though it’s not collected. Also in this course more history of mathematics will be presented than in other courses so you see where these different theorems came from. Most of the time they’re from Riemann, Cauchy, and Weierstrass but you’ll hear others too. If these four classes are taken together, the University of Connecticut gives the student with a 2.0 or better in these classes a minor in mathematics. So even if you’re not majoring in math, it would be very good to have a minor in mathematics on your resume, just a little extra to push yours to the top of the list. The only problem about these courses is that there is only one section so have your advisor help when you are trying to fit this class into your schedule. If you can’t take it freshman year, consider taking it another year because if you really are the student who gets called to help people with their math homework then I think this is the class for you.

Another wrote:

The Advanced Calculus sequence is demanding, to say the least. Exposure to some high school Calculus is recommended [Ed. actually, it is required] but is nothing like what this course is like. This course goes into the theory of the mathematics and some more advanced topics. The course demands that one looks at mathematics and understands the math in a completely new way that will be beneficial in further and more advanced studies in mathematics. Since the mathematics presented will be very new and the course is so demanding, one is not advised making it through alone. The professor is always available for help during office hours and often the best resources are one’s classmates. There are projects throughout the year and the professor sometimes even encourages cooperating and collaborating with your peers.

To sum (no pun intended) up, the course will be unlike any math class you’ve ever seen before and will demand a lot. But it will open a whole new way of seeing, understanding, and comprehending mathematics.

Actually, I really like the email in which he promised the detailed statement quoted above. Among other things, he wrote:

This sequence was like nothing I’ve ever been through before and I’ll be frank, it kicked my ass. But I’m really glad I took it …