Teaching Calculus (2024)

Oliver Knill, 5/11/2024, posted

Let me comment about some points made while completing Math 1a with PDF. I think that the course was a big success. Many of the students made enormous progress. They can be proud but I also feel proud. I also had seen happy visitors who came in and looked at this course. The admission office would send students to my class (and not to calculus courses where worksheets are worked on in groups). And this brings me to a major point I want to make here: teaching is an art which needs to be celebrated and which requires a lot of work. It should not deteriorate to fast food, even if the later is cheaper and less risky. An other reason to write here is because we are at a pivotal moment in education, where we do not know how things will continue. A few years ago, we were afraid that massive open online courses MOOC's would take over and replace higher education. Within a few years, this has collapsed. Some argue that EDX died because it was sold. The reality is probably that massive online education has shown problems during the pandemic and that AI will probably soon destroy it completely; because we can technically very soon no more distinguish between real students and bots who pretend to be human. In any case, we had been worried 10 years ago about MOOC's taking over and this did not happen. (See this written down in 2013 this document which is largely about technology developements.) Now, we are worried that AI will take over. It is maybe good to write this down now for me to see, how I will reassess this in 10 years. Maybe an other tsunami revolution will happen.

About teaching

I myself made good experience with investing heavily into each lecture and deliver fresh content. I feel that if this is not done, teaching will soon be replaced by bots. At the moment I myself am highly pessimistic. Higher education might delegate basic intro courses to bots soon. This is cheaper. We recently voted about unionizing of faculty here. I voted "yes", but I'm also worried about the teaching profession. Once unionized, the schools will think harder about hiring and try to cut costs if this is possible. This will especially happen of a time of reputation decline of universities (which means revenue decline). When I came here, Harvard had been on the top of rankings overall. Now it dabbles around place 5. I fear that with recent brand- and reputation issues that it might sink even lower internationally. Reputation and ranking is largely based on peer judgment which is largely game theoretic. We know from dynamical systems theory that systems depending on network rankings can undergo rapid changes. Therefore, universities need to focus on excellence in teaching and research and not have to engage in politics. Campus guidelines should be enforced. I know that during the last big crisis (the housing market crash 2008), one has actively looked into getting rid of our calculus teaching program. This might happen sooner than later if there is no money for it any more. Higher education came also under attack by big tech companies who want to eat from this sweet cake. What our leaders of universities seem not to understand that like any other industry, it is in danger of becoming obsolete. Cambridge could become Detroit. I hope this does not happen and personally try to give my best every day.

Let me say more about delivering new and original content. As teachers, we only have a limited amount of time to prepare and need to balance work and life. I'm lucky to have been given also this semester enough time to prepare for each class. I think it averages to about 10 hours per class, which is actually pretty steady over the years. More recently I also make short 1 minute videos for example underlying some point of the course. Of course, we can not always reinvent the wheel due to time constraints but I think, a teacher needs to deliver and be allowed to provide new content. Why should a student go to class otherwise? I can today solve an old worksheet also at home and if needed ask a bot to give me hints. A PDF can be submitted to Chat GPT and I can ask it to formulate the answer that a 12 year old does understand it. And it does a good job! AI tools not only solve problems perfectly, they can also slowed-down and detailed solutions, better than many teachers can do today already. And without the social tension which can come with excessive forced group work. I myself still believe that working together in teams is valuable and that some time in the classroom should be dedicated to work together. I especially believe in the Socratic method by asking questions and encourage discussion and that some part of homework benefits from discussed in groups. But I do not think that the majority of class time should be dedicated to "forced group work". Forcing people to breakout sessions has even taken over even professional meetings and information sessions. I'm old enough that I can run away from it.

Meeting in classes and seeing fresh content still makes sense in a time of AI. And in the future this could make even more sense, because online options become less and less possible. Anybody with mediocre tech skills can now pretend to be a student, even discuss in real time by just having a chat box open as an "assistant". The video feeds pretending to be a person are not yet developed but will soon be available if Zuckerberg with his Meta world should succeed. Already a few years ago, many students during the pandemic opted not to show their faces (which we allowed because there could be personal circumstances like students needing to take care of family members or work from small apartments with other family members working). Serious online assessments are not possible any more in a time when bots can pretend to be humans without being detected as such. The good old Turing test has clearly passed. All metrics one could build, have shown this. The last, fastest progress came in math competitive Olympiad type problems.

Maybe I should also comment about "teaching from the worksheet" which is in vogue these days. Some educators justify this with slogans like "active learning" while it is just an excuse not having to write your own lectures any more. Of course, one has to give a definition of what "active learning" is. But it usually means that less content is taught but that students engage directly with the material through worksheets. It sounds good, but it is fast-food, especially if the worksheets are recycled year by year and some content is delivered from videos recorded decades ago (fast food). It is not better than ``teaching from the book" which is despised. The fact that a worksheet is not printed in a book is a technical detail. Some few students download the worksheets ahead and prepare so that they can "shine" in the classroom, leading of course to "frustration" of students who have not done so. The argument, that each teacher "is free to change the worksheet as they like" is unrealistic, first of all since these worksheets are already in an "equilibrium" meaning quite well proof-read and also because some students take them directly from the website and expect each section to experience the same worksheet. Lesson plans are detailed and provide little room for creativity. A teacher who would do anything different would violate the team spirit.

I have recently seen some push-back by students about this. They either do not come any more to classes or then they openly rebel, when addressed directly in class, with statements like "I have engaging anxiety and prefer not to voice my opinion in a group". I myself am a rather slow thinker and with many worksheets floating around, I have at first absolutely no clue what has been asked. As teachers (also course assistants) one has access to solutions and then "of course!" the worksheet make sense. The set-up is popular because as a teacher, one does not have to work and be creative any more. (I actually have a big problem as I can not solve at first most of the tasks asked in these worksheets). I predict that the push back of students to "forced interaction" in the classroom will become even larger in the future, especially in a time when teachers need to navigate a mine field, especially when engaging in group dynamics in the classroom. And especially during a time when the school openly allowed to voice hate and racist slogans and allow them to be displayed in the yard. Racism and violating school policies has been allowed, but humor in the classroom has been banned as it is considered offensive! There is still great talent at our schools but they are drowning in the mud slung around by a few.

Teaching culture is developed over longer period of time. It can deteriorate in a very short amount of time. The change even can become catastrophic. Some is structural: at the moment, teachers are forced to reuse old material under the pretext that otherwise, quantitative reasoning with data requirements QRD are violated. This preposterous interference of other departments is one of the biggest scandals in the teaching landscape at Harvard, in my opinion. I have been in war with it (and repetitively been punished to speak out against it), since the beginning meaning that the "big data overlords" would a priory dismiss anything I wrote about it or de-list courses I teach from QRD. For me, mathematics is about elegance and beauty and effectiveness. Rewriting lectures should be encouraged, especially if it improves the outcome. The current culture prevents calculus teachers to be creative. I'm not anti-data. I just think it should be done with taste. Here are 4 examples that I have written and used: [Data 1 PDF] and [Data 2 PDF] and [Data 3 PDF] and [Data 4 PDF]. The "big data" hype has now been overtaken by "AI" which is "ultra big data". No human wants to deal with these large structures themselves any more. Tuning trillions of weights is not something you want to engage yourself. [I have even difficulty to find space to store on my computers training data used for these AI tools and have hundreds of Terra bytes of hard-drives on my personal computers. It dwarfs my personal book collection of maybe 50000 books and personal movie collection of maybe 2500 movies.] What you want as a human to do is "understand" why mining and learning these structures work. And this is what we mathematicians still do best. Maybe look at this Handout on the Universal approximation theorem. This is what I would call poetry! It is much more valuable than our QRD bean counters wants us to do, adding up some numbers in a spreadsheet is not human.

About exams

Exams should serve as a fair overall assessment and also motivation to review and solidify the material. I believe that a final exam should cover all aspects of a course. By experience, anything which is marked to be left out will not be reviewed. [This semester, I for some reason forgot to include "linearization" in the final exams. No practice exam mentioned it and of course, I also did not ask this in the actual exam. I believe, none of my students reviewed linearization.] Our students are effective in preparations. They have learned this, otherwise they would not be here. I myself made the "mistake" as a first year math student to "learn what I liked" and not "learn what I needed" and "go beyond the course". My grades after the first year were not good. I then adapted my techniques and got better in the next years exams.

The average score of the 2024 spring Math 1a final exam was 75 when scaled to 100. This is exactly my target point I aim for in any course. Having 40 years of teaching experience, and having seen the class working during office hours, I am able to pre-assess the abilities of a class well. Exams provide the best learning moments. Students work in a controlled, quiet environment without distraction. It is like a competition or tournament in sports. A real match, a real game, a real climbing tour is the best exercise. Of course, it also needs preparations for that in the form of strength training, endurance training and watching nutrition. From my own experience, I know also that exams are "memorable". You remember problems which have been asked during an exam. It is one of the best teaching moments overall. It is also nice for slower students or students who are more shy to show what they can do on their own. In a group, it is often the fastest thinker or loudest who can answer problems first and have the others just follow and not go through the process of finding the solution on their own. I'm sure that if we would be able to make a chart about the math abilities of a particular student over time, then right after having done the exam and then seen the solution, the ability of the student would peak. Later of course one forgets again and would have to relearn.

Even the most progressed students should have to think hard to crack some of the harder problems and fail in some problems and afterwards learn from the mistakes and get even better. But also slower students should have a chance and learn to do the problems. If the average score is too high, every little point become an issue. Assume you make an exam with 100 points where the average is 95. Then missing one point is a big deal. With an average of 75/100, this is not a big deal. I like if some students manage to get everything right (meaning that the exam is doable) and also if it is not a matter of luck. Luck comes in when asking only a few select problems.

An anecdote: when I had been tested orally in numerical analysis (the course had been taught by Peter Henrici but he developed a brain tumor at the end of the course and died soon after and could not test us; a replacement prof examined us), I was asked about Chebychev polynomials and I did not recall that.. It was part of the course which I thought was "fringe" and would not be tested. The oral exam was a disaster as the inexperienced prof did not ask anything else. There was just silence. The written exam was good so that I passed. It had been a matter of luck. If I had been asked about Runge-Kutta, I would have been fine. When I had been tested orally in probability and statistic by Hans Foellmer, he asked a cascade of questions, which got more and more advanced until I passed out, then started with an other cascade of questions until I could no more answer. I don't recall the exact questions but it went like this: "what is a probability space? What is a random variable? What is the variance of a random variable? What is the law of large numbers? Why is the central limit theorem important?" "What is entropy" and then an other cascade of questions in statistics. With such an approach, one does sample the knowledge space of a student along different lines like in tomography. The earlier example is like taking a needle and take a random sample from your brain and see whether it is good or bad.

About AI in 2024

When Chat GPT hit the masses in the fall of 2022, almost instantaneously, millions of users used it. Already in 2023, there was hardly anybody I know, who has not regularly used it. Now the entire world is in. With previous revolutions, like email (I use email since 1983), computer algebra systems ( I used Reduce/Macsyma/Cayley in 1985 and Mathematica since 1987 (before the official version was out) the web (I made my first course website for a dynamical systems course at Caltech in 1994), the adaptation of AI was almost instantaneous. It took only weeks to sweep. The intelligence boost which these tools obtained since the fall of 2022 is enormous. Initially, it would still fail in some math problems. Today, there are almost no constraints any more. I could still laugh about errors done in 2023. In the summer of 2023, it had been fun to watch Chat GPT fall into traps and do errors. This is no more the case. This year, humans were surpassed also in Math Olympiad type math questions. Soon, AI might do research better than humans. What does it mean for teaching? This is hard to say. This year (spring 2024) I asked in homework that students use AI, also for creative tasks and explained some of the math behind it. The universal approximation theorem telling that every continuous function can be approximated by neural nets.

The first official Harvard recommendations for AI from 2023 were probably written by Chat GPT 3: recommendations sent to faculty included advise like ``write problems in such a way that AI can not solve it". This was so out of touch: we see these days that AI solves these ambiguous problems especially well. AI can solve Chaptas better than humans. I get regularly banned from websites because I'm unable to solve the Chaptas which are ambiguous. No problem for AI however. It seems that preventing access to AI means forbidding access at all. If you give a vague, conceptual or exploratory problem to an AI, it does solve it already better than humans. In 2023, there were still problems with reliability and basic arithmetic issues (and was good for laughs), but this is no more the case today in general. In doubt, the AI gives you a python program to calculate what you need to know.

The new guidelines (2024) and here are less specific and essentially posts alerts about protecting data and to follow academic integrity (which is probably the best one can do today). They also recommend to use sandboxed AI tools that are procured on behalf of Harvard (and which also can be monitored). At the moment, it is depressing to see AI progress outpace human tasks so quickly. It is depressing because there is little one can do to prevent it. Forbidding or regulating it would not an option since one can not globally forbid it. Unlike atomic weapons, which were difficult to produce, the production of AI could be done by anybody. One would have to forbid computers.

The preface to 1A

Math 1a Spring 2024 with The PDF containing all lectures, worksheets and exams. I add the preface (status May 10, 2024) here.
  • This is an introduction to calculus. I have taught this class from 2011-2014 from 2020-2021 and in the spring 2024 at the Harvard college. In order to fulfill some formal requirements about quantitative reasoning with data (QRD), I was forced to include some problems from the fall. This includes problems 1.1,1.2,1.3, 2.1,2.2,2.3,2.4, 3.1,3.2, 4.2,4.3,4.4, 5.2, 6.1,6.5, 7.5, sometimes simplified and adapted to the style of this course. I needed this year to correct course after three weeks. Even in a standard calculus course like this, course material, difficulty and especially complexity needs to be adapted to the class. These parameters fluctuate from semester to semester. The student preparation levels in 2021 for example were much different than now in 2024. The pandemic has heavily damaged high school education temporarily. Student feedback collected on paper in class as well as work with individual students during office hours helped me to navigate the ship. Autopilot in the form of ``teach from the book" or ``teach from a previous script" would not have worked. Teaching can be brutal: we can easily get 95 percent of all right. But 5 percent that are rotten can kill a course. The present variations in student preparation levels are astounding. In our time, gigantic changes happen in learning and teaching. I myself like to experiment and involve new elements and keep me excited. Since a few years, AI is the ``new kid on the block". This course included AI since 2011 which was after the AI winter and built on math chat bots built in 2003-2004 here at the college. This semester included a lesson about the sigmoid function, the king of functions in AI. A neural network is function f(x)=σ(ax+b). We can build any function from such neural nets. The universal approximation theorem has risen within a few years from obscurity to something more important than rival approximations like Taylor or Fourier series. This is not said lightly. Fourier theory for example is important too as all image, sound or movie compression algorithms use it. But transformer models in their core depend on the fact that any continuous transformation can be approximated by neural nets.
  • Something about the ``difficulty" and ``preparation level" of this course: while no previous calculus exposure is expected, basic pre-calculus skills in geometry and algebra are assumed. This semester, a large part of the class showed initial deficit in these requirements and had to learn it during the semester. Even when having seen some calculus before, a college single variable calculus course like this one will lead to a deeper, more conceptual understanding of the subject. It allows to see the beauty and elegance of a mathematical theory and appreciate its applications. While concepts and applications are important, the mastery of skills is pivotal too. Especially when doing the first steps in a new field, one has to focus on skills. Fortunately, procedures are faster to learn and teach than insight which requires more time as it requires experience and the ability to connect the dots and see similarities, patterns and being able to ask questions.
  • A good strategy is not to worry at first too much about reaching the ultimate picture but to focus on mastering isolated skills. The unfortunate dogma ``you need first to understand before you can do" can lead to learning blockades. Whenever one embarks into a new area of knowledge or activity, one always faces a similar challenge: at first one has to get acquainted with language and jargon, then one learns how to work out things and finally one sees patterns and gains insight and can build bridges between different already grown knowledge patches. It is important at first to just enjoy the learning while doing it. The understanding will come naturally. This is the same for all learning: if you learn to play a music instrument, a new language, to cook or to climb mountains, you first want to know the notes, the words, the ingredients and the gear. Then you learn how to play, to speak, to follow recipes or climb following a mountain guide before you start to improvise, write your own text, to create new meals or discover uncharted climbing routes. I myself constantly keep learning new mathematics. What I do when learning a new theorem is to see it work in an example, learn it by doing. After I'm comfortable with what it can "do", I get motivated to learn "why it works". Entire generations of teachers have been indoctrinated by Bloom taxonomies pretending that one has to first "understand" before one can "do". This is often wrong. You usually do not learn how a car works before you drive a car.
  • You might wonder why it is necessary to learn a single variable theory. Isn't the world vastly multi-dimensional? It turns out that the one-dimensional point of view, the development of a single quantity over time, is extremely important. Many high dimensional phenomena can be reduced to observing a single quantity evolve in time. If you study the motion of the universe for example, you are interested in its expansion rate, which is a function of time and so subject to single variable calculus. If you study the spread of a virus, you are interested in the number of infected people as a function of time. If you study climate or weather, you can be interested in the average global temperature over time, like decades or centuries. These are functions of one variable, despite the fact that the underlying mechanisms are complex systems given usually as partial differential equations. If you are interested in finance, you might be concerned in the stock prize of a single company over time. If you probe a probability space with a random variable, then you are interested in the distribution. These are all functions of one variable again. Time is one dimensional. Single variable calculus therefore is a window to some of the deepest secrets of knowledge.
  • Calculus also is a large part of our scientific cultural heritage. Knowing about the historical development helps us to see today, where the difficulties really are. A student of calculus essentially faces the same challenges than our ancestors who developed the field. [H. Eves, Great moments in mathematics, MAA, 1981] History also produces story lines and add a dramatic element. Only a dozen generations ago, humanity has not yet understood the notion of limit. In antiquity, the subject had been part of sometimes controversial philosophy. Paradoxa formulated already in antiquity illustrate the confusion of that time. Today we have a crystal clear picture. While the ideas of calculus already trace back to the time of ancient Greece, in particular to the time of Archimedes, the subject has exploded into a powerful tool during last few centuries. It is now a pivotal theoretical foundation for other mathematical areas and scientific fields. Without exaggeration, it is safe to say that calculus is one of most amazing scientific and cultural achievements of humanity.
  • Calculus consists of differential and integral calculus. Differential calculus studies ``change", integral calculus deals with ``accumulation". The fundamental theorem of calculus links the two. The subject is very applicable to problems from other scientific disciplines. Calculus is not only important because of its content and applications like life sciences (example: tomography), data science (example: correlations), internet (example: networks), artificial intelligence (example: machine learning), geography (example: data visualization), movie and game industry (computer graphics), the ideas of calculus also enter in disguised form, in statistics, economics, computer science, in art or music theory. Do not forget that we primarily want to learn the nuts and bolts and down-to-earth techniques.
  • The use of computers and computer algebra systems or online tools and apps and large language models help understanding the mathematical structures. The use of laptops or tablets in class to take notes is of course perfect even so, I would encourage also today still to write by hand as exams will be done on paper by hand. Modern AI has killed online-testing. Modern AI passes the Turing test. We are fortunate that the 2017 paper "Attention is all you need" was not written 3 years earlier. It would have made online education during the pandemic a joke. Today, a computer can take an online course and be undetected because the machine can be instructed to write in the style of a certain person with a certain background and level of sophistication. It is trivial today to let your homework done by one of the large language models and stay undetected. AI tools are now especially good in problems which are complex, cryptic or convoluted. The college recommendations on the use of AI were so out of touch until recently that one must have assumed these recommendations were written by 2022 level AI. In 2023, there were still issues with basic arithmetic, but this seems have been ironed out. I strongly discourage giving the problems to AI. It will solve the problem well and write it in a way to become undetected, but you do hardly develop problem solving skills during the process. Computers, phone or tablets of any type can in these days no more be permitted during exams. Computer assistance for homework must be acknowledged in the homework as writing using AI is like plagiarizing. Write problems out on paper or electronic paper. Electronic ink is not really different from ink in this respect. The material sticks better when you write mathematical formulas and procedures by hand. The tactile process of writing by hand enhances long term retention and prepares for exams. One can imagine apps in which you photo an assignment page and out comes a handwritten solution written in your handwriting, in your style and with your level of expertise. Anybody who has seen the developments in the last two years and thinks this is not possible is delusional. [I can ask Chat GPT 4 to write like "Oliver Knill" and it does it.] The future of testing is in-person or by doing oral examinations. I myself was even in basic intro math courses tested in person in an interview process with the professor. This old school testing technique will certainly come back and be effective until Chat GPT 20 has been implanted into our brains. You will be able to chose your AI level depending on how much cash you can afford. Already now, we pay to access higher intelligence AIs.
  • This course does not follow any book or previous course. There are many good resources. Most textbooks have been proof-read quite well but they are usually huge and messy. A popular text which has proven to be useful in the last decades is ``Single Variable Calculus: Concepts and Contexts", by James Stewart. I can recommend small books or the "Cartoon guide to Calculus" by Larry Gonick. Our homework problems are of similar difficulty than in classical textbooks; some of them are trickier and intended to trigger discussion with other students or teaching staff. I recommend to attack each homework problem first on your own. This helps you to develop independent thinking and problem solving skills and prepare for the exams. You should get comfortable with the situation of being stuck at first. But even if you can solve a problem, it is helpful to discuss them with others. Mathematics is also a social activity.