This is the official syllabus for the class. I will have a pdf version of this syllabus made soon and linked here, once some details are settled. You can find links to assignments, materials, and everything else (as they become available) on the Calculus course webpage. If you are new to my classes, please also take a look at my statement of teaching principles.
Instructor
Andrew McIntyre. Most reliable contact method is the Bennington Math Slack; you can also email me at amcintyre, the “at” symbol, bennington.edu. At the moment, for Spring 2021, I am fully remote, and not in the office. I do not have set office hours for this term. However, you can contact me any time with questions, and I am happy to answer over Slack or to set up a Zoom meeting. During the term, I aim to always answer within 24 hours; please feel free to pester me if I don’t.
Class times and location
The class is Mondays and Thursdays, 1:40–3:30pm. It is currently planned to be held remotely and synchronously, over Zoom. (I have sent you calendar invites for the class, please let me know if you have any difficulty.) The first class is Thursday, February 18, 2021, and the last class is Monday, May 24, 2021.
Note on recording
I’m planning to record the synchronous classes, and make the recordings available on an unlisted YouTube channel. I will not record group work or in-depth class discussions, but I am planning to include student questions. If you have concerns about being recorded, please contact me.
Textbooks
There is no required textbook. We will be working mainly from the lecture and problem sets; when I ask you to read a selection from a text or other source, I will provide a pdf or a link.
I do have some recommended texts, if you would like to read further. (Although please see the note below about outside sources!)
This class is quite unconventional in its approach. The closest book to our class—and the inspiration for this way of doing the subject—is
• Calculus Made Easy, by Silvanus P. Thompson
This is a charming book, and very pleasant to read. I will be assigning some reading from it. It is available as a free pdf from Google Books. I did not include bibliographic information because there are many editions; I prefer the older editions (the new one updated by Martin Gardner takes out a little bit of the charm). This book does not cover quite as much as our class, and in some different ways.
One other book that has a similar approach to this class, but which goes much further and in much more detail, is:
• Calculus: An Intuitive and Physical Approach, 2nd ed., by Morris Kline, Dover 1998, ISBN 978-0486404530, Amazon price $21.70 (used starting at $9.01)
Kline is quite long-winded with his explanations, which can be both good and bad. He covers many interesting topics that we will not have time for. Unfortunately there does not seem to be a legal free version available.
Another book which takes a similar approach to mine in its emphasis on differ- ential equations, but is otherwise more modern, is
• Calculus in Context: The Five College Calculus Project, 2008 ed., by Callahan, Cox, et al., Five Colleges Inc. 2008, ISBN 978-0716726302, Amazon price $49.00 (used starting at $17.00)
You can find this text as a free pdf by searching for the title and authors.
If you would like to see a more standard calculus text for reference, two good ones that are available as free pdfs are
• Calculus, 3d ed., by Gilbert Strang, Wellesley-Cambridge 2017, ISBN 978- 0980232752
• Calculus I, II, and III, 3d ed., by Jerrold Marsden and Alan Weinstein, Springer 1998
Strang is one of the better standard texts: it is very clear and to the point (many modern textbooks are very overstuffed). For calculus at a slightly higher level than this class, Marsden and Weinstein is one of the best standard texts.
Unfortunately, we will not be able to have the texts on reserve, due to COVID restrictions. There are copies in the Dickinson Reading Room; I ask that you use those only in the building.
WARNING: The Kindle versions of Thompson and of Kline are, (at least the last time I checked), TERRIBLE. They are essentially unusable. Do not purchase the electronic versions of these books.
Calculators and computers
You will of course need a computer and reliable internet to participate in the synchronous classes. Please contact me and IT if you anticipate any problems, and we will do our best to sort it out.
Other than that, a simple scientific calculator (or phone calculator app) will be sufficient for all the work in this class. I will sometimes suggest graphing things on the computer; for this purpose, the web app Desmos is the nicest graphing tool I know of.
Communications
There will be three main routes of communication: the synchronous classes, the Slack channel, and the webpage. The Slack channel is the main place to look for announcements: any new problem sets, written notes, or other class communications will be posted there. Please either activate notifications for Slack on your computer or phone, or check it regularly. Anything that I announce in the synchronous classes, I will repeat on the Slack. The classes will be recorded and posted on an unlisted YouTube channel (see the Slack for the link). I will collect all links to problem sets, written notes, and other resources on the webpage.
What this class covers
This course covers the breadth of university calculus: differentiation, integration, infinite series, and ordinary differential equations. It focuses on concepts and interconnections. This is an advanced course; Calculus AP or IB are not equivalent to this class. On the other hand, the course does treat the concepts in a logically independent way, so if the other prerequisites are met, no prior experience with calculus is required.
Topics covered include volumes of cones, spheres, and other solids; the meaning of volume and area; computing differential changes; computing volumes by converting to a problem of change (“Fundamental theorem of calculus”); differential equations for projectile motion; exponential growth models; logistic growth models; differential equations for harmonic motion; sines and cosines as infinite series; Newton’s solution of the Kepler problem for planetary motion; logarithms and exponentials; Euler’s formula for complex exponentials; harmonic motion with damping; the Gregory-Leibniz series for π; the Basel problem.
However, this course will not cover some standard computational techniques, which usually make up a large part of the standard Calculus sequence: for example, we will not cover the product or quotient rules, optimization using the derivative, or integration by parts. Other standard topics we will cover, but only in the very simplest particular cases we need, without much practice: the chain rule, integration by the FTC, substitution. (In our course sequence, all these important computational tools are covered in future courses, such as Differential Equations, Dynamical Systems, and Fourier Series.)
If you took Calculus in high school, and were able to do the computations but were unclear about the what was really going on, then this class was designed for you! If you haven’t taken Calculus before, and want an introduction which lets you see the subject as a whole, this class is also for you. My hope is that, upon completing this class, 1) you will be able to see the point of calculus as a historical subject and a coherent whole; 2) you will be able to recognize when a problem in the sciences is a calculus problem and be able to set it up correctly; 3) you will be able to interpret the sort of differential equation models that appear in science papers; and 4) you will be well-prepared to move ahead with calculus-based mathematics classes in future. However, you will not receive extensive training on how to solve such problems in this class.
I would like to emphasize again that this class is not a repetition of AP calculus, nor is it even a repetition of more advanced high school calculus courses (like BC, IB or A-levels). It covers more material in some respects, and less in others; and it does so in a very different way. There is much more emphasis on why things are the way they are. This emphasis is important in learning how to really apply these ideas in unfamiliar situations.
Prerequisites
One prior Bennington College mathematics class, or permission of the instructor. The ideal prerequisite course is MAT 2244: Quantitative Reasoning and Mathematical Modeling, (that is also an ideal sequel to this class!), but other good options are MAT 2482: Linear Algebra or MAT 2410: Logic, Proofs, Algebra, and Set Theory.
The point of the prerequisites is that you are familiar with a certain way of approaching unfamiliar problems, and of thinking about mathematics in a conceptual way. In terms of particular background skills, you will need to be comfortable with polynomial algebra (multiplying out polynomials and simplifying), exponent rules (including fractional and negative exponents), graphing (making a graph of a function and interpreting it), and straight line functions (finding equations, graphing). Some exposure to logarithms and trig functions would be helpful, but not necessary, since we will cover them in this course from scratch in a different way than you have seen before.
Objectives
Every class aims at both learning and evaluation; however, these goals are often in tension. My primary concern is that this class is an effective learning environment; evaluation, while sometimes necessary, is secondary.
In terms of content, I will repeat what I said above: my hope is that, upon completing this class, 1) you will be able to see the point of calculus as a historical subject and a coherent whole; 2) you will be able to recognize when a problem in the sciences is a calculus problem and be able to set it up correctly; 3) you will be able to interpret the sort of differential equation models that appear in science papers; and 4) you will be well-prepared to move ahead with calculus-based mathematics classes in future.
More generally, I’d like you to improve your flexibility and breadth in the way that you think about mathematical problems and concepts.
Work expected
The primary work of the class will be completing problem sets, which will be assigned at a rate of approximately one per week. I will ask you to submit written work on the Slack, and to discuss it with me in Progress Meetings (see below). I will also expect you to do some work during classes (usually collaborative work on problems). In addition, I encourage you to spend time helping your classmates; I will attempt to explicitly credit you for this if you do so. There are no tests, exams, or final projects (although you may choose to expand one of the problem set questions into a larger project).
Some of the problems that I ask you to think about are going to be much more open-ended than you may be used to. For these problems, I am not expecting a complete final solution right away (or maybe ever). What I would like you to do is to write down, and explain to me, your thinking so far on the problem, what you have tried, where you think you are getting stuck.
The time commitment for this class should be about 10 hours per week total, 4 hours in synchronous classes and 6 hours outside of class. This should be fairly constant through the term.
How to submit work
Please submit questions as individual files (one file per question), with a descriptive file name (something like “PS2-Q4”). Please submit them in a Slack direct message to me; I have found that the easiest way for me to keep track of your work. (If you prefer to have a Google Drive folder to keep things organized for yourself, you can make a link to your Google Drive files in the Slack direct message.) I may ask you to submit only selected problems from a problem set; if so, I will let you know.
To submit your work, please use a pdf scanner app to make a pdf. There are several good free ones available; iOS Notes has one built in, and Adobe makes a free one. Please do NOT just take pictures with your phone; these are difficult to read.
Due dates and late policy
I will have suggested due dates for each problem set, to help keep you on pace. However, I do not have strict due dates, and I do not make any penalty for late submissions. The important thing for me is the level of understanding you show by the end of the term; it does not matter how or when you get there. That said, it can be hard to catch up if you let things get too far behind; if you are getting behind, I may send a concern form to that effect.
Notebooks
It is very important to keep a clear and organized notebook for this class. You are going to be doing problems that stretch over several classes, and I am often going to ask you to refer back to solutions that you completed weeks ago.
I recommend keeping a bound notebook (like a composition notebook), for this class only. Write your thinking about the problems in a journal format. Also write class notes, solutions to problems, and summaries. This will help keep your thinking in one place. Number the pages, and put a table of contents at the end.
You may prefer to have a different system: perhaps you could have a bound notebook for notes and journal work, and write problem sets up separately. Or you could use a looseleaf binder, or a system of folders. Whatever you choose, your work should be accessible, clearly written, and understandable to both me and you. Don’t assume that you will remember what you were doing later—write your explanations down in full as well as your computations.
Note that, since you are scanning to submit work, if you like, you can have your problem set solutions in the same bound notebook that you use for notes and thinking.
Feedback
The main instrument of feedback is the “Progress Meeting”. I will be scheduling a time to meet with each student individually for 15 minutes, about once every 2-3 weeks (a total of 4-5 times during the term). The purpose of this meeting is for me to look over your work with you, to give feedback on it, and to ask you questions.
During the meeting, I will tell you if I see problems, either with the answers to specific questions, or more generally (e.g. if you need to be writing more clearly). If problems are incomplete or need correction, I will ask you to redo them and resubmit. The aim is for most of the problems to be done completely and correctly by the end of term. However, with some of the harder problems, I would like to give you some flexibility, to perhaps not complete all of them, and/or to spend more time on the topics that are of most interest to you.
A note on collaboration
For much of this class, I will be expecting you to work with other students. This might seem like it would be unproductive, if neither of you know what is going on—“the blind leading the blind”. But in fact it can be very useful to talk through your thinking with another person. You say what you are thinking, explain where you are stuck, the other person does the same; and in articulating your thinking and difficulties aloud, it becomes clearer where to go next to resolve the problems.
This is not unlike how research mathematicians work together. We don’t have a teacher who knows the answer; perhaps no-one does. We throw out ideas, and then try to test them: e. g. “if that was true, then how would it work in this simple example?”. We try to generate more examples to test ideas, and if we think we have a solution, we try to verify it from different directions.
A large amount of class time will be devoted to collaborative work. I would also encourage you to work with other students outside of class. This is particularly helpful when you are having trouble persisting with problems when you are stuck— having someone else there can really help you push through.
However, it is also important to be able to do work on your own, and to be sure that you truly understand things yourself and are not merely following along. Here is a good rule of thumb: collaborate with other students verbally, and on a whiteboard or chalkboard. But when it comes to write down the solution, do this on your own, without referring to what is on the whiteboard. Write the solution from your own thinking, do not just copy.
A note on lecturing
The emphasis in this class is to get you to figure things out yourself in your own way, as much as possible. This will make you more able to work independently on this type of material after the course is over. It is also the most efficient way to learn new conceptual tools (as opposed to merely learning new information).
For this reason, I will be lecturing infrequently. I will sometimes lecture to give an overview, or if everyone is stuck on a certain point, or if there is something not well-covered by the texts that you can’t figure out on your own.
My view is that the class should primarily offer things that you could not get by staying home reading a book, or watching lectures on the internet. There are two elements to this: one is collaboration with others (see the note above). The other is feedback as you are working. I will get you to start on problems on your own. If you reach a point where you are stuck, I will use my experience to try to give the best possible hint or suggestion appropriate to the point you are stuck at. As much as possible, I will try to only make suggestions that you could have thought of yourself. In this way I hope to help you improve your ability to figure out this sort of material on your own.
Usually mathematics problems consist of many parts that are somewhat rou- tine: you do the next logical thing, or you follow the same strategy as a previous problem—things you can figure out yourself. That alone may solve the problem. Or, it may require a new “trick”. By getting you to do as much as you can on your own, we can focus on the central “trick”, the new bit. When mathematicians remember proofs, they don’t remember all the routine details; they just remember the central “trick”. If I lecture you through the whole thing, it is not clear what you could have done yourself—it is not clear what is routine, and what is the new idea.
I will not tell you if things are correct. A research mathematician has no answer key to tell if things are correct; she must check the result on simple examples, verify that the result is consistent with the facts that she already knows, and try to look for holes herself (preferably by doing the computation or proof in a different way, if possible). I won’t answer questions that you could find in the text or figure out yourself, but I will make suggestions. I will answer questions that are just convention, and which would be impossible to figure out on your own (e. g. “is it sine or cosine that is opposite over hypotenuse?”).
A note on outside sources
NO OUTSIDE SOURCES!
Please, please, do not refer to outside sources when doing the work of this class. Do not even look ahead in the textbooks, beyond what I have asked you to read. There are several reasons for this.
One reason is that I want to use the most simple, classical examples and problems. For example, in discussing volumes, I want to use solids like spheres and cones. Since these are easy to look up, a more typical calculus class will ask about some crazy shapes instead, so that they can ask homework and test problems you haven’t seen. But simple cases like spheres and cones are more interesting, easier to understand, and more useful. If you figure out spheres on your own, it’s not hard to go on to crazy shapes if needed.
Another reason I want to avoid you looking things up, is that it is critical that you work out why things work, for yourself, in your own way. This is more interesting than taking formulas for granted from a book, and it is more useful. In applications, you will never be given a calculus problem and have to just apply a given formula; and if you were, it would be easy to look it up. Rather, in applications you have a physical or biological problem, and you have to recognize whether it could be modeled by a calculus problem, and how that translation could be made. That skill comes from understanding how the formulas are built. And that understanding comes from building them yourself, not looking them up.
On a related note, if you have done any calculus before, please don’t spend a lot of effort trying to solve a problem by remembering something. Chances are, what you are trying to remember is a recipe, and using a recipe without understanding it is just what we are trying to avoid.
Evaluation
My primary goal is for this class to be an effective learning experience for everyone. Evaluating is a secondary concern. But we do need to have some sort of evaluation!
There will be two summary evaluation points, at the midterm and at the end of term. At those points, I will ask you to make a written self-evaluation, and I will also write an evaluation for you. If you are choosing to get a letter grade, you and I will both propose a letter grade as well. The evaluation will be based on the following criteria:
- whether you are spending sufficient time and attention on this class
- how thoroughly you are understanding the concepts
- how thoroughly and correctly you are solving the problems
- how systematic you are at correcting errors or gaps
- how clearly and carefully you are writing your reasoning
- how persistent you are with difficult problems
- how creative you are with challenging problems
- how much of the work you are completing
- how well you are contributing to collaborative work
- how well you are contributing to class discussions (in whatever form)
- how well you are helping other students in the class, if applicable
- how consistent your presence in the class is (see below)
There are no point totals; both your and my evaluations of your work will be holistic judgments based on the listed criteria.
I’d of course like to encourage you to try to do well in all aspects, but you do not have to do well on all the criteria in order to get a good evaluation or grade. For example, if you did not participate much in collaborative work or class discussions, but your written work was good, I might write something like “XX was somewhat quiet, but had a comprehensive understanding of the concepts, based on their clear and thorough written solutions of problems…” Or your balance of strengths might be the opposite. Everyone is different, and there are different ways to do good work!
For the final narrative evaluation and grade, I will take your evaluation and mine both into account. If your evaluation and mine are widely different, we will talk about it and try to come to a mutual understanding of your performance in the class. If we still don’t agree, I reserve the right to have the final say, but I have found that such major disagreement almost never happens in practice.
Attendance and presence policy
The expectation is that you will attend all the synchronous class sessions. This is important since the class activities are often collaborative. Being remote, I also ask the you maintain a consistent “presence” in the class, beyond just attendance: not missing progress meetings, submitting work, asking questions and contributing to class discussions on Slack when appropriate, and working with and helping other students.
However, I completely understand that this may not always be possible. If you cannot attend class for some reason, I ask that you catch up with the class recording (the link to the YouTube channel will be posted on the Slack), and that you try a little harder to maintain class presence in other ways.
If you can’t be present in class (both attendance and other ways), I ask that you keep in touch with me about it, and we can try to make arrangements together to keep you on track. I do not have any particular number of allowed absences, and I do not intend to penalize you on evaluations for absences. However, if you are not present for more than the occasional class, I will send a concern form; and, as a matter of College-wide policy, being absent for two weeks or more in total may lead to a marginal pass or failure even if your other work is good. That said, please talk with me about any gaps in your presence in class, and we should be able to work out a solution. My main concern is that it is difficult to do well if you are not regularly present and engaged. (In particular, I would caution you against the temptation to “skip this one class so I can get caught up”: this never works out well.)
Inclusivity
Unfortunately, mathematics has been and is often tied up with excluding people: both in that people are excluded from mathematics, and in that mathematics is used as a vehicle of exclusion. This is a big, complicated problem, with many facets. The aspects I want to address here are the ingrained attitudes about who is “supposed” to be “good at math”, and who is assumed to be “not good at math”. We have all grown up surrounded by these attitudes to a lesser or greater extent, and they have affected us all, both by the assumptions people make about us, and by the assumptions (perhaps unconscious) that we make about ourselves.
Let me be clear: based both on all the research people have done, and on my own years of personal experience teaching, I see no evidence of any systematic difference in “math ability” (however you measure that) between people based on gender, race, or any other grouping you might make.
Going further, I would assert (again, based both on the research and on my personal experience) that the idea of “math ability” is highly oversold anyway. Most anyone can excel in mathematics, up to whatever level they choose, given a supportive environment.
Although I strongly believe that there are no “math people” and “non-math people”, and that everyone can succeed, there ARE very strong systematic differences in how people have been treated in and around mathematics, and also consequently in how people see themselves. If you have had obstacles put in your path, and have been made to doubt yourself, please know that you are capable, you can excel, and that I will do whatever I can to make a supportive environment for you to do so. On the other hand, if you have been on the other side of these ingrained attitudes—if you have had the advantage of not being systematically minimized or doubted or excluded—then I ask that please recognize the responsibility that comes with that advantage: to not make these unfair assumptions about others, to consciously make space for others, and to help create a culture of mutual respect.
Classroom norms
Everyone is welcome in this classroom, and in mathematics. Please be kind and respectful to one another.
IF YOU ARE NERVOUS ABOUT BEING IN THIS CLASS, please know that, in my experience, everyone can do mathematics, and do it well. You may have had negative experiences with math, and you may have had assumptions made on your ability for invalid reasons. It’s normal to be nervous. You can do this, and you will be supported here.
IF YOU FEEL CONFIDENT ABOUT BEING IN THIS CLASS, (you might sometimes belong to both categories!), please be kind and respectful to your classmates. Do not talk over them, or make it seem like they should know things that they don’t. You will find, as the class goes on, that we are all struggling with math problems in our own way. We are all in this together.
IF YOU ARE MADE UNCOMFORTABLE IN THIS CLASS for any reason, please talk to me about it. (Aside from being uncomfortable about hard problems, but you can talk to me about that too!) I will do what I can to solve the issue. Please do not hesitate to be direct with me if you are having a problem, including if the problem is me: I promise not to be offended or upset. If I do not resolve the issue, or if I am the one making you uncomfortable, please talk to a college administrator (e. g., from The Office of Diversity & Inclusion, Student Life, or Academic Services) or other faculty member.
Academic Accommodations and Health Resources
I am happy to work with you on any needed accommodations. Please let me know the situation.
Here is the official statement: Bennington College provides reasonable accommodations to students with documented disabilities when such accommodations are requested and necessary to ensure equal access to College programs and facilities. If you believe you are entitled to an accommodation speak with Katy Evans, the Academic Services and Accommodations Advisor, about any disability-related needs. If approved, you will receive a memo detailing your specific accommodations; it is your responsibility to provide me with the memo and discuss the implementation of accommodations. Note that I will not be aware of your needs if you do not share this memo with me. Accommodations are not retroactive, so the sooner we meet to discuss your needs, the better. Also, students experiencing mental and/or physical health challenges that are significantly impacting their academic work are encouraged to speak with their faculty advisor and member of Academic Services (academicservices@bennington.edu or 440-4400) about the impact and to connect with resources through health and psychological services (440-4426 or 440-4451).
Basic needs
At Bennington College, we understand that basic needs (food, housing, and wellness) have a direct impact on academic performance, mental-emotional-physical health, professional development, and holistic success of our students. If you have a personal circumstance or need that will affect your learning or performance in this course, please let me or your faculty advisor know so that we can direct you to available resources to help support you during the term.
Conclusion
You can find a more general statement of my teaching principles here. Please contact me if you have any questions at all!
