Look, this adder-subtractor is totally working, not blocked at all (unlike my writing life right now)! Image by en:User:Cburnett [GFDL (http://www.gnu.org/copyleft/fdl.html) or CC-BY-SA-3.0 (http://creativecommons.org/licenses/by-sa/3.0/)], via Wikimedia Commons)
Sometimes I love writing. Sometimes I hate everything I write: mathematical, recreational, shopping lists—sometimes all the words I produce just seem icky. Writer’s block is not something that should matter in math life, right? It’s not adder-and-subtractor block! Alas, I am becoming suspicious that writing might actually be the skill most central to academic life. I write lesson plans and assignments, I write on the board, I write comments on student papers. I write math papers and grant proposals. I write so many emails. I write endless reports on how I am doing great things and should get to keep my job. And of course, I write a blog. But right now writing is difficult, and I am trying to work out how to deal with that.
However, thinking about how much writing I do (and how I still struggle with it) strengthens my belief that it’s important to write in math classes, at all levels. Computational abilities are useless without conceptual understanding and the ability to communicate. Written communication is necessary to effectively convey understanding and justify reasoning in both academic and “real-life” settings. I believe that writing is also a tool to build understanding—by working to express concepts we come to understand them better, by working to express confusion we see a way to clarify. So writing assignments are part of almost all of my courses. However, creating these assignments (and grading the submitted work) is hard!
Actually, making the assignments can be easy: “Write 500 words about the origin of calculus.” Done, right? Sure. Until you start reading the papers that you get, and later reading the comments that students will make on the end of semester evaluations about an assignment like this. This is a terrible assignment. I know because I gave this assignment to a class when I was in graduate school. The best submitted assignments were paraphrased from Wikipedia articles. Grading was a nightmare, as you can imagine—I felt guilty taking any points off, since I had given so few directions and had so little confidence in grading writing. So read a lot of awful papers, agonized, finally gave everyone most of the points, then felt creepy letting these essays pass for college-level writing.
If I didn’t think that writing is really important I would have stopped there. However, I learned from this mistake and got some help. There is plenty of advice and guidance out there on creating and grading writing—a Google search on “writing assignments for college math” turns up many excellent resources. As a general resource here is a collection from the MAA, and here are a bunch of wonderful ideas from Annalisa Crannell. There seems to be a little less material available for more advanced classes. So I thought I would share some of the writing prompts I used for my Intro to Proofs and Modern Algebra courses last semester.
Note that these were designed to be blog posts, complementary to but very different than the formal proofs that they wrote. I wanted these writing assignments to be informal and not research- or problem-based. Mostly I asked students to respond to something or explain something. This may not be what you are looking for, but it has been a useful tool for my courses. I should also mention (again) that whole idea of doing blogs in my math courses was inspired by reading this blog, and there are also some great prompts there. However, I originally used some of these as non-blog assignments in other courses and they went well. I created all of these prompts, but I do not claim that these are all my own brilliant ideas–some were inspired by colleagues or other sources. You can certainly blame me for the ones you don’t like, though. And of course, please share your good writing ideas and thoughts on these assignments in the comments!
For both classes, these first two prompts were always allowed:
1) Explain some idea from the class up to this point in language that a non-mathematician could understand. You could pretend that you are talking to your grandmother or your art-major roommate. Make your explanation as intuitive and non-technical as possible, while bringing across something actually cool about the idea.
2) Write about anything you found especially interesting or puzzling about the material or course so far. Your classmates are your audience here–you can assume that your audience has a similar mathematical background to yours.
3) This was a popular and useful first week assignment:
Imagine that you are a famous mathematician and have written an autobiography. Write a 200-500 word excerpt from this book, focusing on some aspect of your mathematical life up to this point. Your audience is a general reader, at least somewhat interested in math but who may not have taken calculus.
4A) I used the following for Intro to Proof:
There are a ton of math blogs out there, with a huge range of goals and aimed at very different audiences. Steven Strogatz is a professor and math writer who blogged for the New York Times. He aims for a broad audience with the goal of entertaining while explaining real higher mathematics. Here is a link where you can access several of his blog entries:
Read a couple of Strogatz’s blog entries and write 200-500 words on one of the following:
a) Critique a piece–what does he do well? What techniques does he employ to make the mathematical idea clear to a general audience? Are there any things that you think he could do better?
b) Write a continuation of his piece, using a similar style and expanding on what he wrote.
4B) For Algebra, I gave a very similar prompt but focused on this group theory post:
5) In Foundations, we were studying logic and sets, so it seemed worthwhile to bring up this guy…
Bertrand Russell (1872-1970) is one of the greatest figures of modern logic. With Whitehead, Russell wrote Principia Mathematica, a nearly 2000-page tome that rebuilt the foundations of mathematics in terms of set theory. He was also an important figure in philosophy, and generally a public intellectual in many arenas. His personal philosophy led him to a highly controversial lifestyle.
Russell wrestled with some of the same issues of language and logic that we are working with in class. For example, in the following piece, he replies to a reader’s letter regarding his use of the word “implies”. He has used it in the strict mathematical sense we have been considering in truth tables, while the reader took it in the more informal sense that most people use in speech. His reply is enlightening: http://www.users.drew.edu/~jlenz/br-on-if-and-imply.html
For further reading, here is a very large collection of Russell’s work.
Read Russell’s reply and let it influence you as you write 200-500 words on one of the following topics:
a) What is the difference between the mathematical use of “implies” and the usual use of the word? Can you use this to explain what truth tables and logical equivalence do (and don’t) mean?
b) My favorite math joke: Three logicians walk into a bar. The bartender says “Do you all want a drink?” First logician says “I don’t know.” Second says “I don’t know.” Third says “Yes!” Why is this joke funny? Explain.
6) Pierre de Fermat was a French lawyer and mathematician who lived from 1601-1665. He discovered many of the main ideas of differential and integral calculus before Leibniz or Newton, and developed many of the main ideas of analytic geometry before Descartes. He often wrote mathematical ideas and conjectures in the margins of his copy of Diophantus’ Arithmetica (a 3rd century Greek text). Later, mathematicians proved many of the statements in these notes, and also disproved a few, until at last only one remained unresolved. This statement, known as Fermat’s Last Theorem, stood without proof for over 300 years, until the early 1990s when Andrew Wiles, with help from his former student Richard Taylor and others, finally completed a proof. “The Proof” is a documentary about the remarkable process that led to this proof.
Your assignment: Watch “The Proof” [PBS or BBC version]. It is approximately 45 minutes long. After watching, write 500-1000 words on one of the following topics.
a) Mathematicians work in many different ways. Often, popular culture and accounts of some very famous mathematicians give us the sense that great mathematics must be the work of lone geniuses, working alone, in isolation from the distracting outside world. The account in “The Proof” in some ways fits that narrative. However, Wiles was not able to complete the proof entirely alone. Though he developed many of the main ideas, he needed others to carefully check his work, and Richard Taylor made many essential contributions to fixing the proof. He also built on the work of many others who worked on the problem in the previous 300 years.
To what extent is collaboration necessary and useful in mathematics? Is it important to solve problems entirely on your own? What is attractive about working alone? Working with a group? How do you work on mathematics? How do we know when mathematical work is correct?
b) You may have thought a bit about the difference between “pure” and “applied” mathematics. Pure mathematics is often thought of as math for it’s own sake, while applied mathematics is done with an application (practical or otherwise) in mind. One of the main areas of mathematics that arises in the proof of Fermat’s Last Theorem is the study of elliptic curves. These are cubic curves, the points of which form a group under a nifty geometric operation. These curves are also important objects in modern cryptography, the science of sending messages over public channels while keeping the content secret from eavesdroppers. When people first began studying elliptic curves, neither of these applications were anywhere in sight. Often, mathematics that initially seems uselessly “pure” finds amazing application as times and technology change. What is your interest in pure vs. applied mathematics? What kinds of problems do you find exciting? Do you think that the division between pure and applied math is valid? Does the idea that pure mathematics can become applied make it more or less exciting to study pure mathematics?
7) For Algebra:
The Rubik’s cube is an example of a game that can be understood using group theory. At the most basic level, every transformation of the cube can be accomplished by some combination of 6 basic moves and their inverses. In cube circles, these moves are often known as F, B, U, D, L, and R, and they correspond to turning the front, back, top, bottom, left, and right sides of the cube clockwise by 90 degrees. Thus the group of possible transformations of the cube is generated by these 6 elements. This group of transformations has order more than 10^19. However, through group theory, it has been proven that every cube can be solved in a sequence of at most 26 basic moves. For your blog entry, think of another game that relates to algebra in some way. Explain the game and how we can view it through the lens of algebra. Explain what the objects in your set are, and the operation that combines them. Though there does not have to explicitly be a group, you should explain why the group properties do or do not hold.
What I do when I can’t write anything good (gratuitous fishing with my dad pictures):