Advice for new college students

1. Get to class on time.
2. Read your email, but not during class.
3. Spend a summer on campus. Work for a professor, be a tour guide, do research, etc.
4. Use proper grammar and capitalization in the email messages to your professors. The email shorthand that may be appropriate between friends is not appropriate when corresponding with your professor (e.g., “hey, prof. when r u going 2 b in yr office?”).
5. Call your teachers “Professor —” not “Mr. —” or “Mrs. —.” Almost all of your professors have the highest degree in their field (usually a PhD). (Addressing them as “Dr. —” is appropriate too, although it isn’t common at our school.)
6. Get to know your professors and let them get to know you. They’re nice people. Ask your professors about their research, their family, their schooling, etc. Tell them about your summer research projects, your internships, etc. Down the road you may want to ask them for a letter of recommendation and they will be able to write you a much better letter if they know you. Besides, they are human beings, if you are rude to them, they will be less enthusiastic about helping you.
7. Don’t skip class. Either you won’t be able to learn the material that you missed or the “free hour” that you gained will be lost several times over trying to catch up. If you do skip class, DON’T ask the professor what you missed—get notes from a classmate.
8. Take classes outside of your comfort zone.
9. Be protective of your online identity. Don’t post photos on Facebook that you wouldn’t want your parents, your professors, your future inlaws, or your future employers to see.
10. Don’t sell your books back, especially for classes in your major.
11. Don’t be a member of a clique. For many of you college will be the most diverse living experience of your life. Get to know as many people as possible and not just those with the same background as you.
12. Be organized, use a calendar, and pay attention to due dates.
13. Find a good distraction-free place to study.
14. Learn to write well. I’ve seen far too many mathematics and science students avoid writing courses. They are under the impression that it won’t be relevant to them. Writing is an extremely important skill that is a prerequisite for almost all careers. You will be amazed at how much you will need to write.
15. Learn from your mistakes. Look over your assignments when you get them back. The professor put those comments on there for your benefit. If you don’t understand the comments, ask.
16. Do the assigned work. And the related…
17. Don’t ask for extra credit. I don’t give extra credit and neither do most other college professors; if they do, they would give it to the entire class not just to you individually. Extra credit is great for the strong students—it can boost their grades from an A to an A+. Weaker students who need a grade booster should spend their time doing the assigned work (which they often haven’t done—that’s why their grade is in trouble in the first place). Doing the assigned work is the best preparation for the exams in the class—it gives the best “bang for the buck.”
18. Start assignments early and start studying early. Related: don’t email the professor late the night before (or worse, the day of) an exam or the due date for an assignment asking for help.
19. Admit when you are wrong. It may be difficult, painful, or embarrassing, but it is liberating. Living with a lie or a guilty conscience is worse than coming clean.
20. If you choose to drink alcohol, do so in moderation. Not all college students drinks alcohol. According to a survey given here last year at fall break, approximately one fourth of the first year students had not consumed any alcohol in the past year.
21. Stay healthy: eat well, exercise, and get enough sleep.
22. Take the courses you want to take, not the ones your parents want you to take.
23. Beware of technology such as video games, movies, social media, etc. They can be unhealthy, addictive time sinks.
24. Don’t read or send text messages in class.
25. Try new things (clubs, sports, volunteering, etc.), but don’t spread yourself too thin.
26. Call home, but not too often.
27. Get off campus and explore the area. Eat in restaurants, go for a hike, see a movie, visit a museum, etc.
28. Study abroad.
29. Do not beg your professor for more points on your graded work. If you have doubts about the grading, ask the professor to explain his or her reasoning. Most likely, if there was a grading error, your professor will fix it.
30. Show up for appointments and be punctual.
31. Don’t let your parents fight your battles. Professors cannot speak with your parents anyway (without a FERPA release).
32. If things start going wrong, see a counselor. Each year the counseling center is used by 15–20% of the student body. The service is completely confidential; they won’t notify your parents, your professors, your friends, or your insurance company.
33. Let go of your high school anxieties. Your classmates didn’t know you in high school. Make new friends, wear new clothes, listen to different music, and try new things.
34. Don’t lie to your professors; they’ve heard them all (otherwise known as the “dead grandmother rule”). (A retired professor I know used to send a condolence card to the student’s parents every time a student informed him of a death in the family.)
35. Be considerate of the neighbors. Not everyone in town is a college student. Keep this in mind when you are returning from a party at 2:00 AM.
36. Be a good roommate.
37. Don’t cheat. The penalties are steep if you are caught. If you are not caught you will have to contend with a guilty conscience. Cheating will produce a short-term gain and a long-term loss. Besides, it is a slippery slope—this is not the way you want to conduct the rest of your life.
38. Become a novice. You’ll learn more and get more out of college if you don’t hold onto the attitude that you know everything already.
39. Go on a road trip.
40. Look at your final exam schedule before scheduling your flight home.

So, I decided to bring them all together into a one-size-fits-all LaTeX cheat sheet. I kept it to two pages, so it can be printed (double-sided) on one piece of paper. (I have also posted the LaTeX code. Feel free to take it, edit it, and use it.)

It doesn’t have everything. As I said, I’ve left out all information about headers, etc. Also, since these students will probably not be using figures or tables, I’ve left them out. [Update: I added information about figures. I also added links to some online resources.]

Please let me know in the comments if there is anything that you think I should add. I still have a week and a half to tinker with it before classes begin. Also, please let me know if you find any errors. Thanks!

[Note: When I began this project I intended to modify this cheat sheet by Winston Chang to suit my needs. But in the end, I wiped it clean and started from scratch. (I did use his very nice three-column format though.)]

For a fun exercise I had my multivariable calculus class compute the volumes of various balls using multiple integrals. The surprising results inspired this post.

First some terminology. An -dimensional hypersphere (or -sphere) of radius is the set of points in satisfying (I’ll place the center at the origin for simplicity). For example, a 0-sphere is the two-point set on the real number line, a 1-sphere is a circle of radius in the plane, and a 2-sphere is a spherical shell of radius in 3-dimensional space.

An -dimensional ball (or -ball) is the region enclosed by an -sphere: the set of points in satisfying . For example, a 1-ball is the interval , a 2-ball is a disk in the plane, and a 3-ball is a solid ball in 3-dimensional space.

It is possible to define “volume” in —in it is length, in it is area, in it is ordinary volume, and in it is hypervolume. Let denote the volume of the -ball of radius .

It turns out that the volumes of -balls satisfy the following remarkable recursion relation. (I’ll prove this relation at the end of the post.)

It is not difficult to use this recurrence relation to obtain a formula for . In particular, when is even and when is odd . (If you know what the gamma function is you can express this as a single function, )

The volumes of the -balls in the first 15 dimensions are given in the following table.

If you look at the volumes of the unit balls you’ll see they increase at first, reaching a maximum in dimension 5. Then they decrease and tend to zero as the dimension goes to infinity. Strange!

First, what is special about dimension 5? Why is the maximum achieved in this dimension? It turns out that there is nothing special about dimension 5. Below is a GeoGebra applet that allows you to adjust the radii of the balls. As we can see, the maximum volume is not always attained by the ball in dimension 5. Indeed, as the radius increases, the maximum volume occurs in higher dimensions. As John Moeller points out, the powers of in the numerator try to make an increasing function, however the factorials in the denominator always dominate in the end.

Second, what is the intuition behind this limit of zero? One way to see this is to observe that to be on the boundary of the unit -ball, we must have , but for this to happen when is large, most of the ‘s must be very close to zero. For example, the line intersects the -sphere at . On the other hand, the corresponding corners of the hypercube that inscribes the sphere are at , units from the origin. Thus the sphere fills up less and less of the hypercube that contains it. (Notice that the circumscribed hypercube has volume , while inscribed hypercube has volume .)

My colleague informed me that this zero limit is related to the curse of dimensionality in statistics. Volume increases rapidly as dimension increases, so it requires many more data points to get a good estimate. As Wikipedia points out, “100 evenly-spaced sample points suffice to sample a unit interval with no more than 0.01 distance between points; an equivalent sampling of a 10-dimensional unit hypercube with a lattice with a spacing of 0.01 between adjacent points would require sample points.”

Proof

Now I will prove the recurrence relation that I gave above.

Clearly the relation is true for and . Suppose .

First recall that if a solid in -dimensional space is scaled by a factor of , then its volume increases by a factor of . In particular, this implies that

Observe that the intersection of the -plane with the -ball is a disk of radius centered at the origin (see image below). Use polar coordinates to describe points in this disk. Then the perpendicular cross section of the -ball at the point is an -ball of radius .

Thus we can compute by integrating over the disk. We do so using polar coordinates.

The first video was an excellent 50-minute lecture by Fields medalist Curt McMullen from October 11th, 2006. It is part of the  Science Center Research Lecture Series at Harvard. It was intended for a general audience. I highly recommend watching this.

Then, for fun, I showed them Steven Colbert’s plea for his own Fields medal and a rap about Perelman’s proof.

While you’re at it, check out his other interviews as well as his other podcast series, Combinations and Permutation (he discussed my book Euler’s Gem on a recent episode).

A few weeks ago he showed how to make a hyperboloid of one sheet out of 32 shish kabob skewers and 176 hair rubber bands. (Here is a direct link to the instructions.)

We just finished talking about quadric surfaces in the Calculus III class that I’m teaching, so the timing was perfect to make my own.

A hyperboloid of one sheet has the amazing property that it is a ruled surface. That is, it is the the union of straight lines. Actually, it is doubly ruled—there are two straight lines through every point on the surface. The fact that the hyperboloid is doubly ruled allows us to make it out of skewers.

Here are photos of the final product. Left on its own the rubber bands pull the sculpture into a solid column. Twist the top and bottom (being careful of the pointy ends) and it opens into a hyperboloid.

(I am a little concerned about the lifespan of this object since rubber bands degrade over time. I’m thinking of putting a drop of wood glue in each joint and taking the rubber bands off.)

A few years ago I made another physical model of a hyperboloid. I used a jig saw to cut two circles out of wood, drilled holes in them, and threaded string between them. Then, when the two circles are twisted opposite directions the strings stay straight and form a hyperboloid.

He said that I could talk about whatever I wanted. Wow, the possibilities!

So I was thinking about giving a biography of Euler followed by “Euler’s greatest hits.” This is where you come in.

You probably know my favorite theorem of Euler’s. What are your favorites? Please leave them in the comments. List as many as you want. The more, the better.

Thanks!

a website that allows the easy creation and editing of any number of interlinked web pages… [Wikis] are often used to create collaborative websites, to power community websites, for personal note taking, in corporate intranets, and in knowledge management systems.

I have used wikis in three of my classes: two Discrete Mathematics courses (Fall 2008 and Fall 2009) and one Knot Theory course (Spring 2009). In the Discrete Mathematics class the students collaborated to create a homework “solutions manual” on the wiki. In the Knot Theory class each student had a “pet knot” and created a wiki page containing biographical information about their knot.

I used Wikidot to host the wikis. Their wikis are free and ad-free for educational use. You can make the wiki private if you want to. (I did this for Discrete Math because I figured the textbook author wouldn’t want the solutions available on the web.) Also, one of the deciding factors for me was that you can write mathematics in Wikidot using a modified version of LaTeX. (Here’s a Wikidot LaTeX help page that I created.)

Discrete Mathematics

Discrete Mathematics is our gateway course to the mathematics major, so it is in this course we teach the students how to write mathematical arguments (proofs).

I used the wiki as a place for the class to create an online solution manual for the homework. I made a page for each section of the textbook. As the semester progressed I listed the problems that needed solutions. I chose only problems that did not have answers in the back of the book. All of the problems were ones that I had assigned for homework, but I waited until after the students turned in their homework to list the problems on the wiki.

Every couple of weeks (once we’d accumulated enough problems) I required each student to contribute one major edit and two minor edits to the wiki.

Examples of major edits include:

• Typing in the statement of a problem and a complete solution.
• Making substantial changes to an existing solution. Fixing or cleaning up a proof is not sufficient for a major edit, it must be modified in a fundamental way.

Examples of minor edits include:

• Typing in the statement of a problem and a complete solution to a non-assigned problem.
• Correcting all mathematical errors in a solution.
• Correcting all typographical errors in a solution.
• Fixing badly-coded text.
• Modifying text to conform to our style guidelines.
• Starting a new solution, but not making much progress.
• Adding a picture that they created.

Periodically I went through the wiki and read the solutions. I wrote comments on the “talk page” and added banners (complete with goofy clipart) underneath each solution. The banners would say:

• Error. There are one or more errors in this solution. There is probably a discussion of the error on the talk page. When you have corrected the error, you should remove this banner.
• Messy. This text is messy. It might need better math formatting or structural repair. There may be typographical errors. Look on the talk page for more information. Once you have cleaned things up, you should remove this banner.
• Suggestion. I have suggested improvements for this text. The suggestions are described on the talk page. Please read the discussion there for ideas on improving the writing. Once you make the changes, you should remove this banner.
• Not started. This homework problem (or a part of this homework problem) has not been started. After you have written the solution, you should remove this banner.
• Checkmark. (If the solution to a problem is correct, I put a red checkmark underneath it. Once I signed off on it, they do not have to edit it further.)

I graded the students on participation, and not on the quality of their edits. Thus I only kept track of whether they did the required major and minor edits. Even for this I trusted their honesty. Each student had their own page on the wiki where they kept a log of which major and minor edits they made, and the dates they made them. Of course, wikis have excellent version histories, so I could always go back and see who edited what, when.

I was surprised (although in retrospect it isn’t too surprising) to discover that many students would rather type the solution to a non-assigned problem for their minor edit than to read through an existing solution in search of an error. I ended up not allowing this type of minor edit in the latter part of the semester because there were so many lingering non-corrected errors.

The students quickly realized that there was incentive to make their edits early. If they waited until the night before the edits were due, then only the more challenging (or more time-consuming) problems were available.

When we got close to exam times I would assign extra minor edits so that the solutions could be used as a reliable study guide.

Did it work and was it worth it? I think it worked pretty well. By the end of the semester we had a pretty good solution manual for the course. This was a nice resource for the students to study from. However, I wouldn’t say that the students were overly excited by the process. Most students did the bare minimum that was required of them. In neither year did I find any students who went out of their way to clean up the wiki beyond what was required of them. On the other hand, I think that it wasn’t too time consuming for any individual student.

Knot Theory

I taught a low-level mathematics elective on Knot Theory. There were seven students in the class of varying mathematical abilities.

The wiki for this class (which is still available online) was not collaborative. At the beginning of the semester each student chose a “pet knot” and created a wiki page for it. At first they just chose a name for their knot and used KnotPlot to generate a picture of it. Then, as we progressed through the semester and learned about new knot invariants (tricolorability, unknotting number, crossing number, etc.) the students would compute the invariants for their knot and write about them on their pet knot page. For example, here is one student’s pet knot page and another student’s.

One might imagine that some students would have trickier knots than others. That was true for certain invariants, but since they were doing so many different things with their knots, I think it all averaged out in the end. It was fortunate that there were seven students in the class and there are exactly seven 7-crossing knots, so I assigned each student a 7-crossing knot. This made things more fair.

One sticky point was drawing the drawing of knots, graphs, and surfaces and putting them online. In the end the students used a variety of drawing programs: EazyDraw, InkScape, PowerPoint, Adobe Illustrator, etc.

I think the wiki worked pretty well in this class. I could have accomplished the same thing without a wiki—have students turn in solutions for their pet knot on paper during the semester or at the end of the semester—but the online approach worked well. It was nice to see the knot biographies fill up as the course progressed.

They came back with the answers I expected. For example, the trivial topology has only two open sets and all the others have infinitely many, so the trivial topology must be different from the rest. One-point sets are open in the discrete topology, but not in any of the others, so it must be different from the rest.

One student pointed out that the discrete topology is the power set of the real numbers, , and we know (by Cantor’s Theorem) that has a different cardinality than . So he wanted to know what the cardinalities of the other topologies were. For example, if the usual Euclidean topology on , which we’ll denote , has the cardinality of the continuum (the cardinality of ), then this would be another way to show that these two topologies aren’t the same.

My first reaction was joy—I was thrilled with this excellent observation. Then I realized that I didn’t know the cardinality of . I told the class that I strongly suspected that it had the cardinality of the continuum, but that I couldn’t say for sure on the spot.

This is probably “a fact well known to those who know it well” (this was a quote from a book I once read), but I wasn’t one of those people. So I decided to prove it (with the help of my colleague Jeff).

Theorem. and have the same cardinality.

By definition, every set in can be written as the union of -balls, . In fact, it is not difficult to see that every open set can be written as the union of -balls with rational center and rational .

In particular, there is a surjective function . Namely, for ,

.

We know that is countable, and the power set of a countable set has the cardinality of the continuum. Thus there is a bijective function . Hence is surjective.

On the other hand, there is a surjective function . For example, for , let

.

Thus by the Cantor–Bernstein–Schroeder theorem (or, technically, the dual version of the theorem), and have the same cardinality.

I’m using this applet for displaying parametric curves. You can use predefined curves or enter your own. Although the applet is on my web page, it was created by Marc Renault, a friend who teaches down the road at Shippensburg University. I only tweaked it slightly when I posted it on my page.

For fun, I created this applet. It has four parametric curves which, when drawn together, produce a famous logo. Our college is in Pennsylvania, so it went over well.

Marc also created this nice applet illustrating the creation of the cycloid.

Speaking of the cycloid, after the deriving the parametric equations for the cycloid I spent 10 minutes telling my class about the tautochrone and brachistochrone problems.

They loved the story about the history of the brachistochrone problem—the way that Johann Bernoulli taunted Newton, Newton’s 12-hour after-work solution, Newton’s annoyance (“I do not love to be dunned [pestered] and teased by foreigners about mathematical things…”), his anonymous solution, and Bernoulli’s reply (“I recognize the lion by his paw.”).

This afternoon, after teaching the class, I discovered that the tautochrone problem is mentioned in Moby Dick: “[The try-pot] is also a place for profound mathematical meditation. It was in the left-hand try-pot of the Pequod, with the soapstone diligently circling round me, that I was first indirectly struck by the remarkable fact, that in geometry all bodies gliding along a cycloid, my soapstone, for example, will descend from any point in precisely the same time.” Cool!