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It’s hard to take ratemyprofessors.com seriously

August 25, 2009 at 8:33 am | In Life in America, Teaching | 6 Comments

My two most recent reviews on my ratemyprofessors.com profile (a hugely biased sample of four students out of about 150).

Simply he’z a great teacher. he makes Calculus as an easy material you ever expected. Just study for review study and you’ll do fine. An A is acheivable since his exams is a pretty easy.

I sincerely hope that this student speaks English as a second language.

Martin is a mathematical genius. His stories of kangaroos killing people are funny too. The class is tough, tests are tricky, study and an A is achievable. Good teacher!

I don’t recall telling any stories of kangaroos killing people.

Also, if any readers want to add fake reviews (are those two above fake?) I’d appreciate one of those little chili pepper icons saying i’m ‘hot’.

My students think I’m a pot smoking hippy

June 12, 2009 at 5:13 pm | In Life in America, Mathematics, Teaching | 1 Comment

A link one of my students sent me

http://www37.wolframalpha.com/input/?i=PolarPlot[(1+%2B+0.9+Cos[8+t])+(1+%2B+0.1+Cos[24+t])+(0.9+%2B+0.05+Cos[200+t])+(1+%2B+Sin[t]),+{t,+-Pi,+Pi}]

In his defense, I did spend one class just plotting wacky parametric equations.

A story

June 6, 2009 at 11:57 am | In Cricket, Life in America, Teaching | 2 Comments

I received an email from a student who wanted me to add them to my class roster above the 35 person limit. Normally I refuse such requests summarily but I’m the only person teaching this class in summer session II so I agreed. Over a series of emails I organized to meet this person at 1:00 on Friday to go over the paperwork – this should have given me time to watch England play the Netherlands in the T20 world cup.

Unfortunately rain delayed the game so I was forced to set my DVR to record the last 6 overs with the game delicately balanced and left my house at 12:40. I deliberately didn’t take my computer so I wouldn’t be tempted to check the result. I arrived at my office at about 12:50 and waited for the student to arrive, attempting to read a paper on expander codes.

When there was no sight of of my student at 1:30 I left a message on the door and went to a computer lab to check my email. There I received an email sent at 12:30 explaining that the student had managed to enrol through the online system and didn’t need to meet me. Somewhat annoyed with this turn of events I walked home.

There I turned on my computer and unconsciously, automatically, horrifically, went to cricinfo.com. The Netherlands had won.

This semester I am an above average teacher!

May 13, 2009 at 11:45 am | In Mathematics, Teaching | 1 Comment

My class’s average on the final was 67% and the course average was 65%.

Final grades: 4 A’s, 7 B’s, 10 C’s, 3 D’s, 5 E’s.

So far I’ve had one request for extra credit to bump a D up to a C which I summarily denied.

Calculus without limits

March 26, 2009 at 9:59 pm | In Mathematics, Teaching | 13 Comments

or: my descent into crackpottery

So I am quite unhappy with the foundations of calculus as it’s currently taught, in particular with how complicated the proof of the product rule is. Let’s recap one proof:

$(fg)'(x) = \lim_{h \to 0} (f(x+h)g(x+h)-f(x)g(x))/h$
$= \lim_{h \to 0} (f(x+h)g(x+h)-f(x)g(x+h)+f(x)g(x+h)-f(x)g(x))/h$
$= \lim_{h \to 0} g(x+h)(f(x+h)-f(x))/h + f(x)(g(x+h)-g(x))/h$
$= g(x)f'(x)+f(x)g'(x)$ .

It’s pretty awful. Maybe it isn’t that bad a proof as analysis goes but really, it’s unnecessary to subject normal people to such things: that can wait until a real analysis class.

The concept of limit is really difficult for a lot of people. So the solution (or one possible solution) is differentials. This is just replacing one vague concept (limits) with another that is even harder to formalise (differentials are really elements of the cotangent space to something, which is a bit scarier than epsilons and deltas).

But it’s historically accurate and as long as you are willing to accept a little bit of magic works quite well.

If y depends on x (basically is a function locally) then an ‘infinitesimal’ (really small but positive) change of x, called dx, leads to an infinitesimal change in y, called dy.

For example if $y=x^2$ then
$y+dy=(x+dx)^2$
$x^2+dy=x^2+2xdx+(dx)^2$
$dy=2xdx$

Notice $(dx)^2=0$ . You can axiomatise this sort of behaviour (this is really $dx \wedge dx = 0$ or maybe this computation is in $k[dx]/((dx)^2)$ ) but this is expected behaviour for infinitesimals: they’re small enough that only linear terms matter.

The above calculation is ‘using the definition’. After you’ve done derivative rules you would just take d of both sides to go from
$y=x^2$
to
$dy=2x dx$

One property of the derivative is now that $f(x+dx)=f(x)+f'(x)dx$ . This is (arguably) the real meaning of the derivative: it gives the best linear approximation of a function and infinitesimally this approximation is correct.

Now the proof of the product rule is much nicer:
if $y = f(x)g(x)$
$y+dy=f(x+dx)g(x+dx)$
$dy = (f(x)+f'(x)dx)(g(x)+g'(x)dx)-f(x)g(x)$
$dy = f'(x)g(x)dx+f(x)g'(x)dx$

This is a little more difficult to extend to integrals but who ever needs to use the definition of Riemann integral anyway? You can use whatever kind of approximations you want to find areas under curves: the only time you use the Riemann integral is when you have an antiderivative so can use the fundamental theorem of calculus.

The other big problem I have with calculus is that we pretend that functions are important when in science and geometry what you really care about are relations between things. We can do calculus on circles, and who knows whether distance is a function of velocity or the other way around….

Thoughts of a few other people on the matter:
Putting Differentials back. A very interesting paper on the issue with good references.

Calculus without Limits – Almost. A rather crazy textbook.

Other ways to teach calculus:
- These notes from Karl Heinz Dovermann define differentiability by a Lipschitz condition with exponent 2.
- Non standard analysis as seen in Keisler’s book
- Apparently there is a book that does everything with differential forms but I can’t find anything that is truly a single variable calculus text like this.

Precalc results

December 17, 2008 at 8:16 pm | In Teaching, Uni | Leave a Comment

Results are finalised. Will be released to students tomorrow.

5 Bs, 15 Cs, 6 Ds, 7 Es, 2 Ws

Precalculus-metrics

December 17, 2008 at 10:05 am | In Teaching, Uni | 2 Comments

The precalc final exam grading took about six hours yesterday. I graded a question about a rational function that had a slant asymptote.

More interesting were the statistics that we got in a big spreadsheet later that night. The overall average on the final was 110/200. To get a D in this course you need to average 60% overall. Let it never be said that the UofA math department doesn’t have high standards.

My section averaged 106/200, but casting around for something to blame other than myself I came up with the idea that other people just force their poorly performing students to drop. I only had 2 students drop out of the original 35 while other sections had much larger numbers leave the course. Taking the final number of students enrolled as a proxy for number of students who dropped (this isn’t perfect but it’s kind of close) I did some regression:

regression1

There’s definitely some kind of correlation there: what does R^2=0.06 mean?

Anyway, if I put x=33 into this equation I get about 108.5. So I’ve explained away 1.5 points from the average from the fact that I didn’t force many people to drop. I’ve got ideas for the other few points so I feel happy that I did an average job.

Let’s make this blog a little less serious

October 8, 2008 at 7:38 pm | In Life in America, Teaching | 6 Comments

A nice little email from one of my students:

> Martin,
> I want to apoplogize for missing class again, I know I’ve missed t a
> lot lately.
> This realy isn’t an excuse, but I keep sleeping through my alarm clock, I
> honestly didn’t even hear it go off this morning, I woke up litterally 10
> minutes ago. This is becoming a problem and I’m trying to fix it.
> Again, I’m terribly sorry, and I’ll turn my alarm up even louder and hopefully that will help.
> Sincerely,
> Sleepyhead

Another of my students also asked me today if there were kangaroos everywhere where I grew up. I said yes.

My semester so far

October 2, 2008 at 6:08 pm | In Teaching, Uni | 9 Comments

This semester is the first here in Arizona in which I get to do things that I’m interested in. I am struggling to be interested in the things that I’m interested in, so that’s a bit of a problem, but anyway here’s what I’m doing…

My courses:

Algebraic Geometry. We’re using Shafarevich’s textbook which is quite nice although difficult to find actual definitions or anything in. The course is at a slightly higher level than the text (schemes and such are mentioned occasionally) and is going very quickly but is well taught.

Tate’s Thesis. This is a topics course in number theory, but so far we’ve done topological groups, profinite groups, functional analysis, fourier analysis and any number of other topics without seeing any number theory. But this was to be expected: the textbook is called Fourier Analysis on Number Fields and we need to develop all kinds of theory before we start to see what this has to do with number theory.

Research Tutorial Group. This confusingly named course is an introduction to research (in a group). I’m working with my Colombian friend from Colombia, Enrique under the supervision of Doug Ulmer. At the moment I’m reading Lorenzini’s Introduction to Arithmetic Geometry. We’re aiming to do something to do with ranks of quadratic twists of elliptic curves over function fields, based on a paper of Gouvea and Mazur.

Teaching:

I’m teaching MATH120R: Precalculus. This is functions, exponentials and logs and trigonometric functions. I know it sounds really easy but the level of performance required to pass is (slightly) higher than when I learnt this stuff in Year 11 in Australia. The method of teaching is also quite different to in Australian universities: my class has 35 students and so far I have assigned 21 homework assignments, had 2 quizzes, 1 test and 2 computer based assessments. If you think that sounds like a lot of busywork imagine what my students think of me . My section’s website can be found here.

Things I’ve created this week

May 15, 2008 at 4:48 pm | In Mathematics, Teaching, Uni | 1 Comment

College Algebra final grades: 12 E’s, 7 D’s, 6 C’s, 5 B’s, 1 A, 2 W’s, 2 WP’s, 1 I. Ouch!

Analysis take home final exam. Part of question 6 is wrong and question 4 is incomplete but apart from that I did a reasonable job.

Topology take home final exam (lacking hand drawn pictures). Question 8 is backwards and 2,5,6,… could be wrong if I wasn’t thinking about them right.

Topology paper on cobordism. This was kind of mediocre. I didn’t put enough time into it so the first three pages are nice enough then it starts to become less complete. I finished off by putting all the results I wanted to prove and then failing to flesh them out. If you’re reading Greg, sorry I didn’t send you a copy – I’m sure you could have improved it (I don’t really know what I’m talking about in parts) – but I was writing it up until 10 minutes before it was due.

Algebra exam tomorrow and then I can take some time off!

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