College Math Teaching

August 1, 2017

Big lesson that many overlook: math is hard

Filed under: advanced mathematics, conference, editorial, mathematician, mathematics education — Tags: — collegemathteaching @ 11:43 am

First of all, it has been a very long time since I’ve posted something here. There are many reasons that I allowed myself to get distracted. I can say that I’ll try to post more but do not know if I will get it done; I am finishing up a paper and teaching a course that I created (at the request of the Business College), and we have a record enrollment..many of the new students are very unprepared.

Back to the main topic of the post.

I just got back from MAA Mathfest and I admit that is one of my favorite mathematics conferences. Sure, the contributed paper sessions give you a tiny amount of time to present, but the main talks (and many of the simple talks) are geared toward those of us who teach mathematics for a living and do some research on the side; there are some mainstream “basic” subjects that I have not seen in 30 years!

That doesn’t mean that they don’t get excellent people for the main speaker; they do. This time, the main speaker was Dusa McDuff: someone who was a member of the National Academy of Sciences. (a very elite level!)

Her talk was on the basics of symplectec geometry (introductory paper can be found here) and the subject is, well, HARD. But she did an excellent job of giving the flavor of it.

I also enjoyed Erica Flapan’s talk on graph theory and chemistry. One of my papers (done with a friend) referenced her work.

I’ll talk about Douglas Arnold’s talk on “when computational math meets geometry”; let’s just say that I wish I had seen this lecture prior to teaching the “numerical solutions for differential equations” section of numerical analysis.

Well, it looks as if I have digressed yet again.

There were many talks, and some were related to the movie Hidden Figures. And the cheery “I did it and so can you” talks were extremely well attended…applause, celebration, etc.

The talks on sympletec geometry: not so well attended toward the end. Again, that stuff is hard.

And that is one thing I think that we miss when we encourage prospective math students: we neglect to tell them that research level mathematics is difficult stuff and, while some have much more talent for it than others, everyone has to think hard, has to work hard, and almost all of us will fail, quite a bit.

I remember trying to spend over a decade trying to prove something, only to fail and to see a better mathematician get the result. One other time I spent 2 years trying to “prove” something…and I couldn’t “seal the deal”. Good thing too, as what I was trying to prove was false..and happily I was able to publish the counterexample.

March 24, 2015

The Death of the math major at smaller and non-elite universities …

The situation: like many small. private, decent but non-elite universities, we are facing a student shortage. (1100 new freshmen in our peak year; 1030-1040 was more typical, last year was 950). Of course, our university president decided this was a good time to give athletics 8 million dollars from our university budget and, well, she was encouraged to “retire”.

But many of her hires remain and the deans/administrators are on a kick to “make professors more productive”: that is, to have us teach larger classes. They wanted the following enrollment rules: a minimum of 10 students in every “graduate” class and 15 in every upper division class.

Now our department put forth a proposal for a statistics major. One of the proposed courses (advanced statistical modeling) got this feedback from one of the deans (hired by the outgoing president):

Do I understand this correctly? MTH 438 has a prerequisite of 437, which has a prerequisite of 327, which has a prerequisite, etc. Here is my take on the situation, which you are welcome to set me straight on if I have this wrong. It appears to me that a student starting in MTH 121 (calc 1) must take calc 1, calc 2, calc 3, S&P1, S&P2, and applied statistics, all in sequence, before being eligible for 428. L

Let’s be clear: evidently he thinks that 3 semesters of calculus (the standard at most places) and the basic 2 semester sequence in calculus based probability and statistics, along with a basic modeling class is too much for and advanced statistical modeling class. And: ALL IN SEQUENCE. Wow. One must learn to differentiate before calculating a gradient and one must learn multi-variable integration before calculating the expectation of a function of joint random variables. Oh noes, not that!

If that’s the case, then it would seem there is virtually no room for error, no room for taking a semester “off” to dig into a second major (particularly important for students interested in applied statistics, say in a discipline), no room for something not to be taught off cycle due to sabbatical leaves or other faculty leaves. By the time students are eligible to take this course, they will be seniors or perhaps the odd junior who came in with calculus credit. Thus, the course will need to be taught every year, not every other year, and to very few students. Wouldn’t a more open, less prerequisite-laden curriculum afford students more pathways to complete the major, allow them more flexibility to pursue a second major or minor in a related discipline, and grant you and your department the capacity for more nimble course scheduling and enrollment management?

Uh, maybe, but the students in these “prerequisite free” classes wouldn’t be learning anything. Remember he is complaining about having a CALCULUS and “basic probability and statistics” class as being too much.
It doesn’t help that the associate dean is a biologist who assumes to know more than she does.

But that is the way that the “let’s run the university like a business” is shaping up. There is more money in teaching larger sections of courses which aren’t intellectually demanding. It is becoming about “bodies in the class room”, at least at places like ours.

The bottom line: undergraduate mathematics, at least at the junior/senior level, can’t really be taught in a no-prerequisite fashion. But these prerequisites required courses are going to be lower enrollment courses, at least at places like ours.

So, my prediction is that, by the time I retire (15 years? If I am lucky), the math department will be a “service courses only” department and our mathematics major will die.

Now it is possible that the “math major” might be replaced by a “mathematical sciences” major in which students can cobble together a degree by taking some “heavy applied math content” courses in other disciplines and perhaps an “about math” course in our department in which we teach a few “show-and-tell neat stuff from Scientific American” in which we sling around a few neat words and perhaps present a few power points with neat photos.

But we’ll have bodies in chairs…maybe.

I believe that the math major will continued to be offered at Stanford, MIT and places like Big Ten universities but at places like ours, not so much.

I sure hope that I am wrong.

August 26, 2014

How some mathematical definitions are made

I love what Brad Osgood says at 47:37.

The context: one is showing that the Fourier transform of the convolution of two functions is the product of the Fourier transforms (very similar to what happens in the Laplace transform); that is \mathcal{F}(f*g) = F(s)G(s) where f*g = \int^{\infty}_{-\infty} f(x-t)g(t) dt

August 7, 2014

Engineers need to know this stuff part II

This is a 50 minute lecture in a engineering class; one can easily see the mathematical demands put on the students. Many of the seemingly abstract facts from calculus (differentiability, continuity, convergence of a sequence of functions) are heavily used. Of particular interest to me is the remarks from 45 to 50 minutes into the video:

Here is what is going on: if we have a sequence of functions f_n defined on some interval [a,b] and if f is defined on [a,b] , lim_{n \rightarrow \infty} \int^b_a (f_n(x) - f(x))^2 dx =0 then we say that f_n \rightarrow f “in mean” (or “in the L^2 norm”). Basically, as n grows, the area between the graphs of f_n and f gets arbitrarily small.

However this does NOT mean that f_n converges to f point wise!

If that seems strange: remember that the distance between the graphs can say fixed over a set of decreasing measure.

Here is an example that illustrates this: consider the intervals [0, \frac{1}{2}], [\frac{1}{2}, \frac{5}{6}], [\frac{3}{4}, 1], [\frac{11}{20}, \frac{3}{4}],... The intervals have length \frac{1}{2}, \frac{1}{3}, \frac{1}{4},... and start by moving left to right on [0,1] and then moving right to left and so on. They “dance” on [0,1]. Let f_n the the function that is 1 on the interval and 0 off of it. Then clearly lim_{n \rightarrow \infty} \int^b_a (f_n(x) - 0)^2 dx =0 as the interval over which we are integrating is shrinking to zero, but this sequence of functions doesn’t converge point wise ANYWHERE on [0,1] . Of course, a subsequence of functions converges pointwise.

August 1, 2014

Yes, engineers DO care about that stuff…..

I took a break and watched a 45 minute video on Fourier Transforms:

A few take away points for college mathematics instructors:

1. When one talks about the Laplace Transform, one should distinguish between the one sided and two sided transforms (e. g., the latter integrates over the full real line, instead of 0 to \infty .

2. Engineers care about being able to take limits (e. g., using L’Hopitals rule and about problems such as lim_{x \rightarrow 0} \frac{sin(2x)}{x} )

3. Engineers care about DOMAINS; they matter a great deal.

4. Sometimes the dabble in taking limits of sequences of functions (in an informal sense); here the Dirac Delta (a generalized function or distribution) is developed (informally) as a limit of Fourier transforms of a pulse function of height 1 and increasing width.

5. Even students at MIT have to be goaded into issuing answers.

6. They care about doing algebra, especially in the case of a change of variable.

So, I am teaching two sections of first semester calculus. I will emphasize things that students (and sometimes, faculty members of other departments) complain about.

September 14, 2013

Reality of modern college teaching: students with Asperger’s syndrome

Filed under: academia, mathematics education, student learning — Tags: — collegemathteaching @ 4:43 pm

One of the major changes I’ve encountered since I started college teaching (first as a teaching assistant in 1986; then as a new professor in 1991) is that students with Asperger’s syndrome have been showing up.

Most of the time, it isn’t a big deal; the worst I’ve had is one of these students became completely disoriented when he got to class and someone was sitting in “his” seat (no, I don’t make seat assignments; this is college).

This semester, I have a transfer student (not sure why he transferred); in spots he is “disruptive to a minor degree”: you have to remind him that there are 34 OTHER students in the class; this isn’t a one-on-one dialogue just for him.

Also, I sometimes make side remarks (to explain a point to another student) and use analogies; that just confuses the heck out of him. But I am not going to stop being effective with the other 34 students just for him; I just tell him “see me in office hours” or “don’t worry about this”.

On the other hand, he is relatively easy to work with in office hours; the one-on-one exchanges are usually reasonable and pleasant.

Hence, when I see he is getting confused, I tell him “for this point, see me for office hours.”

I’ve searched the internet to see what is out there; most of it is what I already know and much of it is a series of tired cliches, finger wagging, etc. I haven’t found much of the following: “I had these issues in my calculus class; here is how they were resolved” or “these issues COULDN’T be resolved.” Sometimes they aren’t up to the task of being in college.

But, overall, it seems to be this way: we are told to be “more productive” which means more students per semester (105 students in 2 sections of calculus and 1 of differential equations). So no, one cannot tailor lessons and work to the learning style of a specific student, especially if that student is an outlier. One has to teach to a type of average or to the class as a whole; one can adjust for a class full of, say biology students, or one full of engineers or one full of computer science majors.

These students require time, more attention and resources and these COST MONEY. This is where some of the increased educational expense is coming from (some from technology as well). At times, it appears as if colleges and universities are being tugged in different directions.

June 22, 2013

About teaching continuity: a math ed talk

Filed under: calculus, editorial, elementary mathematics, mathematics education, pedagogy — collegemathteaching @ 12:11 pm

There was a talk about students and how they understand the concept of “continuity” of a function. That is a good topic.
One of the examples that was brought up was someone in a graduate program who didn’t understand why the function
f(x) = 2x if x is rational and f(x) = x^2 otherwise”
is continuous at x = 0 . The graduate student said that she couldn’t draw the graph without “lifting the pencil”.

I don’t think that this is a problem with calculus teaching; this person shouldn’t have made it through analysis.

But yes, I agree; sometimes students have trouble with the concept of continuity. So we went on; the idea is that when asked about “what it means for a function to be continuous” students often struggled. Fair enough. The answer that was looked for was: “the function f is continuous at x = a if lim_{x \rightarrow a} f(x) = f(a) which, of course, means that f is defined at x = a to begin with.

Instead, students responded with “keep the pencil on the paper”, “has the same formula (as opposed to a conditional formula)”, “connected graph”, etc.

So I asked “how is the concept of limit defined to begin with” and….see the previous sentence! Such nonsense.

Seriously, if you are going to wave your hands at “limit” (and it may be appropriate to do so) then what is the problem to doing that with continuity?

There is more. Consider the frequently quoted idea that a function can be defined by “formula, text, or a *table of values*”, etc.

We were given something like:

x | 1.98 | 1.9908 | 2.001 | 2.051
y | 8.94 | 8.9671 | 9.003 | 9.023

And the first row is considered to be in the domain. The question: “it is reasonable to expect f(2) = “. You know what the expected answer was, but my question was immediately: “why is it reasonable to expect f to take the integers to the integers?”.

Deer in the headlights look.

Then “good point”.

My larger point: a “table of values” only defines a function IF the domain is restricted to the entries in the appropriate row (or column) OR if there is an associated interpolation scheme to go with the table.

Then we moved on.

Example: students were given two examples:
1. Example one: say the temperature at 6 am was 60.0 F and the temperature at noon was 75.0 F. So, was there a time between 6 and noon when the temperature was, say, 68.5 F? Ok, that is reasonable, though students might be confused by digital readouts and maybe a physics student might talk about quantum effects.

2. Example two: The winning team in a basketball game scored 81 points. Does it mean that, at some point in the game, that team had 45 points? Ok, “no” is the correct answer but THIS HAS NOTHING TO DO WITH CONTINUITY, at least as defined by the topology that the calculus students have seen. Example: in a volleyball game (new rally scoring), it takes 25 points to win a game. So the winning team must have had 1, 2, 3, 4,….24 points at one time or another, and that is because in volleyball, scores can only be made in 1 point increments and that is NOT true in basketball.

No wonder students are often confused!

Note: this is not necessarily an attack on the intellect of the person giving a talk. For example, there was a research mathematician at a division I research university who gave the following problem on a calculus exam: f(x) = x + 1, x \le 1, f(x) = x^2 -x +2 elsewhere. The question: “is f differentiable at x = 1? She told TAs to mark the problem “wrong” if the students said yes, because the function changed formula at x =1 !!! Note: the question asked “differentiable” and not “smooth”.

Vent over…

April 30, 2013

Slate Post on Math Teaching

Filed under: mathematics education — Tags: , — oldgote @ 1:41 am

Ben Orlin in Slate:

As a math teacher, it’s easy to get frustrated with struggling students. They miss class. They procrastinate. When you take away their calculators, they moan like children who’ve lost their teddy bears. (Admittedly, a trauma.)
Even worse is what they don’t do. Ask questions. Take notes. Correct failing quizzes, even when promised that corrections will raise their scores. Don’t they care that they’re failing? Are they trying not to pass?
There are plenty of ways to diagnose such behavior. Chalk it up to sloth, disinterest, out-of-school distractions—surely those all play a role. But if you ask me, there’s a more powerful and underlying cause.

Math makes people feel stupid. It hurts to feel stupid.

Aw. So he goes on to relate his experience as an undergraduate at Yale, in a topology class:

So I did what most students do. I leaned on a friend who understood things better than I did. I bullied my poor girlfriend (also in the class) into explaining the homework problems to me. I never replicated her work outright, but I didn’t really learn it myself, either. I merely absorbed her explanations enough to write them up in my own words, a misty sort of comprehension that soon evaporated in the sun. (It was the Yale equivalent of my high school students’ worst vice, copying homework. If you’re reading this, guys: Don’t do it!)
I blamed others for my ordeal. Why had my girlfriend tricked me into taking this nightmare class? (She hadn’t.) Why did the professor just lurk in the back of the classroom, cackling at our incompetence, instead of teaching us? (He wasn’t cackling. Lurking, maybe, but not cackling.) Why did it need to be stupid topology, instead of something fun? (Topology is beautiful, the mathematics of lava lamps and pottery wheels.) And, when other excuses failed, that final line of defense: I hate this class! I hate topology!

You can read the rest.
Here is his conclusion:

Teachers have such power. He could have crushed me if he wanted.
He didn’t, of course. Once he recognized my infantile state, he spoon-fed me just enough ideas so that I could survive the lecture. I begged him not to ask me any tough questions during the presentation—in effect, asking him not to do his job—and with a sigh he agreed.
I made it through the lecture, graduated the next month, and buried the memory as quickly as I could.
Looking back, it’s amazing what a perfect specimen I was. I manifested every symptom that I now see in my own students:
Muddled half-comprehension.
Fear of asking questions.
Shyness about getting the teacher’s help.
Badgering a friend instead.
Copying homework.
Excuses; blaming others.
Anxiety about public failure.
Terror of the teacher’s judgment.
Feeling incurably stupid.
Not wanting to admit any of it.
It’s surprisingly hard to write about this, even now. Mathematical failure—much like romantic failure—leaves us raw and vulnerable. It demands excuses.
I tell my story to illustrate that failure isn’t about a lack of “natural intelligence,” whatever that is. Instead, failure is born from a messy combination of bad circumstances: high anxiety, low motivation, gaps in background knowledge. Most of all, we fail because, when the moment comes to confront our shortcomings and open ourselves up to teachers and peers, we panic and deploy our defenses instead. For the same reason that I pushed away topology, struggling students push me away now.
Not understanding topology doesn’t make me stupid. It makes me bad at topology.


First of all: it IS in part, about natural intelligence. The really smart math people, in general, don’t have trouble with undergraduate math classes, even those at Yale. I mean, of course, REALLY smart people (no, I am not one of those. 🙂 ).

Now he has an interesting observation about student “employing defenses”; at least some of them do.

But there are a host of other reasons too: some just don’t like the material, some ARE lazy (e. g., they won’t do what isn’t fun) and yes, some aren’t up to the task intellectually. Seriously: there are some subjects that many will never be able to master, even at an undergraduate level.

But one issue: someone on Facebook made the following comment:

Oh boohoo. If you’re bad at something you either make an attempt to improve at it or direct your attention to things you’re better at. Everyone is not good at everything and feeling stupid is not something people should be protected from. If you don’t get told you make mistakes or aren’t made to realize that some things take effort then you’re not improving. Learning disabilities aside, especially in higher math, the kids that are failing aren’t showing up to anything, aren’t doing the work, aren’t asking questions, aren’t studying and they don’t repeat to try and do better the next time. Children need to know what failure feels like (and math teachers were children and they do all likely know what failure feels like) so that they learn to try.

Emphasis mine.

My Opinion:
1. There are things that are too difficult for most of us to learn (e. g. quantum field theory).
2. It is useful to have a grasp of one’s intellectual limitations. All too often I see average people dismissing expert findings because those findings “don’t make sense to them.” People need MORE intellectual humility, not less of it.
3. If you haven’t failed at something, then you haven’t tried enough difficult things.

That said: this article is useful because it does give at least one angle of approach that might work with some students.

April 15, 2013

Google Doodle 15 April 2013

Filed under: mathematician, mathematics education — Tags: — collegemathteaching @ 1:06 pm

Screen shot 2013-04-15 at 8.03.05 AM

Which famous mathematician is being honored? 🙂

February 19, 2013

A Message to Undergraduate Math Majors

Filed under: advanced mathematics, editorial, mathematics education, pedagogy, topology — collegemathteaching @ 8:08 pm

Ok, you are taking, maybe an analysis class or perhaps your first abstract algebra class. You are learning how a proof works. Of course, you might be studying a proof of, say, one of the Sylow Theorems in group theory, or perhaps a convergence proof in analysis.

That proof is elegant and to the point, isn’t it? But here are some things to remember:

1. You are seeing “what worked”; you aren’t seeing the scores of attempts that failed.

Example: one of my papers contains a counterexample to something I thought “for sure” was true; in fact I spent 2 years trying to “prove” the conjecture that I ended up publishing the counterexample for!

2. You are seeing a polished proof.

Example: right now, I am finishing up a paper on wild knots (simple closed curves in 3 space that are not deformable to smooth simple closed curves). I spend 3-4 days on one step of a construction, only to realize that not only were my steps not convincing, they WEREN’T at all necessary!

Here is what lead me to realize I was headed toward a dead end: I was proving something that directly depended on specific properties of the type of knots that I was studying, yet my construction was not using those properties. I was doomed to fail if I kept on this path!

For the record, here is the mistake that I was making:

Suppose you have an annulus in the plane; example: A = ((x,y,0) | \frac{1}{4} \le x^2+y^2 \le 1) . Now suppose you take another annulus B in the region above the plane and attach it to A along its two boundary circles. You get a torus T. But is T necessarily unknotted in 3 space? Hint: we knot that in S^3 = R^3 \cup {\infty} , T bounds at least ONE solid torus, but does it bound two of them?

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