College Math Teaching

February 9, 2016

An economist talks about graphs

Filed under: academia, economics, editorial, pedagogy, student learning — Tags: , — collegemathteaching @ 7:49 pm

Paul Krugman is a Nobel Laureate caliber economist (he won whatever they call the economics prize).
Here he discusses the utility of using a graph to understand an economic situation:

Brad DeLong asks a question about which of the various funny diagrams economists love should be taught in Econ 101. I say production possibilities yes, Edgeworth box no — which, strange to say, is how we deal with this issue in Krugman/Wells. But students who go on to major in economics should be exposed to the box — and those who go on to grad school really, really need to have seen it, and in general need more simple general-equilibrium analysis than, as far as I can tell, many of them get these days.

There was, clearly, a time when economics had too many pictures. But now, I suspect, it doesn’t have enough.

OK, this is partly a personal bias. My own mathematical intuition, and a lot of my economic intuition in general, is visual: I tend to start with a picture, then work out both the math and the verbal argument to make sense of that picture. (Sometimes I have to learn the math, as I did on target zones; the picture points me to the math I need.) I know that’s not true for everyone, but it’s true for a fair number of students, who should be given the chance to learn things that way.

Beyond that, pictures are often the best way to convey global insights about the economy — global in the sense of thinking about all possibilities as opposed to small changes, not as in theworldisflat. […]

And it probably doesn’t hurt to remind ourselves that our students are, in general, NOT like us. What comes to us naturally probably does not come to them naturally.

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February 8, 2016

Where these posts are (often) coming from

Filed under: academia, linear albegra, student learning — Tags: , — collegemathteaching @ 9:57 pm

books

messydesk

theoffice

Yes, my office is messy. Deal with it. 🙂 And yes, some of my professional friends (an accountant and a lawyer) just HAD to send me their office shots…pristine condition, of course.
(all in good fun!)

Note: this semester I teach 3 classes in a row: second semester “business/life science” calculus, second semester “engineering/physical science” calculus and linear algebra. Yes, I love the topics, but there is just enough overlap that I have to really clear my head between lessons. Example: we covered numerical integration in both of my calculus classes, as well as improper integrals. I have to be careful not to throw in \int^{\infty}_{-\infty} \frac{dx}{1+x^2} as an example during my “life science calculus” class. I do the “head clearing” by going up the stairs to my office between classes.

Linear algebra is a bit tricky; we are so accustomed to taking things like “linear independence” for granted that it is easy to forget that this is the first time the students are seeing it. Also, the line between rigor and “computational usefulness” is tricky; for example, how rigorously do we explain “the determinant” of a matrix?

Oh well…back to some admin nonsense.

February 16, 2015

Topologist’s Sine Curve: connected but not path connected.

Filed under: student learning, topology — Tags: , , — collegemathteaching @ 1:01 am

I wrote the following notes for elementary topology class here. Note: they know about metric spaces but not about general topological spaces; we just covered “connected sets”.

topsincurve

I’d like to make one concession to practicality (relatively speaking). When it comes to showing that a space is path connected, we need only show that, given any points x,y \in X there exists f: [a,b] \rightarrow X where f is continuous and f(a) = x, f(b) = y . Here is why: s: [0,1] \rightarrow [a,b] by s(t) = a + (b-a)t maps [0,1] to [a,b] homeomorphically provided b \neq a and so f \circ s provides the required continuous function from [0,1] into X .

Now let us discuss the topologist’s sine curve. As usual, we use the standard metric in R^2 and the subspace topology.

Let S = \{(t, sin(\frac{1}{t}) | t \in (0, \frac{1}{\pi} \} . See the above figure for an illustration. S is path connected as, given any two points (x_1, sin(\frac{1}{x_1}), (x_2, sin(\frac{1}{x_2}) in S , then f(x) = (x, sin(\frac{1}{x}) is the required continuous function [x_1, x_2] \rightarrow S . Therefore S is connected as well.

Note that (0,0) is a limit point for S though (0,0) \notin S .

Exercise: what other limit points does S that are disjoint from S ?

Now let T = S \cup \{ (0,0) \} , that is, we add in the point at the origin.

Fact: T is connected. This follows from a result that we proved earlier but here is how a “from scratch” proof goes: if there were open sets U, V in R^2 that separated T in the subspace topology, every point of S would have to lie in one of these, say U because S is connected. So the only point of T that could lie in V would be (0,0) which is impossible, as every open set containing (0,0) hits a point (actually, uncountably many) of S .

Now we show that T is NOT path connected. To do this, we show that there can be no continuous function f: [0, \frac{1}{\pi}] \rightarrow T where f(0) = (0,0), f(\frac{1}{\pi}) = (\frac{1}{\pi}, 0 ) .

One should be patient with this proof. It will go in the following stages: first we show that any such function f must include EVERY point of S in its image and then we show that such a function cannot be extended to be continuous at (0,0) .

First step: for every (z, sin(\frac{1}{z})), there exists x \in (0,\frac{1}{\pi} ] where f(x) = (z, sin(\frac{1}{z}) ) Suppose one point was missed; let z_0 denote the least upper bound of all x coordinates of points that are not in the image of f . By design z_0 \neq \frac{1}{\pi} (why: continuity and the fact that f(\frac{1}{\pi}) = (\frac{1}{\pi}, 0) ) So (z_0, sin(\frac{1}{z_0}) cuts the image of TS into two disjoint open sets U_1, V_1 (in the subspace topology): that part with x-coordinate less than and that part with x-coordinate greater than x = z_0 . So f^{-1}(U_1) and f^{-1}(V_1) form separating open sets for [0,\frac{1}{\pi}] which is impossible.

Note: if you don’t see the second open set in the picture, note that for all (w, sin(\frac{1}{w})), w > z_0 one can find and open disk that misses the part of the graph that occurs “before” the x coordinate z_0 . The union of these open disks (an uncountable union) plus an open disk around (0,0) forms V_1 ; remember that an arbitrary union of open sets is open.

Second step: Now we know that every point of S is hit by f . Now we can find the sequence a_n \in f^{-1}(\frac{1}{n \pi}, 0)) and note that a_n \rightarrow 0 in [0, \frac{1}{\pi}] . But we can also find b_n \in f^{-1}(\frac{2}{1 + 4n \pi}, 1) where b_n \rightarrow 0 in [0, \frac{1}{\pi}] . So we have two sequences in the domain converging to the same number but going to different values after applying f. That is impossible if f is continuous.

This gives us another classification result: T and [0,1] are not topologically equivalent as T is not path connected.

August 25, 2014

How to succeed at calculus, and why it is worth it!

Filed under: calculus, student learning — Tags: , — collegemathteaching @ 2:06 pm

This post is intended to help the student who is willing to put time and effort into succeeding in a college calculus class.

Part One: How to Study

The first thing to remember is that most students will have to study outside of class in order to learn the material. There are those who pick things up right away, but these students tend to be the rare exception.

Think of it this way: suppose you want to learn to play the piano. A teacher can help show you how to play it and provide a practice schedule. But you won’t be any good if you don’t practice.

Suppose you want to run a marathon. A coach can help you with running form, provide workout schedules and provide feedback. But if you don’t run those workouts, you won’t build up the necessary speed and endurance for success.

The same principle applies for college mathematics classes; you really learn the material when you study it and do the homework exercises.

Here are some specific tips on how to study:

1. It is optimal if you can spend a few minutes scanning the text for the upcoming lesson. If you do this, you’ll be alert for the new concepts as they are presented and the concepts might sink in quicker.

2. There is some research that indicates:
a. It is better to have several shorter study sessions rather than one long one and
b. There is an optimal time delay between study sessions and the associated lecture.

Look at it this way: if you wait too long after the lesson to study it, you would have forgotten much of what was presented. If you study right away, then you really have, in essence, a longer class room session. It is probably best to hit the material right when the initial memory starts to fade; this time interval will vary from individual to individual. For more on this and for more on learning for long term recall, see this article.

3. Learn the basic derivative formulas inside and out; that is, know what the derivatives of functions like sin(x), cos(x), tan(x), sec(x), arctan(x), arcsin(x), exp(x), ln(x) are on sight; you shouldn’t have to think about them. The same goes for the basic trig identities such as \sin ^{2}(x)+\cos ^{2}(x)=1 and \tan^{2}(x)+1 = \sec^{2}(x)

Why is this? The reason is that much of calculus (though not all!) boils down to pattern recognition.

For example, suppose you need to calculate:

\int \dfrac{(\arctan (x))^{5}}{1+x^{2}}dx=

If you don’t know your differentiation formulas, this problem is all but impossible. On the other hand, if you do know your differentiation formulas, then you’ll immediately recognize the arctan(x) and it’s derivative \dfrac{1}{1+x^{2}} and you’ll see that this problem is really the very easy problem \int u^{5}du .

But this all starts with having “automatic” knowledge of the derivative formulas.

Note: this learning is something your professor or TA cannot do for you!

4. Be sure to do some study problems with your notes and your book closed. If you keep flipping to your notes and book to do the homework problems, you won’t be ready for the exams. You have to kick up the training wheels.
Try this; the difference will surprise you. There is also evidence that forcing yourself to recall the material FROM YOUR OWN BRAIN helps you learn the material! Give yourself frequent quizzes on what you are learning.

5. When reviewing for an exam, study the problems in mixed fashion. For example, get some note cards and write problems from the various sections on them (say, some from 3.1, some from 3.2, some from 3.3, and so on), mix the cards, then try the problems. If you just review section by section, you’ll go into each problem knowing what technique to use each time right from the start. Many times, half of the battle is knowing which technique to use with each problem; that is part of the course! Do the problems in mixed order.

If you find yourself whining complaining “I don’t know where to start” it means that you don’t know the material well enough. Remember that a trained monkey can repeat specific actions; you have to be a bit better than that!

6. Read the book, S L O W L Y, with pen and paper nearby. Make sure that you work through the examples in the text and that you understand the reasons for each step.

7. For the “more theoretical” topics, know some specific examples for specific theorems. Here is what I am talking about:

a. Intermediate value theorem: recall that if f(x)=\frac{1}{x} , then f(-1)=-1,f(1)=1 but there is no x such that f(x) = 0 . Why does this not violate the intermediate value theorem?

b. Mean value theorem: note also that there is no c such that f'(c) = \frac{f(1)-f(-1)}{2} = 0 . Why does this NOT violate the Mean Value Theorem?

c. Series: it is useful to know basic series such as those for exp(x), sin(x), cos(x) . It is also good to know some basic examples such as the geometric series, the divergent harmonic series \sum \frac{1}{k} and the conditionally convergent series \sum (-1)^{k}\frac{1}{k} .

d. Limit definition of derivative: be able to work a few basic examples of the derivative via the limit definition: f(x) = x^{n}, f(x) = \frac{1}{x}, f(x)=\sqrt{x} and know why the derivative of f(x) = |x| and f(x) = x^{1/3} do not exist at x = 0 .

Having some “template” example can help you master a theoretical concept.

Part II: Attitude
Your attitude will be very important.

1. Remember that your effort will be essential! Again, you can’t learn to run a marathon without getting off of the couch and making your muscles sore. Learning mathematics involves some frustration and, yes, at times, some tedium. Learning is fun OVERALL but it isn’t always fun at all times. You will encounter discomfort and unpleasantness at times.

2. Remember that winners look for ways to succeed; losers and whiners look for excuses for failure. You can always find those who will be willing to enable your underachievement. Instead, seek out those who bring out your best.

3. Success is NOT guaranteed; that is what makes success rewarding! Think of how good you’ll feel about yourself if you mastered something that seemed impossible to master at first. And yes, anyone who has achieved anything that is remotely difficult has taken some lumps and bruises along the way. You will NOT be spared these.

Remember that if you duck the calculus challenge, you are, in essence, slamming many doors of opportunity shut right from the get-go.

4. On the other hand, remember that Calculus (the first two semesters anyway) is a Freshman level class; exceptional mathematical talent is not a prerequisite for success. True, calculus is easy for some but that isn’t the point. Most reasonably intelligent people can have success, if they are willing to put forth the proper effort in the proper manner.

Just think of how good it will feel to succeed in an area that isn’t your strong suit!

February 25, 2014

The potential harm in using outliers as examples…..

Filed under: academia, editorial, student learning — Tags: — blueollie @ 9:52 pm

I think that this is common in this day and age: I have some students who are struggling in our “elementary conceptual calculus” course. They come to class, but work a large number of hours at a job in order to make ends meet. So…they are often left with very little time to study.

And yes, IN THIS COURSE, most of the students need to study quite a bit in order to have a chance at even a “C”.

In short: most students need to have a certain number of hours in order to sleep and to study..in addition to making the classes and their part time jobs.

Now, some might say that this is nonsense.

I remember a professor I had at the Naval Academy. He said that when he was an undergraduate he studied very little for his math classes as he paid his own way through school by waiting tables. He made up for it by PAYING ATTENTION IN CLASS.

That is well and good…..but then remember that he had an earned Ph.D. in mathematics from MIT.

Most of us don’t have that type of natural ability.

Yes, Mohammed Ali could break the conventional rules of boxing (dangle his arms, lean away from punches):

But most, including most other professional boxers, don’t have that kind of ability.

Yes, there are people who can run a 2:15 marathon on 35 miles a week of training:

Following the 1976 trials he trained by running 35 miles per week and ran “a 2:14:37 for second place at the Nike-Oregon Track Club Marathon in Eugene in 1978. After that, he ran 2:15:23 for 15th place in the Boston Marathon in 1979.”

But most of us aren’t that gifted (this was Tony Sandoval, cowinner of the 1980 US Olympic Trials Marathon)

Yes, some can make a successful film while being stoned on marijuana, but most of us aren’t as talented as the Beatles.

The list can go on and on. The bottom line: you can gain inspiration from the incredibly successful, but you won’t be able to get away with taking the short cuts that many of them got away with. Neither you nor I are outliers.

December 21, 2013

Rant: please stop with the teaching of “gimmicks for calculation”

Filed under: calculus, editorial, integrals, student learning, Uncategorized — Tags: — collegemathteaching @ 1:05 pm

I finished teaching calculus II (our course: techniques for integration, applications of integrals and infinite sequences/series) and noticed that some of our freshmen students came in knowing how to do many of the calculations…did well on the first exam…then didn’t do so well in the rest of the course.

Evidently, they were well versed in calculation tricks learned in high school; give them \int x^3 sin(x) dx and they could whip out a table.

So here is my rant: we teach integration by parts not so much to calculate integrals like \int x^3 sin(x) dx (which can be rapidly done with a calculator) but rather so they can understand the technique of integration by parts.

Why? Well, there are many uses of integration by parts and I’ll just display a few uses of them:

1. Taylor Polynomials. How do we get these? If we assume that f has enough derivatives, we proceed in the following manner: calculate \int ^x_0 f'(t) dt in two different ways: use the Fundamental Theorem of calculus on one side (to obtain f(x) - f(0) and use integration by parts on the other side: u = f'(t), dv = dt, du = f''(t), v = t-x (yes, we are being choosy about which anti derivative of dv to use).
This means: -\int^x_0 f''(t)(t-x)dt +f'(t)(t-x)|^x_0 = f(x)-f(0) so f(x) = f(0) + f'(0)x -\int^x_0 f''(t)(t-x)dt =f(x) and one proceeds from there.

2. Differential equations: given y' + p(x)y = f(x) one seeks to find an integrating factor (which is e^{\int p(x)} so as to get:

e^{\int p(x)}y' + p(x)e^{\int p(x)}y = f(x)e^{\int p(x)} which can be written as \frac{d}{dx}(e^{\int p(x)}y) = f(x)e^{\int p(x)}. That is, the left hand side is just the product rule for derivatives, which, as you know (if you are a calculus teacher), is really all integration by parts is!

Sure, one can jazz it up (as we subtly did in the Taylor Polynomial calculation); the integration by parts formula is really \frac{d}{dx} (f(x)g(x)) = \frac{d}{dx}(f(x)+ C) g(x) + f(x)\frac{d}{dx}(g(x) + D) where C, D are arbitrary constants. But, my main point is that integration by parts should be UNDERSTOOD; short cuts to do tedious calculations are relatively unimportant, IMHO.

Now if you want to ask students “why does tabular integration work”, then….GREAT!

October 15, 2013

What do you mean “you don’t know”???

Filed under: calculus, integrals, student learning — Tags: — collegemathteaching @ 6:38 pm

goatprof

I am grading calculus II exams and the above photo looks a bit like me. I’ll calm down before I hand the exams back to the students.

But I swear: I had one student DURING THE EXAM say “hey, you can’t do \int^1_{\frac{1}{2}} \frac{1}{x} dx  because \frac{1}{x} = x^{-1} and x^{-1+1} \frac{1}{-1+1} is undefined. Yes, this person passed calculus I and yes, we did that integral in calculus I (some schools hold off and develop ln(x) = \int^x_1 \frac{1}{t} dt using the Fundamental Theorem of Calculus). And yes, previously THIS SEMESTER we did integrals like \int \frac{1}{(x-1)(x+1)} dx .

(facepalm).

October 1, 2013

I’ve lost any reason to live…

Filed under: student learning — Tags: , — collegemathteaching @ 7:45 pm

Today, I was grading differential equation papers. Some were really good, others were not in that category.

A calculus 2 student came in (we teach techniques of integration, applications of integration, and infinite series).

She, in a very polite manner, complained that the course was. too. easy.

Too. Easy.

I wish that I drank.

But there is no denying it: we have strong freshmen and some very weak, “take the class multiple times” students in the same class, and heaven help you if your flunk out rate is too high.
I might encourage her to take her complaint to the dean and put it in writing.

#$#@!!!

September 19, 2013

What we mean about poor algebra skills…

Filed under: basic algebra, calculus, student learning — collegemathteaching @ 4:47 pm

Yes, mathematics professors have been complaining about their students lack of algebra skills as long as there have been calculus courses.

No, we aren’t talking about a student who, in a moment of panic, decided to write \int \sqrt{x^2+1} dx = \int \sqrt{x^2} + \sqrt{1} dx because they were stuck on an exam. And yes, I once saw a professor walk into an analysis class, write \sqrt{x^2+1} dx \ne  \sqrt{x^2} + \sqrt{1} on the board (while grinding the chalk into the board) while saying “the next person who makes this mistake will get an F for this class, ON THE SPOT! 🙂

But the weakness is more of the following: in class today, I wrote
\int (sec^2(x) - 1)tan(x) dx = \int (sec^2(x)tan(x) -tan(x)) dx = \frac{1}{2}tan^2(x) - ln(|sec(x)|+C

The student actually understood the integration, but didn’t understand where the first equality came from! I said “it is just algebra” and he STILL didn’t get it.

I have a hard time believing that this student doesn’t understand the distributive axiom of algebra; what I think is going on is that they don’t have the concept as a regular working part of their math/science/engineering mind.

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.

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