West Virginia Math Department and trends..

First of all, I’ll have to read this 2016 article.

But: it is no secret that higher education in the US is in turmoil, at least at the non-elite universities. Some colleges are closing and others are experiencing cut backs due to high operating losses.

This little not will not attempt to explain the problems of why education has gotten so expensive, though things like: reduction of government subsidies, increased costs for technology (computers, wifi, learning management systems), unfunded mandates (e. g. accommodations for an increasing percentage of students with learning disabilities) and staff to handle helicopter parents are all factors adding to increased costs.

And so, many universities are more tuition dependent than ever before, and while the sticker price is high, many (most in many universities) are given steep discounts.

And so, higher administration is trying to figure out what to offer: they need to bring in tuition dollars.

Now about math: our number of majors has dropped, and much, if not most, of the drop comes from math education: teaching is not a popular occupation right now, for many reasons.

Things like this do not help attract student to teacher education programs:

Virginia argues teacher is not entitled to extra compensation after being shot because the shooting was a "hazard of the job." The school board's attorney says it's an "unfortunate reality" that teaching is "not without its dangers."https://t.co/t3H7Sf70dz

— Celeste Headlee (@CelesteHeadlee) August 11, 2023

One thing that hurts enrollment in upper division math courses is that higher math has prerequisites. Of course, many (most?) pure math courses do not appear to have immediate application to other fields (though they often do). And, let’s face it: math is hard. The ideas are very dense.

So, it is my feeling that the math major..one that requires two semesters of abstract algebra and two semesters of analysis, is probably on the way out, at least at non-elite schools. I think it will survive at Ivy caliber schools, MIT, Stanford, and the flagship R-1 schools.

As far as the rest of us: it absolutely hurts my heart to say this, but I feel that for our major to survive at a place like mine, we’ll have to allow for at least some upper division credit to come from “theory of interest”, “math for data science”, etc. type courses…and perhaps allow for mathy electives from other disciplines. I see us as having to become a “mathematical sciences” type program…or not existing at all.

Now for the West Virginia situation (and they probably won’t be the last):

https://twitter.com/AnonymousF59605/status/1690077384282607616

I went on their faculty page and noted that they had 31 Associate/Full professors; the remainder appeard to be “instructors” or “assistant professors of instruction” and the like. So while I do not have any special information, it appears that they are cutting the non-tenured..the ones who did a lot (most?) of the undergraduate teaching.

Now for the uninitiated: keeping current with research at the R-1 level is, in and of itself, is a full time job. Now I am NOT one of those who says that “researchers are bad teachers” (that is often untrue) but I can say that teaching full loads (10-12 hours of undergraduate classes) is a very different job than running a graduate seminar, advising graduate students, researching, and getting NSF grants (often a prerequisite for getting tenure to begin with.

So, a lot of professor’s lives are going to change, not only for those being let go, but also for those still left. I’d imagine that some of the research professors might leave and have their place taken by the teaching faculty who are due to be cut, but that is pure speculation on my part.

So, you want our tenure-track academic math job…

Background: we are a “primarily undergraduate” non-R1 institution. We do not offer math master’s degrees but the engineering college does.

Me: old full professor who has either served on or chaired several search committees.

I’ll break this post down into the two types of jobs we are likely to offer:
Tenure Track lecturer

Tenure Track Assistant Professor.

Lecturer

No research requirement; this job consists of teaching 12 hour loads of lower division mathematics classes, mostly “business calculus and below”; college algebra and precalculus will be your staples. There will be some service work too.

What we are looking for:

Evidence that you have taught lower division courses (college algebra, precalculus, maybe “baby stats”) successfully. Yes, it is great that you were the only postdoc asked to teach a course on differentiable manifolds or commutative ring theory but that is not relevant to this job.

So hopefully you have had taught these courses in the past (several times) and your teaching references talk about how well you did in said courses; e. g. students did well in said courses, went on to the next course prepared, course was as well received as such a course can be, etc. If you won a teaching award of some kind (or nominated for one), that is good to note. And, in this day and age..how did the online stuff go?

Teaching statement: ok, I am speaking for myself, but what I look for is: did you evaluate your own teaching? What did you try? What problems did you notice? Where could you have done better, or what could you try next time? Did you discuss your teaching with someone else? All of those things stand out to me. And yes, that means recognizing that what you tried didn’t work this time…and that you have a plan to revise it..or DID revise it. This applies to the online stuff too.

Diversity Statement Yes, that is a relatively new requirement for us. What I look for: how do you adjust to having some cultural variation in your classroom? Here are examples of what I am talking about:

We usually get students from the suburbs who are used to a “car culture.” So, I often use the car speedometer as something that gives you the derivative of the car’s position. But I ended up with a student from an urban culture and she explained to me that she and her friends took public transportation everywhere…I had to explain what a speedometer was. It was NOT walking around knowledge.

Or: there was a time when I uploaded *.doc files to our learning management system. It turns out that not all students have Microsoft word; taking a few seconds to make them *.pdf files made it a LOT easier for them.

Other things: not everyone gets every sports analogy, gambling analogy (cards, dice, etc.) so be patient when explaining the background for such examples.

Also: a discussion on how one adjusts for the “gaps” in preparation that students have is a plus; a student can place into a course but have missing topics here and there. And the rigor of the high school courses may well vary from student to student; some might expect to be given a “make up” exam if they do poorly on an exam; another might have been used to be given credit for totally incorrect work (I’ve seen both).

Also: if you’ve tutored or volunteered to help a diverse group of students, be sure to mention that (e. g. maternity homes, sports teams, urban league, or just the tutoring center, etc.)

Transcript: yes, we require it, but what we are looking for is breath for the lecturer’s job: the typical is to have three of the following covered: “algebra, analysis, topology, probability, statistics, applied math”

Cover letter: Something that shows that you know the type of job we are offering is very helpful; if you state that you “want to direct undergraduate research”, well, our lecturer job will be a huge letdown.

Assistant Professor

This job will involve 9-12 hours teaching; 10-11 is typical and we do have a modest research requirement. 2-3 papers in solid journals will be sufficient for tenure; you might not want to have your heart set on an Annals of Math publication. If you do get one, you won’t be with us for long anyway. There is also advising and service work.

What we are looking for: teaching: we want some evidence that you can teach the courses typically taught by our department. This means some experience in calculus/business calculus for our math track, and statistics for our statistics track. For this job, some evidence for upper division is a plus, but not required nor even expected; is is an extra “nice to have.”

But it is all but essential that your teaching references talks about your performance in teaching lower division classes (calculus or below); if all you have is “the functional analysis students loved him/her”, that is not helpful. Being observed while teaching a lower division course is all but essential.

Teaching and Diversity statement : same as for the lecturer job. An extra: did you have any involvement with the math club?

Research: the thing we are looking for is: will you “die on the vine” or not? Having a plan: “I intend to move from my dissertation in this direction” is a plus, as is having others to collaborate with (though collaboration isn’t necessary). Also, a statement from your advisor that you can work INDEPENDENTLY ..that is, you can find realistic problems to work on and do NOT need hand holding, is a major plus. You are likely to be somewhat isolated here. And of course, loving mathematics is essential with us. Not all candidates do..if you see your dissertation as a task you had to do to get the credential then our job isn’t for you.

Another plus: having side projects that an undergraduate can work on is a plus. We do have some undergraduate research but that won’t be the bulk of the job.

Transcript: same as the lecturer job.

Introduction to infinite series for inexperienced calculus teachers

Let me start by saying that this is NOT: this is not an introduction for calculus students (too steep) nor is this intended for experienced calculus teachers. Nor is this a “you should teach it THIS way” or “introduce the concepts in THIS order or emphasize THESE topics”; that is for the individual teacher to decide.

Rather, this is a quick overview to help the new teacher (or for the teacher who has not taught it in a long time) decide for themselves how to go about it.

And yes, I’ll be giving a lot of opinions; disagree if you like.

What series will be used for.

Of course, infinite series have applications in probability theory (discrete density functions, expectation and higher moment values of discrete random variables), financial mathematics (perpetuities), etc. and these are great reasons to learn about them. But in calculus, these tend to be background material for power series.

Power series: , the most important thing is to determine the open interval of absolute convergence; that is, the intervals on which converges.

We teach that these intervals are *always* symmetric about (that is, at only, on some open interval or the whole real line. Side note: this is an interesting place to point out the influence that the calculus of complex variables has on real variable calculus! These open intervals are the most important aspect as one can prove that one can differentiate and integrate said series “term by term” on the open interval of absolute convergence; sometimes one can extend the results to the boundary of the interval.

Therefore, if time is limited, I tend to focus on material more relevant for series that are absolutely convergent though there are some interesting (and fun) things one can do for a series which is conditionally convergent (convergent, but not absolutely convergent; e. g. .

Important principles: I think it is a good idea to first deal with geometric series and then series with positive terms…make that “non-negative” terms.

Geometric series: ; here we see that for , and is equal to for ; to show this do the old “shifted sum” addition: , then subtract: as most of the terms cancel with the subtraction.

Now to show the geometric series converges, (convergence being the standard kind: the “n’th partial sum, then the series converges if an only if the sequence of partial sums converges; yes there are other types of convergence)

Now that we’ve established that for the geometric series, and we get convergence if goes to zero, which happens only if .

Why geometric series: two of the most common series tests (root and ratio tests) involve a comparison to a geometric series. Also, the geometric series concept is used both in the theory of improper integrals and in measure theory (e. g., showing that the rational numbers have measure zero).

Series of non-negative terms. For now, we’ll assume that has all (suppressing the indices).

Main principle: though most texts talk about the various tests, I believe that most of the tests involved really involve three key principles, two of which the geometric series and the following result from sequences of positive numbers:

Key sequence result: every monotone bounded sequence of positive numbers converges to its least upper bound.

True: many calculus texts don’t do that much with the least upper bound concept but I feel it is intuitive enough to at least mention. If the least upper bound is, say, , then if is the sequence in question, there has to be some such that for any small, positive . Then because is monotone, for all

The third key principle is “common sense” : if converges (standard convergence) then as a sequence. This is pretty clear if the are non-negative; the idea is that the sequence of partial sums cannot converge to a limit unless becomes arbitrarily small. Of course, this is true even if the terms are not all positive.

Secondary results I think that the next results are “second order” results: the main results depend on these, and these depend on the key 3 that we just discussed.

The first of these secondary results is the direct comparison test for series of non-negative terms:

Direct comparison test

If and converges, then so does . If diverges, then so does .

The proof is basically the “bounded monotone sequence” principle applied to the partial sums. I like to call it “if you are taller than an NBA center then you are tall” principle.

Evidently, some see this result as a “just get to something else” result, but it is extremely useful; one can apply this to show that the exponential of a square matrix is defined; it is the principle behind the Weierstrass M-test, etc. Do not underestimate this test!

Absolute convergence: this is the most important kind of convergence for power series as this is the type of convergence we will have on an open interval. A series is absolutely convergent if converges. Now, of course, absolute convergence implies convergence:

Note and if converges, then converges by direct comparison. Now note is the difference of two convergent series: and therefore converges.

Integral test This is an important test for convergence at a point. This test assumes that is a non-negative, non-decreasing function on some (that is, ) Then converges if and only if converges as an improper integral.

Proof: is just a right endpoint Riemann sum for and therefore the sequence of partial sums is an increasing, bounded sequence. Now if the sum converges, note that is the right endpoint estimate for so the integral can be defined as a limit of a bounded, increasing sequence so the integral converges.

Yes, these are crude whiteboards but they get the job done.

Note: we need the hypothesis that is decreasing (or non-decreasing). Example: the function certainly has converging but diverging.

Going the other way, defining gives an unbounded function with unbounded sum but the integral converges to the sum . The “boxes” get taller and skinnier.

Note: the above shows the integral and sum starting at 0; same principle though.

Now wait a minute: we haven’t really gone over how students will do most of their homework and exam problems. We’ve covered none of these: p-test, limit comparison test, ratio test, root test. Ok, logically, we have but not practically.

Let’s remedy that. First, start with the “point convergence” tests.

p-test. This says that converges if and diverges otherwise. Proof: Integral test.

Limit comparison test Given two series of positive terms: and

Suppose

If converges and then so does .

If diverges and then so does

I’ll show the “converge” part of the proof: choose then such that This means and we get convergence by direct comparison. See how useful that test is?

But note what is going on: it really isn’t necessary for to exist; for the convergence case it is only necessary that there be some for which ; if one is familiar with the limit superior (“limsup”) that is enough to make the test work.

We will see this again.

Why limit comparison is used: Something like clearly converges, but nailing down the proof with direct comparison can be hard. But a limit comparison with is pretty easy.

Ratio test this test is most commonly used when the series has powers and/or factorials in it. Basically: given consider (if the limit exists..if it doesn’t..stay tuned).

If the series converges. If the series diverges. If the test is inconclusive.

Note: if it turns out that there is exists some such that for all we have then the series converges (we can use the limsup concept here as well)

Why this works: suppose there exists some such that for all we have Then write

now factor out a to obtain

Now multiply the terms by 1 in a clever way:

See where this is going: each ratio is less than so we have:

which is a convergent geometric series.

See: there is geometric series and the direct comparison test, again.

Root Test No, this is NOT the same as the ratio test. In fact, it is a bit “stronger” than the ratio test in that the root test will work for anything the ratio test works for, but there are some series that the root test works for that the ratio test comes up empty.

I’ll state the “lim sup” version of the ratio test: if there exists some such that, for all we have then the series converges (exercise: find the “divergence version”).

As before: if the condition is met, so the original series converges by direction comparison.

Now as far as my previous remark about the ratio test: Consider the series:

Yes, this series is bounded by the convergent geometric series with and therefore converges by direct comparison. And the limsup version of the root test works as well.

But the ratio test is a disaster as which is unbounded..but .

What about non-absolute convergence (aka “conditional convergence”)

Series like converges but does NOT converge absolutely (p-test). On one hand, such series are a LOT of fun..but the convergence is very slow and unstable and so might say that these series are not as important as the series that converges absolutely. But there is a lot of interesting mathematics to be had here.

So, let’s chat about these a bit.

We say is conditionally convergent if the series converges but diverges.

One elementary tool for dealing with these is the alternating series test:

for this, let and for all .

Then converges if and only if as a sequence.

That the sequence of terms goes to zero is necessary. That it is sufficient in this alternating case: first note that the terms of the sequence of partial sums are bounded above by (as the magnitudes get steadily smaller) and below by (same reason. Note also that so the sequence of partial sums of even index are an increasing bounded sequence and therefore converges to some limit, say, . But and so by a routine “epsilon-N” argument the odd partial sums converge to as well.

Of course, there are conditionally convergent series that are NOT alternating. And conditionally convergent series have some interesting properties.

One of the most interesting properties is that such series can be “rearranged” (“derangment” in Knopp’s book) to either converge to any number of choice or to diverge to infinity or to have no limit at all.

Here is an outline of the arguments:

To rearrange a series to converge to , start with the positive terms (which must diverge as the series is conditionally convergent) and add them up to exceed ; stop just after is exceeded. Call that partial sum . Note: this could be 0 terms. Now use the negative terms to go of the left of and stop the first one past. Call that Then move to the right, past again with the positive terms..note that the overshoot is smaller as the terms are smaller. This is . Then go back again to get to the left of . Repeat.

Note that at every stage, every partial sum after the first one past is between some and the bracket and the distance is shrinking to become arbitrarily small.

To rearrange a series to diverge to infinity: Add the positive terms to exceed 1. Add a negative term. Then add the terms to exceed 2. Add a negative term. Repeat this for each positive integer .

Have fun with this; you can have the partial sums end up all over the place.

That’s it for now; I might do power series later.

Big lesson that many overlook: math is hard

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.

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.

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 where

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 defined on some interval and if is defined on , then we say that “in mean” (or “in the norm”). Basically, as grows, the area between the graphs of and gets arbitrarily small.

However this does NOT mean that converges to 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 The intervals have length and start by moving left to right on and then moving right to left and so on. They “dance” on [0,1]. Let the the function that is 1 on the interval and 0 off of it. Then clearly as the interval over which we are integrating is shrinking to zero, but this sequence of functions doesn’t converge point wise ANYWHERE on . Of course, a subsequence of functions converges pointwise.

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 .

2. Engineers care about being able to take limits (e. g., using L’Hopitals rule and about problems such as )

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.

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

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.

About teaching continuity: a math ed talk

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
“ if is rational and otherwise”
is continuous at . 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 is continuous at if which, of course, means that is defined at 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 “. You know what the expected answer was, but my question was immediately: “why is it reasonable to expect 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: , elsewhere. The question: “is differentiable at ? She told TAs to mark the problem “wrong” if the students said yes, because the function changed formula at !!! Note: the question asked “differentiable” and not “smooth”.

Vent over…