Tall, Dark, and Mysterious


Drinking (tea) with mathematicians

File under: I Made It Out Of Clay, Queen of Sciences. Posted by Moebius Stripper at 9:02 pm.

This morning, I made myself some tea in one of the matching mugs from my dinner set, waited an inordinately long time for it to start to cool down, and then downed the rest of it in a few gulps because it was quickly becoming too cold to drink. Not the most satisfying tea-drinking experience, but that’s what I get (I think) for throwing narrow cylinders.

So I started to wonder what shape of mug, made out of clay of constant thickness, would be ideal for drinking hot beverages. Specifically: I would like to pour myself some tea, wait a fixed time t for the surface of the tea to cool down to a drinkable temperature before beginning to sip, and then sip at a fixed (and preferably constant) rate until the tea is gone, with the tea remaining at a constant temperature. It seems like this should be doable; at the very least, I don’t think it’s not a ridiculous thing to ask for: the surface of the tea cools much faster than the tea below, and so there should be a way to sip tea from the appropriate vessel in such a way that each sip exposes another layer of tea that cools down to the temperature of the previous layer, just as I’m drinking it. It also seems clear that the top should be wider than the bottom, as the tea at the bottom will not be completely insulated by the tea above. Also, the heat transfer through the mug cannot be ignored: the specific heat capacity of clay is around a third of that of water, and only a third more than that of air.

…that’s as far as I got: I don’t have the physics background to set up this problem, though I spoke with someone who did, and he came up with some rather grisly equations that I don’t have the calculus background to solve. I also don’t know what simplifications could reasonably be made. But if anyone does, and can come up with a reasonable shape of mug that is conducive to tea-drinking, I’ll make you such a mug.


Your stupid misconceptions addressed

File under: Righteous Indignation, Those Who Can't. Posted by Moebius Stripper at 9:42 pm.

Things are pretty busy here at TD&M Headquarters: I’m away on business working long hours; I’ve been spending a lot of time in the studio; and, oh yeah, I’ve been taking care of everything that needed to be taken care of before my new condo officially became mine. Why, I’d have nothing worth blogging if it weren’t for the fact that one of my former students recently took me to task in the comments section on my most recent post!

Just kidding! Reader Carolyn wasn’t one of my students; rather, she’s a genetically engineered composite of every student that every academic blogger has ever complained about. O Lord, thank you for this bounty.

Hi, I just happened to come across your website while searching for something on Google.

Welcome! Make yourself at home, and above all, don’t be shy about telling me what you think of me and my blog!

I have read a lot of your posts and some of them are pretty interesting, while others are kinda depressing.

I am currently an undergraduate student, and therefore your posts about education interests me the most. I’d like to say that some of them are not totally fair, this is, of course, from a student’s perspective, and as we all know, a students’ perspective is often different from that of the teachers. :)

Yes, different. Bear in mind, though, that all teachers once were students, whereas few students once were teachers.

It’s called perspective.

First of all, I’d like to state that I am not a bad student, nor a lazy one. In fact I’m a pretty good student, if not the best in my school. Still, I found that playing bingos made of students’ mistakes is a cruel idea.

You should see the bingo card I made from students’ complaints about my blog.

Well, yes, I can see the humor in it, but still, in my homeland, there is an old saying: there are no bad students, there are only bad teachers.

And you left your homeland, yes?

I understand a teacher’s frustration when he can not make a student understand something, I also understand that certain students can be irritating, but, it’s not very professional to make fun of your students that way, well, not even on a website. You know not everybody in this world is very bright, but if you are not tolerant of this, you shouldn’t be teaching in the first place.Maybe I am taking it too seriously , but I feel very sad when I think about the possiblity that my own teachers may do the same with our mistakes.

Let me provide some never-before-published context to Precalculus Bingo, one of whose squares is “(x+y)^2=x^2+y^2″. Students are taught to expand (x+y)^2 in grade nine, if not grade ten; they then use it again in grades eleven and twelve, both of which were prerequisites for the college math class I taught. I had them expand such a quantity on their first quiz. Three quarters of them made the mistake on the bingo card. I spent ten minutes the next day going over it; I explained how they could substitute values for x and y and see that their fabricated identity didn’t hold; I showed them how to expand it.

A week later, I gave the same question on the next quiz. Nearly as many students got it wrong. Again I went over it in detail; again I explained how to expand algebraic expressions. I announced that I’d spent over half an hour on that one question, plus they’d had plenty of similar ones in the homework; I told my students that if I ever saw that mistake again, they’d get a mark of zero on the question. Consider yourselves warned, I said.

Five students made that mistake on the test. This is the sort of mistake I put on my bingo card.

Secondly, about grade inflation. I’d like to say that grades are not accurate indicators of one’s ability and will never be. I am not saying that people who always get D’s can actually be bright, though many famous scientists used to be terrible students when they were young (Albert Eistein for one). Therefore, I see no point for a teacher to take it to the extreme and say “oh, I am gonna be a super strict teacher and give the whole class a D average just to prove I am a committed teacher who cares about my students”.

Oh, and I said that where? Nowhere. I did, however, say this:

Precalculus I is a prerequisite for Precalculus II, which many of my students, such as you, are going to need to take. And Precalculus II is harder than this course, and builds upon it. A mark of C+ or higher, from me, means that you have the background that you need to pass Precalculus II. If I just increase marks of D’s to B’s, that doesn’t mean that a D student has the understanding they need for Precalculus II - they’ll still fail it. So I wouldn’t be doing anyone any favours if I made this course, or my tests, easier. It’s only by showing me that you have C+ understanding, or more, of this class, that I will be able to see that you’re prepared for Precalculus II.

I, and every single other teacher who does not run an independent school, have a curriculum to follow. Mathematics is cumulative; I need to know that my students have attained the level of understanding that they need to move on. Which kind of flies in the face of your “marks don’t mean anything, so give me an A” theory, so feel free to disregard. After all, students often have a different perspective from teachers.

From my experience, the real learning does not take place in school anyways.

And this translates into a mandate to give higher marks…how, precisely?

If a person is really commited to learning, he/she can always borrow books from library and study on his/her own without going through any kind of formal education. Since you are a college professor, I would assume your students are old enough to do this if they want to.

That’s two uses of the hypothetical syllogism in as many sentences, so let’s address both antecedents right now:

One - here’s a real-life conversation (abridged) that I had last term:

Student: I was never good at math and haven’t taken any math courses for the first three years of my degree. What do you suggest?

Me: Well, I’ve placed some high school math books on reserve in the library, and -

Student: Where’s the library?

Two - speaking of learning stuff on one’s own: sidebar sez, erstwhile college instructor. Run along, look it up on your own outside the classroom; I’ll still be here when you’re done.

Then, what’s the real point of going to university or college? learning? well, maybe, a little bit. I don’t think that learning something in three months and then getting tested on it(whether it’s math or statistics) can make one more educated.

Then get out of the classroom; you’re wasting your time.

Therefore, in my humble opinion, an important aspect of going to school is getting regonized for one’s abilility. That’s what the grading system is for. It is to show the future employer that a person is intelligent enough to perform certain tasks. Therefore, grades are important to students and will always be.

A summary:

  1. Grades are meaningless.
  2. Therefore I should give higher grades.
  3. You don’t really get educated in school, remember? Grades are meaningless.
  4. Grades are a means of recognizing students for their ability.
  5. Which was not acquired or honed in the classroom in which those grades were assigned.
  6. Making grades kinda disconnected from the grader and the material being graded.
  7. Still, because they determine employment prospects, grades are super important to students.
  8. But still meaningless.

Good to know.

Many would be idealistic and say university is about learning, I would beg to differ and say it’s 50% about personal growth and 20% about learning and 30% about the degree.

Well, that’s a load of 95% bullshit.

Seriously: I hate, hate, hate it when people think that their unfounded theories gain legitimacy by virtue of having made-up numbers attached to them. That right there is a reason that every member of a democratic society should learn statistics: so that they’re not so dazzled by numbers that they accept uncritically every statement that invokes them. Nonetheless, even if we accept your made-up statistics, how does the fact that university is 50% about personal growth mandate me giving higher grades?

This leads to the whole “students complaining about marks” question. I wouldn’t lie and say I have never complained about a teacher giving us a unreasonably hard test or about an irresponsible TA marking over-strictly. I actually do it a lot, though not always directly to the professor or the TA’s.

And we take your complaints very very seriously. Sometimes we even blog about them!

I have to say that students, as payers of their education, should have the right to question an unfair mark, if they indeed have the reason to.

And they do have that right; it’s a free country. Hell, I even let my non-paying scholarship and bursary students question their marks. The Canadian Charter of Rights and Freedoms guarantees them the right to ask questions; who am I to rescind it?

I have found, from my undergraduate experience, that professors and TA’s can be totally unreasonable and unfair when marking students’ tests. I can understand that from a teacher’s point of view, you may not think you have done anything wrong. But from the students’ point of view, we have not done anything wrong neither. And there is no reason why you must be absolutely right, and we must be absolutely wrong.

Yes, there is a reason: I have two degrees in the subject I teach, and you have none. I am familiar with the curriculum; therefore, I know what students need to know in order to advance to the next course, and you don’t. I have a broad background in my subject, and I know how the various threads of it fit together; consequently, I know what types of mistakes are serious, and which types are minor - and you don’t. I bring a decade of study and experience to the table, and I base my judgements on that. You bring only a sense of entitlement. There’s a chance you’re right and I’m wrong about a test or a grade, but frankly, the odds are against you.

One important thing about academia is the freedom of thoughts, the freedom of argument and sharing our different views. I’m sorry if I have sounded rude, but that’s the way it is.

You don’t sound rude; just ignorant. After all, you’re espousing the view that learning is only a minor function of the academy, whereas providing a forum where people wave around their baseless claims so that they can compete with one another on equal footing is “one important thing” about it. You don’t need university to “share [your] different views”; you can do that at a coffee shop, a park, or a party. You can even complain about how unfair your teachers are in all of those forums.

If you are going to outright dismiss the superior expertise and background of the people charged with expanding your worldview and not merely validating it, then there is no point in you going to university.

People are biased creatures you know, even the brightest man can make mistakes, and I don’t think every professor belongs to the brightest catogary.

Yes, everyone can make mistakes. I have certainly done so in setting tests or grading, and I have made amends - successfully, from what I’ve been told - whenever that came to my attention.

A low grade or a difficult test, however, is not prima facie a mistake.

Finally, I want to agree with you that the Canadian, and actually the whole North American education system doesn’t prepare a lot of people well for an university education. . Since I have studied for quite some time in East Asia, I have a comparison. It’s true that the education here is pretty slack. From your posts I can see that you are quite fond of the education of “your time”, and I am kind of suspicous of this.

Well, “my time” had its advantages and disadvantages. On the one hand, I’m feeling more than a shade nostalgic for those halcyon days when we didn’t sass our elders like you’re doing now. On the other hand, walking to school barefoot in the snow kinda sucked.

In fact, I find this generation of students to be a lot brighter on average than the previous generations, in many aspects (technology for one).

I tutored a grade 12 student this summer. He was weak in basic algebra, so we spent some time going over the basics. I went through one linear equation step-by-step, and then pointed to the last line and said, “So - x equals five times two,” and paused.

“Holdonasec,” said my student, and darted upstairs. I waited for two minutes until he returned with his backpack. He threw the bag on the floor beside him, opened it, and withdrew a pencil case. Slowly, he opened the pencil case, and pulled out a fucking graphing calculator. He then keyed in, 5, x, 2, = before triumphantly declaring, “Ten.”

Every generation has geniues and idiots, and it’s hard to compare.

No, it’s easy to compare: a first-year statistics course’ll provide you with the tools to identify trends. See your local university for a statistics class near you! If you think that you can’t analyze data because it can’t always be linearly ordered, then your university education has been for naught.

Easter education is probably the strictest and the most rigorious in the world, but they produce far less Nobelist than North America, and there is a reason for that. This is where a rigorous education is simply not enough.

In any first-year statistics course - you know, the one whose content you obviously don’t get at all - one learns that one can’t compare data sets by looking at the outliers.

Go learn about what that means - independently, in a library - and then we’ll talk.

well, this was a long rant. Just wanted to share a student’s point of view, somewhat different from yours. Maybe you find these ideas unimpressive, but at least it will help you understand better what your students may be thinking. I see you are a committed and serious teacher, these are what you want to know, right?

Well, inasmuch as gazing into the abyss can be educational, sure.


Impulse Purchases I Have Made

File under: Oh, Give Me A Home. Posted by Moebius Stripper at 7:52 pm.
  • South-facing condo in the eastern part of the city.


Driving with mathematicians

File under: Queen of Sciences. Posted by Moebius Stripper at 6:49 pm.

“How many gas stations do you think we’ll pass on the way home?”

“You mean on the side of the road, or at exits?”

“Whichever - places where we’d consider buying gas.”

“I don’t know, six, eight…why?”

“Well, if we pass the first 1/e gas stations and then stop at the first place that offers cheaper gas, then we’ll maximize the chance that we’ll end up buying the cheapest gas. I mean, assuming that gas prices are random and don’t follow any trend with regards to location, which obviously isn’t the case.”

“Oh. Ha! But that’s not the algorithm we want - that one maximizes the probability of getting the cheapest gas; it doesn’t minimize the expected price of the gas we buy, which is what we want.”

“Oh, yeah, you’re right: if we apply that first algorithm, then in the likely event that we don’t get the very cheapest gas, there’s still a reasonable chance that we’ll end up with pretty expensive gas.”


“So what’s the strategy for minimizing the expected price of gas?”

“I don’t know.”

So, is there a probabilist in the house?


Where textbooks come from

File under: Those Who Can't, When We Were Young. Posted by Moebius Stripper at 9:51 am.

When I was a kid, my father told me about a strange phenomenon, which was later explained to him, that he’d observed while eating on airplanes. Back in the seventies and early eighties, Dad travelled a lot on business, and he noticed that the slices of hard-boiled eggs he received never included any ends. Each egg slice that made its way onto his tray consisted of yellow and white concentric circles - and, even more curiously, they all appeared to be congruent. At first, my father chalked this up to coincidence - maybe he just always happened to get the middles of the eggs? - but soon, he noticed that his seatmates also received only egg middles. It was nearly a statistical impossibility that such a large random sample of egg slices would never contain the end pieces. Were the airline chefs throwing out the yolkless ends? It didn’t make any sense.

Dad’s question was answered some time later, when he had the opportunity to watch a video on the production of airline food. Included was a segment specifically devoted to the preparation of eggs. At last, a chance to settle this question!

The documentary showed a huge, industrial-sized kitchen, where several chefs were diligently cracking eggs, separating the whites from the yolks. One large mixer processed hundreds of egg yolks; a second contained the whites. The chefs then poured the yolks into a long, hollow tube, where they were boiled. Then they formed the whites into a sheet half an inch or so thick, where they were hardened just enough to be moved without collapsing. Then, the whites were positioned to surround the stick of yolk. The entire apparatus was cooked one final time, resulting in a symmetric, several-foot-long cylinder of hard-boiled egg - which a machine then sliced into the discs that would be served to airline passengers.

This is why my father never got the ends of eggs.

In related news, friend and reader oxeador sent me a link to The Muddle Machine, a textbook editor’s firsthand account of why elementary and high school texts used in the United States are bland, incoherent, expensive, and updated with every new phase of the moon - even as they offer little new content with each edition. Tamim Ansary has written an appalling exposition of how a lot of bureaucracy, a lot of money, a lot of politics, and hardly any pedagogical or subject expertise has given rise to books that serve their creators and their financial backers rather than the students and teachers that use them.

I got a hint of things to come when I overheard my boss lamenting, “The books are done and we still don’t have an author! I must sign someone today!”

Every time a friend with kids in school tells me textbooks are too generic, I think back to that moment. “Who writes these things?” people ask me. I have to tell them, without a hint of irony, “No one.”

Last year, I did some contract work writing and editing a textbook - elementary school math for college students, more or less. “Writing” and “editing” such a text consists of taking perfectly functional texts and guidelines, and processing them in a manner that is disturbingly similar to the way that our airline chefs of yore mass-produced hard-boiled egg slices. My result, arrived at after hours of poring through specs and sources, differed from the existing texts about as substantially as the egg tube different from its (hard-boiled) constituent parts. Ansary’s experience is similar:

[A]t each grade level, the editors distill their notes into detailed outlines…[later], they divide the outline into theoretically manageable parts and assign these to writers to flesh into sentences.

What comes back isn’t even close to being the book. The first project I worked on was at this stage when I arrived. My assignment was to reduce a stack of pages 17 inches high, supplied by 40 writers, to a 3-inch stack that would sound as if it had all come from one source. The original text was just ore. A few of the original words survived, I suppose, but no whole sentences.

To avoid the unwelcome appearance of originality at this stage, editors send their writers voluminous guidelines. I am one of these writers, and this summer I wrote a 10-page story for a reading program. The guideline for the assignment, delivered to me in a three-ring binder, was 300 pages long.

There’s so little I can add to this piece; it’s a bit long, but it’s an easy read. And Asmary does provide some constructive suggestions at the end, the second of which I especially support:

Reduce basals to reference books — slim core texts that set forth as clearly as a dictionary the essential skills and information to be learned at each grade level in each subject. In content areas like history and science, the core texts would be like mini-encyclopedias, fact-checked by experts in the field and then reviewed by master teachers for scope and sequence.

Sounds a lot better than the current state of affairs, where politicians and lobbyists play those roles.

[Related article that I’ve linked before: Underwood Dudley’s “review” of a calculus textbook.]


Math class: now with more social justice (and less math)

File under: Character Writ Large, Righteous Indignation, Those Who Can't, Queen of Sciences. Posted by Moebius Stripper at 8:54 pm.

The adage that addresses the issue of judging books by their covers counsels unambiguously against, but I’ve always rejected it on the grounds that it assumes, generally incorrectly, that authors have no editorial control over the presentation of their work. I unabashedly judge books by their covers in the figurative sense; but at the moment I’m being quite literal. Specifically, I am judging Rethinking Mathematics by its cover:

The authors of this tome aim to “provide examples of how to weave social justice issues throughout the mathematics curriculum and how to integrate mathematics into other curricular areas”, and if the cover is one such example, then we can safely conclude that the integral of SOCIAL JUSTICE ISSUES + MATHEMATICS CURRICULUM equals GIBBERISH. It’s an equation! No, it’s an inequality! No, wait…it’s bullshit! Seriously -what are the units of MULTICULTURALISM * POVERTY / INEQUALITY?

I haven’t read the book. My local library doesn’t carry it, and if I were to pay the $16 cover price to purchase it from pair of white guys who write about economic racism, well, I fear that that would make me part of the problem.

Nevertheless, even those of us who haven’t read the book can find plenty to criticize (cf “critical thinking”) in the introduction alone, which decries the “unfortunate scarcity of social justice connections” that are to be found in conventional high school mathematics curricula. Sadly, some old fuddy-duddies think that math classes should deal with stuff like trigonometry, as opposed to, say, the War on Iraq Boondocks Cartoon. Those naysayers, however, are just party poopers who totally don’t get it, but they can easily be convinced that a a social justice approach is consistent with your (yes, your) state’s mathematics standards:

Occasionally, a teacher needs to defend this kind of curriculum to supervisors, colleagues, or parents. One approach is to survey your state’s math standards (or the national standards) and to find references to “critical thinking” or “problem solving” and use those to explain your curriculum. Also, the NCTM clearly states that “mathematical connections” between curriculum and students’ lives are important.

There’s a valuable lesson to be learned here, actually, and it is this: if the description of your curriculum is impenetrably vague and long-winded, then people can use it to justify anything. For instance, the paid-by-the-word folks behind the Illinois math curriclum talk about a goal, within the math class, for students to “express and interpret information and ideas”, from which one could argue that it follows that we should be teaching interpretive dance in lieu of geometry.

This book pisses me off. It pisses me off a lot. Not because I think that math and life should be disjoint - quite the contrary, as I’ve frequently argued in this space. Hell - during the first ten minutes of the statistics-for-social-scientists course I taught last year, I stated, in so many words, that no one can accurately claim to be a fully-functioning member of a democratic society if they can’t interpret quantitative data. (Nor am I the first to make this argument; John Allen Paulos, author of the marvellous - and bestselling - book Innumeracy, says as much himself. The fact that the authors of Rethinking Mathematics are so unfamiliar with the literature that exists on the topic of mathematical literacy that they claim that theirs is the only resource of its kind, does not speak well to their expertise on the subject.) I also don’t think that the social-justice-based math curriculum is the dominant force behind students’ appalling inability to work with quantities. Sure, it’s not helping, but to claim that students think that 2/3+5/7=7/10 because their teachers are ideologues who have rethought mathematics is to diminish the roles played by innumerate elementary school teachers, innumerate curriculum developers, absent fathers, working mothers, fucking graphing calculators, sugary breakfast cereals, sex on TV, and shock rock in bringing about that sorry state of affairs. Hell, a good many of the topics in this book look quite worthwhile: the section on how unemployment figures are reported, for instance, seems like a nice topic for the “how you present data impacts what people think” section that appears in every single statistics chapter in every single high school math text, not just this “first of its kind” resource, but anyway! No, I am not opposed to mathematical literacy, and I wish that folks who are more politically-inclined than I would invoke it more often.

No, what bothers is this: is anyone familiar with a movement among social studies educators in secondary schools to use math in their courses, or does the movement toward interdisciplinary studies of social justice only go in the other direction? I am aware of none. Why are the educators who are motivated by political issues - and who see numeracy as a means to that end - injecting those issues into the math curriculum, rather than injecting math into social studies classes - which seems more natural to me? If I think that potters would improve their craft by learning some elementary Newtonian mechanics, I’d sooner give impromptu physics lessons at the pottery wheel than drag my physics classmates to the studio.

Is the overall effect to the high school curriculum, a net reduction of mathematical content?

The authors of Rethinking Mathematics are unabashedly politically-driven, and from the table of contents it is apparent that the math they present in their book leads students, none too subtly, to such conclusions as the one that capitalism is a fundamentally damaging economic system. Leaving aside for the moment the validity of this conclusion - I personally dispute it - let’s consider just how very involved a topic economics is. To come to any conclusion about capitalism requires one of two things: 1) a great deal of in-depth studies of economics and related issues, issues that Ph.D. students have written theses about; or 2) some superficial examination of pre-selected data (is this the Global Capitalist Economy Cartoon mentioned in the book’s table of contents?) that leads directly to the desired conclusion. In the context of a high school math class, (1) entails a huge use of the mathematics class’s time to teach and learn economics, while (2) constitutes brainwashing.

Given how ill-prepared the majority of high school students are to either do mathematics or think (let alone “think critically”, and the first person to point out case of that phrase being used by anyone who doesn’t have an ideological axe to grind, gets a cookie), you’ll forgive me if I can’t get on board with either of those two options.

This book, if used more than very sparingly, will give innumerate high school students highly skewed foundations on a wealth of complicated topics, and direct them to predetermined conclusions. Judging from the table of contents, it might prepare students for jobs preparing statistical expositions of positions espoused by lefty think-tanks. And, hell, that’s more than a lot of high school classes prepare students for, so I can’t even find fault with that; the problem is that while grooming students for that path, the social-justice math class will inevitably omit, because of time constraints, some other topics that might prepare students for further study in other areas. Will students whose teachers are motivated by social justice concerns learn enough trigonometry to hold their own in a university engineering course, should they wish to pursue that path? Will they learn enough algebra to succeed in the chemistry courses required by every medical school? The authors of this text talk about using mathematics to “potentially change the world”, which is hardly the exclusive domain of the social justice activists: anyone who thinks that engineers and doctors haven’t used math to change the world, has spent too long at rallies and is brainwashed beyond salvation. Engineers and doctors have changed the world for the better, even if measured in social justice terms. A robust, demanding, contentful high school mathematics curriculum, even one that suffers from an “unfortunate scarcity of social justice connections” (yes, they did use that phrase to describe the standard high school math curriculum, because you know that when I see the “how to use your fucking graphing calculator to plot a straight line” unit, the first thing I think is “but where’s the social justice?”) will leave the door open for students to acquire the tools they will need to use math to change the world - whether or not they later choose to become social justice crusaders. A curriclum designed to “guide students towards a social justice orientation” will cripple them if they choose any path other than that one.

And given how deluded high school students seem to be about the nature of equations , it can’t be a good idea to let them anywhere near that horrible cover.

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