What Do I Need To Pass This Class?


I haven’t gone in for reblogging my own posts, but, I realized I have two that are somewhat timely. That’s my post about how to calculate the minimum grade you need on the final exam to get a desired final score, and then a set of tables that work out the minimum scores for people who aren’t confident they’re going to get the formula worked out right. I hope people will be patient with repetitions of these postings. This is the post with the complete formula, and instructions how to work it out, so, it’s to be used by those who felt they knew what was going on in pre-algebra and regular algebra.

nebusresearch

I admit a bit of difficulty in identifying with people who are very worried about their grades. I stopped worrying about my grades somewhere around fifth grade, when I missed a straight-A report card by one question on one spelling test and decided the stress wasn’t worth it for changing one B+ into an A-. But it’s a question instructors get, increasingly, as the final exam approaches, and students are almost never satisfied with the correct answers, especially when circumstances require a time machine. I suppose I understand the despair in that case.

Anyway, working out the minimum grade you need to pass — or what you need to get an A, or to get a B, or whatever you like — is an easy enough problem it’s surprising when students don’t work it out on their own. Maybe they don’t realize this is what they learn algebra for. But…

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Reading the Comics, February 1, 2014


For today’s round of mathematics-themed comic strips a little deeper pattern turns out to have emerged: π, that most popular of the transcendental numbers, turns up quite a bit in the comics that drew my attention the past couple weeks. Let me explain.

Dan Thompson’s Brevity (January 23) returns to the anthropomorphic numbers racket, with the kind of mathematics puns designed to get the strip pasted to the walls of the teacher’s lounge. I wonder how that’s going for him.

Greg Evans’s Luann Againn (January 25, rerun from 1986) has Luann not understanding how to work out an arithmetic problem until it’s shown how to do it: use the calculator. This is a joke that’s probably going to be with us as long as there are practical, personal calculating devices, because it is a good question why someone should bother learning arithmetic when a device will do it faster and better by every reasonable measure. I admit not being sure there is much point to learning arithmetic, other than as a way to practice a particular way of learning how to apply algorithms. I suppose it also stands as a way to get people who are really into mathematics to highlight themselves: someone who memorizes the times tables is probably interested in the kinds of systematic thought that mathematics depends on. But that’s a weak reason to demand it of every student. I suppose arithmetic is very testable, but that’s an even worse reason to make students go through it.

Mind you, I am quite open to the idea that arithmetic drills are useful for students. That I don’t know a particular reason why I should care whether a seventh-grader can divide 391 by 17 by hand doesn’t mean that I don’t think there is one.

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Reading the Comics, December 29, 2013


I haven’t quite got seven comics mentioning mathematics themes this time around, but, it’s so busy the end of the year that maybe it’s better publishing what I have and not worrying about an arbitrary quota like mine.

Wuff and Morgenthaler’s WuMo (December 16) uses a spray of a bit of mathematics to stand in for “something just too complicated to understand”, and even uses a caricature of Albert Einstein to represent the person who’s just too smart to be understood. I’m a touch disappointed that, as best I can tell, the equations sprayed out don’t mean anything; I’ve enjoyed WuMo — a new comic to North American audiences — so far and kind of expected they would get an irrelevant detail like that plausibly right.

I’m also interested that sixty years after his death the portrait of Einstein still hasn’t been topped as an image for The Really, Really Smart Guy. Possibly nobody since him has managed to combine being both incredibly important — even if it weren’t for relativity, Einstein would be an important figure in science for his work in quantum mechanics, and if he didn’t have relativity or quantum mechanics, he’d still be important for statistical mechanics — and iconic-looking, which I guess really means he let his hair grow wild. I wonder if Stephen Hawking will be able to hold some of that similar pop cultural presence.

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September 2013’s Statistics


And as it’s the start of the month I have a fresh round of reviewing the statistics for readership around here. I have seen a nice increase in both views — from 367 to about 466 total views — and in visitors — from 175 to 236 — which maybe reflects the resumption of the school year (in the United States, anyway) and some more reliable posting (of original articles and of links to other people’s) on my part. (Maybe. If I’m reading this rightly I actually only posted nine new things in September, which is the same as in August. I’m surprised that WordPress’s statistics page doesn’t seem to report how many new articles there were in the month, though.) My contrarian nature forces me to note this means my views-per-reader ratio has dropped to 1.97, down from 2.10. I suppose as long as the views-per-reader statistic stays above 1.00 I’m not doing too badly.

The most popular articles the past month were:

  1. From ElKement: Space Balls, Baywatch, and the Geekiness of Classical Mechanics, which is really just pointing and slightly setting up ElKement’s start to a series on quantum field theory which you can too understand;
  2. How Many Trapezoids I Can Draw, which is a persistent favorite and makes me suspect that I’ve hit on something that teachers ask students about. If I could think of a couple other nice little how-many-of-these-things problems there are I’d post them gladly, although that might screw up some people’s homework assignments;
  3. Reading the Comics, September 11, 2012, which is another persistent favorite and I can’t imagine that it’s entirely about the date (although the similar Reading the Comics entry for September 11 of 2013 just missed being one of the top articles this month so perhaps the subject lines are just that effective a bit of click-baiting);
  4. What Is Calculus I Like?, about my own realization that I never took a Calculus I course in the conditions that most people who take it do. I’d like more answers to the question of what experiences in intro-to-calculus courses are like, since I’m assuming that I will someday teach it again and while I think I can empathize with students, I would surely do better at understanding what they don’t understand if I knew better what people in similar courses went through;
  5. Some Difficult Math Problems That You Understand, which is again pointing to another blog — here, Maths In A Minute — with a couple of mathematics problems that pretty much anyone can understand on their first reading. The problems are hard ones, each of which has challenged the mathematical community for generations, so you aren’t going to solve them; but, thinking about them and trying to solve them is probably a great exercise and likely to lead you to discovering something you didn’t know.

I got the greatest number of readers from the United States again (271), with Canada (31) once more in second place. The United Kingdom’s climbed back into the top three (21), while August’s number-three, Denmark, dropped out of the top ten and behind both Singapore and the Philippines. I got a mass of single-reader countries this time, too: Azerbaijan, Bangladesh, Belgium, Cambodia, the Czech Republic, Indonesia, Israel, Italy, Mexico, Norway, Poland, Qatar, Spain, Sri Lanka, Switzerland, and Thailand. Bangladesh and Sri Lanka are repeats from last month, but my Estonian readership seems to have fled entirely. At least India and New Zealand still like me.

Feynman Online Physics


Likely everybody in the world has already spotted this before, but what the heck: CalTech and the Feynman Lectures Website have put online an edition of volume one of The Feynman Lectures on Physics. This is an HTML 5 edition, so older web browsers might not be able to read it sensibly.

The Feynman Lectures are generally regarded as one of the best expositions of basic physics; they started as part of an introduction to physics class that spiralled out of control and that got nearly all the freshmen who were trying to take it lost. I know the sense of being lost; when I was taking introductory physics I turned to them on the theory they might help me understand what the instructor was going on about. It didn’t help me.

This isn’t because Feynman wasn’t explaining well what was going on. It’s just that he approached things with a much deeper, much broader perspective than were really needed for me to figure out my problems in — oh, I’m not sure, probably something like how long a block needs to slide down a track or something like that. Here’s a fine example, excerpted from Chapter 5-2, “Time”:

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What Is Calculus I Like?


Although I haven’t got a mathematics class to teach this term, at least not right now, I have thought a bit about it and realized that I’ve surprisingly missed a nearly universal affair: I haven’t had a Calculus I course, the kind taught in big lecture halls capable of seating hundreds of students, literally several of whom are awake and alert and paying attention. The closest I’ve come is a history-of-computation course, with a nominal enrollment of about 130 students, and a similarly sized Introduction to C; but the big mathematics course college students are supposed to get through so they learn they really don’t like calculus, I haven’t done. While I was teaching assistant for some Calculus I courses, I never had professors who wanted me to attend lecture as a regular thing, and I just came in to do recitations.

More, I never had Calculus I as a student. I was in a magnet program in high school that got me enough advanced placement credit that I skipped pretty near the whole freshman year of the mathematics major sequence, and I could jump right into the courses with 30-to-40 student enrollments like Vector Calculus and Introduction to Differential Equations. That was great for me, but it’s finally struck me that I missed a pretty big, pretty common experience.

So I’m curious what it’s like: what the experience is, what students are expecting from their professors, what professors expect from students, how those expectations clash. I know the sorts of class methods I liked as a student and that I like as an instructor, but not how well that fits the attempt to teach a hundred-plus students who are just there because the school requires the passing of some mathematics courses and this is the one they offer 140 sections of.

Where Do Negative Numbers Come From?


Some time ago — and I forget when, I’m embarrassed to say, and can’t seem to find it because the search tool doesn’t work on comments — I was asked about how negative numbers got to be accepted. That’s a great question, particularly since while it seems like the idea of positive numbers is probably lost in prehistory, negative numbers definitely progressed in the past thousand years or so from something people might wildly speculate about to being a reasonably comfortable part of daily mathematics.

While searching for background information I ran across a doctoral thesis, Making Sense Of Negative Numbers, which is uncredited in the PDF I just linked to but appears to be by Dr Cecilia Kilhamn, of the University of Gothenburg, Sweden. Dr Kilhamn’s particular interest (here) is in how people learn to use negative numbers, so most of the thesis is about the conceptual difficulties people have when facing the minus sign (not least because it serves two roles, of marking a number as negative and of marking the subtraction operation), but the first chapters describe the historical process of developing the concept of negative numbers.

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How Long Is A Bad Ellipse Question?


Something came to mind while thinking about that failed grading scheme for multivariable calculus. I’d taught it two summers, and the first time around — when I didn’t try the alternate grading scheme — I made what everyone assured me was a common mistake.

One of the techniques taught in multivariable calculus is how to compute the length of a curve. There are a couple of ways of doing this, but you can think of them as variations on the same idea: imagine the curve as a track, and imagine that there’s a dot which moves along that track over some stretch of time. Then, if you know how quickly the dot is moving at each moment in time, you can figure out how long the track is, in much the same way you’d know that your parents’ place is 35 miles away if it takes you 35 minutes of travelling at 60 miles per hour to get there. There are details to be filled in here, which is why this is fit in an advanced calculus course.

Anyway, the introduction of this, and the homeworks, start out with pretty simple curves — straight lines, for example, or circles — because they’re easy to understand, and the student can tell offhand if the answer she got was right, and the calculus involved is easy. You can focus energy on learning the concept instead of integrating bizarre or unpleasant functions. But this also makes it harder to come up with a fresh problem for the exams: the student knowing how to find the length of a parabola segment or the circumference of a circle might reflect mastering the idea, or just that they remembered it from class.

So for the exam I assigned a simple variant, something we hadn’t done in class but was surely close enough that I didn’t need to work the problem out before printing up and handing out the exams. I’m sure it will shock you that an instructor might give out on an exam a problem he hasn’t actually solved already, but, I promise you, sometimes even teachers who aren’t grad students taking summer courses will do this. Usually it’s all right. Here’s where it wasn’t.

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Complex Experiments with Grading Mathematics


While I’ve never managed to attempt an experimental grading system as the one I enjoyed in Real Analysis, I have tried a few more modest experiments. The one chance I’ve had to really go wild and do something I’d never seen before, sadly, failed, but let me resurrect it enough to leave someone else, I hope, better-informed.

The setting was a summer course, which the department routinely gave to graduate students as a way of keeping them in the luxurious lifestyle to which grad students become accustomed. For five weeks and a couple days I’d spend several hours explaining the elements of vector calculus to students who either didn’t get it the first time around or who wanted to not have to deal with it during the normal term. (It’s the expansion of calculus to deal with integrals and differentials along curves, and across surfaces, and through solid bodies, and remarkably is not as impossibly complicated as this sounds. It’s probably easier to learn, once you know normal calculus, than it is to learn calculus to start. It’s essential, among other things, for working out physics problems in space, since it gives you the mathematical background to handle things like electric fields or the flow of fluids.)

What I thought was: the goal of the class is to get students to be proficient in a variety of techniques — that they could recognize what they were supposed to do, set up a problem to use whatever technique was needed, and could carry out the technique successfully. So why not divide the course up into all the things that I thought were different techniques, and challenge students to demonstrate proficiency in each of them? With experience behind me I understand at least one major objection to this, but if the forthcoming objection were to be dealt with, I’d still have blown it in the implementation.

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Reading the Comics, April 28, 2013


The flow of mathematics-themed comic strips almost dried up in April. I’m going to assume this reflects the kids of the cartoonists being on Spring Break, and teachers not placing exams immediately after the exam, in early to mid-March, and that we were just seeing the lag from that. I’m joking a little bit, but surely there’s some explanation for the rash of did-you-get-your-taxes-done comics appearing two weeks after April 15, and I’m fairly sure it isn’t the high regard United States newspaper syndicates have for their Canadian readership.

Dave Whamond’s Reality Check (April 8) uses the “infinity” symbol and tossed pizza dough together. The ∞ symbol, I understand, is credited to the English mathematician John Wallis, who introduced it in the Treatise on the Conic Sections, a text that made clearer how conic sections could be described algebraically. Wikipedia claims that Wallis had the idea that negative numbers were, rather than less than zero, actually greater than infinity, which we’d regard as a quirky interpretation, but (if I can verify it) it makes for an interesting point in figuring out how long people took to understand negative numbers like we believe we do today.

Jonathan Lemon’s Rabbits Against Magic (April 9) does a playing-the-odds joke, in this case in the efficiency of alligator repellent. The joke in this sort of thing comes to the assumption of independence of events — whether the chance that a thing works this time is the same as the chance of it working last time — and a bit of the idea that you find the probability of something working by trying it many times and counting the successes. Trusting in the Law of Large Numbers (and the independence of the events), this empirically-generated probability can be expected to match the actual probability, once you spend enough time thinking about what you mean by a term like “the actual probability”.

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Real Experiments with Grading Mathematics


[ On an unrelated note I see someone’s been going through and grading my essays. I thank you, whoever you are; I’ll take any stars I can get. And I’m also delighted to be near to my 9,500th page view; I’ll try to find something neat to do for either 9,999 or 10,000, whichever feels like the better number. ]

As a math major I staggered through a yearlong course in Real Analysis. My impression is this is the reaction most math majors have to it, as it’s the course in which you study why it is that Calculus works, so it’s everything that’s baffling about Calculus only moreso. I’d be interested to know what courses math majors consider their most crushingly difficult; I’d think only Abstract Algebra could rival Real Analysis for the position.

While I didn’t fail, I did have to re-take Real Analysis in graduate school, since you can’t go on to many other important courses without mastering it. Remarkably, courses that sound like they should be harder — Complex Analysis, Functional Analysis and their like — often feel easier. Possibly this is because the most important tricks to studying these fields are all introduced in Real Analysis so that by the fourth semester around the techniques are comfortably familiar. Or Functional Analysis really is easier than Real Analysis.

The second time around went quite well, possibly because a class really is easier the second time around (I don’t have the experience in re-taking classes to compare it to) or possibly because I clicked better with the professor, Dr Harry McLaughlin at Rensselaer Polytechnic Institute. Besides giving what I think might be the best homework assignment I ever received, he also used a grading scheme that I really responded to well, and that I’m sorry I haven’t been able to effectively employ when I’ve taught courses.

His concept — I believe he used it for all his classes, but certainly he put it to use in Real Analysis — came from as I remember it his being bored with the routine of grading weekly homeworks and monthly exams and a big final. Instead, students could put together a portfolio, showing their mastery of different parts of the course’s topics. The grade for the course was what he judged your mastery of the subject was, based on the breadth and depth of your portfolio work.

Any slightly different way of running class is a source of anxiety, and he did some steps to keep it from being too terrifying a departure. First is that you could turn in a portfolio for a review as you liked mid-course and he’d say what he felt was missing or inadequate or which needed reworking. I believe his official policy was that you could turn it in as often as you liked for review, though I wonder what he would do for the most grade-grabby students, the ones who wrestle obsessively for every half-point on every assignment, and who might turn in portfolio revisions on an hourly basis. Maybe he had a rule about doing at most one review a week per student or something like that.

The other is that he still gave out homework assignments and offered exams, and if you wanted you could have them graded as in a normal course, with the portfolio grade being what the traditional course grade would be. So if you were just too afraid to try this portfolio scheme you could just pretend the whole thing was one of those odd jokes professors will offer and not worry.

I really liked this system and was sorry I didn’t have the chance to take more courses from him. The course work felt easier, no doubt partly because there was no particular need to do homework at the last minute or cram for an exam, and if you just couldn’t get around to one assignment you didn’t need to fear a specific and immediate grade penalty. Or at least the penalty as you estimated it was something you could make up by thinking about the material and working on a similar breadth of work to the assignments and exams offered.

I regret that I haven’t had the courage to try this system on a course I was teaching, although I have tried a couple of non-traditional grading schemes. I’m always interested in hearing of more, though, in case I do get back into teaching and feel secure enough to try something odd.

Reblog: Mathematical Creativity: Multiple Solutions to the Pencil Sharpener Problem


TheGeometryTeacher has here the four kinds of results gotten from a class given a word problem (about the time needed for a certain event to occur). I like not just the original problem but the different approaches taken to the answer. It seems to me often lost to students, or at least poorly communicated to them, that nearly any interesting problem can be solved several ways over. Probably that’s a reflection of wanting to teach the most efficient way to do any particular problem, so showing more than one approach is judged a waste of time unless the alternate approach is feeding some other class objective.

Given the problem myself, I’d be inclined toward what’s here labelled as the “guessing and checking” approach, as I find a little experimentation like that helps me get to understand the workings of the problem pretty well. If the problem is small enough this might be all that I need to get to the answer. If it’s not, then the experience I get from a couple guesses and seeing why they don’t work would guide me to a more rigorous answer and one that looks more like the graph depicted.

Guessing and checking gets little respect, probably because when you’re trying to train the ability to calculate like “what is eight times seven” it’s hard to distinguish informed guessing from a complete failure to try. (The correct answer is, of course, “nobody knows”; the sevens and eights times tables are beyond human understanding.)

But when you’re venturing into original work for which there may be no guidance what a correct answer is (or whether there is one), or when you’re trying to do something for fun like figure out “What are the odds my roller coaster car will get stuck at the top of a ride like Top Thrill Dragster?” guessing and correcting from that original guess are often effective starting points.

Mr. Shauver - Learner Educator

I enjoy watching students exploring a problem that forces them to come up with their own structure for solving it. Today, a group got a chance to mess around with The Pencil Sharpener Problem, which is a problem I posted a month or so ago. (I’ll leave you to read it if you are curious what the problem is.)

From my perspective, what makes this problem interesting for the students is the ease with which it is communicated and the complexity with which is it solved. It seems quite easy. The answer is fairly predictable, but the students quickly found out that if they were going to solve this problem accurately, they were going to need two things:

1. A way to organize their thoughts and,

2. a way to verify their answer.

As long as the solution process included those two things, the students ended up fairly successful…

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How Big Was West Jersey?


A map of New Jersey's counties, with municipal boundaries added.

A book I’d read about the history of New Jersey mentioned something usable for a real-world-based problem in fraction manipulation, for a class which was trying to get students back up to speed on arithmetic on their way into algebra. It required some setup to be usable, though. The point is a property sale from the 17th century, from George Hutcheson to Anthony Woodhouse, transferring “1/32 of 3/90 of 90/100 shares” of land in the province of West Jersey. There were a hundred shares in the province, so, the natural question to build is: how much land was transferred?

The obvious question, to people who failed to pay attention to John T Cunningham’s This Is New Jersey in fourth grade, or who spent fourth grade not in New Jersey, or who didn’t encounter that one Isaac Asimov puzzle mystery (I won’t say which lest it spoil you), is: what’s West Jersey? That takes some historical context.

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Quick Little Calculus Puzzle


fluffy, one of my friends and regular readers, got to discussing with me a couple of limit problems, particularly, ones that seemed to be solved through L’Hopital’s Rule and then ran across some that don’t call for that tool of Freshman Calculus which you maybe remember. It’s the thing about limits of zero divided by zero, or infinity divided by infinity. (It can also be applied to a couple of other “indeterminate forms”; I remember when I took this level calculus the teacher explaining there were seven such forms. Without looking them up, I think they’re \frac00, \frac{\infty}{\infty}, 0^0, \infty^{0}, 0^{\infty}, 1^{\infty}, \mbox{ and } \infty - \infty but I would not recommend trusting my memory in favor of actually studying for your test.)

Anyway, fluffy put forth two cute little puzzles that I had immediate responses for, and then started getting plagued by doubts about, so I thought I’d put them out here for people who want the recreation. They’re both about taking the limit at zero of fractions, specifically:

\lim_{x \rightarrow 0} \frac{e^x}{x^e}

\lim_{x \rightarrow 0} \frac{x^e}{e^x}

where e here is the base of the natural logarithm, that is, that number just a little high of 2.71828 that mathematicians find so interesting even though it isn’t pi.

The limit is, if you want to be exact, a subtly and carefully defined idea that took centuries of really bright work to explain. But the first really good feeling that I really got for it is to imagine a function evaluated at the points near but not exactly at the target point — in the limits here, where x equals zero — and to see, if you keep evaluating x very near zero, are the values of your expression very near something? If it does, that thing the expression gets near is probably the limit at that point.

So, yes, you can plug in values of x like 0.1 and 0.01 and 0.0001 and so on into \frac{e^x}{x^e} and \frac{x^e}{e^x} and get a feeling for what the limit probably is. Saying what it definitely is takes a little more work.

Ted Baxter and the Binomial Distribution


There are many hard things about teaching, although I appreciate that since I’m in mathematics I have advantages over many other fields. For example, students come in with the assumption that there are certainly right and certainly wrong answers to questions. I’m generally spared the problem of convincing students that I have authority to rule some answers in or out. There’s actually a lot of discretion and judgement and opinion involved, but most of that comes in when one is doing research. In an introductory course, there are some techniques that have gotten so well-established and useful we could fairly well pretend there isn’t any judgement left.

But one hard part is probably common to all fields: how closely to guide a student working out something. This case comes from office hours, as I tried getting a student to work out a problem in binomial distributions. Binomial distributions come up in studying the case where there are many attempts at something; and each attempt has a certain, fixed, chance of succeeding; and you want to know the chance of there being exactly some particular number of successes out of all those tries. For example, imagine rolling four dice, and being interested in getting exactly two 6’s on the four dice.

To work it out, you need the number of attempts, and the number of successes you’re interested in, and the chance of each attempt at something succeeding, and the chance of each attempt failing. For the four-dice problem, each attempt is the rolling of one die; there are four attempts at rolling die; we’re interested in finding two successful rolls of 6; the chance of successfully getting a 6 on any roll is 1/6; and the chance of failure on any one roll is —

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