I learned the following trick for memorizing the value of e from my colleague, Gene White. It never fails to impress calculus students (given a wide enough definition of “impress”).
Start by carefully looking at this picture:
That’s a 20 dollar bill, so memorize “2″ and put down the decimal point.
The picture on the bill is of Andrew Jackson. He was our seventh President, so put a “7″ after the decimal point to get 2.7.
Jackson was elected in 1828, so put down “1828″ next. Since there’s a 2 in front of the decimal place, put “1828″ a second time. We’re now up to 2.718281828.
Now look at the red square over Jackson’s face. The diagonal creates two congruent right triangles with angle measures 45, 90, and 45. So, add on 459045 to get 2.718281828459045. And that’s e to 15 places.
I’m doing some research, if you can call it that, right now that involves looking at past editions of popular and/or influential calculus books to track the evolution of how certain concepts are developed and presented. I’ll have a lot to say on this if I ever get anywhere with it. But in the course of reading, I have been struck with how little some books change over the course of several editions. For example, the classic Stewart text has retained the exact wording and presentation in its section on concavity in every edition since the first, which was released in the mid-80’s. There’s nothing wrong with sticking with a particular way of doing things, if it works; but you have to ask yourself, does it really work? And if so, why are we now on the sixth edition of the book? I know that books need refreshing from time to time, but five times in 15 years?
Anyhow, it occurred to me that there’s something really simple that textbook companies could do that would both help out students who have a hard time affording textbooks (which is a lot of students) and give themselves an incentive not to update book editions for merely superficial reasons. That simple thing is: When a textbook undergoes a change in edition, post the old edition to the web as a free download. That could be a plain PDF, or it could be a Kindle or iBooks version. Whatever the format, make it free, and make it easy to get.
This would be a win-win-win for publishers, authors, and students:
By charging the regular full price for the “premium” (= most up-to-date) edition of the book, the publisher wouldn’t experience any big changes in its revenue stream, provided (and this is a big “if”) the premium edition provides significant additional value over the old edition. In other words, as long as the new edition is really new, it would cost the publisher nothing to give the old version away.
But if the premium edition is just a superficial update of the old one, it will cost the publisher big money. So publishers would have significant incentive not to update editions for no good reason, thereby costing consumers (students) money they didn’t really need to spend (and may not have had in the first place).
All the add-ons like CD-ROMs, websites, and other items that often get bundled with textbooks would only be bundled with the premium edition. That would provide additional incentive for those who can afford to pay for the premium edition to do so. (It would also provide a litmus test for exactly how much value those add-ons really add to the book.)
It’s a lot easier to download a PDF of a deprecated version of a book, free and legally, then to try your luck with the various torrent sites or what-have-you to get the newest edition. Therefore, pirated versions of the textbook would be less desirable, benefitting both publishers and authors.
Schools with limited budgets (including homeschooling families) could simply agree not to use the premium version and go with the free, deprecated version instead. This would always be the case if the cost of the new edition outweighs the benefits of adopting it — which again, puts pressure on the publishers not to update editions unless there are really good reasons to do so and the differences between editions are really significant.
The above point also holds in a big, big way for schools in developing countries or in poverty-stricken areas in this country.
Individual students could also choose to use the old edition, and presumably accept responsibility for the differences in edition, even if their schools use the premium edition. Those who teach college know that many students do this now already, except the old editions aren’t free (unless someone gives the book to them).
All this provides publishers and authors to take the moral high road while still preserving their means of making money and doing good business.
Classes started for us this week. It’s gotten me thinking about what profs do on the first day of class and their overall concepts for how to approach the first few days of a class, where students form those crucial first impressions about the course and the instructor. Here’s my overall approach:
I prefer a quick, energetic launch directly into the course material. I spend maybe the first 7-10 minutes on course structure. Then we start right into the course content through a lecture/activity combination.
To help with the first point, I will often create screencasts for some of the course management stuff (like this screencast for how to navigate Moodle) and email students the links to these, often before the first class meets.
I do not go in for icebreakers, get-to-know-you activities, exercises intended to discover students Myers-Briggs types or learning styles, or any of that. Not that I think such things are not useful. But I’d rather the students get to work and get to know themselves and each other in the context of working, rather than get to know each other instead of working.
I give a full-bodied assignment on the first day of class to do for the second day of class — something that would really take about two hours outside of class to do, if the class meeting took one hour. Here’s the assignment list, for example, for my calculus class. That’s about 2 hours worth of work, although if you look closely, a lot of it is watching instructional screencasts and playing around with course software, so it’s less work than it looks like. But still, students have to do stuff.
Students form their conceptions of the class — and keep that conception through the whole semester — in these first few moments of the course. I want to give students the impression that the class is something they need to take seriously, and there’s a workload that has to be managed carefully, and they cannot expect to succeed if they hold the course at arms’ length. I think jumping in, rather than easing in, to the coursework is a good way to accomplish this. A potential downside of my approach is that students often get shellshocked by the initial workload and give up before they even get started. I always get a few students coming by with drop forms, saying “I just don’t think I’m going to have the time for this course.”
How do you approach the first day, and next few days, of a course? Or, if you are not a teacher, what was the best or worst approach you’ve seen to the initial few days of a course?
Friday music time again, and just about the only thing I’ve had time to post this week due to classes starting back:
Texas Flood (Stevie Ray Vaughan & Double Trouble, Greatest Hits)
40 Days (Third Day, Come Together)
Who’s Been Talkin’ (Howlin’ Wolf, His Best: Chess 50th Anniversary)
Man in the Green Shirt (Weather Report, Best of Weather Report)
Waiting on the World to Change (John Mayer, Continuum)
Where You Are (Rich Mullins, The World As Best As I Remember It v. 1)
Heavy On My Mind (Back Door Slam, Roll Away)
Try (John Mayer Trio, Try! (Live))
Living Loving Maid (She’s Just A Woman) (Led Zeppelin, Led Zeppelin II)
Doing It To Death (James Brown, The CD of JB)
Normally I would take one of the entries in the list that gets my attention and do a video focus on it. This time… Well, the classic Led Zeppelin chestnut “Living Loving Maid” (#9) makes me think of the fantastic cover done by Dread Zeppelin. You know — that band that does Led Zeppelin covers, only they’re done in a reggae style and using a late-70’s era Elvis impersonator as their lead singer. Sadly, I couldn’t find a video for that. So instead, here’s the video for their version of “Your Time Is Gonna Come”, which Robert Plant once said he preferred to the original.
There seem to be two pieces of technology that all mathematicians and other technical professionals use, regardless of how technophobic they might be: email, and . There are ways to typeset mathematical expressions out there that have a more shallow learning curve, but when it comes to flexibility, extendability, and just the sheer aesthetic quality of the result, has no rival. Plus, it’s free and runs on every computing platform in existence. It even runs on WordPress.com blogs (as you can see here) and just made its entry into Google Documents in miniature form as Google Docs’ equation editor. is not going anywhere anytime soon, and in fact it seems to be showing up in more and more places as the typesetting system of choice.
But gets a bad rap as too complicated for normal people to use. It seems to be something people learn only in graduate school, with few undergraduates — and even fewer high school students — ever seeing it, much less using it. There is a grain of truth there; is not a WYSIWYG word processor, and the near-programming aspect of using can overwhelm users used to pointing-and-clicking for everything.
But I think that the benefits of using outweigh the costs, and undergraduates and high school students can, and ought to, learn how to use as fluently as they use a word processor for other courses. A couple of years ago, I put together a series of twelve screencasts for use in our sophomore “transition-to-proof” class on learning . I put these screencasts online, but mainly they were only advertised to my students and colleagues. Now, however, I’d like to throw these out there for everyone to use.
All twelve of these are done on a Windows system running MiKTeX and the free IDE known as TeXNicCenter. This provides students with as close to a point/click interface to as you could expect to get. Within that context, there are two basic intro videos:
These two videos are enough to learn how works and will allow you to make a simple file with uncomplicated math and text in it. The remaining 10 videos follow from these two. Some are prerequisites for the others — and those prereqs are stated explicitly at the beginning of any video that has them — but if you watch them in the following order there will be no dependency problems:
Some of these are pretty long, but all totalled (including the two “basics” videos) this is less than two hours of viewing.
When I’ve used these in class, I give students some printed instructions on how to download and configure MiKTeX and TeXNicCenter, and then I have them watch these videos out of class. They are instructed to work along with the videos. I give them about a week to do so. Within that week, if there’s a problem set or something else in the class that could be done with , I’ll offer extra credit to students to do so, to incentivize their learning the system. After the end of that week, I will insist that all major assignments have to be done in , or else the assignment gets a grade of “0″.
Students have sometimes struggled to get up the learning curve, but if they’re allowed and encouraged to help each other, everyone eventually gets to the point where they are quite fluent writing up homework and so on. Students have even elected to use on assignments in other courses, even non-math courses.
I’m going to use these videos in linear algebra this semester (our transition-to-proof course is now defunct) and I’ll be making up a new screencast on MATLAB and . Later, probably during the summer, I’ve been thinking about redoing the entire video series; I now have better screencasting tools than I used to have, and I’d like to keep all the videos under 10 minutes so they can go on YouTube.
So feel free to use these (attributing authorship to me is appreciated but not required), and if you have suggestions or comments, please email them or leave them below.
We start classes this week, a bit later than most other folks thanks to our January term. That means the long-awaited MATLAB course will be formally kicking off. I’ve had a few people ask me about what we’re doing in this course, so here’s an update.
This has been a tricky course to plan, because the audience is definitely not the usual one for an introductory MATLAB course. Almost all the introductory textbooks and materials I reviewed for the course, and all the introductory MATLAB courses I looked at from other schools, have a particular student demographic in mind: they are all engineering majors; they are all freshmen or sophomores with either a semester of programming under their belts or at least a very high level of comfort with computers and the “guts” of programming; and they are all attending large universities in which the particular academic makeup of the institution plays little to no role in how the class is designed.
By contrast, of the 15 students enrolled right now in my course:
9 are freshmen; 5 are sophomores. (At least that demographic fits the profile.)
5 are Education majors; 3 of those are secondary education majors, the other 2 elementary education majors. There is one lone Pure Mathematics major. 9 have no declared major at all.
Of the 9 undeclareds, two of those are students who are pursuing our dual-degree program in Engineering and just haven’t gotten around to filling out the paperwork yet.
But that’s it: Two undocumented engineering majors; NO science majors. The plurality are Education majors, and once the undeclareds get around to declaring, this may become the majority.
I don’t have data for this, but I am pretty sure less than half of the students have ever had any exposure to programming whatsoever. I wouldn’t be surprised to find out it’s more like 1/5 of the class with no programming experience.
Again, no data, but I think a good portion of the students would not consider themselves comfortable around computers once we start talking about something besides basic office apps and web pages. For example, anything having to do with typing stuff into a command line. (Like much of MATLAB.)
So this is not the audience that is “supposed” to be taking a MATLAB course. They are not (for the most part) scientists and engineers, and if you start throwing the details of fprintf, memory addressing, floating point arithmetic, and so forth to them you will likely lose them.
And yet — and this has been the frustrating thing — almost all introductory MATLAB materials assume students are engineers and scientists who have no problem being thrown directly into learning about fprintf, memory addressing, and floating point arithmetic. I won’t name names or authors, and these are not bad books, but they are written with a particular assumption in mind about who is using them. And that assumption does not work for my class.
Therefore I have had to do a lot of remixing and retrofitting of existing materials in order to deliver what I think is a solid intro MATLAB course, one that I honestly think satisfies the same learning objectives as the course at our partner university (which uses one of those books I mentioned above, and it works for them because they have the “right” audience for those books), but also one that really works for the decidedly non-MATLAB-like group of students I will have.
I’ve tried to adhere to a few basic design principles when drawing up this course:
Get students comfortable with the software and how it works before throwing them into programming. But: Don’t wait too long to begin programming.
Connect use of the software back to the mathematics courses they know: namely Calculus I and Calculus II. (The course is a prerequisite for Calculus III and is usually taken alongside Calc II.) And take it very easy on any other kind of math in the course.
Get them doing plots and working with data from Excel files early and often. Pictures and data: Students dig these.
Make heavy use of the Symbolic Toolbox and get them using it early as well. And by “Symbolic Toolbox” I really mean MuPad, not the nearly-indecipherable symbolic manipulation done inside the MATLAB command line by calling the Symbolic Toolbox. This is actually quite different than how most intro MATLAB materials do it; if you see the Symbolic Toolbox at all, it’s the last chapter of the book and MuPad is never mentioned.
Take it easy on the science content of the course and instead emphasize use of MATLAB on topics with which the majority of the class is familiar and comfortable. In the canonical audience, that would mean precisely that a lot of science ought to show up. For us, not so much.
Have fun first, foremost, and throughout. Remember that many of the students in the course are not only unfamiliar with computers (aside from using them to check Facebook or use a word processor) but are actually scared of them — probably a greater portion of the class than will let on to it. Keep it light. If something looks more confusing than useful, it probably is; and unless there’s some compelling reason to hack through it, just drop it and cover something else.
I think that with these principles in mind, a lot of MATLAB purists would look at my course and sniff at it, thinking it’s not a “real” MATLAB course. It certainly will not walk or talk like the MATLAB course you’ll find at the typical Big Engineering School. But we’re not a big engineering school; we’re a liberal arts college, and the Liberal Arts informs what we do in every course, including a MATLAB course. I guess I am not trying to create a course that in turn creates more MATLAB purists. I am trying to create a course that shows students that programming is a great application and instantiation of critical thinking and problem solving. If they don’t know all the command line options to fprintf, but later on when confronted with a problem they first thing they think is to try out some MATLAB code to solve it, I think we’ve had a successful experience.
That’s an overview of the philosophy and design of the course. In another post, I’ll talk about the course schedule, assessments, and plans for what we’ll do in the class. This gets interesting because, as mentioned above, we’re not really using a book but rather just McGyvering a bunch of pre-existing resources to fit our particular needs.
Students and faculty at University Preparatory School in Redding, CA have created the world’s largest Sierpinski triangle constructed entirely out of Doritos. (Well, it’s probably the only one, but still.) It is 64 feet long and made out of 12,000 Doritos. This was done as an entry to the Doritos Crash the Superbowl contest. Watch, and be awed:
Can a 128-, 256-, etc. foot long Dorito Sierpinski triangle be far behind? I bet the parent company for Doritos would seriously consider some corporate sponsorship.
Thanks to Cory Poole, math and physics teacher at U-Prep, who sent this in. That’s a great, creative way to get students interested in math. (And you can eat it when it’s done.) There’s more on the video here.
Without The Light (Kelly Joe Phelps, Roll Away the Stone)
Partita #3, Menuet II (Paul Galbraith, Bach: The Sonatas and Partitas)
Tenderoni (Chromeo, Fancy Footwork)
Ob-La-Di, Ob-La-Da (The Beatles, White Album)
Jump Up! (Imagination Movers, For Those About to Hop)
Birdland (Weather Report, Best of Weather Report)
Get Up, Stand Up (Bob Marley, Legend)
Territories (Rush, Power Windows)
The Remembering (High the Memory) (Yes, Tales from Topographic Oceans)
With My Own Two Hands (Jack Johnson + Ben Harper, Sing-a-Longs and Lullabies)
Lots of good stuff to feature here this week — the Bob Marley piece is an especially welcome reminder of warmer climates right now, as it’s 15 degrees and snow on the ground here in Indiana. But in the spirit of 80’s music started last week, here’s a live version of Rush doing “Territories” (#8). Watch it for no other reason that to see Geddy Lee doing three things simultaneously — playing a hard bass line, playing intricate keyboard hits, and doing vocals — any one of which would give most musicians (<raises hand>) fits.
The LA Times reports on a study suggesting that female elementary school teachers who are anxious about math transmit that anxiety to the girls in their classes:
Girls have long embraced the stereotype that they’re not supposed to be good at math. It seems they may be getting the idea from a surprising source — their female elementary school teachers.
First- and second-graders whose teachers were anxious about mathematics were more likely to believe that boys are hard-wired for math and that girls are better at reading, a new study has found. What’s more, the girls who bought into that notion scored significantly lower on math tests than their peers who didn’t.
The gap in test scores was not apparent in the fall when the kids were first tested, but emerged after spending a school year in the classrooms of teachers with math anxiety. That detail convinced researchers that the teachers — all of them women — were the culprits.
It’s no surprise that teachers who are weak in or nervous about a subject do not inspire confidence, or performance, in that subject among their students. What’s different here is the gender connection — female teachers having a pronounced effect upon girl students — and the subject area. It would be interesting to see just how many elementary school teachers view themselves as “anxious” about teaching math, and then to see how that self-description breaks down by gender. Do a lot of female elementary teachers feel anxious about math? Is it more than male elementary school teachers? I don’t know, but that is certainly the stereotype.
At any rate, the opposite seems to be implied by this study too — female teachers who are strong with math and comfortable with teaching it to kids will have an enhanced positive effect on girls’ perceptions of math and their performance with it. And it seems like a no-brainer that elementary education curricula ought to stress a strong degree of math content mastery among all preservice teachers — of both genders — and demand a high level of fluency with doing and teaching math.Teaching math to little kids is hard, and you have to know a lot of math outside of what you are going to teach if you’re going to do it well. We need to have done with another stereotype: that you major in elementary education because “you just love kids” (you need more than sentimentality to be a good teacher) or because it’s supposedly an easy major (it isn’t, or at least shouldn’t be).
My 6-year old is in kindergarten and fascinated by school and schoolteachers. Last week she asked me: “Daddy, are you a teacher?” I told her I was. “What’s your school?” I told her I teach at a college. “What’s a college?” I told her: “A college is a school for grown-ups.” And in that off-the-cuff answer, we have an economical way of describing the difference between college and pre-college education, and of encapsulating the hopes and goals of higher education.
College students, even the wide-eyed freshmen who show up every fall, are not kids. They are emerging adults, having worked 12 years for a high school education and who now enter a 4-5 year buffer zone before entering into the world with nothing more than the things they know, the experiences they’ve had, and the people around them. Therefore we college professors aren’t serving students if we treat them like kids, refer to them as “kids”, or in any way give them a reason to conceive of themselves as children. If we do any of these, students will simply model what we do and stay children. Indeed, I hear my own students talk much more frequently about “the other kids in my class” than “the other students” or even “the other people“.
What we hope to do in higher education is not so much to convey academic subject matter but rather, when you boil it all down, we are trying to teach people how to think like grown-ups. Of course we want our students to retain the best aspects of childhood — curiosity, energy, and so on — but we also want them to temper their child-likeness with adult sensibilities. We want them to think about other people besides themselves; to be self-motivating and responsible; to judge information objectively; to trust but verify what they see; to draw freely from a depth and breadth of experiences to make sense of what they encounter; to be able to think and act for themselves.
Since this is what higher education aims for, I’d like to give a good-natured challenge to my colleagues in higher ed, and by extension all those high school teachers who teach “college preparatory” courses. Very simply: From now until the end of classes this spring, don’t refer to your students as “kids”. Think of them instead as adults, “emerging adults” if you like, and refer to them accordingly. And take a look at your course policies and the ways you make decisions about how to deal with and treat students. If these are set up in a way that places grown-up behavior as the basic assumption, or at the very least provides a road map for younger students to ramp up into grown-up behavior, then I’d say that’s on the right track; otherwise think about ways to change.
I did this recently. I was visiting a high school class I had been working with as part of a dual-enrollment course. During my visit, the teacher allowed the students to have some open Q&A time with me, and several of the students asked me about the differences between high school and college. After spelling out some specifics, I told them, “At my college, we’re going to treat you like men and women when you come in. Not like kids. You won’t be kids and you shouldn’t be treated like kids.” Those young men and women got the message immediately — they straightened up in their chairs and more than a few got smiles on their faces. If we college profs all agree to treat students like men and women — daring students to believe in their own adulthood — I think we’ll see the same positive effect.
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