Calculus on the Brain
I don’t know what it is about calculus, but every time I’ve had to deal with it, it really gets into my brain.
Like…it permeates my dreams and becomes something that my brain just can’t let go of. I remember this happening the first time I took an actual calculus class during that first summer at UBC (yeah, I know, first calculus class was during grad school. Shut up.). When I went to Boston for the APS conference, all I could think of was calculus. All my brain produced in my dreams was calculus-related.
Same thing now. Now that I’m teaching it, all my brain does is produce calculus-related thoughts and dreams. I’m waking up thinking of limits.
It’s kind of a nice throwback to when I was first learning calc.
And, you know, it always makes me think of Leibniz, so there ya go.
ZoomaZoomaMathsGear
Heyo.
Y’all want some cool math-related stuff? Check out this website.
Favorites:
- Skew dice.
- Shapes of constant width.
- Symmetry groups wrapping paper (I don’t know why, but I love the symmetry groups. I’ve blogged about them a few times.)
- Torus balloons.
- Math stack playing cards.
- RGB/CMYK earrings.
- Galton board (Matt, I think you showed this to me once?)
WOO!
Yo Dawg, I Herd U Like Factorials
Have you ever wondered why zero factorial (0!) equals 1? One of my students sent me a link to this explanation. It’s pretty intuitive when explained this way.
Psst…
…you like mathematicians?
Yeah, you do. Here’s a timeline showing when many major math figures lived and what their main contributions were.
(Leibniz FTW.)
AND SPEAKING OF LEIBNIZ…
CALCULUS!
So this guy, 3Blue1Brown, posts all sorts of math-related videos on YouTube. His series “Essence of Calculus” is really interesting and a good way to get into calc if you’re interested in it but you don’t know quite where to start (or if it’s been a while since you’ve learned it).
Nummers
Yo, this is super interesting. Infinities!!
HIGH FRUCTOSE CORN HERO
Y’all want some cool math stuff?
Of course you do.
Nate was reading a Reddit thread titled “What is the coolest mathematical fact you know of?” The two that were the coolest to me are the following:
An explanation of just how many different ways there are to uniquely shuffle a deck of 52 cards.
Graham’s number. Ridiculous.
Seriously, read these. Numbers are amazing.
Poosers
Holy crapples, this is fantastic.
I think they should have assigned this as required reading to all first-year grad students who had to TA as part of their funding, and then made them re-read it at the beginning of every subsequent year so as not to forget important stuff. It’s also still relevant as an instructor. At least, most of it.
Highlights:
“Many instructors assume that students will read what is handed to them; I think this is incorrect.”
Oh my god, yes. This wasn’t something I ever did as a TA, but as an instructor (both at UI and U of C), I like to take time during the first lecture to actually go over the syllabus and any other important hand-outs. I particularly like to do this in the form of a PowerPoint so that I can really focus on the big things. I think it really helps emphasize what’s important to the students rather than making them wade through a two- or three-page document that includes a little information on every aspect of the class.
“People never learn course material as well as when they have to explain it to others.”
U of C has a thing up here for their 200-level stats classes called “continuous tutorial.” This is kind of like drop-in homework help where a TA staffs a computer lab for an hour, and during that hour students from STAT 213 and STAT 217 can drop in, work on homework, and ask questions of the TA if they have them. During my first continuous tutorial, I botched the hell out of a really simple probability question while helping a student. It wasn’t because I didn’t know how to do that type of problem, but because I hadn’t done that type of problem in quite some time, I blanked on the very simple solution and really confused the student. Brilliant, right? It is super important, both as a TA and as an instructor, to actually work through the homeworks assigned to the students and make sure you know how to do them. Because there’s not a lot of things more embarrassing than blanking on a question covering a subject that you supposedly know well enough to teach to the students.
“To me, motivating means addressing the history, culture, and usefulness of mathematics.”
LAKJSDFLASKFJALKF ASDFYADJFSDJ YES YES YES YES YES YES YES A THOUSAND TIMES YES
If you can put the topic into some sort of “non-computational” context, I think students are apt to be more open to it, approach it with less fear, and maybe even get excited about it. This is such an important idea to me, you have no idea.
Yay.
Well, they WERE two doctors (yeah, I know, stupid joke)
This is super coolio.
Well, at least I’m consistent.
Hey, doods.
Back in 2012, I took an Approximate Number System aptitude test that I found online.
Then, in 2014, I took it again to see if my score had changed (since I’d done so much more math between 2012 and 2014 than I had prior to 2012). My score didn’t change at all.
So it’s 2016 now…wanna guess what I did?
I took it again!
And my score is still the same!
2012
2014
2016
Yeah. I guess the amount of math/numbers I deal with doesn’t affect how good I am at this test. Pretty cool result, though, nonetheless.
Calculus is God
Today I had to go invigilate a MATH 277 final as part of my TA requirements (we each have to invigilate/proctor two final exams; sometimes we get ones we’ve actually been TAs for and sometimes we don’t. This was a case of the latter). It turns out that MATT 277 is University of Calgary’s version of MATH 275, or multivariate calculus. The test involved about 20 or so questions.
Our job as TAs, apart from making everybody sign in on the little attendance sheet, was mainly to just walk around in order to discourage cheating and to help anybody out who raised their hand.
So let me just quickly set the scene for you: a large gym full of 250+ students, a 2-hour exam, and lots and lots of calculus.
I bet you can guess what I was thinking about.
I was thinking about Leibniz!
I was wondering, as I walked down the aisles of seats, watching students write the elongated “s” for integration and the dx/dy (or variations of that) for differentiation, what Leibniz would think if he saw a roomful of people, in 2016, still using some of his original symbols. Like, how ridiculous is that? Calculus has been studied, expanded upon, and extended to a ton of different fields/uses since it was first developed, but we’re still using some of Leibniz’ original symbols.
And what would he think about calculus being taught as basically standard curriculum at universities? What would he think about the tons of different uses of calculus today?
I know I kind of talked about this in a previous post, but I actually think about this quite a bit. Especially today.
Yay calculus! Yay Leibniz!
Baa Baa Black Sheep, Have You Any Blogs?
GUYS I HAVE PURCHASED A SLIDE RULE
Story: Okay, so it’s been like four days since I’ve gone walking, which is four days too many, so I decided to walk down to Chinook Centre (about 13 miles round trip). However, once I got down there, I changed my mind and decided to check out the Value Village close to the mall instead. I was just going to look around since I have practically no money at the moment, but when I was digging through the miscellaneous baskets on one of the shelves I came upon THIS:
And since it was only $2, I had to get it. So I did.
It didn’t have a manual or anything with it and I’ve never used a slide rule before in my life, so I went to teh internetz to see if I could figure out how it works. And it’s super cool! Let me show you a few basic things.
Multiplication: Say I wanted to multiply 1.2 by 3. What I’d do is find 1.2 on the “D” scale and slide my “C” scale so that its 1 is right above 1.2.
Then I would find 3 on my “C” scale. Whatever number on the “D” scale is right below the 3 on the “C” scale is the product of 1.2 times 3: 3.6!
Division: Let’s do 6 divided by 4. You take the divisor, 4, on the “C” scale and set it above the dividend, 6, on the “D” scale like so:
The quotient is whatever number is on the “D” scale right below the 1 on the “C” scale: 1.5!
Finding a square root: Say I wanted to find the square root of 5. I would find 5 on the “A” scale. Whatever number on the “D” scale that is below that 5 is the square root: approximately 2.23!
Super cool. You can also do things like cubes/cube roots, proportions, logarithms, and sines and tangents of angles, but I’m still learning.
WOO!
Mental Math: The Struggle is Real
Alright fools, sit your butts down. Today’s blog post is an important one.
I’ll start this whole thing off with a confession. You’ve all heard me say that I can’t do math in my head, right? Well, that’s a lie. I am perfectly capable of doing math in my head.
I just can’t do it when others expect me to be able to do math in my head.
Elaboration: like a lot of people, I’ve always equated math ability with intelligence. I know that’s a narrow and inaccurate way to define intelligence, but for the longest time, math was my go-to smarts-o-meter. That’s probably because I used to be hella afraid of it and thus considered anyone who wasn’t hella afraid of it to be way smarter than I was.
But anyway.
I’ve long since redefined how I view intelligence. Namely, it’s very obvious to me now that people can easily be “intelligent” in a wide variety of things (think Gardner’s theory of multiple intelligences). A dude who’s fantastic at painting but horrible with numbers, for example, can be just as intelligent as a dude who’s amazing with numbers but not so much with paint. And people who are not “book smart” (or “school smart” or whatever) can be ridiculously intelligent in other aspects of existence that just aren’t captured by that book smartness/school smartness.
I’m sure most if not all of my readers would agree with this.
However, if you’re someone who likes math and are around people who know you like math, they’re probably going to expect you to be good at mental calculations. That’s always been my experience, at least.
And that makes me panic like you wouldn’t believe.
Especially since going into the quantitative/statistics side of things, my ability to do math in my head—“on the fly”—has gotten worse. And I think that’s because if the people I’m around know I’m into stats, I suspect they automatically assume I’m some sort of human calculator. And if I can’t prove my amazing calculating abilities, then I’m too stupid to be studying something like stats. After all, who wants a statistician who can’t add 23 + 27 in their heads?
Here’s the thing. I can add 23 + 27 in my head. It’s super easy to do. But if you just ask me to do it, I will panic and not be able to because I’m too busy freaking out about being judged on if I’m doing the calculation quick enough or what would happen if I make an error.
That sounds really stupid and maybe a bit unclear. Let’s use pictures to clear it up a bit.
Here’s what I would suspect loosely happens in the head of a person without this “math on the fly” anxiety when they’re asked to add 146 + 279:
And here’s what happens to me and, I suspect, a good deal of others:
I’m not exaggerating. When someone poses a math question—even something simple like basic addition—I automatically lose focus on the numbers and start freaking out about how dumb they think I am if I don’t answer it right away.
Ridiculous? Yes.
Reality? Yes.
And I can’t be the only one. However, most of my friends (based on just watching them answer impromptu math questions) don’t experience this, so I just wanted to show you how it is for me.
So there you go.
Mathin’
I FOUND SOMETHING ELSE I WANT.
IT IS A MATH STENCIL.
IT IS GLORIOUS.
CAPS LOCK.
Look at this. Look at those lower case Greek letter stencils. I’ve needed those so many times in the past.
Freaking zeta.
(Image from Tumblr.) And the curly braces. THE CURLY BRACES, PEOPLE.
Edit: Okay, found the motherlode.
This is glorious.
Slippery Slopes
So remember that post I did awhile back that talked about why we use “m” for slope? Well, I mentioned in it that some other countries use different letters. Here’s a handy dandy map of those different letters, based on this info.
So…consider this a follow-up blog?
Swiggety swog, what’s in the blog?
Today we learned how to use complex analysis to solve real-values integrals that would otherwise be very difficult to solve.
Example:
No complex variables in sight in that integral, right (assuming x is real-valued, haha)? Well you can CONVERT THIS TO A COMPLEX-VALUED INTEGRAL AND HAVE AN EASIER TO SOLVE PROBLEM!
That freaking blew my mind this morning in class. I’d go through the details of how to do this, but I’m a lazyass and don’t want to use Word’s equation editor to make like 30 different equations showing the steps to solve. Instead, I’ll link to Dr. Datta’s notes from class. Go to page 10 in the PDF (the page labeled “161”) for this example.
FREAKING. AWESOME.
Side note: if any of you ever end up going back to UI or know anyone who will be taking some upper-division math classes there, I highly recommend Dr. Datta. She’s very clear at explaining things, good at giving examples, gives reasonable homework, and is always willing to help.
Do ghosts enjoy Boo-lean algebra?
Remember my post on John Napier awhile back?
Well check out the Genaille-Lucas rulers, a variant of Napier’s bones. They’re used to carry out multiplication and have a really snazzy way of visually representing the “carry” part of multiplication.
Check out this example on Wikipedia.
I’m totally printing out that PDF of the rulers at the bottom of the Wiki page. Snazzy!
The Cones of Rebelling
HAHAHA, oh, Yale.
I read about the Conic Sections Rebellion quite some time ago, but I’d forgotten about it.
The rebellion refers to two incidents that occurred at Yale in 1825 and 1830. Both were students’ responses to having to draw out their own conic sections diagrams for exams rather than being able to refer to the diagrams in their textbooks.
A large number of students refused to take their final exams because of this, resulting in about 50% of the students in both the ’25 and ’30 incidents being expelled. That’s awesome.
Binary!
I’m a really visual person. That’s how I learn best, by studying diagrams or remembering the processes of things. So seeing binary addition visually is really, really helpful to me. I wish I’d found this back when I first learned this—it probably would have been less confusing, haha.
How cool is that?
Hey baby, let me expand your series
This is a really interesting read.
I am someone who has very little mathematical intuition. I mean, I think some people just have a knack for thinking about math and “math things” and for piecing bits of different types of math together. I don’t. Like, even at the most basic levels—simplifying factorial expressions, the logic behind summation rules, all that stuff. I mean, I know I’m a total idiot, but still. At least with other topics I have some degree of intuition.
And I’ve always wondered if others who actually have a more intuitive understanding of math—or at least have delved into it far enough—see advanced math (or math in general) in a different way.
Anyway. Interesting read, check it out.
Mathymaths
Do you like math?
Do you like fiction?
DO YOU LIKE THEM BOTH??
Go here! It’s a pretty comprehensive list of math-related fiction. If you so desire, you can search by keyword, genre, topic (calculus, chaos, fractals, statistics, etc.), motif, or rating in terms of literary value or math involvement.
Just a quick little blog today!
RESPECT THE TRIANGLE
So remember this Approximate Number System aptitude test I did back in 2012? I decided to try it again. I’ve been doing so much more math since then, I wondered if that would affect my performance at all.
Results:
OLD (2012)
NEW (today)
HAHA, nope. My Weber fraction (w = .13) is exactly the same. That’s hilarious.
Though to be fair, it’s 4:30 in the morning and I’m kinda hyper, so maybe that’s affecting stuff.
But I’m always like that, so maybe not.
WOOOOOOOOOOOOOOOOOOOOOOOOOOO!
“When Will I Use That?” – Calculus Edition
Alternate title: Claudia Makes Things Way More Complicated than They Need to Be Because She Sucks
We had this bonus question on our homework for Probability today:
Suppose X has a density defined by
Let FX(x) be the cumulative distribution of X. Find the area of the region bounded by the x-axis, the y-axis, the line y = 1, and the curve y = FX(x).
And I was like, “Aw, sweet! Areas of regions! CALCULUS!”
So first, I had to find the cumulative distribution function (cdf) of X. Easy. It’s just the integral of the density fX(x) from negative infinity to a constant b. In this case:
With 2 ≤ b ≤ 3. So that’s my curve y. The area I’m looking for, therefore, is this (the red part, not the purplish part):
Now anyone with half a brain would look at this and go, “oh yeah, that’s easy. I can find the area of the rectangle formed by the two axes, the line y = 1, and the line x = 3, then find the area of the region below the curve from 2 to 3, and subtract the latter from the former to get the correct area.”
Which works. Area of rectangle = 3, area of region below FX(x) = .25, area of region of
interest = 2.75.
Or they could remember the freaking formula that was explicitly taught last week. Such areas can be calculated using:
But did I see either of those? Nooooooope.
I looked at the graph and was like, “how the hell do you find that?” I tried a few things that didn’t work, then realized that it would be a lot easier to figure out if I changed the integral from being in terms of x (or b, rather) to being in terms of y.
So then I just had to integrate. This gave me the right answer: 2.75!
Moral of the story: don’t complicate things. But if you do complicate things, you might actually end up in a scenario where you’ll use something that you were taught back in calculus I but didn’t ever suspect you’d actually use. I had appreciated learning the handy-dandy technique of changing variables, but I didn’t think I’d be in a situation where I’d apply it. Shows what I know, eh?
It was a nice refresher, at least. I’ve missed calculus.
I’M HYPER AND YOU CAN’T STOP ME
How complex numbers are made:
Seriously, someone take Flash away from me.
Eigenblogger 