Continuous Everywhere but Differentiable Nowhere

I saw a tweet that sumidiot posted on the parametric equations and and spent a good 30 minutes thinking about the value of which makes the graph intersect itself exactly once. I was going to post about my solution, but I was beaten to the punch. And thank goodness, because there are graphs and everything on sumidiot’s solution, which you should read here.

I still haven’t fully evaluated twitter, which I will post about later once I come to some solid conclusions. But I’m leaning towards liking it. I’ve got lots of good links, anyway. If you want to be my twitter friend, I’m here!

I don’t know exactly why, but this comic got me all giggling hilariously. For those who had today as their last day (me too!), congratulations! 

I handed back a graded problem set back to my multivariable calculus class last week. One student emailed me to meet to go over it with him. We did that today. 

But the best part? One of my comments was that there were not enough words and motivation for each mathematical step. His response — totally unprompted by me — was something to the effect of “Now that I had to do all that writing for the Kepler paper, I now think I know what sorts of things I’m needing to explain, that I wasn’t explaining before.”

If there was any smidgen of doubt left in my mind that our Kepler “unit” wasn’t useful, it has been eradicated. The skills I wanted them to pick up? They can now apply them to the rest of the course. HUZZAH!

It’s college admissions time again — and for those who applied early action / early decision, the results are coming in this weekend and next week. I try my best to ignore all of this in the classroom. I know that some students are going through devastating times, while others are so elated they can’t contain themselves. But I don’t know which is which, and honestly, I don’t want to know. 

Why? Because I can’t do anything about it. 

College acceptances and rejections are, unfortunately, a trial that all seniors have to endure [1]. Having books, counselors, and teachers around telling them “it’ll all be okay” is fine and dandy, but it doesn’t do them any good. Yes, we know it’s true. We know that these students will get in somewhere. We know that years down the road, they won’t be able to imagine going to a different college than the one they went to, because they’ll have made all these friends and had these amazing experiences. We might even say all that to them, in addition to the “it’ll all be okay.” But our words won’t make a difference to them, and we know that too. We say it because we can’t say anything else, because we’re helpless to help in this trying time. Really we say it for ourselves, our own contrived pretense that we can help in a situation where we have no control.

Their hurt is real, immediate, and prevents a broad outlook.

My own story shapes how I feel about this.

As a senior, I applied early to Harvard. I told everyone that it was a long shot and I didn’t think I would get in, and that I didn’t care if I did or didn’t. I applied early just to get it over with. And part of me wanted to believe all that, but a deeper part of me thought that I couldn’t not get in. This wasn’t because I was vain or conceited or thought I was something special. (Trust me, I was incredibly self-effacing in high school.) But the truth of the matter was I honestly couldn’t think of anything I could have done to make my application stronger [2]. Although I said I wouldn’t get in, I thought I would.

You know where this is going… I didn’t get in. I got the small envelope and was devastated. 

And the honest truth was: I wasn’t crushed because I didn’t get into Harvard. It wasn’t some place I had always longed to go, or had a sweatshirt from, or anything like that. Harvard was just one of the five schools I applied to. I was devastated because:

(1) I was judged and was deemed not be good enough
(2) I didn’t know why I wasn’t accepted
(3) I had worked so hard in high school and I felt it was all for nothing
(4) What would my friends think of me, because I believed they all expected me to get in

Those thoughts rattled around in my brain for months. Seriously. I tried to shake them off, but couldn’t. All the nice things that people said to me slid right off, because “it’s all going to be okay” didn’t address any of my concerns. It wasn’t until months later when I saw the inner workings of the MIT admissions office — working there as an undergraduate — that I saw the insane number and quality of applications that were coming in. By that time, of course, I was over the college admissions fiasco. I was enjoying my freshman year. But I finally understood the arbitrariness that sneaks its way into college admissions, and comprehended the statement that I had heard from colleges way back when I was a senior in high school: if we threw out all the applications of the students we admitted, and picked another freshman class from the remaining students, we would get a class as strong and successful as the first. 

All of this is to say: this time can suck for seniors emotionally. Rejection or deferral from college is complex, because it deals with how we perceive ourselves, how we think others perceive us, and toys with our own notions of self-worth. Our grades aren’t applying. We are. We make it into a story of morality: have we been deemed good enough to enter the hallowed gates? 

I wish I could give all my senior who were rejected / deferred / waitlisted early the gift of hindsight and perspective. But I can’t. So I’ll just say “everything will be alright, I promise” and continue on with my job. 

[1] Well, actually, I wrote a letter of recommendation for a junior last year to get into a summer program at one particular school, and if she got in this summer program and did well, she was guaranteed acceptance to that school. So this one senior got a free pass.

[2] If you really want to know: near perfect SATs and SAT IIs, straight As in all my classes, 5s on all my APs (which spanned a variety of disciplines), going to the local state university to take multivariable calculus my senior year, doing a number of extra curricular activities which focused on community service, two summers of mathcamp, and participating in a number of different math competitions. (I also had a pretty good social life, believe it or not. I loved high school. And honestly, for the most part, I didn’t join things for college applications… I joined specific clubs because I loved what we did… or because all my friends were involved with them…)

I can check parent conferences off my to-do list. I had around 25 this year, and most of them were enjoyable 10-15 minute conversations. About half were parents of students who are doing fabulously in my class. About half were parents of students who need to recognize that their previous method of doing the bare minimum and not seeking help wasn’t going to work anymore. Some parents have a reasonable and healthy outlook on grades. All the parents — this year — recognized that we were both on the same side. One parent even brought me a tin full of candy!

Overall: success.

I closed up a unit in calculus on position/velocity graphs. Most of my students had horrific memories of physics their freshman year. That teacher, needless to say, is gone. Last year, a number of my calculus students just shut down when we encountered this topic.

This year, I focused a lot on the concepts. One day I showed dy/dan’s graphing stories and that night, I had them each come up with their own problems. For these problems, they needed to draw the velocity versus time graph and the position versus time graph.

Initially I was going to use my favorite on the assessment, however, there were so many hilarious, exemplary problems, I had to type them up and spend the next day using them.

Some of my favorites:

1. Dare-devil Mr. Shah one day decides to go Bungee Jumping. At the top of a mountain, scared, he hesitates 2 seconds, then jumps. He [falls] quickly, eventually reaching terminal velocity at 100 m/s . At the bottom the rope reaches its limit pulling him back up, [coming] to a stop. Mr. Shah smiles.

2. Mr. Shah is riding the elevator to the 4th floor. He waits for the elevator for a bit and then gets on. The elevator goes to the basement to make sure no one is waiting down there [and, of course, no one is there, as always]. It quickly goes back up to the first floor, where 15 seniors try to crowd on. When everyone is in the elevator it heads up to the 4th floor stopping at the 3rd floor to let people off. Finally Mr. Shah reaches the 4th floor and comes over to our calculus class.

3. The Jonas Brothers are walking down the streets of New York City at a strolling rate of 2 mph for 10 minutes as they composed a new song. Suddenly, [student 1] and [student 2] began running at them screaming, at 7 mph. Struggling to find a hiding spot, the brothers run down the block at 8 mph for 5 whole minutes, when they lost the crazy groupies-in-training. Stopping for a break, the boys catch their breath for 5 minutes on a stoop. Walking away when the coast was clear at the same strolling rate as they began with, Nick remarked, “Sorry guys. I’ll try to be less attractive.”

4. A helicopter is taking off. It rises constantly at 200 ft/minute. After rising for five minutes. It stops for one minute to survey the surrounding area. After rising again for 2 minutes, the helicopter is abruptly blown up by a terrorist missile.

5. A man runs from a tiger going at a constant velocity of 3 mph for 1 hour. The tiger gets tired so the man catches his breath for 20 minutes. A rhino appears and begins to chase again and the man picks up speed to 5 mph.

6. You are in an elevator on the top floor (6th floor). Each floor, it picks up more people and it goes slightly faster each time. When it stops on the 2nd floor, so many get in that it breaks and crashes to the basement. People die.

This was a fun class. And almost all my class got pefect scores on the conceptual part of the latest assessment, on this material. They got it1 They really got it!

When I introduce an idea and have students practice a few problems in class to see if they are getting it, I walk around and individually help students. Neighboring students other also help each other out. 

What do you do, however, when a few students who “get” things right away finish before the others? The rest of the class seems to get it, but it just is taking them a while to work out the problems. What do you do with these kids? Do you just have them sit quietly?

Sometimes I’ve had them: (1) make up their own problems for me to possibly use on a future assessment, or (2) walk around and ask if anyone else needs help. Most of the time I let them sit. I feel like I should have a good way to deal with this. But I can’t always ask them to make up their own problems or walk around. It’s a temporary solution.

Ideas? Strategies? Do we all struggle with this?

Currently, I’m thinking that each time I put problems up, I should put a couple up, and then put one “challenge problem” too — and then have the expectation that students finish the standard problems, and say that if you don’t have time to get to the challenge problem, that’s totally okay.

I found — when searching for something else — this page on the calculus of love. It’s actually really cute, and totally accurate mathematically. Both big plusses in my book.

The article analyzing these graphs is here. Definitely something to check out if you’re teaching precalculus or calculus.

Okay, it is 6:54 and I have to run to school for a 7:30 meeting! Yikes!

Lates.

Right now, I’m about to start teaching Partial Derivatives in my multivariable calculus class. I’m going to teach them in a traditional way, to build a sense of what they are. However, I really want to create a project that has students take actual data and find something useful with it.

To take you down my train of thought, look at this applet:

So of course we will soon relate partial derivatives to the gradient which will get us to exploring topological maps. Pretty standard stuff.

However, wouldn’t it be neat if each student could pick a place on the globe and create a topological map for it? (And then, using some simple computer tools or a protractor and ruler, come up with estimations about the steepness or flatness of the terrain at various points?) Well, I can easily make this happen! Because now GoogleMaps has a Terrain feature, and if you zoom in enough, you get to see the level curves with the height of the land marked. And you can use sites like this to calculate the distance between two points!

Here’s some random place in Alaska.

I’m thinking that having my students actually work to calculate some of these values by hand might really hammer home what these strange calculus concepts are. It’s easy to take the derivative with respect of of . It’s less easy to understand what that means, or what the gradient means, or how they are calculated. 

I don’t know if I’ll have time to whip this up, but I think it could be a really great activity.

I’m about to teach Related Rates in my Calculus class. And the book and the Internets aren’t helping me. Supposedly, related rates are so important because there are so many “real world” applications of it.

Like a  snowball melting, a ladder falling, a balloon being blown up, a stone creating a circular ripple in a lake, or two people/boats/planes/animals moving away from each other at a right angle.

Weird exemplars — I wonder where they got started and why they still hold so much water in every textbook? Because seriously?!, a ladder sliding down a wall — when is anyone truly going to need to know the rate of change of the angle over time? Same with the melting snowball.

I’m not someone who needs a real world application to justify everything I teach. In fact, I rarely do. But when we’re teaching something and hold it up as “calculus in the real world,” I refuse to believe that this is the best we can come up with.

I am searching high and low for one true real world problem. No contrivances, but something where I can point to and say: “this calculation needed to get done and because it was, we now have ____.”

I am thinking that maybe figuring out how a radar gun calculates the speed of a car, especially if it is being used from a moving car, might have something good there.

So far, though, the closest I can get is here:

Rockets: A camera is mounted at a point so many feet from a rocket launching pad. The rocket rises vertically and the elevation of the camera needs to change at just the right rate to keep it in sight. In addition, the camera-to-rocket distance is changing constantly, which means the focusing mechanism will also have to change at just the right rate to keep the picture sharp. Related rates applications can be used to answer the focusing problem as well as the elevation problem.

A number of AP Calculus classes have their students make videos with related rates problems. But those problems are just like the others: contrived. It’s like using integration to do simple addition. This video is the exception; I love it.

Anyway, holla below in the comments if you got anything.