Author: samjshah

The Calculus of Friendship

In the past few weeks, I’ve read a few books about math.

I don’t have a lot to say about the first book. I learned a few interesting vignettes and a few interesting facts, but overall, I’m not sure I would recommend it to others. The second book was actually incredibly fascinating, and I will maybe write a little somethin’ somethin’ about that later. However, I just finished The Calculus of Friendship and wanted to give it a major shout out.

Let me first tell you how I came upon this book.

Prof. Strogatz emailed me in October of 2008.

I happened across your blog today (isn’t the Internet amazing?) and felt compelled to try contacting you for many reasons.  You seem like a great (former) student who has now turned into a great teacher. That’s wonderful.  Sorry we didn’t overlap at MIT.  And it’s very admirable that you’re now bringing your enthusiasm and training to help inspire high school kids.

The email was longer than that, and incredibly sweet, but what was more amazing than getting this email was the timing of it. I explained in my reply:

You won’t believe how coincidental your email is! In one of my calculus sections this past Friday, we finished the material we needed to go over ten minutes early and we got — somehow, don’t ask me how — on the topic of chaos. So of course I go off on this mini-lesson on the chaotic waterwheel. We watched youtube videos and talked about what it means for something to be chaotic, and how they should understand why weather is so unpredictable from this.

One student came to my office after class, and I showed him your textbook (which I hold up as one paragon of what college math textbooks should strive to be; it was far and away the best math textbook I’ve used, besides the calculus textbook that I used in high school, which will always have a special place in my heart and on my bookshelf).

I loved that.

Then this year, Prof. Strogatz emailed me asking if he can send me a copy of his new book The Calculus of Friendship. The timing of that email was strange too. In my reply email, I said:

Wow! Thank you so much for this super unexpected and thrilling surprise. Talk about things that brighten the day. Something in the zeitgeist must be in sync (ha, groan) because just yesterday I was looking around in our math department for your Chaos DVDs (I asked my department head purchase them last year). They were nowhere to be found. Turns out one of the math teachers took them home over the summer to watch, and her boyfriend then got hooked on them, and that’s why they were missing. Too good! And what a compliment to your digital teaching presence.

I looked at the first few pages of your new book on Amazon.com, and it can’t but help but be an emotional read. Because I suspect that ensconsed in the pages is a portrait of the teacher that I strive to be.

Enough prelude. I finally did get to sit down and read this book over winter break. It’s a short read, only about 150 pages. And it is broken up into sweet little vignettes. Although I could have polished this book off in a few hours, I wanted to savor it, let it linger. So I limited myself to only a dozen or so pages each day.

I was introduced to two characters: a precocious high school student (Strogatz) and a veteran teacher (Joffray). And as I slowly devoured the book, I was taken on an emotional journey about two minds which played off each other, and two lives which slowly and inexorably intertwined with each other. Strogatz has written an honest and critical autobiographical piece, while at the same writing a sublime elegy for his former high school calculus teacher.

The Calculus of Friendship is crafted by the author and narrator, Steve, by analyzing his epistolatory relationship with his teacher, Joff. The letters started after high school and focused on interesting mathematical questions. These letter exchanges continued for decades. What’s interesting is not only the contents of the letters (which I will talk about below), but the changing role that the letters played in the writers’ lives. The meaning of the correspondance between Steve and Joff changed, although the content itself was often intensely and narrowly focused on interesting mathematical problems and solutions. This book is indeed, as the publisher’s blurb says, “an exploration of change. It’s about the transformation that takes place in a student’s heart, as he and his teacher reverse roles, as they age, as they are buffeted by life itself.”

As expected from the author of one of my favorite college textbooks, the actual math is explained clearly. The math problems the two worked on through the years are interesting (chase problems, some fun integrals and series, the gamma function, dimensional analysis, etc.). The problems were different and interesting enough, or the approaches out-of-the-box enough, that I wasn’t bored and didn’t skip any of the math explanations.

Because the epistolatory nature of Steve and Joff’s relationship, and because they were each egging each other on with questions and observations, the puzzle-y aspect of problems solving came to the forefront. Some problems were attacked in a number of different ways, with a few different approaches. (My favorite one was finding \frac{\sin 1}{1}+\frac{\sin 2}{2}+\frac{\sin 3}{3}+\frac{\sin 4}{4}+....)

So yeah, I give the book two thumbs up.

As an aside, this book gave me a thought: a textbook (or unit) written entirely via letters. A fresh back and forth exchange. A little back story to draw in the reader. This approach could showing how math really unfolds, how questions get raised and answered, how some approaches work while other approaches fail, etc. Basically a textbook showing the messy nature that math evolves, because it is written as a dialogue between two people trying to figure something out. Where everything isn’t presented in a sterile, whitewashed way. Where the driving question for a unit is something like “so I was wondering if you can find a curve that goes through the point (2,1) and (4,-2). I’ve figured out how to find a line that goes through these points (which I will explain in this letter below). But what about some other curves? I mean, I can draw an infinite number of curves between these two points. [figure included.] How do I find their equations?”

Okay I should get to bed now. The twilight of my winter break is nigh and my alarm goes off in 7 hours to wake up for my first day back.

Insolvability of the Quintic

One day a few weeks ago I had a day to kill with one of my two multivariable calculus students. So I decided to talk a bit about something which intrigued me when I first learned about it.

If you have any linear polynomial (ax+b=0), then it is easy to come up with the algebraic solution for any a and b. (Obviously it is x=-b/a.)

If you have any quadratic polynomial (ax^2+bx+c=0), then it is still pretty easy to come up with the algebraic solution for any a, b, and c. (Obviously it is x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}.)

What about the cubic? quartic? quintic? higher?

I had my student take out his laptop and we used SAGE online to see what we got. We entered something like this:

I then asked if we could, for any coefficients, find all zeros (real or complex) to the polynomials.

Before pulling out the laptops, we had already answered the first two. Which we  confirmed with SAGE:

Then I asked about the cubic.

He said he knew how to find the solutions to some cubics. Like he could graph them and if they hit the x-axis three times, he could get the solutions. Not good enough. We wanted formulas for the solutions. For any coefficients. Without graphing. Like we had for the quadratic.

He (naturally) said he didn’t know.

So we turned to SAGE and found:

So there is a formula. And it’s messier than the quadratic, which is messier than the linear [1].

So we checked the quartic.

And indeed, SAGE found a solution. I’m only copying the first few lines here since it goes on forever.

So here we are, and there’s a pretty good chance things will continue — there’ll be a solution but it’ll be more and more complicated as the polynomial degree gets higher and higher.

But when we try the quintic and the sixth-degree polynomials, we get:

Um… it won’t solve it for us? Is SAGE just not powerful enough to help us? Do the solutions get *that* much harder?

Could be.

But it isn’t that SAGE was broken. It turns out there is no “formula” to get the zeros of any given quintic or higher polynomial. Sure, you can solve some quintic polynomials. Heck, they might even be factorable to (x-1)(x-2)(x-3)(x-4)(x-5) or something. But that isn’t what we’re asking. We’re asking if you have any quintic (or higher) polynomials, can you come up with an algebraic formula for the exact roots.

No.

And the reason I wanted to show my student that is because it was learning that fact in high school, the insolvability of the quintic, that got me even more interested in math. It raised the huge question: what broke down after 4? Why is 5 the magic number? Is it truly impossible for any degree polynomial greater than 5? How can anyone show that a degree 1021 polynomial won’t have a “formula” solution for its zeros? No one could explain it to me, but my math teacher swore it was true.

It seemed so crazy to me! Heck, it still does. Interesting tidbits like that lit a fire under my feet [2] to take college level algebra (Abstract Algebra) to help me understand it. It was one of the most glorious days of college when in our Abstract Algebra class we finally got to tackle and solve this problem.

Do I remember how we did it?

Sigh. No. I have Flowers For Algernon syndrome.

But I at least know that the solution is out there, and given enough time and patience, I can understand it once again.

I doubt my student got out of it the same level of “WHAT THE HECK?!” as I did, nor do I think it lit the curiosity fire under his feet. But heck if I didn’t show him that our intuition breaks down without cause sometimes, and there are answers to be found. Maybe not in our class, but in some other class if he ever wants to solve the mystery.

[1] Well, when I was in high school, I had my dad’s worn, cover-falling-off CRC Book of Mathematical Tables and Formulae. In it was the solution of how to solve any cubic, and how there is indeed a formula (like the quadratic) for the cubic. (You can see it nicely typed here.) I suspect it also had a paragraph or two about the Cardano/Tartaglia dispute.

[2] Godel’s incompleteness theorem was another one.

a stubborn equilateral triangle

My sister is a teacher too. And she’s smart. And sometimes she poses questions which stump me. She posed a good physics problem on Facebook a while ago.

In case you can’t tell, the three fixed, point masses have masses 1, 4, and 9. She wants to know where you can place a mass so that it won’t move! So that the net gravitational force on it is nil.

Just in case you forgot Newton’s Law of Gravitation between two bodies: F=G\frac{m_1 m_2}{r^2}

Before starting, I thought this problem would be so easy. If the three masses were equal, we’d have a simple geometry problem. Since they aren’t, it turns out we have something more tricky. I thought the solution would come easy. For me, it didn’t. But I think I got an approximate solution.

Just so we can compare solutions, let’s put our masses on the cartesian plane as below:

As you can tell, I placed the three points on the unit circle.

I don’t want to give much away, so I’m just going to leave you to it. Throw your thoughts in the comments below. If you’re dying for the answer, I’ve hidden what I got on this site somewhere in some not-hard-to-find spot.

If you get stuck, look after the jump for some encouragement.

Just a note: I don’t know if I got the right answer… I think I did, when checking it, but I’m not totally sure. I got tired of working it. That’s why I wanted to throw this up there to see if anyone could corroborate, and also to see your approaches!

(more…)

the evolution of a student teacher

I’m doing a huge shoutout to justagirl24, who has finished her student internship.

I don’t know her, and I’ve tried posting a few (long) comments on her blog — only to have them disappear into the internet ether. But I’ve been following her journey as a student teacher for the past four months. I think her blog, from August 2009 to December 2009, should be suggested reading for beginning student teachers everywhere. I remember having a lot of the same thoughts that she did when I did my teaching practicum at a public high school in Cambridge, MA.

I’ve linked to her blog a couple times on Twitter, but I don’t know if anyone clicked. So to induce you to read, let me whet your appetite by spoiling the narrative a bit. I’m going to show you the beginning and the end of this story.

September 7, 2009:

I’ve come to realize that this is all a chore to me. I don’t want to drive in the morning to school. I hate it. I can’t quit. I wish there was a way to make it better. Sure I can work with other people to come up with solutions, but you know what it’s all up to me to enforce those solutions. And it usually ends in failure for me and my students… That’s what sucks about this, I hate being alone. No one can be there to defend me. I’m the one who needs to stand up and do it. Can I go in and say “I’m still learning, guys and girls.. I’m new at this.. give me a chance to experiment with you”… maybe I can say that.. but I dunno.. right now I have so many things floating in my mind. All I can think about is school and this whole experience. Nothing else.. I need something in my life to make me happy and right now I’ve got nothing. Nothing to keep me distracted.. Nothing to keep my mind from thinking about this “chore”.

December 17, 2009

So it’s the last week.. it’s bittersweet.. I’m sad to go but happy to be done. I’m definitely going to miss these kids. I wanted pictures of all of them, so I have a group shot of every class. For my math10 class, I was handing some last minute stuff back.. and was kind of giving a mini speech.. about how I really enjoyed this internship.. I was starting to choke up.. so I stopped talking.. lol.. It’s bizarre to think that 3 months ago I would make it to the finish line.. But I have.. I survived and endured.. Surely not without any struggles.. This is surely one of the best experiences in my life thus far. I just can’t believe I was considering quitting in the beginning. I think it’s a phase that everyone goes through when they’re thrown into a different and new environment.

This was four months. That blows me away. Four long, hard, rewarding, frustrating, emotional months. So congrats, justagirl24. If you happen to see this… thanks for sharing your thoughts with me. I’ve been there, silently rooting you on. Not in a creepy “he’s stalking me” way. In a “you can do it!” way. And you did.

Now don’t forget to return your classroom key.

Natural Philosophers as Puppets

Two of my calculus students today approached me at the end of class (this was the last time I was going to see them before winter break) and handed me

as a present. Indeed, if you identified this as a finger puppet of Newton, you would be correct! They said they looked for a Leibniz too, but one didn’t exist. Of course if one did exist, things might have gotten a little dangerous. Goodness knows I might have had to create a few history of calculus videos (a la Potter Puppet Pals).

All I have to say is: hooray for my “histoy of calculus” webquest!

Teachers Say (and do) the Darndest Things

We all have catch phrases. Things we say, purposefully or accidentally, enough times that the kids have taken note. You know, these are things kids probably mimic when doing impressions of us. Which I know they do. I mean, don’t they?

I bet someone doing an impression of me teaching any of my classes would say “ooooh, CRUST!” as an expletive a lot. That’s my curseword in class, whenever I lose track of time or make a mistake. I also often deny mistakes, jokingly. A brave student will note “you forgot the negative sign there.” I’ll carefully add it in and say “Um, hell-O, no, I DIDN’T. I have no idea what you’re talking about.”

We also have catch phrases related to math. In calculus, you all know my motto, which gets said at least once a week if not two or three or four times a week:

turn what you don’t know into what you do know

And in Algebra 2, I have one class rule for safety. Which I pull out of my pocket a lot:

don’t divide by zero! if you do, the world BURSTS INTO FLAMES!

[And then I take the red smartboard marker and draw flames around the thing that would have a zero in the denominator.]

Kate Nowak’s recent post talked about the changing of traditional teaching phrases in her class[1]:

Today in Algebra 2 we reviewed negative exponents and the children acted like they had never seen it before. I told them about the phrase “move it, lose it” for dealing with a negative exponent, as in, move the term to the other side of the fraction, and lose the negative sign. A student who moved here from another state (where, you know, they get to spend enough time on things to actually learn them) told us about the phrase she learned “cross the line, change the sign.” Which the kids liked better. “You know, because it actually rhymes, Miss Nowak. Unlike yours.” Um, last I checked “it” rhymes with “it.” I’m not an English teacher! You can tell because I’m not wearing cool shoes and I don’t give hugs.

Okay, a big giant *grin* for the best two lines I’ve ever read on a blog (the last tines, obvi). But it got me thinking more about these techniques we use to teach kids to remember things. Yes, I think kids should know the reason why particular algebraic manipulations / formulas work. But once they show me that they “get” it I have no problem with them using phrases and shortcuts to help them remember things.

I mean, how many of you always rederive the quotient rule when taking a derivative of a rational function in calculus? Or do you sing a little sea shanty like:

low de-high less high de-low
and down below
denominator squared goes

Or for the quadratic formula? In your mind you’re definitely saying the formula in a very specific way each time. Think about it.

THINK, I said.

Yup, I thought so. So inspired by Kate, I thought it would be a neat exercise to chronicle three of the ways I get kids to remember things or do hard things.

(1)

The “pop it out” rule for logarithms. When we are learning how to expand \log(x^3) to get 3\log(x), I always say “POP IT OUT!” and do a raise the roof hand gesture. I don’t know why I do that. I don’t know how it started. Maybe the raising of the roof is popping out of the exponent. Then when students are working and get stuck, I sometimes help ’em out by quickly doing a mini raise the roof. Then they always exclaim “Oh! Pop!”

(2)

[Note: I am pretty sure I stole this from someone from a couple of years ago.] When teaching how to visually find the domain and range of a function, I tell students: “Guess what? You don’t know this but besides your calculators you have another a highly sensitive and powerful mathematical instrument. It’s kind of awesome. It’s called a domain meter.” I throw a relation on the board:

I tell them to hold out their index finger way to the left of the graph. Then slowly move it rightward across the graph. As they do it, I do it with them, and as soon as my finger hits x=-2 I start annoyingly sounding “BEEEEEEEEEEEP” while continuing to move my finger. Finally when my finger reaches x=2, I stop beeping and I silently continue to move my finger. I tell them: “my domain meter only beeps when it hits the graph. What’s the domain?” (They get it.)

Then I say “Believe it or not, you have another amazing mathematical instrument. You guessed it, a range meter.” I then hold up my vertical index finger (“domain meter!” I exclaim) and turn it horizontal (“range meter!” I exclaim). Then I take my horizontal finger and start at the bottom of the graph and move it upwards. I only start beeping at y=0 and continue until y=2. They all can state the range at this point.

I’ll end up finally throwing something more complicated on the board:

and they’ll get it, first try.

(3)

In calculus, I want my kids to be able to see derivatives quickly. My first year teaching it, I focused a lot on u-substitution to take the derivative of \cos^4(x). Why? Because my kids just couldn’t get the hang of “seeing” the answer. So I came up with the “box method” of teaching the chain rule, which works great. (And yes, of course, I always teach u-substitution first and we talk about why this “box method” works.)

I have my kids first rewrite the function so that they can see “inner” and “outer” functions. So for example, they have to rewrite \cos^4(x) as (\cos(x))^4. That way they can see the “inner function” easily. Similarly, they need to rewrite \sqrt{\cos(x)} for the same reason. I then ask them to put a box around the inner and outer functions respectively. If there are more than two (functions within functions), they should make all the boxes.

So let me show you with a simple example from class today:

I have students write the functions in terms of “outer,” “inner,” “more inner,” etc. until they get to the gooey center of a composition of functions. Then I tell them to look at the outermost function, ignoring everything from the boxes inside (in our example above, they’d say “sine of blah”). I asked them “what’s the derivative of sine of blah?” and they all say “COSINE!” So I write

\cos(stuff)

and ask “what do I put in here?” They say “don’t touch the innards!” So I fill it in:

\cos(\cos(x^{1/2})).

Then I put a check next to the outermost function and say “we’ve dealt with you, so we’re done with you.” I then go to the middle function and say: “What’s the derivative of cosine of blah?” and they all say “NEGATIVE SINE!” So I go to the board and add

\cos(\cos(x^{1/2}))*-\sin(stuff)

and ask “what do I put in here?” They say “don’t touch the innards!” So I fill it in:

\cos(\cos(x^{1/2}))*-\sin(x^{1/2})

Then I put a check next to the cosine function and say “we’ve dealt with you, so we’re done with you.” I then go to the middle function and say: “What’s the derivative of x^{1/2}?” and they all say “\frac{1}{2}x^{-1/2}. So I stick that on at the end.

\cos(\cos(x^{1/2}))*-\sin(x^{1/2})*\frac{1}{2}x^{-1/2}.

And fin, we’re done. It goes pretty fast once they get the hang of it. And they actually secretly love having equations that are scrawled across a whole page.

[1] Kate, forgive me for cribbing so much wholesale. But I needed to have the last sentence in there!

2009 Edublog Nominations

My nominations for the 2009 Edublog awards

Best new blog
David Cox’s Questions? [http://coxmathblog.wordpress.com/]
(Anyone who has read this blog won’t question why it is top of my list.)

Best resource sharing blog
Kate Nowak’s f(t) [http://function-of-time.blogspot.com/]
(Her resources are more than nice lessons, but also interesting activities that can be used to group kids and differentiate instruction.)

Most influential blog post
Dan Meyer’s What I Would Do With This: Pocket Change [http://blog.mrmeyer.com/?p=4905]
(This was one of the best blog posts I read all year. It was inspirational. It didn’t get a lot of comments, so maybe not a lot of press, but it — and Dan’s whole What Can You Do With This series — is an exciting framework for how math can be taught.)

Best teacher blog
Dan Meyer’s dy/dan [http://blog.mrmeyer.com/]
(If you’re reading this, chances are slim to nil that you’ve not been to his blog. If you are in that slim to nil category, shame on you.)

Best elearning / corporate education blog
Maria Andersen’s Teaching College Math [http://teachingcollegemath.com/]
(I look up to her. I always find ideas and thoughts, and get exposed to some resources out there in the edutechnoblatespheroid, waiting for me at her blog.)

Best educational use of video / visual
Rhett Allain’s Dot Physics [just moved to http://scienceblogs.com/dotphysics/; originally at http://blog.dotphys.net/ ]
(Sweet design aesthetic and, oh yeah, he does some pretty sweet physics too.)