Month: October 2009

Desk Banging in Calculus

I have two calculus classes, and one of them is deafeningly quiet. I know or have taught many of these kids before, and they are just a shy and reticent bunch. Enthusiasm and a lighthearted atmosphere has always worked in the past, as has groupwork, but not with these kids.

I normally can break through this, but so far it’s still a little weird. Today, though, I had a glimmer of hope. Just a glimmer, but that was enough encouragement!

Today, we were talking about average and instantaneous rate of change, and how these ideas relate to the slope of the secant and tangent lines. And of course I wanted to relate this to position/time graphs.

I wanted my kids to see that having a position of an object was so… powerful. I always assumed that my kids in years past understood the awesomeness that came out of understanding that knowing the position at all times tells you the velocity at all times. That knowing information about the position of an object was giving us information about the motion of the object.

Maybe to us this is obvious. It is an idea only truly half formed for them. They have an intuition about it, sure, but that’s about it. I really wanted to drive home the idea that we could see so much from a position versus time graph.

So I showed them this graph [1]

and I ask them where the object is at various times.

Then I say, let’s actually act this out.

I start banging my hand on the desk nearest to me, BANG BANG BANG BANG… I hold up my other hand in a fist 6 inches above the desk, and say “it starts out at 6 inches.”

As I did this, all my students joined in by banging on their desks. (I didn’t ask them to.) They all put their hands up. They all joined in. It was a true glimmer of life! of community!

For three BANGS we held our fist still, and then after three bangs, we moved it down to the desk and held it there for another bang, when all hands shot in the air and then stayed there for an extra bang.

We acted it out. And then I said: let me tell you what makes calculus so powerful. It allows us to look at this graph which gives us just the position an object — that’s all it gives us — and it lets us understand the MOTION of the object. What we have here is information about where the object is, but we now can find out how the object is moving. We could actually act out the MOTION of the object from that. It seems so obvious, but the connection is so deep.

Then we constructed the velocity-time graph. And I pointed to the [4,5] interval and said “oh, negative. what does that mean?” (Our fist was going down.) And I pointed to the [5,6] interval and said “oh, positive. what does that mean?” (Our fist was going up.) And I pointed to the [6,7] interval and said “oh, zero, what does that mean?” (Our fist wasn’t moving. It was high in the air, but standing still.)

I’m building the concepts here. We do the math later.

[1] Pre-emptive footnote: Yes, we had a discussion about how this graph could not actually be representing something in the world, because of the sharp edges.

Writing in Algebra 2

One of my pushes this year is to get my Algebra II students to write math better. Last year I put “explain this” problems on a few exams and wasn’t so impressed with their responses. This year I am teaching my kids to write responses.

On their first assessment, I put a question similar to one we talked about in class:

Explain to someone who doesn’t know a lot about math why you can never find an $x$ which would make $|25x+5.1|-5=-6$ .

The responses were disappointing across the board. There were bits and pieces of gems, but nothing complete. Not a single student was able to construct a well-written response. Things I received included:

The other side of the equation is negative, leaving no possible solution to the problem.
You can never find x because the answer is negative and an absolute value problem with a negative after the equal sign is not possible.

So what I did was type up the following document and passed it out a few days after the assessment:

View this document on Scribd

We talked about the vagueness of the responses, the use of pronouns like “it” and making references to “the other side of the equation,” and most crucial, the lack of reference in almost every solution to the original equation. How can you answer a question about an equation without even talking about the equation?

My favorite moment of the discussion this generated was when one student raised her hand and critiqued her own solution, and then said: “I wrote this and don’t even know what I meant.”

On the next assessment, without telling them I was going to do this, I threw the exact same question down. It was on. I saw my kids reread their responses after they wrote them, and really pay attention to their writing. Let me tell you: it all paid off. On this second round, most students got full marks. (On the first assessment, almost no one got full marks, or close to it, for that matter.)

Here are some random smatterings of their thoughtful answers:

You could never find an $x$ to make the absolute value equation above true because you would have to subtract -5 from -6, which still gives you a negative number. $|-25x+5.1|=-1$ . An absolute value equation cannot equal a negative number because absolute value is the distance from zero and is always positive [my correction: or zero].
In this absolute value equation there is no solution because any number in the absolute value has to be 0 or a positive number. And if you subtract 5 from 0 or a positive number, there is no possible way that can equal -6. So there is no solution to this equation.
An absolute value of anything can never be equal to a negative number, since it expresses a distance. When this equation is simplified, it becomes $|-25x+5.1|=-1$ . If the ‘-1’ were replaced with a positive number, you could find the answer [for] $x$ . But since it is a negative, you already know that is impossible.

I am continuing to ask them to express themselves through writing. On that same assessment where I asked them to repeat the absolute value problem, I also asked the following two questions, to which I got some really nice writups.

The following two questions build upon each other. The solution to part (a) will very much help you explain part (b).

(a) Explain why $a^2a^4=a^6$ without using your exponent rules. Explain it to someone so they can understand it simply!

(b) Explain why $a^ma^n=a^{m+n}$ is true. You can assume $m$ and $n$ are positive integers. Explain it to someone so they can understand it simply!

I still have to do more work with this, but I just wanted to say: it is worth it to talk with your kids about writing. One 15/20 minute conversation has already yielded great dividends for me.

NCTM, day 2

The second day was a disappointment. Of the four talks I went to, three of them were bad. If they were a smell, I would be passed out. So bad. I actually felt angered by two of them, because the description was so fascinating that I felt betrayed. Talks in sheeps clothing.

I feel bad listing the three terrible talks, so instead I thought I’d at least point to the one good talk:

#201: Linear Functions: Much More than y=mx+b

The major thesis of this talk was that we might want to invert our traditional way of teaching linear functions. We tend to teach:

1. $y=3x+4$
2. make table of x-y values
3. plot
4. connect the points. oh my gosh! a line!

But students find the equation $y=3x+4$ to be the abstract part. The numbers and working with them is the easy part. So the speaker provided some ways to say let’s END with the equation and have it make sense to the kids, rather than START with the equation.

What was nice is that he started with some easy problems — that I couldn’t use in my classes — but then went to more advanced and more interesting problems — including one that would be great for an independent research project for a kid, and one that just blew my mind relating Pick’s Theorem to… systems of equations. Seriously.

But what was great is that he focused on student learning, and eschewed ed jargon and talked about why he made his choices for each lesson, and what his students got out of it. It was sweetness.

UPDATE: Commenter “m” below has prompted me to flesh things out a bit more. The easy part is with Pick’s Theorem… the speakers said he stole his connection to systems of equations from somewhere else… I suspect here! (He also showed a second way to derive Pick’s Theorem, which I am too lazy to do here. I remember first learning about this theorem in high school and spending days trying to prove it. I did eventually prove it and proudly showed my writeup to my math teacher.)

As for motivating simple linear functions, he basically had students engage in pattern recognition and play around with numbers.

White blocks in the first picture? The second picture? The third picture? What about the 5th picture? The 27th picture? He also talked about relating the blocks to tables to graphs really explicitly, as well as making explicit the connection between the “slope” (I put that in quotations because the speaker hates the term slope – he thinks it obfuscates) and the pattern, and the “y-intercept” and the pattern. His thesis was actualized: being explicit and very visual, and having students start with numbers and then come up with the equation out of these numbers provided a more natural and more deep way of motivating linear functions.

NCTM, day 1

So right now I am sitting in Hynes Convention Center – room 109. In case you aren’t in the know (for shame!), I am at the National Council for Teachers of Mathematics (NCTM) conference in Boston. I just finished Day 1. I spoke to a total of three strangers, one of them who I recognized (and who recognized me) from the Phillip Exeter conference from this past summer. I don’t do well with meeting new people, which is such a shame in such a math-teacher-rich environment. But hey, three isn’t bad.

The sessions I went to today were:

#14: Identifying and Remediating Misconceptions [about CAS/TI-Nspire and developing numerical intuition]
#46: Show me the Sign! [about using sign analysis effectively in 9th, 11th, and 12th grades]
#79: Helping Students Read Math [about how to teach students to read their textbooks]
#142: Discovering Trigonometry [on how Exeter uses problem solving to teach their courses, using trigonometry as the vehicle to talk about that]

This was my first NCTM conference. Let me put this one piece of information about me: I don’t like my time wasted, so I tend to be critical of speakers [1]. I expected to really appreciate one or two of the sessions, and politely sit through the others. I thought I’d be inspired maybe once or twice.

You can see where this is going. I really, really enjoyed all four sessions. The speakers were prepared, and focused – for the most part – on concrete things in the classroom. It wasn’t about giving us the most difficult but interesting mathematical problems to work on. In other words, it wasn’t about mathematics. It was about teaching mathematics. We talked about topics and skills we work with everyday, and the speakers spoke about their approaches. None of them were zealots, saying “you should do it my way because it is the best.” It was “this is what I do, this is why I do it, and maybe you can use bits and pieces of what you hear here in your own classrooms.” I appreciated that.

I don’t know if I will have time to post about each individual session, but I will hopefully post some interesting bits later. (I said that about things I learned at the Exeter conference this past summer, and never did, though. So I can’t promise.) But maybe if (when) I actually apply some of what I’m getting to the classroom, I’ll feel more inspired to write.

(FYI, if you feel like you just absolutely need to know more about one of the sessions I went to, throw that down in the comments. You know I can’t deny you.)

[1] Yes, yes, I know our kids feel the same way, and we should always keep this in mind when we enter a classroom.

Mr. Sandman

I don’t know what it is but thing have been so busy in the last two weeks that I don’t remember a time I left school before 7pm (most days 8)… and I continue doing work at home until 10 or 11. I honestly don’t know why. I’m not getting any more work done that I have previously. I’m just grading and lesson planning. Maybe it’s that all my free time at school is now being taken up by meetings with students. (Note to self: do you really want to encourage students to seek out help? [1])

It’s a little bit crazy. Actually totally wackadoodle crazy. I mean, anytime that I don’t read my “back twitter feed” (the tweets that happen when I’m in school or working), it’s just wackadoodle crazy. And I’ve had a couple of those days where I thought “Can it be winter vacay already, please, Mr. Calendar, because I seriously need to lie down and sleep for hours.”

Unfortunately, I won’t have time to do that for a while. Because today right after school I am getting on a bus to go to Boston for the NCTM conference. I don’t return until Sunday evening. And what do I have waiting for me on Monday morning? Yes, the substitute is administering exams in 3 of my 4 classes. So I totally win the prize by having tons of stuff on my plate when I return.

AWESOME.

Followed quickly by the end of the quarter, and comment writing time.

Please, sir, I’d just like some sleep.

[1] In case it wasn’t obvious, sarcasm. Obviously.

Flowers for Algernon?

A couple o’ days ago, I posted a question about how to come up with a set of parametric equations equivalent to an implicit equation. It seemed to me like the general solution to this broad question would be like differential equations. There would be certain tools you could pull from the toolbox, once you saw what “kind” of equation you were dealing with. There isn’t a one-size-fits-all algorithm for solving differential equations (at least, not that I learned).

I got to thinking… didn’t I learn how to convert between parametric and implicit equations some time years ago? And, in fact, the answer was: yes. I took a class on algebraic geometry. The book we used (one of my favorite math textbooks when I was an undergrad) was:

The way this course was designed (it was officially a “seminar”) was that each day, two students would “teach” a section from the book to the rest of the class. We somehow made it through the whole book. It was a great experience, having to learn a section well enough to teach it the my classmates. The class was — however — a bit of a failure. The desks were in a row, people rarely asked questions, and no one engaged with each other. (Much like most of my college math classes, actually.) For something so student-based, it was strange that I didn’t make a single friend in that class. Plus, there was no “teaching” us how to teach well. Some some of us were great teachers, but most of us sucked. I can’t say what I was, really. I don’t remember. Regardless, I remember thinking: this textbook was incredible because I pretty much had to teach myself the subject. (I have to give major kudos to the instructor because he forced me to learn an entire course by reading a textbook.)

So upon reminiscing about this class and this book, I pulled it down from my bookshelf.

I am so dumb.

I will revise: I am so dumb now.

I look at the pages, and read theorems like Theorem 9 on page 241

“Two affine varieties $V \subset k^m$ and $W \subset k^n$ are isomorphic if and only if there is an isomorphism $k[V] \cong k[W]$ of coordinate rings which is the identity on constant functions.”

and see words like “Nullstellensatz,” and wonder how I ever got to the point where this stuff made sense, and how I got to the point where I see a bunch of gibberish now. Seriously, it’s disturbing. I mean, I don’t expect to be able to pick up a book I learned from years ago and know everything in it, but I do expect that it is in a language I can read.

I’ve figured it out. I am Charlie in Flowers for Algernon.

I don’t know how to feel about the loss of my mathematical mind, besides sad.

Maybe I’ll try teaching myself some math again, to either prove to myself I still have it somewhere in me, or to know that my brain has truly atrophied to a giant anti-intellectual morass.

Folium of Descartes

Today, actually just an hour or so ago, another math teacher asked me if I knew a way to parametrize the following:

$x^3+y^3=3xy$

It is also known as the Folium of Descartes and looks like:

Her purposes was just trying to find a quick and easy way to graph it on the calculator. It was just a small, unimportant question. She didn’t need to know the answer, if it wasn’t easily doable. But to me, I needed to know! How do we find the parametric equations which define this? I just don’t know. The answer I found online is:

$x(t)=\frac{3t}{1+t^3}$ and $y(t)=\frac{3t^2}{1+t^3}$

And it’s pretty easily verifiable when you work backwards with the parametrization.

Is there something I’m missing? Is there a method to working from implicitly defined 2D functions to parametrizations for them?

Continuous Everywhere but Differentiable Nowhere

I have no idea why I picked this blog name, but there's no turning back now