Month: May 2010

Topic Lists, Reprise: Obvious and yet, I never would have thought of it

This idea totally came from someone else, and I’m awful for not remembering who from the math-teacher-edu-blogosphere came up with it. But it’s just such an awesome idea, and I wanted to spread the love. If this is your idea, just throw the original post down in the comments, and I’ll be sure to add a huge giant link to it so you can have credit.

It could be really useful if you’re trying to help kids get organized for an end-of-year exam.

I wrote a while ago (causing some chafing for a few) about how I give my kids topic lists before major assessments.

They used to look like this:

Now, I’ve added a single image, in order to help students more effectively learn how to study:

So you can see what it looks like in it’s final glory…

It’s a little late in the year to make this effective, but I’m hoping it’s helped a few kids identify where they should focus their (precious and limited) time studying. If a student bombs an assessment, when I meet with them, I can ask them to pull out their topic list with these little boxes filled out, and we can start a conversation correlating their assessment with their filled out topic list.

(Of course, this is after the all important question: “Tell me how you prepared for the assessment. In detail. Don’t leave anything out.”)

Weights! Goldsmiths! Optimization!

I am in a problem solving group at my school, and I took 45 minutes of one of our sessions to lead a mock class. Not really mock, to be fair. I assumed I’d have 3 math teachers and 2 science teachers as my class, and I wanted a problem which would get them to think, work together, and also let me guide without leading (or is it lead without guiding).

The problem I chose was exactly the problem that Brent just wrote about on The Math Less Traveled: the broken weight problem.

A merchant had a forty pound measuring weight that broke into four pieces as the result of a fall. When the pieces were subsequently weighed, it was found that the weight of each piece was a whole number of pounds and that the four pieces could be used to weigh every integral weight between 1 and 40 pounds. What were the weights of the pieces? [I gave the problem with ounces.]

I have to say that I was really thrilled that I was able to get them to a solution, with very little nudging. I let them take their time. I started them out by giving them slips of paper of various sizes with corresponding weights written on them, and asked them to use those weights to be able to weigh something like 10 ozs. I helped them organize their thoughts with observations, and I helped them latch onto key ideas once they emerged. I never gave the key ideas, and I didn’t push. It was awesome to witness them work together.

It was also surprising in two other ways:

1. I had the pathway in mind that I thought they were going to take — basically a recursive approach. They did not go that way, and it was afterwards — when examining the problem once they had the solution – that they saw the recursion.

2. I had prepared two “hint cards.” They were written on origami paper and folded up — because, why not? I told ’em that if they all agreed, they could take the first hint card, and if they felt they really needed it, they could have the second hint card. They didn’t take any of ’em. I thought they would. In fact, I predicted that they would get frustrated and take the first one pretty quickly, so I put on the first hint card: “YOU CAN DO IT! Keep working at least for another 5 minutes.” It wasn’t a hint, but a “work through frustration” note. The second card had a hint leading them to recursion (saying something like “What if you only had any 2 weights… what would they be so that you can weigh the most: 1 oz? 2 ozs? 3ozs? 4 ozs? …”)

As a result of watching them operate, and places they struggled (including understanding the problem!), I wanted to challenge myself.

How could I create a formal lesson plan for this? A lesson plan that guides without leading.

Here’s my first crack at it (PDF here):

PS. Yes, I know there’s a typo in question 1.

Solution to the “what curve is this?” problem

So a while ago I posted a problem that me and another teacher worked on in our problem solving group. We didn’t have the most elegant solution (that honor goes to Jake), But I think it is slightly qualitatively different than the solutions posed in the comments of the original post. Our solution involved systems of equations and parametric equations and L’Hopital’s rule.  Yup, believe it or not, L’Hopital arose naturally in the wild, and when I was coming up with my plan of attack, I suspected it would if things were going right.

To remind you, I wanted to find the equation for this blue curve:

(If you want more details, just check out the original problem.)

So here it goes.

The crucial question we asked ourselves is: if we drew all the red lines, where would the blue line come from?

The answer, which was fundamental for our solution, was: if we drew two red lines which were infinitessimally close to each other, their intersection would give us one point on the blue curve. Think about that. That is the key insight. The rest is algebra. If we could find all these intersection points, they form the line.

So we picked two points close to each other: one with endpoints (a,0) and (0,5-a) and the other with endpoints (a+\epsilon,0) and (0,5-a-\epsilon).

Notice that as we bring \epsilon closer and closer to 0, these two lines are getting closer and closer to being identical. But right now, \epsilon is just any number.

So the first line is (in slope-intercept form): y=-\frac{5-a}{a}x+5-a (any of the red lines)
And the second line is: y=-\frac{5-a-\epsilon}{a+\epsilon}x+5-a-\epsilon (any of the other red lines)

We want to find the point of intersection. So setting the ys equal to each other and solving for x, we get:

x=\frac{\epsilon}{\frac{5-a}{a}-\frac{5-a-\epsilon}{a+\epsilon}}

Of course now we want to see what happens to the intersection point as we bring the two lines infinitely close together. So we are going to take the limit as \epsilon approaches 0.

x_{blue}=\lim_{\epsilon \to 0} \frac{\epsilon}{\frac{5-a}{a}-\frac{5-a-\epsilon}{a+\epsilon}}

Notice you’ll see that we get a 0/0 form if we just plug in \epsilon=0, so we must L’Hopital it!

When we do that (remember we take the derivative of the numerator and denominator with respect to \epsilon), we find that:

x_{blue}=\frac{a^2}{5}.

And plugging that into our equation for the first line, we find that the y_{blue} coordinate is:

y_{blue}=\frac{(a-5)^2}{5}

At this point, we rejoyce and do the DANCE OF JOY!

GAAAK! Almost. You silly fools. You’re like my kids, who get so proud when they do the hard part of a problem, that they forget what the question is asking and move on to the next problem. We still don’t have an equation. And what does (x_{blue},y_{blue}) mean anyway?

To start, that point represents the intersection point of two lines infinitesimally close to each other in our family of red lines above. But this a business? It’s confusing. I like to think of it like a parameter! As I move a between 0 and 5, I am going to get out all the points on the blue curve.

So how do I find this curve? Exactly how I would if these were parametric equations:

x=\frac{a^2}{5} and y=\frac{(a-5)^2}{5}.

I take the first equation and solve it for a: a=\sqrt{5x}.

I then plug that value into the second equation for y: y=\frac{(\sqrt{5x}-5)^2}{5}.

And we’re done! We graph to confirm:

And now, indeed, we may do the dance of joy!

Guest Post: On Being Yourself While Doing Math

I got an email from Rebecca Zook, who is a fellow math blogger, who was partially inspired by a post from forever ago (“Don’t Judge a Book By…“), asking me if I wanted a guest post on the same theme. Well, I’ve never had a guest post, and this is a darn good one, and now is as good a time as any! So without any more fuss and muss…


On Being Yourself While Doing Math­
by Rebecca Zook

When I met my new math tutoring student and her mom wearing my celery green pantaloons and a dress that made me look like a fluffy yellow daffodil, I wasn’t sure how they would react.

They looked at each other.  They smiled with relief.  And then, they beamed at me.  Next we got down to business and had a massively productive tutoring session.

This experience caused me to reconsider my entire philosophy of teaching attire.

My whole life I’ve had my own distinct style, whether that meant wearing galoshes without socks regardless of the weather (preschool) or making a dress printed with the solar system from an old curtain I found at Goodwill (high school).

But I started my career as a math educator by teaching SAT math for a big corporate test prep company.  So, despite the fact that they hired me when I was wearing a homemade miniskirt printed with text from a French nursery rhyme and pictures of chickens, once I got into the classroom, I seriously curtailed my exuberance and dressed for my teacher-role.  My efforts to wear business casual mainly consisted of me wearing the same pair of black slacks almost every time I taught.

Later, when I left the corporate world and started my own math tutoring practice, I still felt the need to dress “professionally” when I began to meet one-on-one with my own clients.  I worried that if I really dressed like myself—instead of some idea of what a female math tutor should look like—students or parents would get turned off or distracted by my clothes.

But I started to ask myself, why was it “educational” to pretend to be less fabulous than I really am?  Why not wear a really awesome outfit to teach in instead of trying to “look normal”?

In that moment, walking towards my new student and her mom, what I experienced was nothing like I’d feared.  It was something totally different: appreciation, excitement, and even recognition.

What the heck was going on here?   Why were they so clearly excited about me looking so different than they expected?  They might just be relieved that I was a female math tutor that wasn’t afraid to be girly, or something.

Or, it might just be because my new immaculately student, who frequently came to tutoring sessions wearing a pristine corset paired with bloomers, high-heeled knee-high boots, and antique goggles perched on her head, just approved of my weird style.

But then I remembered some of my different students’ styles: the 18-year-old homeschooler with blond dreadlocks and a torso-length tattoo of a Buddhist goddess; the fifth grader wearing her private school uniform who fervently professed her love of Abercrombie; the seventh grader who wore classic rock t-shirts and cherished her florescent vintage sunglasses from the ‘80s.

I realized it was something completely different than my students liking my style.

It’s about being yourself while doing math.

I want to create a space where my students feel they can create their own solutions and find what’s best for them.   I want to help my students gain true confidence in who they are, whether that means how they dress or how they think.

When a student spontaneously makes up a song about even numbers to the tune of Michael Jackson’s “Beat It”, or leaps out of their chair to spin a certain number of degrees to solve an angle measurement problem, I feel I’ve succeeded in helping them be comfortable with themselves and asserting their own choices.

Maybe finding and embracing your own learning style isn’t that different from finding your own fashion style.

Maybe I’m crazy here, but when students only see one kind of person doing math, maybe they’re getting a message that they have to be a certain kind of person or dress a certain kind of way to be good at math.

I believe that you do not need to restrain your awesomeness/exuberance to kick butt at math.  In fact, the same passion, sense of adventure, and endurance that may lead someone to get a full torso tattoo of a deity can also serve them while mastering math problems.

You can learn math no matter what you look like, whether you’re into dressing like a daffodil, a Buddhist punk, or an Abercrombie fiend.  And you definitely don’t have to act, or look, boring.
BIO: Guest blogger Rebecca Zook is an online female math tutor who has been helping students get math into their brains for seven years.  Her blog, Triangle Suitcase, is about unpacking the process of learning in all its complexity, frustration, and delight.

MY THOUGHTS: If you’ve ever met me in real life (well, only two or three of you have), you know I am a lot of … something. And I love that about me. In high school, I was big into shopping at thrift stores and pairing together plaid golf pants with a pair of silver spray painted shoes with a ratty old tshirt. I was a clothing bricoleur. There are lots of reasons for that, but the consequence was that my sense of self was tied up in how I dressed. I saw myself as an unconventional almalgam, and my clothes were a conscious reflection of that. Well, because of that, I fully support students expressing that sense of self. And Rebecca has tied this same thing — this sense of individuality and choice and confidence — to how students do math. I like that. I also really was struck by this line:

Maybe I’m crazy here, but when students only see one kind of person doing math, maybe they’re getting a message that they have to be a certain kind of person or dress a certain kind of way to be good at math.

Holla! That gets to the crux of why diversity is important. It’s why we need female science teachers teaching AP courses, and female leaders in student goverment, and a diversity of ethnicities, religions, and sexual orientations in visible places in schools. You don’t need to be a guy to be great at math, and you can be gay and good at sports, and you can be a girl and in charge of student government. So yeah, it’s just clothing. But it’s kind of something more, if you look at it from a slightly different angle.