Oreos, Barbies, and Essential Questions: framing projects for differentiated learning

The Oreos lesson/unit has been going swimmingly. Students loved the project and the poster, though I'm feeling a little bored with my project ideas right now. Not every project culminates in making a poster. I have a bunch of other ideas I want to write about in another blog post, but that is not where I am headed in this post.

What I want to ponder is, I wonder if we have not been making our Essential Questions (EQs) in mathematics too small. Too narrow. Ever since my Global Math presentation (the one where I had the epic #micfail that left me playing Harpo to Daniel and Tina's Groucho and Chico), I have been thinking about Understanding by Design more and more, and that has led me to ask myself if I don't need to make my Essential Questions in math lessons a whole lot bigger and deeper. There are so many ways to bring the real world into my math classroom, and one of those ways is to frame our work using questions that adolescents are obsessed with thinking about in their everyday lives — questions such as, How dangerous is too dangerous? How do we define what is fair? truthful?

These EQs can form a frame around the activities we do to connect the mathematics to the real world around us. They help provide a situational motivation for learning — and for wallowing in — the mathematics that starts from a place where all students are naturally. And they also make the work we do more, well, essential.

Some of my "major" Oreo lesson EQs blossomed into, Are Nabisco's claims about their Double Stuf Oreo products fair? Are they truthful? just? And my "minor" EQs started revolving around, How can systems of linear equations in two variables help us to model and assess the validity of this claim in the real world?

I am finding more and more that when I frame our work in this way, I hear less and less of the question, "When am I ever going to use this?" And frankly, that's less wear and tear on my soul as a teacher.

With Barbie Bungee, in addition to creating an occasion for more practice in reading aloud and practicing decoding and interpretation skills, I used my situation set-up to raise the EQ, how dangerous is too dangerous? This is a question every adolescent has had to wrestle with since the dawn of time (or at least, since the dawn of puberty as a social construct). In their effort to keep him safe, Siddhartha's parents built him a golden cage of pleasure palaces and theme parks so he would marry and have a life there and never want to leave home. And I can only imagine what Moses' parents must have gone through ("Put down that rod! You're going to put someone's eye out with that thing!").

And the things that matter about that question are (a) the assertion you come up with and (b) the way you marshall concrete evidence and interpretive scaffolding and support to persuade someone else (such as your parents) of the validity and rightness of your assertion.

Coming from a writerly and an entrepreneurial background, I often find that the math of a thing — the essential mathematics of a thing — comes down to what I can persuade someone else of. 

For example, How dangerous is too dangerous? 

Well, it turns out that 28 rubber bands can be empirically demonstrated to be one rubber band too many. With 27 rubber bands, Barbie can have a thrilling — but still safe enough — ride, but at 28, she cracks her head open on the sidewalk, the lawsuits begin, and her parents return to her graveside frequently to tell her "I told you so!" throughout eternity.

And isn't that something we ALL dread?

With the Oreos experiment, the EQs were, Am I being cheated? Are Double Stufs, in fact, double? Is this fair? Is this a good deal? and of course, also, "Does this seem universally and predictably true?"

These are questions every adolescent wallows in every day of their lives. How many times a day do YOU hear, "But that's not FAIR!" or "Mr. C decided such and such. Do you think that is FAIR?"

Fairness is about our own personal beliefs and interpretations of the evidence in light of our own experiences in our world. And only you can answer a question like that for yourself.

Oreos on Trial — Expert Witness Edition (or The Curious Case of the "Double Stuf" Oreos)

This is the week when everyone in our eighth grade class needs to give his or her formal presentation as the final part of their already super-huge culminating assessment project. This public speaking trial by fire will take place this week in English classrooms full of other eighth grade students, eighth-graders' parents and grandparents, school board members, our superintendent, principal, and assistant principal, seventh graders being introduced to the expectations they will have to meet next year, and any other interested parties, politicians, or luminaries from the community who wish to stop by.

The net effect of this situation is that the eighth graders are all nervous wrecks this week, the seventh graders are all bundles of hyperkinetic rubber-band energy, and our mornings will be filled from start to finish with formal presentations and — for me, as their teacher — high-stakes tech support for PowerPoint and Keynote slide shows that have been tinkered with more than is recommended.

For all these reasons, I have realized that this is the PERFECT week for us to do the great Double Stuf Oreo investigation project in my mixed 7th and 8th grade Algebra 1 classes during the afternoons.

I have created a minimally scaffolded version of this lesson that has an embedded literacy component in the situation set-up (a whole-group reading aloud activity) because (a) I have finally started to understand this year how much even the strongest adolescent readers can benefit from practicing their reading and decoding skills in a low-stakes, whole-class setting, (b) this is in keeping with my understanding of Common Core's cross-curricular demand for literacy activities in every subject area, with significant practice in reading, writing, speaking, and thinking along the way, and (c) I have also realized that writing these situation set-ups is fun for me and that reading them aloud is fun for the kids and a welcome change of pace from what they are used to. They love being "written into a situation," and this experience gives them a healthy receptivity to practicing the reading and decoding skills they will still be developing well into their high school and college years.

So here is a link to the PDF version of my Oreo investigation on the Math Teacher Wiki. And here's a link to the Comprehensive Oreology reference site curated by the inestimable Christopher Danielson.

Also, you should know that Nabisco is having a HUGE sale on Oreos this week (at least in Northern California), and between my Safeway club card and the coupons I got from the Nabisco rep at Safeway this afternoon, I feel that the company has gone a long way toward repairing the damage done by some blockhead in customer service who clearly didn't get the importance of Christopher Danielson's and Chris Lusto's attempts to receive an answer that was worthy of the mighty Oreo itself.

More news as it breaks!

An act of wisdom

"One thing I know for sure is that when you are hungry, it is an act of wisdom each time you turn down a spoonful if you know that the food is poisoned."                            

— Anne Lamott, Operating Instructions

There are some truths you have to live, even when that path is hard. For me, this is one of those times. I have this quote hanging over my desk, which is helpful because I have really had to live it this school year. Every morning I need to remind myself of the wisdom and sanity of this perspective.

For me, this truth is bedrock. 

I resigned from my current school in March to remove myself from a toxic situation that is still unfolding. My conscience told me I could not be a part of the direction that is being pursued.

I had to turn down the spoonful to save my soul because I knew in my bones that the food being offered had been poisoned.

Hence my current job search.

I may have resigned that position, but there is no way on earth I am going to leave this profession.

I am a very effective and highly qualified teacher of mathematics, which is an area of desperate need and critical shortage around here. But we are living through an extraordinary period of economic uncertainty and complete political insanity — a time in which our leaders oscillate between one extreme of grandiose talk about "reforming" public education and its opposite of all-out panic at the crisis-level reality of our schools' current situation. 

Our leaders are lost, and our children are bearing the brunt.

The Serenity Prayer instructs me to accept the things I cannot change, the courage to reach out and change the things I can affect, and the wisdom to discern the difference between these two very different kinds of things. 

So as I apply for new jobs and do interviews and give demo lessons, I am also choosing moment by moment to renew my focus on growing and improving my practice as a teacher of mathematics.

And as I do this — even as I fret or worry about finding a new position — a curious thing keeps happening: I keep falling in love with math teaching all over again.

I've created a really great project-based learning (PBL) version of the Barbie Bungee activity (see here and here and here and of course, here), and I'm doing the same thing for the Double Stuf Oreo measurement extravaganza I plan to guide my students through this week. I am learning a ton about differentiation through teaching problem-solving from the online course I am taking from Max Ray at The Math Forum, even though I feel like I can never do enough of the coursework. And Kate Nowak (now of Mathalicious!) and I are having a blast brainstorming our 'PCMI Problem-Solving, TMC-style' problem-solving session for Twitter Math Camp '13 in late-night Google doc chat sessions.

I am hoping that all of this work will be of benefit to me in the fall, but the reality is, of course, that there are no guarantees.

I remind myself daily of the three great teachings my own teacher Natalie Goldberg passed on to me from her root teacher Katagiri Roshi. These are:

•  Continue under all circumstances
•  Don't be tossed away
•  Make positive effort for the good

I am working on writing up and sharing all these lesson ideas and learnings that I'm figuring out, but to be honest, I am struggling to find the time right now. So I am taking good notes to help me write up these blog posts over the summer.

I also remind myself of my amazing good fortune to have my tribe of math teacher-bloggers in the math twitterblogosphere. You support and inspire me every day, and my gratitude for you is bottomless.

Substitution with stars

This one is for Max, who asked about it on Twitter, and for Ashli, who interviewed me for her Infinite Tangents podcasts.

As Ashli and I were talking about some of the struggles we see as young adolescents make the transition from concrete thinking to abstraction, I mentioned substitution.

For many learners, there comes a point in their journey when abstraction shows up as a very polite ladder to be scaled. But for others (and I count myself among this number), abstraction showed up as the edge of a cliff looking out over a giant canyon chasm. A chasm without a bridge.

This chasm appears whenever students need to apply the substitution property of equality — namely, the principle that if one algebraic expression is equivalent to another, then that equivalence will be durable enough to withstand the seismic shift that might occur if one were asked to make it in order to solve a system of equations.

Here is how I have tinkered with the concept and procedures.

Most kids understand the idea that a dollar is worth one hundred cents and that one hundred cents is equivalent to the value of one dollar. I would characterize this as a robust conceptual understanding of the ideas of substitution and of equivalence.

One dime is equivalent to ten cents. Seventy-five pennies are equivalent to three quarters. You get the idea.

We play a game. "I have in my hand a dollar bill. Here are the rules. When George's face is up, it's worth one dollar. When George is face down, it's worth one hundred cents. Now, here's my question."

I pause.

"Do you care which side is facing up when I hand it to you?"

No one has yet told me they care.

"OK. So now, let's say that I take this little green paper star I have here on the document camera. Everybody take a little paper star in whatever color you like."

Autonomy and choice are important. I have a student pass around a bowl of brightly colored little paper stars I made using a Martha Stewart shape punch I got at Michael's.

Everybody chooses a star and wonders what kind of crazy thing I am going to have them do next.

We consider a system of equations which I have them write down in their INB (on a right-hand-side page):

We use some noticing and wondering on this little gem, and eventually we identify that y is, in fact, equivalent to 11x – 16.

On one side of our little paper star, we write "y" while on the other side, we write "11x-16":

I think this becomes a tangible metaphor for the process we are considering. The important thing seems to be, we are all taking a step out over the edge of the cliff together.

We flip our little stars over on our desks several times. This seems to give everybody a chance to get comfortable with things. One side up displays "y." The other side up displays "11x–16." Over and over and over. The more students handle their tools, the more comfortable they get with the concepts and ideas they represent.

Then we rewrite equation #1 on our INB page a little bigger and with a properly labeled blank where the "y" lived just a few short moments ago:

"Hey, look!" somebody usually says. "It looks like a Mad Lib!"

Exactly. It looks like a Mad Lib. Gauss probably starts spinning in his grave.

"Can we play Mad Libs?" "I love Mad Libs!" "We did Mad Libs in fifth grade!" "We have a lot of Mad Libs at my house!" "I'll bring in my Mad Libs books!" "No, mine!"

It usually takes a few minutes to calm the people down. This is middle school.

I now ask students to place their star y-side-up in the blank staring back at us.

Then it's time to ask everybody to buckle up. "Are you ready?"

When everybody can assure me that they are ready, we flip the star. Flip it! For good measure, we tape it down with Scotch tape. Very satisfying.

A little distributive property action, a little combining of like terms, and our usual fancy footwork to finish solving for x.

Some students stick with substitution stars for every single problem they encounter for a week. Maybe two. I let them use the stars for as long as they want. I consider them a form of algebraic training wheels, like all good manipulatives. But eventually, everybody gets comfortable making the shift to abstraction and the Ziploc bag of little stars goes back into my rolling backpack for another year.

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I'd like to thank the Academy and Martha Stewart for my fabulous star puncher, without which, this idea would never have arisen.

I wore out my first star puncher, so I've added a link above for my new paper punch that works much better for making substitution stars. Only eight bucks at Amazon. What's not to like? :)