Betweenness and non-betweenness: absolute value inequalities and Patrick Callahan

I felt a little nervous about having Patrick Callahan come to observe my classroom yesterday, but in the end, it was fun. I had asked one of our security guards, to bring him down to my room when he arrived at our school. He walked in as he always does, all mathematical open-mindedness and pedagogical curiosity.

And we got started.

I felt anxious about having him observe my conceptual lessons about betweenness and non-betweenness. I have never seen anything even close to how I understand and talk about absolute value and inequalities. I talk about boundary points and betweenness and I have students hold up their fists and point their thumbs to show me their understanding. “Is this a situation of betweenness — or NON-betweenness?” I demonstrate with my own fists, swinging my thumbs inward or outward. “Your fists are the boundary points and your thumbs are how you shade your graph on the number line. So is this a situation of betweenness... or NON-betweenness?”

If it is a situation of "betweenness," then students point their thumbs inward towards each other, touching the tips together. If it is NON-betweenness, then they point their thumbs outward in either direction, like a group of indecisive hitchhikers. And once we have done this analysis, then we can do whatever calculations we may need to find our boundary points.

So much of advanced algebra and precalculus depends on having this kind of deep conceptual understanding and thinking. Am I looking for quantities that are GREATER than...? or LESS than? Is this quantity going to be positive? or negative?

For me, the whole thing is intimately hooked together with the real number line. And with number sense. 

When we started last week, we began with an inquiry into “more than” and “less than” and widened our thinking outward from there.We connected more than and less than to number line thinking. I always emphasize Number-Line-Order and Number-Line-Thinking in my Algebra 1 classes. If they think about the number line, then they can anchor their thoughts in their bodies. LHS (or Left-Hand Side) and RHS (Right-Hand-Side) are fundamental ways of thinking in algebra. These ideas are eternal and unchanging. The number line is the foundation of everything. It gives you the “true north” of the real number system.

So we always ground our thinking in our bodies. I ask, “Left Hand Side or Right Hand Side?” “Is this a situation of betweenness or non-betweenness?” “OK, now that we know that, now what?”

I also anchor this unit in what they know about logical reasoning. They have an intuitive sense of how many possible cases a situation may present. I've been a huge Yogi Berra philosophy fan all my life, so I believe that when you come to a fork in the road, you should take it. When you come to a fork in the road, you can go left or you can go right. Or you can stay right where you are. Three possible cases. Over and over I ask them, “What’s going on here? How do you know?”

Absolute value inequalities are either situations of betweenness or situations of non-betweenness. Figure that out and then everything else will run smoothly. Then all you have to do is to use what you already know.

Once students have gotten that figured out, it’s just one more small step to combining their new knowledge with their existing knowledge. Follow the order of operations and common sense. Plus everything you know about the real number line and multiple representations. Then things can naturally unfold the right way.

But I always come back to number sense to what we know about the real number line. Numbers are the ground, the foundation.

So when Patrick walked in yesterday — this world-class mathematician and math education expert — what he encountered was my bootcamp in algebraic thinking. “Hold up your fists! Is this a situtation of betweenness or non-betweenness?”  "How do you know?" And then my waiting until everybody’s thumbs are pointing in the same direction.

It is Logic 101 and numbers and anchoring our thoughts about numbers in our bodies. Like the ancient Greeks and Babylonians and Egyptians before us.

Our next step is to solidify our thinking through what How People Learn calls “deliberate practice with metacognitive awareness.” We are going to do two days of Speed Dating. Now I have to make up Speed Dating cards and a test to use on Thursday. 

And then to document my thinking.

When the class ended, Patrick came up to my tech podium and was excited. He grabbled a whiteboard marker and started sketching and pouring out ideas.

For me, that was the best possible review I could have gotten on this lesson. A five-unicorn review. A direct hit. :)

"Find What You Love; Do More of It"— #MTBoS Edition, a report from the field

While my friend @TrianglemanCSD Christopher Danielson is holding down the fort at the Minnesota State Fair and the Math On A Stick exhibit, I'm in my third full week of school avec students, and I realized I needed to peel off another layer of my persona and take his advice from his keynote address at #TMC15:

Find what you love; do more of it.

I love storytelling and stories. I come from a family of storytellers, where dinnertime was always a time of sharing stories.

I also love history — ancient history — and I know more about it than I usually give myself credit for.

So I did more of both of those things today. In my Algebra 1 class, no less.

It felt like a huge risk. But I decided to feel the fear and do it anyway.

So I told them what I have long known about the origins of algebra and equations. In its earliest uses, "algebra" means "balance." An equation is a metaphor for what everyone in the ancient world knew and understood with their own inherent sense-making and mathematical reasoning: just as a vendor at a market weighs out what a customer wants to buy — weighs it out with standards-based measures that are sanctioned by governments and universally accepted — so too is an equation a representation of these scales... and our goal is to balance out abstract or concrete quantities using that familiar structure from daily life.

I asked students what they knew about weighing and measuring and they told me honestly what their experiences have been at the farmer's markets all over our city.

We investigated what we knew about what happens when you place a known amount of weights on one side of a balance, and we imagined — and in some cases, acted out — what happens as you pour a continuous quantity of something onto the other side of the balance.

We talked about what it means to bring a scale into balance and we applied what we knew — the best we had — to the ideas at hand.

We talked about national and international standards bodies and about how even a perfect model degrades over time, which is why it needs to be monitored and occasionally replenished.

And so, by the time we got to the addition, subtraction, multiplication, and division properties of equality, we were already deep into our own connections with the metaphor and with the human history of algebra and with our own active and vivid imaginations.

By the end of class, nobody even complained about having to do the 2-2 homework. I am hoping that's because it was a little more deeply connected to their humanness than it ever had been before.

TMC #14 Group Work Working Group Morning Session – Annotated References & Framework

I'm having a lot of fun planning the Group Work Working Group morning session for Twitter Math Camp 2014, and it's time to start sharing.

Here is the background material I'm using for developing the group work morning sessions. Please note that this is NOT required reading!  Recreational reading only! So please don't freak out!  :)

I wanted to give people a sense of the framework and background I'd like us to start from so attendees can decide whether this morning session will be right for them. I also wanted to provide links and titles to valuable materials.

These are listed in order of relevance to the Group Work Working Group morning session — they are not in formal bibliographical form.

National Academies Press, How People Learn (downloadable PDF here)
This amazing free book provides the framework within which we'll consider the use of group work. I am especially keen for us to explore how we can develop and implement tasks that fit within their (approximately) four-stage cycle for optimizing learning with understanding while also fitting with our own individual school and district requirements. In a nutshell, the four stages are as follows:

STAGE 1 - a hands-on introductory task designed to uncover & organize prior knowledge (in which collaboration cultivates exploratory talk to uncover and organize existing knowledge)

STAGE 2 - initial provision of a new expert model (with scaffolding & metacognitive practices) to help students organize, scaffold, & develop new knowledge (in which collaboration provides a setting to externalize mental processes and to negotiate understanding)

STAGE 3 - what HPL refers to as "'deliberate practice' with metacognitive self-monitoring" (in which collaboration provides a context for advancing through the 3 stages of fluency with metacognitive practices)

STAGE 4 - transfer tasks to extend and apply this new knowledge & understanding in new and unfamiliar non-routine contexts

Malcolm Swan, "Collaborative Learning in Mathematics" (downloadable PDF here)
A short and highly readable summary of Swan's instructional design strategy for collaborative tasks, including notes on his five types of mathematical activities that constitute the bulk of the Shell Centre's formative assessment MAP tasks and lessons.

Malcolm Swan, Improving learning in mathematics: challenges and strategies (downloadable PDF here)
An in-depth introduction to Swan's approach to designing and using the kind of rich tasks offered by the Shell Centre and the MARS and MAP tasks.

Chris Bills, Liz Bills, Anne Watson, & John Mason, Thinkers (can be purchased from ATM here)
The richest source book imaginable for ideas for activities to stimulate mathematical thinking. Often credited by Malcolm Swan and Dylan Wiliam.

Anne Watson & John Mason, Questions and Prompts for Mathematical Thinking (can be purchased from ATM here)
The richest source book imaginable for variations on questioning and prompting strategies.

Dylan Wiliam, Embedded Formative Assessment
This book is a gold mine. Don't leave home without it.

New strategy for introducing INBs: complex instruction approach

After months of not feeling like my best teacher self in the classroom, I got fed up and spent all weekend tearing stuff down and rebuilding from the ground up.

INBs are something I know well — something that work for students. So I decided to take what I had available and, as Sam would say, turn what I DON'T know into what I DO know. Love those Calculus mottos.

So I rebuilt my version of the exponential functions unit in terms of INBs. But that meant, I would have to introduce INBs.

As one girl said, "New marking period, new me!" The kids just went with it and really took to it.

Here is what I did.

ON EACH GROUP TABLE: I placed a sample INB that began with a single-sheet Table of Contents (p. 1), an Exponential Functions pocket page (p. 3), and had pages numbered through page 7. There were TOC sheets and glue sticks on the table.

SMART BOARD: on the projector, I put a countdown timer (set for 15 minutes) and an agenda slide that said,

• New seats!
• Choose a notebook! Good colors still available!
• Make your notebook look like the sample notebook on your table 

As soon as the bell rang, I hit Start on the timer, which counted down like a bomb in a James Bond movie.

Alfred Hitchcock once said, if you want to create suspense, place a ticking time bomb under a card table at which four people are playing bridge. This seemed like good advice for introducing INBs to my students.

I think because it was a familiar, group work task approach to an unfamiliar problem, all the kids simply went went with it. "How did you make the pocket? Do you fold it this way? Where does the table of contents go? What does 'TOC' mean? What goes on page 5?" And so on and so on.

I circulated, taking attendance and making notes about participation. When students would ask me a question about how to do something, I would ask them first, "Is this a group question?" If not, they knew what was going to happen. If it was, I was happy to help them get unstuck.

Then came the acid test: the actual note-taking.

I was concerned, but they were riveted. They felt a lot more ownership over their own learning process.

There are still plenty of groupworthy tasks coming up, but at least now they have a container for their notes and reflection process.

I'm going to do a "Five Things" reflection (trace your hand on a RHS page and write down five important things from the day's lesson or group work) and notes for a "Four Summary Statements" poster, but I finally feel like I have a framework to help kids organize their learning.

I've even created a web site with links to photos of my master INB in case they miss class and need to copy the notes. Here's a link to the Box.com photo files, along with a picture of page 5:

We only got through half as much as I wanted us to get through, but they were amazed at how many notes we had in such a small and convenient space.

It feels good to be back!

Collaboration Literacy Part 2 — DRAFT Rubric: essential skills for mathematical learning groups

I have said this before: middle schoolers are extremely concrete thinkers. This is why I find it so helpful to have a clear and concrete rubric I can use to help them to understand assessment of their work as specifically as possible. I'm reasonably happy with the rubric I've revised over the years for problem-solving, as it seems to help students diagnose and understand what went wrong in their individual work and where they need to head. But I've realized I also needed a new rubric — one for what I've been calling "collaboration literacy" in this blog. My students need help naming and understanding the various component skills that make up being a healthy and valuable collaborator.

My draft of this rubric for collaboration, which is grounded in restorative practices, can be found on the MS Math Teacher's wiki. I would very much value your input and feedback on this tool and its ideas.

I don't want to spend a lot of time talking about how and why Complex Instruction does not work for me. Suffice it to say that the rigid assignment of individual roles is a deal breaker. If CI works for you, please accept that I am happy that you have something that works well for you in your teaching practice.

This rubric incorporates a lot of great ideas from a lot of sources I admire deeply, including the restorative practices people everywhere, Dr. Fred Joseph Orr, Max Ray and The Math Forum, Malcolm Swan, Judy Kysh/CPM, Brian R. Lawler, Dan Pink's book Drive, Sam J. Shah, Kate Nowak, Jason Buell, Megan Hayes-Golding, Ashli Black, Grace A. Chen, Breedeen Murray, Avery Pickford, "Sophie Germain," and yes, also the Complex Instruction folks. I hope it is worthy of all that they have taught me.

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.