Compound Inequalities Treasure Map

Never underestimate the power of novelty to help you engage certain students.

I just spent the last hour and a half-long block period with my jaw on the floor, watching in amazement as my most discouraged, 12th grade College Prep Math students worked productively and peacefully on, of all things, the analysis and solving of compound inequalities.

During my prep, I turned a boring worksheet into a treasure map. And that turned a boring requirement into a very peaceful and enjoyable period.

As she was leaving, one girl asked, Could we please do more work like this?

I'll take that as a compliment!

Stalkers and dreamers

I've talked about this before: there are those who learn by stalking — step by step, one day at a time, one skill at a time, little by little. And then there are dreamers: those of us who try and fail, try and fail, try and fail. Carlos Castañeda makes this distinction between stalkers and dreamers, and it has been a useful distinction for me from the moment I encountered it.

I am a dreamer. I first became aware of this learning pattern when I was about five and learning to ride on two wheels. My dad removed the training wheels from my little red bike and I would practice.

I practiced riding day after day for weeks.

And day after day, for weeks on end, I would fail.

I fell everywhere — on the sidewalk in front of our house, in the driveway, on our block at low-traffic times.

I fell on smooth pavement, on concrete, any time I encountered gravel.

I was frustrated and pretty scuffed up. But in my mind's eye at night, I could imagine riding perfectly.

When I dreamed, I could feel myself rolling smoothly and swiftly on two wheels. In my dream life, I was a person who could ride on two wheels, and I could do so successfully anywhere.

A little less so in my waking life.

I skidded, slid, or toppled over after only a few feet. I still remember how it felt to fall at different moments and on different surfaces. I have a vivid and complete felt sense memory of rolling onto a patch of gravel and sliding to the ground at the intersection of Lenox Road and Hershey. I distinctly remember the feeling of gravel biting into the skin of my knee.

I must have wanted it bad to keep on trying.

Then one day, it just happened.

I had steeled myself yet again for the failure that had become my 'normal,' and I readied myself for more cuts and bruises and wounded ego.

But I didn't fall over.

I was so excited I parked my bike and the garage and rushed inside to tell my mother what had happened. It was the most exhilarating thing I could imagine at the time.

Still, though, I assumed it was a fluke. I continued to prepare myself for further failure.

But then it happened again. And again. And again.

The story of the larva that becomes a butterfly had taken hold of my life. Even after you emerge from the transformation, it takes a while for your awareness to catch up with your changed reality. It took several days before I realized I had stepped into a new normal.

I'd been afraid to hope.

Nowadays I wonder how my students experience transformation from people who believe they can't do math into people who understand that they can. It's hard to trust transformation. As A.H. Almaas says, you've been a larva crawling around all your life, and you believe that the best you can hope for is to crawl faster and to become a bigger, fatter, happier, more successful larva. You see butterflies flying around and you classfy them as anomalies. Most of us never automatically think, gee, that's where *I'm* headed too. Most people think, "Wow those are really interesting beings. I wonder where they come from. I wonder what it would feel like to be one of those."

In math, as in learning to ride a bicycle, it never occurred to me that I could take what I know from other areas of my life to help myself become one of those magical creatures who can ride a bicycle or do math. I did not know it was my birthright to be good and successful at those things. I thought I was destined to remain an earthbound larva.

For a lot of us, it's not enough to say, if you can't do these problems fluently after this investigation, then that means you need to seek out more practice. I needed both experience or discovery and also practice. I needed opportunities for practice and maybe a choice of activities that allowed me to seek out the practice I needed while others were ready for more discovery. Maybe this takes the form of a branching of activities — a practice table and an extensions table, for example. All I know is that students need support and opportunities to self-diagnose and to seek out the experiences they need in that moment. Stalkers need space to stalk further while dreamers need space and time to practice and fall down a lot more.

There is a mystical part of this process that cannot be discounted. 

At times when I feel discouraged about my teaching practice, I have to remind myself about all of this. I feel like I am trying and failing, trying and failing, trying and failing. I have a lot of psychic gravel chewing through the skin on my psychic knees from falling down. I have to remind myself that this is my process.

Attending to Precision: INBs and group work (Interactive Notebooks)

I love new beginnings, but I am only so-so with early middles. Now that kids have started their INB journey, we've arrived at that crucial moment between the beginning and the first INB check. This, as the saying goes, is where the rubber meets the road.

I find that kids never understand at this stage why I insist on being so darned nit-picky about their notebooks. Every day someone new asks me why this or that HAS to go on the right-hand side or EXACTLY on page 5.

One of the many reasons why this is important, I have learned, is that it is all about teaching strategies for attending to precision — Mathematical Practice Standard #6, which is defined this way in the standards documents:

Mathematically proficient students try to communicate precisely to others.• They try to use clear definitions in discussion with others and in their own reasoning.When making mathematical arguments about a solution, strategy, or conjecture (see MP.3), mathematically proficient elementary students learn to craft careful explanations that communicate their reasoning by referring specifically to each important mathematical element, describing the relationships among them, and connecting their words clearly to their representations. They state the meaning of the symbols they choose, including us- ing the equal sign consistently and appropriately.• They are careful about specifying units of measure,• and labeling axes to clarify the correspondence with quantities in a problem.• They calculate ac- curately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions. 

The problem, I find, is that this description of precision is precise only at the theoretical level. On the front lines, it's unrealistic because most kids never get to this level of precision.

And that is because their notes and their work are generally quite a mess.

A big part of teaching students to attend to precision is giving them a structure for being an impeccable warrior as a math student — that is to say, taking and keeping good notes, noticing and keeping track of your own progress as a learner, preserving your homework in a predictable place that is not, let us say, the very bottom of your backpack, crushed into a handful of loose raisins.

It means stepping up your game as a student of mathematics and presenting your work in a way that makes it possible for others to notice the care with which you are specifying units, crafting careful explanations, describing relationships, and so on. And it means presenting your work in this way ALWAYS — in all things, in all times, wherever you go.

INBs are an incredibly low-barrier-to-entry, accessible structure for teaching attention to precision. There are no students who cannot benefit from having a clear, common, and predictable structure for organizing their learning. INBs are also a great leveler. For those of us who are focused on creating equity in our classrooms, INBs offer all students a chance to prove both to themselves and others that they are indeed smart in mathematics. As I saw the other night at Back To School Night, my strongest note-keeping students are rarely the top students computationally speaking. But they are the ones who can always find what they are looking for — a major advantage on an open-notes test.

INBs are also a phenomenal formative assessment tool. Flipping through a students INB gives me an incredible snapshot of where and when they were truly attending to precision and where they were fuzzing out. Blank spaces and lack of color or highlighter on specific notes pages give me a targeted spot for further formative assessment. In my experience, it is exceedingly rare for a student who thoroughly understands a topic to write no notes or diagrams on that page. If anything, they are the ones who are most likely to appreciate the chance to consolidate their understanding.

So I am sticking with it and zooming in on some of the areas where kids' understanding fell apart last week. We'll be reviewing how to convert from percentages to decimals and how to document and analyze the iterative process of calculating compound interest because that is where my students' notes fell apart.

I'll be astonished — but will report back honestly — if these on-the-fly assessments prove to have been inaccurate.