The concept of betweenness

I am coming to believe that, much like the concept of substitution, developing a deep understanding of the idea of betweenness is a huge part of the psychological and conceptual work of Algebra 1.

I have been dissatisfied for years now with the fact that we tell students about the geometric interpretation of absolute value, but we don't really get them to live it. And yet that idea of the "distance from zero" on the real number line is not something we give students time to really marinate in.

And you know what? That feels dumb to me.

So this week and part of the next, my students and I are really wallowing in that idea.

I've taken a page from my studies as a young piano student. In the study of the piano, there are certain studies of technique that really force you to slow down and take apart the finger movements. There are specific figures that you have to practice over and over and over so that they become part of your finger memory. How they feel in your fingers is how you come to relate to them.

This is not just about developing automaticity, although that is a side benefit. This is about learning to feel these foundational figures in your bones. In your body. They become so fundamental that as you learn and grow as a musician, you come to feel them when you see them coming up in a new score you are studying.

The technique does not replace musicianship. The technique supports the musicianship.

I've been noticing lately how my own experience of absolute value is about noticing boundary points at the periphery of my mathematical perception. I see them out of the corner of my mathematical mind's eye. And how an inequality is said to relate to them defines how I relate to those boundary points.

So I am taking the risk of sharing this mathematical experience with my students.

As with young piano students, we take this slowly. One figure at a time. Right now we are only dealing with the case of an absolute value being less than a nonnegative quantity. We are dealing with situations of betweenness, where an inequality presents us with a figural situation that is going to wind up with a quantity being between two boundary points.

That is all. And that is enough.

I see the effort in their faces and in their fingers as they rewrite, revise, calculate, solve, and sketch graphs. I see them noticing and wondering whether they need to use a closed dot or and open dot.

And I hear them developing the confidence that comes from experience in developing a relationship with these quantities.

They are not following rules. They are listening to their own deeper wisdom. Everybody knows something about the situation of being "between" other things. Betweenness is one of the most elemental human ideas.

They are making friends with mathematics.

The deeper wisdom of the body in math class

This post is for Malke.

As I was eavesdropping on a recent conversation between Lani Horn (@tchmathculture) and Malke Rosenfeld (@mathinyourfeet), I received a pointer to an article summarizing recent research that shows that kids with ADHD actually need to squirm in order to learn.

This makes sense to me. The deeper wisdom of the body is usually overlooked in thinking about teaching, learning, and assessment in mathematics. And yet, it can provide a vital link for our students in claiming their mathematics as well as their humanity.

I was thinking about this on Friday afternoon when I tweeted out the following:

I love the dulcet tones of compasses, rulers, & pencils during a Friday afternoon constructions quiz. #geomchat

Malke tweeted back:

I love that they are doing all that by hand. And that there are dulcet tones. :)

I responded:

Geo has affirmed my belief in the life made by hand. Huge benefits to Ss [students] from physically constructing their understanding.

I had another in a series of lightbulb moments this past month about what How People Learn says about externalizing our understanding.

And ever the good online learning partner, Malke tweeted back:

have you blogged about it?

So here I am.

In How People Learn, the authors talk about how we can use skits, presentations, and posters in group work to help students externalize their emerging understanding. This makes sense to me. In order to learn something, one first has to notice it, and that means developing a metacognitive self-awareness of the process and how it’s going.

Over the last few years, I have found that teaching students to use foldables, INBs (Interactive Notebooks), guided note-taking, and physical constructions is another extremely rich field of helping students to externalize their emerging understanding — only in these cases, they are externalizing their understanding through physical, kinesthetic processes — not just through talk, listening, and presentation processes.

The physical dimension is a good grounding for conceptual understanding. Teaching students how to literally use their tools can be a multidimensional process of making their learning both physical and tangible. Flipping open a flap or a page in a composition book is a physical manifestation of the process of retrieval or comparison or evaluation. Likewise, the process of using patty paper as a tracing medium to externalize the concept of superposition and projection of a figure to confirm congruence is a way of helping students to slow down their speeding monkey minds and to become present with the mathematics that are right in front of them.

When I tuned in to the clatter of compasses, rulers, and pencils on Friday, I really noticed how deeply engaged my students were with the geometry they were working on. Their body postures indicated how deeply immersed they were in the experience of flow: set your compass opening to an appropriate width and draw an arc across the angle you want to copy. Stab the endpoint of the segment where you want to create a new copy of your original angle and swipe the same arc there. Go back to the first angle. Refine your compass opening so that it now matches the width between the intersections of your arc and the original angle. Shift your paper, stab at the lower intersection of your copied angle-in-process and swipe an arc that will intersect with the arc you just drew there. Drop your compass; pick up your straight edge. Carefully draw a line to connect the endpoint of your target segment with the intersection of the arcs you have drawn, completing the terminal side/ ray you need to draw. Drop the straight edge; position the patty paper over your original angle and use your straight edge to trace it. Drop the straight edge and carefully slide your traced angle over your constructed angle. Does it match your figure perfectly?

The concentration etched on their brows matched the precision of their work on the page in front of them. Bisect an angle. Construct the perpendicular bisector of a segment. Construct a parallel line through an external point by using it to define an angle that you can copy.

Hopping from one stone to the next, you can cross an entire river. By placing one foot on the Earth after another in a pattern of glide reflections, you can complete a journey of a thousand miles or more.

That is one of the lessons of deductive and spatial reasoning at the heart of any good Geometry course. Noticing that it is happening in my classroom — really happening through physical, mental, and whole-hearted engagement — is one of the greatest blessings of being a teacher.


What do you do after Formative Assessment reveals a gaping hole in understanding? More Talking Points, of course. :)

My Geometers took the opportunity to inform me through their Chapter 1 exams that they really don't get how angles are named. So this seemed like a perfect opportunity for more Talking Points, of course. :)

This time I'm giving everybody a diagram of a figure that the Talking Points refer to. They will have to do some reasoning about naming angles in order to do the Talking Points. They love doing Talking Points, but they mostly like coming to immediate consensus. Hopefully this will throw a monkey wrench (so to speak) into those works.

Here is the Talking Points file (they print 2-UP) and here is the set of diagrams (they print 6-UP) to use together for this lesson/activity.

More news as it happens!

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.

Reflection on wallowing after the "Two Faces of 'Smartness'" workshop at the Creating Balance in an Unjust World conference

So yesterday I was at the Creating Balance in an Unjust World conference on math and social justice in San Francisco with Jason Buell (@jybuell) and Grace Chen (@graceachen), and I finally got to meet Brian Lawler of CSU San Marcos (@blaw0013) and Bryan Meyer (@doingmath) in person. They are (of course!) both terrific. I came away so impressed with Brian Lawler — a wonderful math education teacher and researcher as well as a fun guy and a total mensch, in addition to being my friend Sophie (@sophgermain) Germain's mentor. You should definitely follow him on Twitter if you're not already.

He and Jason and I crashed the "Two Faces of 'Smartness'" workshop session yesterday right after lunch, which was beautifully given by Nicole Louie and Evra Baldinger.  It was glorious to do math on the floor with Brian and Jason at the back of a classroom where about fifty math teachers from around the country had crammed ourselves in because we wanted to learn about this and well, honestly, we just didn't care about having to sit on the floor.

What was wonderful about it?

Well, first they are both such amazing, caring, reflective teachers and mathematicians. We were given an Algebra 1-style complex instruction group task and we worked on it deeply, in our own ways, for 15 or 20 minutes. There was so much respect for the others in the group, along with deep listening and amazing mathematical and teacherly thinking.

Some of that wonderfulness was wonderful for me was because of my own issues over the years with shame about my own mathematical thinking processes, which are usually quite different from those of other mathematicians and math thinkers I work with. Even after many years of intensive work, I still have a conditioned habit of abandoning my own thinking in favor of somebody else's — anybody else's — especially if they seem confident about their thinking. I have a sense that this is similar to what our discouraged math students often experience, the ones who prefer English class or music or social studies, because I was one of those students myself.

Jason is a middle school science teacher, so he also has a slightly different way of looking at math than Brian or I do. At one point he suggested that we verify our idea by counting the squares. Brian and I looked at each other dumbfounded for a moment, since that had not occurred to either one of us. I exclaimed, "What a great idea! It's so science-y!"

We laughed, counted, and continued our work together.

There can be such joy in a group task like that, but not if the group's working structure is set up with rigid norms that limit individual students' participation. Frankly, I just hate the "assigned roles" kind of group work because I find that it fails to reflect -- or prepare students for -- the kind of group work that happens in a real-world collaborative setting such as a software design meeting. There, engineers and product marketers and managers come together but are not restricted in their participatory roles. No one is limited to being a "recorder" or a "task manager." One person presents a starting point to kick things off, and from there everybody just jumps in with the best they have.

That's how we were working on the floor together yesterday. It was an organic, free-flowing process, and as a result, both the learning and the mathematical conversation were far more authentic than I've usually experienced in this kind of work.

There were also rat-holes — glorious rat-holes! — we chased down. At one point we had even (temporarily) convinced ourselves of the possibility of a quadratic formulation of the rule, even though we'd been told (by the title of the worksheet) that this was a linear growth function. After a few minutes, we punctured the balloon of that idea, and felt a little deflated ourselves. We'd been working for several minutes straight but unlike most of the other table groups, we had still not called the teacher over for even our first of three check-ins. I hung my head in discouragement. "We are fucked," I said. But then we laughed again, brushed ourselves off, and picked back up where we'd last seen something productive.

The end of the workshop activity involved reflection on what the others in our group had contributed to the process in a method known in complex instruction as "assigning competence." You highlight one of the key competencies that another group member demonstrated and you tie it to a positive learning consequence to which it had led us. For example, I appreciated how Jason had brought in ideas from the real-world thinking of science because it had added rigor and a verification mindset to our process. Brian appreciated how I'd been able to stay with my confusion and keep articulating it in a way that made my process visible and available for investigation.

This pleased me because it is something important I think I have to contribute. I call it my process of "wallowing." I have a deep and self-aware willingness to wallow in my mathematics.

One of the other teachers in the room asked me to say more about what "wallowing" meant, and at first I felt overwhelmingly self-conscious about speaking up. But then I remembered that (a) I was being asked by other math teachers who are passionately interested in understanding different ways of reaching students who have their own unusual relationships to mathematics, and (b) I had Brian Lawler and Jason Buell on either side as my wing men, and come on, who wouldn't find uncharacteristic courage in that situation?

So I told him that for me — and in my classroom — wallowing means learning how to be actively confused and developing a comfort and a willingness to stay present with that confusion while doing mathematics.

In much school mathematics, appearing confused is a frequent and constant source of unspoken shame. And so most people with any amount of self-regard quickly learn how to cover it up and hide it from view. I went through much of high school disguising my confusion and shame as thoughtful reflection. I became a master of avoiding humiliation in class by keeping my confusion well-hidden. If I didn't get *caught* being confused, then I couldn't be humiliated or shamed about it.

Later I would work through my confusion privately so I could show up in class always and only being able to raise my hand about something I felt I understood cold.

So I believe there needs to be a culture of allowing for wallowing in active confusion in our math classrooms. We should not be too quick to dismiss confusion or try to resolve it or spackle over it.  I would even argue we need to consider it a badge of honor and an activity worthy of our time, consideration, and cultivation. The only way to cultivate curiosity is to cultivate an environment that is supportive of wallowing — active engagement and presence in the process of being confused.

It is the deepest form of mathematical engagement I know, and it is thrilling, inspiring, and honoring to be a part of it. When a student experiences that euphoric, lightbulb moment of authentic, personal insight, we can be assured that the understanding it signals is not only deep but also durable. This is true because when you are actively confused about something, you are fully engaged in making your own mathematics. Whether it is a "big" idea or a "little" procedure, these moments of insight are the source of all intrinsic motivation. And isn't this the kind of reflective, metacognitive insight about student learning processes that we are hoping to cultivate?