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. :)

Impromptu Twitter master class on homework strategies

On Friday afternoon Anna (@borschtwithanna) tweeted out this call for conversation:

Hey #MTBoS, teach me how to go over hw. I give answer keys, but it still takes like half the period to answer questions & make connections
— Anna Blinstein (@Borschtwithanna) April 24, 2015

What followed was a virtual master class on handling homework in different settings. It's the sort of conversation that the #MTBoS excels at, pulling in thoughtful responses from teachers at every level of practice.

I've been thinking a lot about homework because I am overhauling my homework strategy for next year.  This blog post is my attempt to capture my strategy overview, along with pointers to resources that have helped me think through what I need to do.

Overall HW strategy for next year

There are four pillars to my homework strategy for next year:

1.  HOMEWORK LAGS CLASSWORK: I am implementing  Henri Picciotto's strategy of having the majority of each night's homework "lag" the current classwork focus of the day; 
2. EACH DAY'S INTRO TASK IS TABLE GROUP REVIEW: the first ten minutes of class will be for students to discuss/review/help each other out on the previous night's homework problems together in their table groups; 
3. REFRAMING THE ROLE OF THE TEACHER: I will take questions on particularly troublesome homework problems, but I will take whole-group questions only; and 
4. COLLECT, STAMP, & GRADE HW PACKETS EVERY 2 WEEKS: I collect, stamp, and grade homework packets every two weeks, grading for completeness of effort (every problem in every night's problem set attempted). 

Here are my elaborations on these pieces.

1. HOMEWORK LAGS CLASSWORK

Henri Picciotto's blog post on this is a classic. There are so many valuable things about having the majority of each night's homework "lag" the current day's classwork. First and foremost, it ensures that most of every night's homework is accessible to every student every day. Secondly, it helps with heterogeneous classes. It gives students multiple at-bats for each kind of homework problem, keeping things meaningful for everybody. New material challenges proficient students, while those who are still working toward proficiency get multiple opportunities to work toward master.

2. DAILY INTRO TASK IS TABLE GROUP HOMEWORK REVIEW

The first ten minutes of class are the time for students to get help on homework — and by "help" I mean helping each other first. A big function of math homework in my view is to help each student cultivate an autonomous and independent approach toward their own struggles with their own problems.

A.H. Almaas describes the problem like this: "Many people... unconsciously act out a desire to be 'saved' by a teacher. But if a teacher 'saved' you, you would lose something. You would lose the value of struggle" (Diamond Heart Book One, p. 123).  In my view, the first ten minutes of class are the place where I expect students to begin to shift their mindset about the value of struggle. If a problem is not pushing beyond their Zone of Proximal Development, I expect them to develop the habit of resolving their own confusions themselves.

3. REFRAMING THE ROLE OF THE TEACHER

In keeping with #2, I want to be conscious and intentional with framing my role in their lives about resolving their own problems with problems. Almaas describes this reframing better than anybody I know:

So there are two ways to approach the teacher. One approach is to hope the teacher will take away your problems; the other is to use the teacher, not with the expectation that she will take away your problems or offer solutions or "make it better" but that she will give you a little push in your struggle. (DH Book One, p. 124). 

This is why I love Dan Anderson's (@dandersod) description of the "mass confusion" rule: unless a problem causes mass confusion, students have to work out their problems independently and help each other out on during that first ten minutes of class.

4. COLLECT, STAMP, & GRADE HOMEWORK PACKETS EVERY TWO WEEKS

This has been the most surprisingly successful thing I've done this year. Students have to turn in a stapled packet of their "Home Enjoyment" problems every two weeks. I collect it, I stamp it, and I give them a grade for it every two weeks. Every problem must have been attempted. They should show effort to have sought a resolution for problems they didn't understand the first time through.

This has been a great accountability practice for students. It's an easy, easy 'A' grade every two weeks plus it gives them that push to keep themselves on track and not fall behind on homework. In a high-achieving school filled with motivated students, I expected not to have to do this, but in fact, my experience has revealed the reverse: students appreciate that little push toward accountability. It triggers their automatic reflexes in a way that supports their autonomy.

It also takes me very little time. I am basically just stamping packets, looking for effort and gaps, and rendering a grade whose default is 100% unless stuff is missing or late. I take off a 10% late fee per day. That is usually the only penalty students incur. I've been astounded by how being an old-school hard-ass about this has simplified and streamlined the process.

I hope this is helpful!