Sometimes It's an Accident

We are learning about angles in grade 7.  Yesterday, I asked students to draw 10 different angles (at least 3 acute and at least 3 obtuse) and measure them with a protractor.  Historically, I've been really bad at teaching students how to use a protractor, but they made their best pass at it.

Today, I drew random angles around the classroom on our whiteboards and was going to ask for volunteers to walk up with their protractors and measure them at the board.  But before doing so, I went around the classroom asking the class to classify the angles as acute, obtuse or right.  When we came to a right-is angle, the class was divided; some said acute, some obtuse and a few said right.

"Ok, so what do we do?"

Sam picks up his protractor, holds it out in front of his face, closes one eye and peers through the hole at the bottom of the protractor.  I stepped back and watched what he was doing.  He was peeking at the vertex into the hole, while lining up one of the rays with the guides at the bottom of the protractor.  He then says, "Mr. Cox, it's pretty much 90 degrees."

Now, I loved this for reasons.  1) This kid invented a hack for doing a little better than estimating 2) The entire class understood what he was doing and started using his hack and 3) I had never thought of doing this before.

But that isn't even the good part.

I had drawn three straight angles that had a second ray breaking it into two supplementary angles.  We argued a bit about whether there were two or three angles shown.  Everyone eventually agreed that there was an acute, obtuse and straight angle represented.  Then we got to the measures.  Glad to say, the pairs they measured were all supplementary.  Then came the two students who made mistakes (on purpose).  We discussed how some students use the wrong numbers on the protractor, but if they classify the angle first as acute or obtuse, that helps them know which number to use.

That wasn't the good part either.

I wanted to start a conversation on supplementary pairs and I was going to use the drawings of straight angles broken into two supplementary angles that were on the board.  But then I thought about Sam and his protractor hack.  Change of plans.

"Ok, take a look at your protractors and look at the pairs of numbers.  What do you notice?"

We created a list of numbers.

170   10

160   20

150   30

140   40

130   50

120   60

110   70

100   80

 90    90

Then I added one more entry:

34   ?

"Without using your protractor, make your best guess about the number that should be paired with 34."

We had three answers. 

154   146   156

Argument 1:  "I think it's 154 because 150 and 30 are paired together.  Since we added 4 to the 30 to get 34, we need to add 4 to 150 to get 154."

Quick check of the class to see who understood the argument. I was careful to let them know that saying they understood the argument was different than agreeing with it.  They understood. 

Argument 2:  "I disagree with 154 because on one side the numbers are increasing and on the other they're decreasing.  34 is between 30 and 40, so our answer needs to be between 150 and 140.  So, I think it's 146."

Homework:  Who do you agree with and why?  If you think the answer is something different, make an argument.

Can't wait until tomorrow. 

I Had My Doubts

Dawson is the oldest.  Aidan is the musician/artist.  Bohdan is the baby.  And Jabin is, well, Jabin.

But my second son, Nevan just finished his first semester of high school.  He hadn't been in a classroom since the end of his second grade year and he hadn't done traditional math course work in years.  Instead, he did a lot of interesting math problems.

So, I has a little hesitant when the local high school recommended he take math 2 and chemistry as a freshman based on a single placement test.

Our conversation went a little like this:

Me:  Nevan, I know they say you're ready for these classes, but remember, you're being asked to take math 2 and there's a lot of content in math 1 that you haven't done.

Nevan:  I know Dad, but I want try it.

Me: Ok, that's good enough for me.

I mean, he wanted to challenge himself.  And even though I have reservations about acceleration in general, I knew Nevan would have support at home.   So, we decided he could give it a shot on the condition that he could always move into a math 1 class if he felt overwhelmed.

Yeah, that didn't happen.

Suffice it to say, he has done very well in both his math 2 and chemistry classes so far.

The takeaway for me here isn't that my wife and I have done anything extraordinary.  The takeaway is that my son has learned how to learn.  He reads everything that he can get his hands on, is curious and is willing to take risks.

The fact that he had some content gaps didn't matter because he was willing to do what he needed to do in order to fill them.

This really makes me wonder about all of the time we spend worrying about "gaps" in student's knowledge.  I realize that claiming this as a mindset win would be shortsighted as Nevan has advantages that most of my students don't.  However, it's a clear indication that habits of mind matter.  They matter a lot.

Amplifying Student Voice

Recently, I had a student "teach" me how to solve a Rubik's Cube.

This experience found its way into our next staff meeting.

This student and I recreated the entire scenario.  He had a cube and I had a cube.  This time I had an audience of my peers, not his.  His instructions were faster than I could follow and I got lost a few times.  He said "move top left", I went right.  He said "bottom right", I went left.  He whispers to me, "now I know how you guys feel.  This is hard."  I couldn't look at him because I was dialed into my failure.   I began to feel flushed and was tempted to just give up and tell the staff, "well, you get the point."  We didn't quit and I'm glad because the tension in the room was important.  This is the same tension our students feel when right answers matter and they don't know them.

I juxtaposed this experience with a visual pattern.  I gave very simple instructions for the staff to demonstrate what figure 100 would look like.  This wasn't a math activity at this moment; it was an opportunity for individuals to describe what they see and understand.  There was no "right way" to describe the 100th figure.  You want to draw a picture? Go ahead.  Use a table?  Sure.  How about a verbal description?  Of course.

The math notation or vocabulary wasn't necessary for everyone to enter into the task, however it could prove useful for explaining to someone else.

There are so many layers to this experience for me.

As we were going through the process of trying to solve the cube, I was incredibly frustrated.

My "teacher" was telling his story without considering mine.  He shared his connections and ignored mine.  He gave many instructions and kept going assuming I heard them and responded appropriately. I didn't.

This is where we fail our students.  We assume we have a shared understanding/experience with our students.  We don't.

A staff member later told me she was frustrated because she wasn't sure what connections I wanted her to make.  But then she said, "Then I realized, that was the point.  We needed to make our own connections."

This is what Max means when he says 2 > 4.  Or Dan when he suggests we cast students as the hero.

At least, that's what I think.

"Figures Never Lie...

...But Liars Always Figure"

I remember a professor saying this to class many years ago.  It stuck with me.

Good

Hey, look!  Since 2009, unemployment rates are going down. Wow, let's graph a regression line and marvel at that negative slope.



Un-good

But wait, over the same period of time, labor force participation has also been on the decline.



How do we help our students make sense of this?

When The Activity Isn't Enough

I love the learn by playing nature of activities like Marbleslides.  In fact, I just visited a classroom yesterday where kids were digging in.  It was interesting to watch as students engaged in this environment.  It was fascinating to try to understanding their thinking.

If we walked into 100 classrooms where students were learning about graphing lines in slope-intercept form, we'd find more than our fair share of lessons where some sort of direct instruction is happening.  We'd likely hear academic vocabulary, see a formula for finding slope and probably even a general equation like y = mx + b.

I'm not against those things.  However, I'm for giving students an experience that can be precisely described by knowing those things. Activities like Marbleslides do this.

The activity isn't enough.

Here are four different students who are all engaged in the same activity.  Consider the following questions:

What do you notice?
What questions would you ask this student?
What could you have offered this student prior to starting this activity?


Student 1

Student 2

Student 3

Student 4

Here's what I see.

Student 1 is WAGging like crazy.  These are just random guesses. No adjusting or learning from feedback.  If this student achieves success, it'd be like a blind squirrel finding an acorn.

Student 2 is an answer chaser.  I mean literally, look at the guesses.  Once this student sees which part of the equation to adjust and the line moving in the right direction, the adjustments are incremental.

Student 3 is a strategic thinker.  Slope? Nah, don't need it.  y-intercept? Yeah, that's the stuff.  Let's trap the answer and close in on it.

Student 4 is engaged, believe it or not.  This student is paralyzed by options.  Just waiting for the correct answer to pop into the brain.


So, how do you respond to each student?

Lessons in Pedagogy With Papa Frank

 AMORIS LÆTITIA 261:

Obsession, however, is not education. We cannot control every situation that a child may experience. Here it remains true that “time is greater than space”. In other words, it is more important to start processes than to dominate spaces. If parents are obsessed with always knowing where their children are and controlling all their movements, they will seek only to dominate space. But this is no way to educate, strengthen and prepare their children to face challenges. What is most important is the ability lovingly to help them grow in freedom, maturity, overall discipline and real autonomy. Only in this way will children come to possess the wherewithal needed to fend for themselves and to act intelligently and prudently whenever they meet with difficulties. The real question, then, is not where our children are physically, or whom they are with at any given time, but rather where they are existentially, where they stand in terms of their convictions, goals, desires and dreams. The questions I would put to parents are these: “Do we seek to understand ‘where’ our children really are in their journey? Where is their soul, do we really know? And above all, do we want to know?”

 When I first read this, I couldn't get it out of my head.  One sentence in particular, stood out.

In other words, it is more important to start processes than to dominate spaces.

As a father of five boys, the struggle with finding that line between holding on and letting go is real. Fortunately, my wife and I have always approached parenting through the lens of "we are preparing them to leave."  However, it doesn't make the struggle any less difficult.

It didn't take very long for my thoughts to extend to education in general.  So much of what we do dominates student spaces instead of helping them start processes.  And even if we "start processes," they're all too often processes we, the adults, determine to be important.

How do we help students determine their own processes?

How do we help them strengthen their own voice?

How often do we pretend to help students start a process when really we're just masking the ridiculous game of "guess what I'm thinking" that we'll publicly reject, but privately use as a default setting?

Questions?  I have many.  Answers? Not so much.

But that's my process.

"What I See Doesn't Matter...

"... All that matters is how you see it."

This is such a difficult thing for students to believe.  But I try to say this in some variation every day to my students.  

Grace nails the sentiment here.

points out diff btwn "show your work" as window into S thinking vs as backwards engineering what the teacher wants me to write

— grace a chen (@graceachen) November 3, 2016

 Once students begin to believe that the way they see something is the currency, then our job is to simply help them refine their communication so their audience can understand them.  Only then does the syntax of mathematics matter.

"Help me understand you."

"Help me see what you see."

These are the things we should say more often.