Saturday, September 10, 2022

Learning from my students

This week, I saw an algebra student try to find the factors of 15 by repeatedly multiplying numbers by 2 to see if the result might be 15.  I saw an advanced algebra student not remember how to simplify an expression using the order of operations.  And I saw the lesson of two very experienced teachers fail miserably.  (Full disclosure: the two teachers were me and my coteacher.)

But I also saw that algebra student excited to learn how to factor trinomials and differences of squares and know when she whether she was correct or not.  I saw that advanced algebra student smile when he realized he could understand function notation and its relationship with a graph when he used his calculator to do the arithmetic. And I saw two experienced teachers put their heads together to create a lesson that engaged their students, and uncovered some of the reasons why the previous one failed.

Teaching is full of large and small disappointments as well as tiny joys and enormous wonder.  And just like my students, sometimes I fail.  And when a lesson designed for an 85 minute block fails, that's a long 85 minutes.  Especially when you realize it's going off the rails in the first 15 minutes, and you keep scrambling to try to pull it together for the remainder of the block, but you keep failing for more than an hour.  Wednesday was rough.

At the same time, Wednesday had many moments of joy, like the two students who learned how to factor and use function notation.  It also included a group in one class becoming gleeful that they solved a problem unexpectedly by thinking about a different question.  And another class all gathered in one corner of the classroom to learn the definitions of sine and cosine.  (And I had forgotten about these last two moments until I was writing this post; it's not always easy to remember the good stuff when I'm trying to figure out where I went wrong.)

A two by three grid with A in the lower left corner, and B in the upper right. The puzzle was to find how many ways to get from A to B, following lines only upward or to the right.

Anyway, my coteacher and I decided to revamp our thinking for Friday's lesson to figure out what went wrong on Wednesday.  We started the lesson with a non-curricular puzzle at the white boards (NPVS's for those of you following along).  The puzzle engaged the students, most groups came up with an answer they were happy with after a few trials, and four groups explained their solving process, two of them inventing notation to keep track of their work*.  Clearly, the students have the ability to think, communicate with each other, and problem solve.

The class then moved the desks out of the way, and put the chairs in a large circle.  We stood in a circle to acknowledge the power of seeing and hearing each other, which took a few minutes, as the kids were feeling a bit squirrelly, probably uncomfortable with the process.  (Who wants to be seen and heard in a math class?!)  After we sat down, my coteacher invited comments about how everyone thought the class was going.  We had asked questions three weeks ago in this format about what everyone hoped for and what success would look like for them in class.  This time, it was quiet for a moment, until one student asked if we really wanted to hear stuff, and could they be honest?  We answered yes, and another student quickly started the conversation with "I don't feel like I'm learning anything in this class."  And that opened the gates.  We talked about all kinds of issues:

  • We don't like the random groups every day.
  • We do like the groups we sit with (which were self-chosen).
  • We don't like working at the white boards.
  • After we work at the white boards, we don't have enough time to practice.
  • We don't know everyone's name.
  • We spend too much time on get-to-know you activities.
  • There's too much homework.
  • We need a break during class.
And so on.  There was one moment that stood out for me:  One of the students said that he didn't think the homework was too much, but he doesn't like doing it, so he just copies the solutions.  A few other kids jumped on him for his opinion about the length of the homework, and he started to retract the statement.  I interrupted to tell him that how he was feeling was perfectly valid, that he clearly spoke only for himself, and that I would not stand for other students ganging up because they shared a different opinion.  As the conversation continued, other students were making general statements about the class, and I continued to push them to speak only for themselves.  More students started using statements that started with "I feel ..." rather than "The class ...".

The circle discussion took longer than I had expected, and we did not get to any math content.  However, the students did agree that sticking with the same random group for the white board work for one week would be okay and that spending less time on the white boards each day and more time consolidating the ideas and then practicing them would be helpful.  The points about taking a break and homework length got tabled for a future discussion.  Everyone helped put the desks back into the pods formation, and we had a few minutes to hang out before the bell rang.  While students were having individual conversations, I checked in with a few students who had not spoken during the circle conversation.  A couple agreed with what was being said, and offered me their viewpoints.  Two students said that they liked the class the way it was, but didn't want to say anything in front of everyone else.  I also thanked the student who commented about his experience with homework for speaking his truth.

After the students left, my coteacher and I agreed that the conversation had been a good one.  While a few students checked their phones during the discussion, the engagement level was much higher that it had been on Wednesday, and the time on phones was significantly smaller.  We discussed the pros and cons of doing a white board problem on Monday, with the shorter class periods, and decided to go for it.  We believe the time at the white boards is some of the most important time we spend, and neither of us is willing to sacrifice that.  Given the length of the conversation with the students, we'll have to reschedule a couple things to account for the missed content time, but overall, we believe the trade-off will be worth it.

We'll see what Monday brings.

*One group used a series of shapes to track their work, another used series of arrows.

Monday, September 5, 2022

Constructive Arguments

Last week felt pretty good.  I'm trying to use "Non-Permanent Vertical Surfaces"* in my math classes on a daily basis, and I'm happy with what I am seeing.  In most cases, I present a problem, send the students to the whiteboards in their random groups, and watch to see where their thinking goes.  Some days, they make wonderful mistakes, and I gather everyone around a particularly interesting whiteboard to quickly talk about the good thinking, the correct paths, and the miss-takes that lead to the very interesting solutions.  Then, I send the kids back to boards, and watch as they discuss what they had been thinking and make revisions.  Some of the best days are when this process goes through a couple iterations, and students thoughtfully revise their work several times.  I always bring them back together afterward to focus on a couple of the solutions, and highlight the vocabulary and formalize the thinking.

It's been great fun to plan and watch, and the kids seem to enjoy the process.  One day this week, I needed some additional space to make some notes, and erased one of the whiteboards, and the students from that group complained that I did so.  They clearly took pride in what they had been thinking (as most kids appear to so, since they often take pictures of their work before the end of class when we erase all the boards).  I apologized to the group for erasing their work prematurely, and promised I would not so that again.

In my Precalculus class, we're starting our unit on the trigonometric functions.  To start class, I sat in the middle of the floor with the kids gathered around, and drew a picture of a bicycle.  Apparently it was a really bad picture, as the pedals were not connected to the frame, and no one was sure which side had the handlebars and which side had the seat.  After straightening that out, I indicated that the bike had ridden over a piece of gum which got stuck to the wheel, and rotated around.  I asked the students to make a graph showing the relationship between the height of the gum from the ground and the time.  Here are some of the results:

Photos of three whiteboards.  The first shows several different types of curvy lines, the second has one curvy line similar to a sine wave, and a circle with some notations, and the third shows a graph with a series of straight-sided v-shapes.
To start the conversation, I asked the students who drew the graph on the left to explain their thinking, as they had a great discussion on whether the horizontal axis represented time or distance traveled.  The other students agreed that I had asked them to graph the height of the gum in terms of time, but spending a little time on the drawings allowed us to preview a cycloid graph, which I plan to discuss later in the course.  

The other two graphs in the picture represent the work that appeared on all the other whiteboards.  The students with the pointy graph stated they believed it would be made of straight lines, since we had said the bicycle was traveling at a constant speed.  Great connection between constant speed and linearity!  That convinced most of the curvier graph groups that they had made a mistake.  A couple curvy graph groups stuck to their ideas and tried to explain that the vertical height of the gum was not traveling at a constant rate, even though the bike was.  It was a great few minutes of argument, after which I suggested we "do some math" to look at the evidence one way or another.
A circle with five radii: one to the bottom, and the other four spaced 45 degrees apart, travelling up the right side of the circle.  The ends of the first four radii are labeled A, B, C, and D respectively.
I drew a version of this diagram on the board, and everyone agreed that since the bike was traveling at a constant speed, the gum would rotate from point A to point C in the same amount of time it would rotate from point B to point D.  The pointy-graph folks were feeling pretty good.  Then, with some special-right triangle geometry (which we had to take a detour to justify, since that was something the students would have seen in Geometry, when they were doing school remotely) we showed that the vertical distance from Point A to point C was shorter than the vertical distance from point B to point D.  Since the BD distance was greater, the gum was moving faster in a vertical line there than along the AC distance.  A small existential crisis began arise among some of the pointy-graph-constant-speed kids, until another student pointed out that if you looked at the wheel edge on, you would see the gum moving only vertically.  Some discussions at the tables ensued, and everyone seemed more comfortable with the graph actually being curvy.

In previous years, these kinds of discussions and willingness to revise work and thinking occurred much later in the year, if they occurred at all.  Using the random groups with the NPVS's, and having a block schedule to give us time to explore made the rich conversations possible.

I can't wait to see what happens this week!

*Non-permanent vertical surfaces and visibly random groups are two of the strategies outlined in Peter Liljedahl's Building Thinking Classrooms in Mathematics.