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.

Sunday, August 28, 2022

Two weeks in

 Students have been back in my classroom for two weeks now, and I have a number of observations ...

  • Most importantly, I don't have my "teacher stamina" yet.  After a day in the classroom, I get home exhausted, and have little energy for doing stuff I enjoy much less any more school work.  Thanks goodness my wife has been taking care of dinner!
  • I'm also thankful for wonderful colleagues to talk to and share ideas with.  The time I spend talking to other teachers is not a lot, but it is good.
  • I am really enjoying block scheduling!  I feel like we can really dig in and the students can have a chance to actually think.
  • I still don't have everyone's first name solid in my mind yet, and last names are still a mystery for now.  Also, I mis-pronouned a student on Friday, so I'll need to apologize on Monday.  Learning is a process.
  • I am really frustrated by the inflexibility of the online gradebook, and the expectations from parents and admin that it work like an ATM.  It's too hard to focus on learning (for me and the kids) when we have to track every last point.  On a related note, the students were visibly relieved when I suggested that we don't put quiz grades in the grade book.  Going gradeless will also be a process.
  • Having students solve problems groups of three while standing at whiteboards has been really satisfying.  There has been lots of thinking and discussion.  There's still a focus on "Is this right?" and I need to reflect on how that's going and how my responses are affecting the kids' view on the process.  (I also need to remember to take a picture; it's been really satisfying to watch them work.)
  • I am a little worried about "getting through the curriculum".  While we are spending time developing thinking, it does seem to be going slower.  On the one hand, we are not giving semester exams anymore, so there isn't that goalpost to get past, but then will the kids be prepared for "the next course"?
  • Many of the kids (especially in my precalculus class) are worried that the pandemic has put them behind.  The fact that the phrases "learning loss" and "students are behind" really bothers me.  This is not a race, and not understanding everything we do in class quickly is not a sign that you are bad at math.
Chalk board with the words "If you stumble, make it part of the dance. -author unknown-"  (Image by Gerd Altmann from Pixabay)

I have lots more thoughts and worries and ideas and fears and likes, but these are at the top of my brain for now.  We'll see how the next week plays out.

P.S. The department secretary got us Hagoromo chalk, and I love it!

Saturday, August 6, 2022

Ideas circling in my mind

This week, I've be reading some articles on "Ungrading" and reviewing my notes about democratic classroom techniques.  (For a beginning article on ungrading, see; and* for democratic classroom.)  I've also received an email from my department chair about how my electronic gradebook must be structured, with categories only for "Summative Assessment" (60%) and "Formative Assessment" (40%).  I have to think about this structure a bit, because I am worried that it still plays into the narrative that the grade matters more than the learning.  Also if "formative assessment" is assessment for learning**, I fear that putting points in the gradebook for this category, it becomes "assessment for points" instead.  My views on points in the gradebook have been evolving ever since I had to start using an online gradebook accessible to students and parents, and I need to think about this new requirement.  I'll write more on that later.

A circle containing a fractal image of purple spirals formed by small squares

As I think about creating a classroom space where students feel like they can trust me to give good feedback and not play "gotcha" with grades, and where they are more interested in learning and doing math than in the points they can earn, I'm recalling the democratic classroom strategy of a classroom circle.  This is an activity that starts with everyone, including teachers and aides standing close together so that everyone can see everyone else, without having to do more than turning their heads.  Ideally, the furniture should be moved out of the way so that the space inside the circle is open; chairs can be placed around the outside so that everyone can sit down at some point.  Creating a good circle is a group effort, and everyone needs to participate to make it as round as it can be.  It's a good format to start a name game or other ice-breakers at the beginning of the year.  Creating the circle is also an opportunity to begin offering feedback, as the teacher (or ideally anyone) can offer comments, such as "The circle looks flat on that side" and the group can make adjustments accordingly.

Given the disconnectedness of the last couple years, I am wondering if creating some kind of script to use every time we move into a circle might provide some consistency and community.  Something along the lines of "We stand in a circle, where we can each see each other and be seen, with nothing to get in the way, with no one more important than another.  We can expand the circle to include others, and if someone is absent, while we can still maintain the circle, it will be diminished."  Still a work in progress ...  One thought I had for the first day, where we only have 15 or so minutes for each class, is to create a circle, and go around so everyone can say their name and pronouns and one thing they would like the rest of the class to know about them.  On the second day, also start in a circle, go around so everyone can say their name and something they know, think, wonder, or imagine about circles, then adjust the script to include some of those ideas.

A circle of handprints, with fingers outward.  Each handprint is made up of several colors.

The part of me that dislikes touchy-feely stuff is asking "Where's the math??!" but I'm willing to ignore that somewhat, for the first week anyway.  I also need to review the new curriculum documents I got this week to see what math I should be including.  More on that next week.

*Most of what I've learned about implementing a democratic classroom has come from the folks at Full Circle Leadership.  Thanks to Chris Fontana for continued encouragement!

**See Dylan Wiliam's Formative Assessment: Definitions and Relationships for more info about this.

Friday, July 29, 2022

About to start a new school year!

My first official day back at school is in two weeks; of course, I'll be there a few days earlier, with the boxes full of materials I took home when I left for my fellowship two years ago.  I am both excited and anxious to return to the classroom.  It's been a busy two years, and both the school and I have changed.

I feel a bit out of practice in front of students, perhaps even more than when I was department chair for three years.  Being away from the building entirely, I feel like I've forgotten what it's like!  I know that's not entirely true and I need to put that bit of worry aside.  Also, I will be teaching the same classes that I taught before the fellowship, so the classes should be familiar.  On the other hand, the school has switched to a block schedule, the curriculum for two of the classes have changed, there has been some turnover in the administration, and I will be coteaching a class for the first time.  So who knows what my year will look like?

I do have some goals.  I want to really work on creating a classroom community using some of the democratic classroom techniques I started trying a few years ago, and adding in some of the team- and relationship-building I have learned in the fellowship.  I also want to use more inclusive and student-centered pedagogy, including what I've learned while researching un-grading techniques, Liljedahl's building thinking classrooms, historical puzzles and games, and, of course, the see-think-wonder and related pedagogies from the Library of Congress.  My more personal goals are to write more about my own practice and about finding the joy of mathematics across the curriculum.  (I'm hoping to write a summary of my work at the end of each week on this blog.)  I also plan to maintain a work-life balance and find time to read and write for myself. 

Picture by Fernando FLeitas from Pixabay 

One big thing I've learned (and continue learning) is that I don't have to get it right the first, or even the nth time.  Living and working remotely (and remotely from home as well) has helped me be more comfortable with failure or with just being in the moment and not worrying about the end result so much.  So my last goal is to remember this when I'm in the classroom as well.

Will it all work out?
Will my worries become reality?
Will I accomplish my goals?
Will the whole thing fall apart?
Who knows?  But if it does come crashing down, I'll just yell "Jenga!" then pick up the pieces and start again.

Monday, June 6, 2022

Out of my comfort zone

There are reasons I chose to teach math.  I like the thought experiments, the proofs, the problems I can think about where all I need is my brain and maybe some paper and a pen.  And I don't expect to have to deal with this:

Okay, I only saw this* in one of the specimen rooms behind the scenes at the Smithsonian Natural History Museum.  I did actually get my hands on a buckler dory, however, as I was assigned that fish to notice and wonder about.  Our latest Fellowship professional development day had us learning about fish at the Museum, with folks who specialize in all things fishy.  So when we arrived at the classroom in the SNHS "Q?rious" classroom, we were each given a card which told us which fish we needed to find in the trays around the room to get to know up close and personal.

I found mine pretty quickly, because my card said "bukler dory".  Since it was named "Dory," I looked for something that might bear a resemblance to the character in the animated movie.  It wasn't bright blue, but it did have the right shape:

We spent some time noticing and wondering about our individual fish.  On my tray, there was a juvenile as well as an adult buckler dory.  Other trays had a puffer fish (not the poisonous kind), a sting ray (again, not poisonous, but still sharp), flounder, angler fish, different types of sharks, and something that looked like an alien brain sucker, that I wish I had written down the name of.

All the fish had been caught off the coast of Massachusetts as part of a fish population survey in the spring, and these had all been frozen until they were brought out for us to study.  They were on their way to be skeletonized and preserved for the museum's collections.

I was a little tentative at first about recording my observations of the dory, and just listed what I could see.  Then I got brave and poked it a bit (with a gloved hand) and found that one part of it was kind of like a balloon, while other parts were firm or downright bony.  I actually picked it up to see the other side, and I opened its mouth to peek inside (look at me being all brave!) and that's when one of the Museum's Fellows came over to see what I had found.  Matt just got his (bare!) hands on the thing and pointed out a number of really interesting features.  He opened the mouth all the way and explained how there are bony protrusions in the fish's throat that look a bit like molars and do the actual chewing of food.  (The tiny teeth around the mouth are all pointed inward to keep the food from swimming back out.)  There's also a mechanism to keep the food from passing over the gills.  Matt stuck his fingers in the gills to show me; they were really red because that's the most oxygenated organ on the fish.  (I decided I had already handled the poor thing enough and did not need to stick my fingers in there as well.)

Here's a picture of dory with it's mouth opened all the way.  It works kind of like a vacuum to suck in as much food as it can.  (I won't be able to watch that fish movie the same way again.)

I walked around a bit to take a look at the other fishes.  (If the group of fish you're talking about has more than one species, the plural is "fishes".)  The biology and life science teachers in our group were completely in their element, happily chatting away about their fish, and hugging them like old friends.  Okay, they weren't actually hugging, but they had no problems getting their hands on (and sometimes inside) their fish.  While I don't think I enjoyed the experience as much as some of the teachers, I did have a good time, I learned a lot about fish, and I got to see the Notice and Wonder strategy in a completely new setting.

I also got to look backstage in the specimen rooms, but I'll write more about that later.  In the meantime, here's a selfie of me and dory.  (I'm the one who took the picture.)

*I didn't catch the name of this particular specimen.  It's labeled "OH MY" because it's part of a group of items that curators bring out to show off, and that's the reaction most people have when they see it.  There were several OH MYs in the specimen room we got to see.  I think they did that on purpose.

Saturday, May 28, 2022

Still Angry, but working on it.

Well, the last week has again been pretty upsetting for me as a teacher.  A school shooting and the political responses to it (or lack thereof) have been really disheartening.  Why does there seem to be a political will to do nothing about the gun epidemic (especially from those proclaiming to be pro-life), even as it affects teachers and children?  Why are teachers not trusted to teach children well, but we're hailed as heroes when we go above and beyond to do more for our students?  Why is it that in some places in this country, incorporating social-emotional learning into math class is seen as unnecessary, when we are having lockdown drills at the same time?  Is teaching the racist parts of U.S. history really so much more traumatic than having a shooting in the classroom?  I'm feeling pretty disrespected right now.

I know I am not alone.  I have talked to colleagues who are leaving the profession because of the toxic culture in education.  And a school can say, "we're not like that", but when administrators are more worried that parents will complain about bad grades than they are about their teachers' well-being or credibility, it's clear there is a link between the culture at large and the individual experiences.  It certainly makes me question my decision to go back to the classroom in August.

I hate that I feel this way.  There have been so many teachers in my family, and I've had so many teachers who have inspired me, that I can remember wanting to be a teacher since I was in third grade.  Being a teacher is so much a part of who I am, that it's hard to imagine doing anything else.

So, I am writing this blog post, reading about building Thinking Classrooms*, researching "ungrading" practices, and working on myself to provide a welcoming classroom for all my students.  I've also written my representatives in Congress about gun control.

*Specifically, Peter Liljedahl's Building Thinking Classrooms in Mathematics, Grades K-12.

Image by athree23 from Pixabay with modifications by me.