Doing It Wrong ...

Just before Spring Break, we were finishing up a unit on sequences and series in my pre-calculus classes, and I started thinking about review problems.  I thought about the Handshake problem, the diagonals of a convex n-gon, constructing a Sierpinski Gasket and related figures, patterns in Pascal's Triangle, and a few other "puzzle-style" problems.  Some of these showed up in some form in the homework problems I had assigned during the unit, but I realized that I had not really used them in class.  Looking back through my plans for the unit, I found I had pretty much followed the textbook, and in the process probably reduced a really important set of ideas to a bunch of formulas, and lost some of the excitement I usually had for the subject.

I had taught the unit wrong.

I didn't teach any of the content incorrectly.  I didn't leave out important ideas that the students will need as they move into the next course.  I tried to make sure everyone learned the content, I checked for understanding as we went along, and I gave opportunities for students to process new ideas.

But I had lost the "story" of the unit.  A colleague once told me that she did a lot of up-front planning on a unit, and on a course, because she really wanted to understand the content as a story, with a beginning, middle, and end, including plot twists and cliff-hangers.  My sequences and series unit had all the important pieces, but lacked any narrative thread to bring the ideas to life.  The result was I had students really worried about memorizing formulas, asking "Is this on the test?" or panicking because they were having trouble with patterns outside the typical arithmetic or geometric ones.

I can't go back and reteach the unit a better way.  We don't have time left for that in the school year, and once the students have seen the major plot points, rearranging them or weaving in story details isn't going to make the story more coherent.  So, I now have a big note on the first page of my planning notes for the unit to get back to the story.  And I'm working on the narration for our next unit.

Musical Chairs

If you'd seen my classroom in the last many years, you saw that I have my students sitting in groups of three or four, with their desks turned inward, toward each other, or even sitting at tables rather than desks.  I set my classroom up like this a few times when I taught junior high back in the early 1990s, then went back to rows for the most part until I decided that having students talk, a lot, in class was the best way for them to learn math.  So I got tables and desks that could easily be arranged in pods, and that became my standard room arrangement.  I did not completely know what I would do to make the situation work, but I thought that if I did not force the issue, I would probably continue to put off really using the collaborative work that I knew would be good for my students.

So the first couple of years there were lots of bumps and scrapes, with plenty for me and the students to learn.  I realized that I needed to do more ice-breaker activities, because if I wanted students to work together, they had to know each others' names and get comfortable with each other.  I had to write lesson plans that included some pretty significant problems that would insure the students would not be able to complete them on their own.  I had to let go of some of my more controlling impulses, and allow for varieties of solutions. I also had to come to terms with some off-task behavior; after all, if the students were going to talk to each other, it was unreasonable to think they would talk about nothing but the math.  Logistically, I had to write different versions of quizzes, and move the desks apart when it came time for tests.  I also had to think about how to divide the students into groups: randomly? based on their current abilities?  their ability to get along with each other?  their comfort level with each other?  (In the end, I usually used random group assignments, and let the students know that I expected and believed that they could learn from and work with anyone.  But I also rearranged the home-groups from time to time, and regrouped students for individual lessons as needed.)

Fast-forward to this year, and visitors to my room now see that I have the students sitting in a kind of double-'U' shape, with the open end of the 'U' facing the screen at the front of the room.  I can still quickly group them as needed, with a pair from the inner 'U' matched with a pair from the outer 'U'.  Why the change?  I was not as happy with the students' interactions or their focus on the lessons while sitting in their pods.  Also, I was having trouble getting a discussion going across the room, as students tended to focus on their small groups, and ignore students in other groups.  I spent part of the semester break thinking about what was happening, and why this year was different.  Perhaps I was out of practice with orchestrating collaborative learning.  Perhaps the attached chairs and desks made it more difficult for the students to comfortably see what their group-mates were doing.  Perhaps the problems I was asking the students to work on were not as engaging as they needed to be.  I wasn't sure of the reason, but I knew that I needed to do something different; I thought all of us could benefit from shaking things up a bit.

The students were a little surprised by the change, but adapted quickly.  Their focus is more on the math and less on social topics, I think.  I do have to be very specific about when I want them to talk to each other, and prompt them to discuss their work with their group-mates.  Whole class discussions seem a bit more focused, and students are starting to respond to each other, even across the room.  I am worried that spontaneous collaboration will suffer, but I have noticed that some groups will start working together without prompting; I need to think about how to encourage that with other groups.  Some students have commented that they like the new arrangement.

I really have three take-aways from my game of musical chairs.  First, the seating chart is a powerful tool that can reinforce the big lessons I want to teach.  Second, just because one strategy has worked well for a long time, it doesn’t mean that it will work forever, or that there isn’t a more appropriate or effective strategy.  And finally, just like my mother used to move the furniture around in the living room when she felt like life was getting a little dull, creating a very visible change in the classroom can shake up an atmosphere that might be starting to feel old and tired.

I might go back to pods eventually, but I want to play with this arrangement a little more.  I also want to get one of the instructional coaches in so I can get another pair of eyes on what’s happening now.

By the way, thanks to Tools for Teaching by Fred Jones, The Skillful Teacher by Jon Saphier, and two colleagues who have been using the double-’U’ for their inspiration and guidance.

How time flies!

I realized the other day that I had not posted a blog entry since school started this year, and almost an entire semester has gone by!  My intent was to capture some of the things I was thinking as I returned to the classroom this year.  Apparently, I was very optimistic about how much time I would have.  Here are some quick bullets listing things I should have written about, and maybe still will ...

• First week jitters are not just for novice teachers.  Having been out of the classroom for a few years, I was feeling really nervous about starting back.  I also realized, about a week into the school year, that I was having to relearn many of the routines and practices that had been second nature to me prior to becoming a department chair.
• The amount of time it takes to prepare good lessons, grade papers, and respond to all the stuff teachers are presented with each day takes a lot more time than I remembered.  (And more time than my family anticipated, I think.)  I also feel like I did not pay enough attention to some really important issues:  More students than I would like have been performing poorly in one of my classes; I am apparently unable to write a test that is doable in one class period; and I still feel like I slide into lecture-mode far too frequently.
• Lots of good things have happened ...  The desmos graphing project I assigned to my 2 Algebra students was successful in getting them to think about functions differently.  A couple of students who had been struggling started coming for extra help, and their grades are starting to improve.  I have developed a pretty good sense of community in my classes.
• I participated on a panel discussion about the Common Core at the College of DuPage and co-presented a one day workshop on teaching Trigonometry through the Metropolitan Mathematics Club of Chicago.
• I tutor only a few students (and none from ETHS), and between them and some of my ETHS students, I am (re)learning that students struggle for lots of reasons, but two reasons appear to be almost crippling, mathematically:  serious misconceptions about how numbers behave and a a really entrenched belief that math is impossible to really understand.  Both of these are fixable, but I haven't figured out how for all my students and tutees, yet. 
• Being "the boss" and then returning to the classroom as a peer includes all sorts of awkwardness.
• A question I had before becoming department chair has resurfaced for me, now with some serious soul-searching about my role in answering it or not:  Everyone expects teachers to inspire their students; whose job is it to inspire the teachers?  
• Winter Break is quite possibly as good as chocolate.

I'm sure there's lots more I can write about.  I'm going to try to write more frequently in the new year.

My Favorite Math Memories

School starts tomorrow, and I am really excited to have my own classes for the year again.  One of the assignments I give my students on the first or second day is to write a math autobiography, including their best, worst, and earliest memories of math.  I wrote about my earliest math memories here.  I thought I would share some of my best memories today.

I recall several moments working with students after I started teaching; I think about these often, but they are more about teaching than about doing math, so I'll save those for another post.  The other good memories that stick in my brain are two episodes from college math classes.  I'll write about the one that occurred later here, and save the other story for another post, I think.

Abstract Algebra was a three-course sequence, of which the first two courses were required of all math and math-ed majors.  The third course was an elective.  This is the course that studied groups, rings, and fields, and included lots of strategies for proving that a particular set of numbers could be classified as a group, ring, or field.  (I wrote about a discussion with my younger son around the topic of groups here.)  Since it was a required course, the first class started with about 30 students, which was about as big a math class as there was at DePaul at the time.  My favorite college professor, Jeff Bergen, was teaching it.  He had a great sense of humor, was very patient about answering questions, and explained complicated ideas well.  Nevertheless, the class was difficult, and by the midterm, only about 24 students remained in the class.

The second course began with 18 students, and was again taught by Dr. Bergen.  He had taught most of us in Calculus and Differential Equations in prior years, and we all knew each other to some extent.  The class was interesting and fun, but still difficult, and only 12 of us remained by the midterm.  I can't recall a set of math classes I have worked harder in, but I asked lots of questions, spent many hours on homework, and did well overall.  Knowing Dr. Bergen would teach the third course, I signed up for it, even though it wasn't required.  Only two other students signed up, and I thought the class would be cancelled.  It wasn't.  The three of us met in Bergen's office, and furiously took notes while one of us or Bergen solved problems on the board.  Talk about pressure!  No place to hide, no other students to answer the hard questions, and it was great!  But still beastly difficult.  By the midterm, only one of us (not me!) had a decent grade, and I had to think long and hard about whether or not I would drop the class.  I spent at least one sleepless night walking around campus trying to weigh my options, and thinking about the different strategies I would have to use for studying if I was going to stay in the class and have any hope of passing with a reasonable grade.  (At that point, I would have been happy with a C, folks.)  I talked to Dr. Bergen about my worry, and he suggested a couple more strategies.  I stayed in the class, along with the one guy who was passing, changed how I was studying, used more office hours to get more questions answered, and had a great time.  I did pass the class, and I will never forget the satisfaction I felt in completing all three courses, working through the difficulties I faced, and coming out with a better understanding of not only the math, but also of my own learning capabilities.  That was probably the lowest math grade I've ever received, and yet that is the one I am the most proud of.

High Expectations

The Fields Medal, viewed as the highest award given to mathematicians, was recently awarded to Maryam Mirzakhani, a mathematician of Iranian descent currently working at Stanford University, and studying the topology of abstract surfaces.  There is an article about Dr. Mirzakhani here.

What struck me when I read the article was that Dr. Mirzakhani did not set out to be a mathematician, but thought early on that she wanted to be a writer.  During her first year in middle school in Iran, Dr. Mirzakhani did poorly in her math class, her belief in her ability stunted by a teacher who "didn't think she was particularly talented".  The article goes on to say, "The following year, Mirzakhani had a more encouraging teacher, however, and her performance improved enormously."

I continue to be amazed at the power teachers have in demonstrating their belief (or lack thereof) in their students.  And I often wonder if I am consistently sending those positive belief messages to each of my students.  

Sometimes I know I have been successful.  I remember Laura who would tell me "I can't do this" whenever we started something new.  At the beginning of the year, I would respond "Of course not, we just started learning it.  But you'll get better at it; I'll help you."  As the year went on, I responded, "You can't do it, yet, but you will."  At one point in the second semester, Laura looked up from the problem she had barely started and again said, "I can't do this."  She looked at me, sighed, and said "Yet," then went back to work on the problem and solved it correctly.

Other times, I don't know if I am as successful, so I am always on the lookout for ways to make sure I am sending the messages, "This is important; you can do it; and I won't give up on you."  Recently, I came across Richard Curwin's article, Believing in Students: The Power to Make a Difference and was reminded about some of the things I can do to communicate positive expectations for my students.

It is my hope and my intention to demonstrate to my students my belief in their ability to succeed, through my words, my grading policies, how I build relationships with them, and through every one of the hundreds of decisions I make in a class period.  Each of my students needs to leave my class feeling like I cared and believed in them.

And not just because one of them could be a future Fields Medal winner.  

Because each one of them deserves it.

My Earliest Math Memories

When I start teaching a new group of students, I often ask them to share with me their "Math Autobiographies", a brief written description of their best, worst, and earliest memories of doing math.  Reading these helps me get a better understanding of where my students come from, and who they are at this point in time as math students.

I don't usually share my Math Autobiography, but I thought this blog would be a good place to start.  So here's one of my earliest math memories.  I'll share some of my best and worst math memories later.

When I was in second or third grade, I remember doing lots of rote problems for classwork and homework.  Multiple-digit addition and multiplication sticks in my brain for some reason.  I also remember doing many long division problems, but those probably came later.  Anyway, I remember not really enjoying doing lots of addition and multiplication, but there were these problems at the bottom of the page in a brightly colored box, with a little cartoon monkey hanging on the side.  These were the "challenge" problems, and I can remember rushing through the rest of my work, just to see what the challenge monkey had in store for me.  I think there was one that showed a line of squares, with common edges, and the question was something like how many sticks are needed to make a line of five squares?  Ten squares?  100 squares?  Maybe I'm just making that problem up from another memory, but the excitement I felt about that challenge monkey and the more interesting problems it signaled was real.

I tried to do an internet search for the text book, but not knowing anything other than the years I might have used the book, I wasn't able to find an actual example of the monkey and his challenge problems.  I do still enjoy challenge problems, however, and I will still happily work through some "grunt work" if I know it will help me reach something interesting at the end.

Skillful Teaching

Last week, I was in Massachusetts attending several days of training through Research for Better Teaching on how to teach their "Studying Skillful Teaching" course.  One of their instructors taught a class of about forty teachers, and my teaching partner and I observed, then debriefed the lessons afterward.  I'd been through the training before and I've taught the class for several years at my high school; my teaching partner will be teaching the class with me for the first time in the fall.

Throughout the week, I was struck, again, about how important it is for teachers to maintain their "learning chops" if they want to do a good job in the classroom.  Ann, the instructor leading the training, always modeled a learning frame of mind when teaching the class.  "Modeled" isn't quite the right word there, since it implies a not-quite-real thing.  Ann really lives the learning frame of mind, and tries to make it explicit to the teachers she works with.

One day, it was apparent that a number of the teachers taking the course had not really understood an idea about formative assessment that Ann had presented earlier in the day.  During the debriefing at the end of the day, Ann and the rest of us observing looked at the responses on the exit slips and tried to come up with reasons for the misunderstanding.  A couple of us wanted to go right to solutions and how to fix the mistake, but as a group we persevered in looking at the teachers' work to really try to understand the misconceptions they had.  We thought about what the teachers who did show understanding might need, and we talked about the context of the next day, in which time for reteaching would be tight.  In the end, because we spent time digging into the errors and trying to match the reteaching strategies to the students, to the constraints of the day, and to the flow of the content, Ann was able to really address the mistakes, and the responses of the teachers the next day showed improvement.

I mention this example because it really illustrated for me the teacher-as-learner frame of mind.  At no point in the discussion did any of us put on our "expert faces" and say "this is how you should teach this".  Rather, we asked lots of questions and listened to each other thinking out loud.  As a result, the analysis of student work, exploration of alternatives, and collaborative synthesis of a reteaching strategy produced a much more powerful and effective lesson the next day.

As I teach my classes, I try to put myself into a learner frame of mind, and examples like this inspire me to keep working at it.  So thanks to Ann, Jon, Ganae, Karen, and Nancy, and to all the teachers who participated in the course last week.  I appreciate the chance to hear from all of you; your questions, challenges, and insights help me deepen my thinking about what it means to be a skillful teacher, and my students will be better for it.