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.

There's No Place Like Home

After serving as Department Chair for the last three years, and not teaching any classes, I am giving up the position to return to the classroom next year.  This week, with the school year over, I have started moving things into the classroom where I will be teaching next year.  I started by rearranging the desks into groups of four (my preferred seating arrangement) from the rows the previous teacher typically used.  I knew the desks would all be moved into the hall so the carpet could be cleaned, and I have no idea what formation the desks will take when they are moved back in, but I felt like I needed to put my house in order.

After twenty minutes of rearranging, checking sight lines and walking space, I stood in the middle of the room, and took a deep breath.  Remember that feeling when you arrive home after a vacation that's been fun, but you're glad to be back on really familiar and comfortable ground?  Standing in the middle of my new classroom, that's what I felt in that moment.  The last year, and especially the last six months as I have been transitioning out of the department chair role and helping someone new take over, have been emotional for me, and I have sometimes questioned my decision to return to the classroom.  In that moment, standing in a classroom bare of decoration, but with the desks arranged the way I like them, the worries of leadership drained away, and the thousand details about teacher schedules, new student placement, and department office construction, all disappeared, and for that moment I was exactly where I belonged.

I am looking forward to my students joining me in my new space.

What I Have Learned from Biology

About seven years ago, I was at a point in my career where I thought I was doing pretty well.  I had been teaching for 18 years; I had been an instructional coach in my department for a couple years; and many of my students would tell me at the end of the year that they enjoyed my class.

Then my older son entered seventh grade.

All of a sudden, his grades started to mean something, and I started paying more attention to what his teachers did or did not do in their classes.  Additionally, I could see his maturing face reflected in the students sitting in front of me.  And I started to ask myself, “If I know my son so well, and I want his teachers to teach him well, am I really doing the best I can for all the kids in my classes?”

My teaching started to change (again).  Looking back, I can identify a few rules I started to use in my classroom, because I wanted to be for my students the kind of teacher I wanted for my son.  By the way, these are rules for me, not the kids, and I am listing them in roughly the order they started showing up more clearly in my practice.  (Not that I wasn’t aiming for these all along, I just became more aware of them and tried to implement them more deliberately.)

Every kid deserves love, attention, and belief (or, Playing to a student’s strengths is more effective than trying to fix his weaknesses.)

My son’s seventh grade Social Studies teacher told me early in the year that he was hoping to have one of the boys near the top of the class when they graduated from eighth grade, and he thought Evan could be that kid.  I was a little surprised.  I knew Evan had been doing well in school, and his primary grade teachers always seemed to have plenty of love for him as well as for the other students.  But primary teachers are supposed to love the students, right?  To hear a junior high content area teacher talk about his belief in and care for a student made me think.  I had taught junior high math for a couple of years early in my career, and after 18 years, I knew I really enjoyed getting to know the students.  But I wondered, did they know how much I cared about them?  

At about the same time, I was teaching a particularly challenging freshman algebra class.  Nikki was an especially prickly student, prone to interpret the slightest gesture or facial expression as an indictment of all she held dear, and ready to defend herself, verbally and physically, at a moment’s notice against perceived attacks from other students or from me.  She was one of several students in the class with all sorts of guarded behaviors about school, about math, about white male teachers.  She could also be very generous and protective of others, loved her grandmother, and had a soft spot for small animals.  And she could really do the math when she took some time and thought about it.  When I started seeing Evan in Nikki, I refocused my efforts with her on the good things I knew, and not just on getting her to cooperate or do the math.  I don’t remember how well Nikki did in the class, and there was no moment that someone would make a movie out of, with her or any of the other students, but I felt more calm about the class, seeing her and everyone else not as problems to be solved, but as individual children, with needs, wants, fears, and hopes, whom I needed to take care of.

The next year, I resolved to call my students’ parents and guardians within the first week of school.  All of them.  I wanted to get to know them better from the start, so I could see those good things right away.  A veteran teacher in the department had been telling the rest of us that she did this every year, because, as she said, “The families in the community are sending us the best children they have; they’re not saving the good ones back.  We need to know what makes each of our kids the good ones.”  I always thought I did not have the time to make this kind of effort, but it really paid off.  The students knew I would call home, and not just about “bad stuff”.  The parents appreciated my reaching out, and the chance to say something good about their children.  Both saw this as a sign that I cared about the students as individual human beings.  And I got to have a better understanding of my students.  I could really see them.

Failure cannot be ignored.

My son did not become his eighth grade class valedictorian, although he did well, and continued to do well in his high school classes.  There were times, however, when my wife or I would get worried about something he was struggling with, and we would intervene, checking on homework, or quizzing him on his vocabulary words, or helping to edit a paper or review for a test.  Then one day, I was watching my students take a test, and I remember the sudden recognition I had.  The way they sat, the looks on their faces, the twiddling of pencils, the scratching of heads -- this was what Evan looked like when he was working hard or concerned about something!  Then, Evan started really having difficulty in one of his high school math classes, losing points for small mistakes, and feeling like his teacher was not really answering his questions.  His teacher’s response was, “This is how it is in college, and I have to prepare the students for that.”  (In the teacher’s defense, he was fairly new to the profession, so I tried not to assert my own experience or my new position as Department Chair at my school.  Honestly, I did try, but he wanted to know what kind of help Evan was getting at home, and whether I thought I was good at math.  The conference ended more quickly than anticipated.)

My take away from all of this was that I could not let a bad test grade, a bad quiz grade, or even some bad class work go by without some intervention on my part.  My sons deserved teachers who would care about their academic welfare like I did, so my students deserved nothing less.

This doesn’t mean I sit down with the struggling students one-on-one and provide the individual help I sometimes give my sons.  All of my students struggle at some point, and for lots of different reasons, and I am only one person.  But I try not to let even small failures go by without some sort of response or intervention.  It might be a probing question to a small group, or an encouraging word to one student, or a new example for the class, or an individual or small group help session, or a quiz retake, or an entirely new lesson the next day.  It may mean that I call home, or talk to the student about studying differently, or consult with the student’s other teachers and counselor.  Am I always successful?  No.  Do students still fail my class?  Sometimes.  But I have to try something; I can’t ignore failure and pretend it didn’t happen.  Failure is something we grow from, not something to hide from.

Being flexible is not the same thing as having low standards.

“Take out the garbage.”

“Okay, Dad.”

A couple hours later, “Did you take the garbage out yet?”

“In a few minutes.”

Just before going to bed, “Pick up day is tomorrow; those bags need to get off the porch tonight.”

“I’m on it!”

Despite the confidence tinged with exasperation of the last statement, the opossum that lived down the block was the one who took care of the garbage overnight, and my son had to clean up the mess strewn across the porch the next morning.  I think the same scenario played out one or two more times, and the local opossum family can still be seen checking out our porch on occasion, but there are only so many times one has to clean up a mix of coffee grounds, vegetable peelings, and other trash in a five foot radius before one realizes that taking out the garbage when asked is actually the easier option.

Natural consequences provide far better motivation for getting things done than “Because I said so.”  And both my sons respond much better when faced with options.  So when a student came and told me that she couldn’t do her homework on any Tuesday or Thursday because she had to work those nights and take care of her baby, I had to make a choice.  Do I stick to my homework deadline policy, or do I allow for some flexibility?  What about the other students?  They did not all give me reasons for why homework was not getting done.  Should I allow for some flexibility with them?  Wouldn’t allowing late homework be lowering my standards?  In the end, I decided to try letting students turn in homework late up to the Monday following the due date.  I explained that this would allow for a couple days on the weekend for them to catch up if they got behind, but it would still be best if they did the homework every night.  And I still pestered them if they didn’t turn in homework every day, and I still recorded a zero in the grade book when they didn’t have it.  But I gave full credit if they turned in the homework the next week.  It became quickly apparent that more students were getting higher homework point averages than before I instituted this policy.  And on the anonymous surveys at the end of the semester, only one student admitted to copying homework under this policy.  I probed this with the students later, and found that since the pressure to complete assignments in one night was off, the students said they could spend more time on the work and didn’t need to copy.

So, building in some flexibility actually allowed my students to take more responsibility for their work, and the standard for production went up.  I still had to remind them about the assignments, but I could focus on the learning they were to get from it rather than on the deadline and the missing points.  I can do the math on this one: having rigid deadlines does not equal having high standards.

Teenagers need to test their freedom.

This is kind of a corollary to the last one about flexibility.  Nothing tested my flexibility or my patience as much as my son learning to drive, taking the car to go out with friends, and calling home at 11:30 to ask if he could stay until midnight, when he was supposed to leave at 11:00 so he could be home by 11:30.  My reasoning, “Because I said so” was not convincing for Evan, although it still seems pretty good to me.  Unfortunately, the resulting power struggle always ended in some sulking on both sides the next day.

Pushing boundaries is one way my sons, and my students learn what they are capable of, and what “society” will allow.  I need my sons to become independent human beings and my students to become independent thinkers and problem solvers.  I need to work to make myself, in some ways, unnecessary.  (For the benefit of my younger son, Ryan, who reads this blog, please note that becoming unnecessary is not the same thing as becoming irrelevant; I can only take so much boundary pushing at a time. :) )  At home, this means I have to let the boys, and sometimes push them, to try new things, and to take responsibility for their choices.  (Evan going away to college was a lesson for him and for me in this regard.)

In the classroom, I have to provide opportunities for my students to try the problems, make mistakes, fix their errors, and own the content and their thinking.  They need to work with and learn from each other as much as from me, to make choices about and use a variety of resources, and to make connections between lots of ideas.  These are not lessons students learn on their own, and if I only think about how I am teaching the math, then it is likely the students will miss those more important lessons.  

Beliefs, persistence, flexibility, and independence.  I don’t think these things were completely missing from the first 18 years of my career, but I don’t think I was as aware of them as I could have been until I saw my sons’ faces when I looked at my students.  Being more aware has made me a better teacher, and I will forever be grateful to the boys for teaching me their lessons.

Dinner with Authors

Last week, I had the opportunity to have dinner with and hear talks from both Carol Dweck and Zal Usiskin.  Carol is the author of Mindset which I have found very relevant to how I teach.  I worked with Zal on the third edition of UCSMP Precalculus and Discrete Mathematics and he is also one of the authors on another book I have enjoyed, Mathematics for High School Teachers: Advanced Perspective.

Carol Dweck had been invited to speak at ETHS, and the speaking engagement last Friday and the dinner on Thursday were sponsored by the Family Action Network.  The "Jeffersonian Dinner" was attended by a couple dozen people with connections to FAN; I was invited by my principal.  We started the dinner conversation, in which everyone participated, around the topic of balancing achievement with ethics, but the direction quickly changed to developing the mindset that intelligence is malleable, and people can get smarter.  (The change was not surprising, given Carol's presence at the table; not that she changed the conversation, but the rest of us really wanted to hear from her on the topic.)  Given the number of educators at the table, we talked at length about how to help students develop grit, to persist in the face of failure, believing that setbacks are common and helpful to the learning process.  We also discussed how most school systems are not designed with this mindset and also how as a society we try to avoid failure.  I wondered, as did others by the end of dinner, to what extent do each of us truly embrace the mindset of malleable intelligence.  Two dozen well-educated people around the table had spent a couple of hours talking about the importance of the mindset, but acknowledged how hard it is to maintain when faced with our own children's, our own students', or even our own failures and difficulties.  It becomes very easy to blame a failure on someone else (because then we don't have to acknowledge the learning we should be taking from it) or to decide that we must not have an aptitude for the thing we failed at (so we shouldn't really bother trying again and be faced with another opportunity for failure).  The danger for me, as a math teacher, is two-fold.  First, if I don't continuously provide my students with the message that their math ability can be grown, and provide the classroom structures and learning opportunities to demonstrate this, then a significant number of my students will face more limited opportunities moving forward.  The second danger is more personal: if I don't truly believe that I can continue learning and that I can get better at teaching, then I face a fairly unhappy and stagnant rest of my career.

In her talk the next day at ETHS, Carol presented some additional research about the power of a malleable mindset.  Studies have demonstrated that if students can be convinced that they can continue learning despite setbacks and failure, then they do.  And not just small children; some of the research Carol presented involved high school and university students as well.  When the efforts and work of students are honored, and they are not just praised for being smart, they are more willing to expend additional effort to continue working in a field even if they have experienced failure.  This is particularly significant for women and Black or Latino/a students.

Completely unrelated to Carol Dweck's talk, I also attended the May meeting of the Metropolitan Mathematics Club of Chicago, which included dinner and a talk by Zal Usiskin.  The MMC is celebrating its 100th birthday this year, and Zal spoke about the history of the organization (which he has been a part of for many years).  The National Council of Teachers of Mathematics actually originated in the MMC, and when the national organization started granting affiliate status to local organizations, MMC was granted Affiliate Charter Number 1.  Zal pointed out that the tension between "traditional" and "progressive" mathematics education existed 100 years ago, which was one of the reasons the MMC was founded.  That tension still exists today, currently embodied in the fight over the Common Core State Standards for Mathematics, and the MMC still has (and will continue to have) the role of providing an opportunity for discussion and collaboration around this tension.

The conversations and talks with Carol and Zal both have given me lots to think about.  I'll post some of my thoughts here as I continue to wrestle with them.

What this blog is about ...

I was reading over the first two entries, and realized that I've been pretty mathy without any indication of what I am actually trying to do.  So allow me to explain ...

When I tell my students to "show the work", what I mean is I want them to show me their thought process.  Whether their answer is right or wrong, I am really interested in knowing what was going on inside their brains when they were thinking about the problem.  If the answer was correct, then I get to see the process the student used to get to that answer, and compare it to the processes other students used to get their correct answers.  If the answer is wrong, then I can better see what misconceptions the students have about the process or the more general concept, and then I can take steps to help adjust that misconception.  Either way, when a student lets me see their work, I get to be a better teacher for that student because I am learning something about them.

By calling this blog "Showing the Work", my intention is to let you see my thought process, and how I am coming to what I think are the right answers.  Along the way, I am likely to have a number of misconceptions, and you are welcome to see those as well.  Based on my first two posts, there is a lot of math going on in my brains, but there is also a lot of pedagogy and educational philosophy.  I'll try to make sure to show my work there as well.

Why I love math

The other day, I came across the following problem:  Given two squares, both with the same side length, where a vertex of one square is fixed at the center of the other, what is the maximum and minimum area of the overlapping region?  I sat with it for a few minutes, thinking about the different ways I might draw the figure, like in the diagram here, and how I could represent the pieces that change, like the length of the part of a side of the red square that is inside the blue square, or some angle of rotation of a side of the red square.  I anticipated using some trig function, which got me a little excited since I will probably be teaching trig next year.  I soon realized the solution, which turned out to be something different from what I thought it would be, and recognized a connection I had not anticipated.

So I started thinking, would the same thing happen with other shapes?  I tried equilateral triangles, and quickly realized there was a different thing happening here.  Which has led to more questions about other polygons, polygons of different sizes, combinations of polygons …  I am still exploring some of these.  (I’ll post some of my thinking on my questions and solutions later.)

Some might think that after being a math teacher for a long time, the topics become a little old, a little stale.  Not so!  I am always finding problems that are interesting and often have surprising or elegant solutions.  (That’s the best part.  An unexpected solution or connection is like discovering a great restaurant or reading a good story.)  Most often, solving these problems needs nothing beyond a little algebra or a little trig, which makes it even more fun, because I can share the problems and discoveries with my students.  The follow up questions I think about are always an added bonus, and while I never seem to quite finish answering all the questions, it’s always nice to revisit them later.