Monday, December 22, 2014

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.

Sunday, August 24, 2014

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.

Wednesday, August 13, 2014

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.

Saturday, July 19, 2014

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.

Tuesday, July 15, 2014

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.

Thursday, June 19, 2014

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.

Saturday, May 24, 2014

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.

Saturday, May 17, 2014

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.

Thursday, May 8, 2014

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.

Tuesday, May 6, 2014

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.

Tuesday, April 29, 2014

Why Abelian Grapes Are Funny and Important

Why Abelian Grapes Are Funny and Important

A couple weeks ago, my son started studying complex numbers in his Advanced Algebra class.  He had caught onto the idea of adding and multiplying them, but he had lots of questions about where they came from, why imaginary numbers are called "imaginary", and what these numbers really were.  We spent some time talking about the history of complex numbers and I showed him how to graph them, making the connection between the real number line and the need for two dimensions.

This idea of two-dimensional numbers got him thinking and he wanted to know about higher-dimensional numbers.  This led us to a discussion (very limited) about quaternions.  We looked them up on Wikipedia, and the second line was about multiplication on quarternions being non-commutative.  This kind of blew his mind, since everything he had studied so far said that multiplication was commutative.  So we talked for a few minutes about defining sets of numbers as groups, rings, and fields.  I had not remembered all the properties that applied in each case, so we had a nice research session online.  I had also just been reading Edward Frenkel's Love and Math which has a nice description of an abelian group using rotations in the first couple of chapters, so I showed him that example.

He's been really excited about our math talks, and I enjoy talking to him about the subject I enjoy.*  Since we've been having these conversations, his grade in his class has gone up, even though we have not spent as much time on the actual classwork.  As I think about our math talks, I am reminded about how important context is for some students.  Simplifying rational expressions was not very interesting for him until we chatted about limits, and the quadratic formula was just a dreary procedure until we talked about Girolamo Cardano and Niccolo Tartaglia.  Frenkel uses the analogy that the math we teach in school is to the actual topic of Mathematics as painting a fence is to appreciating and talking about Art.  (You can see an interview with Edward Frenkel on The Colbert Report.)  Makes me think I need to brush up on my Math History as I start planning lessons for the fall ...

*I am slowly trying to convince my son to think about majoring in math when he gets to college in a few years, but he still thinks engineering is his thing.  He got excited about some notation and game theory ideas a few weeks ago, and when I told him the joke, "What's purple and commutes?  An abelian grape," he laughed.  So I'm winning. :)