A Portrait of a Math Teacher as an Aging Man

Parametrics / Conics

As I have written before, we teach AP Calculus BC here as a second year Calculus course in our school. This gives me loads of time to play and explore with these students. On Monday we start up again – weather permitting – and we start with our study of parametric and polar equations. Our precalculus class does not cover either topic in great depth (a situation I hope that I can remedy starting next year) and a number of our BC kids are ones who start off in AB Calculus when they come to our school. With so many of our students coming from different parts of the world at different times in their career, we have a wide variety of experiences in the BC group. I guess this is a long-winded way of saying that I have to treat this material as if they have not encountered these ideas at all, really. I intend to spend two days in our computer lab working with building up some fluency with Desmos. I have my room set up in a sort of Harkness-style where the kids are facing each other. Being in the computer lab gives me the flexibility of having the students work with Desmos in a hands-on fashion rather than just watching me. That’s the plus. The downside is that they are working in isolation in this room. I’ll have to deal with that downside for a few days. So, I was digging through my memory bank and I remembered that the great Sam Shah had written a lovely post about introducing conics through Desmos. I downloaded his Scribd file and modified it a bit (you can see my version here) but I still need to go back and play with it a bit more. The way the file looks to me now is way too close to plagiarism – though I do give his website a nod of thanks there. I want the language and the feel to reflect my language and the way my students react.

I am making a real commitment to myself to get out of the way more in 2014. There was a lovely piece that was tweeted out by an old colleague named Gayle Allen. It was called ‘Becoming Invisible in My Classroom‘ and it has given me a renewed sense of mission here. I am also thinking of my visit to SLA last year for EduCon. I walked into a physics class and could not figure out who/where the teacher was for a few minutes. I was amazed and humbled. Need to hold on to that feeling…

So, I’ll start on Monday with a bit of leading/lecturing to set the stage. I’ll give them an assignment to play a bit with Desmos Monday night, then we hit the lab. I’ll be giving an update on how it goes. Wish me luck!

PS – I have a fun Desmos file to look at for them as well. You can see it here. It’s fun to animate the slide and see what happens.

2013 in review

The WordPress.com stats helper monkeys prepared a 2013 annual report for this blog.

Here’s an excerpt:

A San Francisco cable car holds 60 people. This blog was viewed about 1,300 times in 2013. If it were a cable car, it would take about 22 trips to carry that many people.

Click here to see the complete report.

Time to Reflect and Regather

With the beginning of my Christmas vacation only 24 hours away now (one class and then a long committee meeting today followed by a committee meeting tomorrow morning) this feels like a natural time to think about what has gone right this academic year and what can be better in the next calendar year.

I was hired at my school four years ago to teach four sections a day and to serve as department chair. For a variety of reasons, during the first three years I taught five classes per day during 8 of the 9 trimesters. This year I started off with five classes again and now I am finally down to four per day. I have had as many as four different preps in the past, but now have two. The difference in energy level required during the day is stunning. I like to think that I was doing alright in the past. I am optimistic that I am a better teacher (day by day) now. I am able to spend more time and energy planning at night and in the morning. I have found some fun activities and problems to explore. I feel sharper and fresher when we have discussions in class. However, there are some big issues I want to address.

Before the year began I purchased a marble notebook for each of my stats students. I wanted them to have more regular feedback from me and I envisioned taking time about once a week to give them the last ten minutes of class to work on a problem attached in their notebook as a formative, non-graded assessment. My hope was that we’d look at it the next day and by the time graded assessments rolled around, they would have a clearer understanding of what they understand. I’d have a clearer sense of what I needed to explain in better detail, or at least have a sense of what points needed reinforcement. What I discovered was that ungraded meant unimportant to most of my students. Even those who were earning A’s by the time of a graded quiz or tests were turning in blanks or sheer nonsense. Frustrated by the time and energy I was spending with little obvious return, I stopped doing this after four or five rounds. I need to grow up, deal with the disappointment, explain myself better, and do what I believe is the right thing to do.

We teach AP Calculus BC as a follow-up to AP Calculus AB. Consequently, we have nowhere near the calendar pressure of other AP courses. I need to take greater advantage of that freedom. In 2014 I want to devote one day per week to some combination of games, puzzles, and cooperative problem sets. These are the sharpest math minds at our school and they deserve to be challenged regularly. We instituted this curricular decision so that we would have more time for reflection for our students. I know that many of them would have been successful by and measurable metric if they raced their way through this curriculum in one year. However, I am convinced that they benefit from the time we allow them to revisit ideas and explore them more deeply. They benefit from some space to breathe and reflect. I do not want to restrict that time and energy only to problems from the AP curriculum. There is a larger world of ideas to play with. the game of Set, the game of Ultimate tic-tac-toe, visual patterns. These are all things I talk about, I visit these sites, I advocate for these activities. However, I too often fall into the trap of just turning the page in our text and worrying about the next test or quiz. They deserve better.

So, even though it’s early these are my New Year’s Resolutions. I have always believed that I am more likely to carry through with them if them if they are public so I am putting myself on notice.

The Beauty of Community

It’s not math that is on my mind tonight – at least not here, been tweeting a bit looking for help!

My wife just recently started a new job. Rather than working at the same school where I work and where we live, she is now working at a Catholic college about a mile away or so. Yesterday morning we took the kids with us to a breakfast with Santa. I was so charmed by the co-workers I met, by the good cheer in the room, by the warmth of the people who interacted with my kids – especially my 4 year old little girl. It was a great way to start our day. It was snowing when we went, a gentle snowfall that ended up lasting all day and into the night. Being a Florida boy, I don’t have much inherent appreciation for the whole white Christmas thing, but I was taken by it yesterday.

Tonight we had our community Christmas dinner at my school. We eat twice a week with students at a sit down mean we call family style. The students rotate through and receive new assignments every three weeks. Our res life director tries to switch it up so there are boys and girls at each table, kids from different dorms, friends and strangers. I really appreciate these dinners for a variety of reasons. Tonight we dressed up, had a lovely meal prepared by our hard working dining staff, had kids sing holiday songs and had a student play Santa for all of the faculty kids (there are about 20 of them on campus!) One of the dining hall staff buys individual gifts for all the campus kids, the students sit and watch the Santa festivities and coo over the youngest ones. It’s a pretty magical night really. We have four more days left – if the weather allows us to! – and this is just one of the terrific traditions of our boarding community. I’m going to go to sleep feeling happy and peaceful tonight.

Looking Ahead

Excited about news at my school. We’re working on a job description for a fall 2014 hire. Looking to bridge between the math and science departments. We are hoping to find someone to teach some upper level math and work with our freshman in our STEM foundations class. Our school has made commitments to cross curricular work in a couple of important courses. We have a two period class that is co-taught by our history dept chair and our English dept chair called Seminar in American Studies. We have a course called Creative Spirit that is taught by our Performing Arts dept chair and one of our studio art teachers. So, we know how to make this work. Our next challenge is to define a job for a teacher who will work with both our math curriculum and our science curriculum. We are in our first year of an exciting 9th grade course called STEM Foundations and as math chair, I am excited to see and hear what they are up to. I know we are going to continue moving forward in this direction and this hire will be an important one. Who out there is interested in being part of this project?

Finding inspiration

Been a rough couple of weeks. Lost my dad to a too short battle with cancer and then found out a dear friend from my FLA days died just last week. Both were too young to go.

So, I’ve been trying to focus on the day to day and look for triumphs right now. There are a few to be savored. In Calc BC we are studying Differential Equations and I was proud of both of my classes this week. I posed the following challenge to them. Suppose I leave the dorm this morning with a cup of coffee (with a lid) in each hand. I place one of them on the sidewalk and take off the lid. I bring the other to my classroom and remove its lid. Now, sketch what the temp v time graph would look like for each. We noticed and wondered for awhile (I”m not good at using that language explicitly, but I am pretty good at generating some noticing through careful questions) and we decided that the slopes were in some ways related to the difference in temperature AND that there had to be a horizontal asymptote. We had a good conversation about the difference between the theoretical and the measurable temp differences between the room and the cup of coffee. We talked about the relative size of the surroundings versus the size of the object at hand. We got hung up on whether the initial temp of the object was all that important in the end. We realized that there was a proportionality constant to be accounted for (one student pointed out that the specific heat of the object mattered – I was impressed and I think he even used that lingo accurately) and we finally arrived at Newton’s Law of Cooling. I’m not naive enough to think that many of them will remember the exact conclusion, but I am pretty convinced that many of them will remember the process. The conversation that gives me hope about this is one I had in AP Stats today. We were introducing the idea of probability distributions and talking about finding means and standard deviations of random variables. We had to look at the ugly standard deviation formula and try to remember what it all meant. In each class kids were able to contribute meaningful memory of our earlier conversations and I even got a great challenge question from one of my students who doubles in BC and Stats. After we decided that we wanted all the differences between scores and the mean to be positive (so that they don’t all sum to zero) this student wanted to know why we bother squaring the difference between a score the mean score instead of just looking at absolute values. I was able to have a nice conversation about the differentiability of the square and square root functions and how they just behave more ‘nicely’ than absolute value functions do. I was just so pleased that there was such good memory about conversations we had weeks and weeks ago.

I’m also finding inspiration from blogs and tweets these days and it feels like something might be big brewing in the back of my head. Don’t have it together yet, but I’ve been so inspired lately by tweets from @gfrblxt, @JustinAion, @wwndtd and others by blog posts from Michael Pershan, Gary Johnston, Sam Shah and a slew of others, by the Nix The Tricks project that Tina has organized, the list could go on and on.

I’m so thankful for all of the inspirations I find in my daily contacts with family, friends, students, and colleagues for all the inspiration I find in my inbox and through my tweet deck, I just hope that I’m adding to that good karma in my own way…

The challenges of Technology in the Classroom

As is true of most of you, I wear quite a few hats at my school. They all pose interesting, and different, challenges. Tonight I want to think out loud about my role as the chair of a digital learning committee. I work with our lower school colleagues on this committee. Three years ago we decided to move toward 1 – 1 implementation in our middle school (grades 5 – 8 here) and we decided – after a great deal of deliberation – to go with a modified BYOB plan. Many of our middle school teachers are Mac proficient and use quite a few of the applications and programs native to the Mac environment. Our BYOD options are Macbook Pro, iPad, or another PC laptop. Parents were encouraged to consider the Apple products since our teachers had more hands-on experience with those. We introduced this program with a required device for 5th grade students during the 2012 – 13 school year. For this year, we require a device for 5th and 6th and we expect to continue to roll forward. As is to be expected, change produces some discomfort. Discomfort for our network as more and more machines are tapped in. Discomfort for our students as they try to balance the fun and distraction that technology introduces with the responsibilities of being a focused student. Discomfort for our faculty as they try to navigate technology glitches in a classroom packed with young students and packed with curricular expectations. We are doing a fine job of helping each other out and sharing our growing pains, but I know that we can improve our focus and do a better job. This, of course, is a perpetual feeling with any aspect of schooling, isn’t it? As I began preparing mentally for our next meeting I ran across the following blog post from Sean Nash . I urge you to flow that hyperlink. It’s an eloquent and thorough discussion of the patterns behind failed technology initiatives. I especially appreciate his telescope/microscope image. This is probably because I use this language to discuss problems students have in their study of calculus. Students struggle and the tighten up and focus in on little details (look through the microscope) rather than step back and really try to get a bigger picture (look through the telescope) of the binding themes of their study of the calculus.

This post has now been shared with my committee members and I am optimistic that it will generate some powerful conversation and help us to keep our focus.

The Spectre of Final Exams

So – yesterday I blogged musing on what I find interesting. Today I am blogging about something I DO NOT find interesting. I have had a troubled relationship with final exams in schools for some time. I believe – I mean, I really deeply believe – that learning should be a cumulative exercise. The ideas we work hard to understand, the skills we practice toward mastery, the meaningful conversations we have together in class are all experiences that we should be able to carry forward and use as building blocks to grow as learners. In this worldview a final exam should not be a cause of deep stress. Especially if the exam is structured in a way that places less of an emphasis on smaller facts and skills that we might ask in a quiz or a short unit test. You know, look for the big ideas and capture the broad strokes of what we have discussed during the term. However, I have been teaching math long enough to know that most of my students aren’t learning in this fashion. They feel pressured, they feel frantic at times, they feel overworked. In these conditions homework is often just something to get done. Studying turns into short, intense bursts of cramming material in short-term memory. Too often their teachers are complicit in concentrating on the day to day progress so that we are not explicit enough often enough to point out connections and remind ourselves and our students of what has happened in the past. Under this worldview final exams are a terrifying mess. In the span of four days this week my students will have up to five exams and those with fewer exams are trading off by having large papers turned in. What I see are tired, stressed kids who don’t seem to be growing as a result of this experience. Surely, there are better ways to use our time together. I am a team player and I administer exams in the fall and spring in my classes. We say as a school that we value these opportunities to show cumulative growth. If we say we believe in this process, then I want to be on board. We say that part of our role as a college preparatory school is to help prepare for the expectations of college and one of those standard expectations is a final exam for many courses. What troubles me is that there is an inverse relationship between the number of exams a student has and the grade level that they are in. Our freshman have the most exams and our seniors have the fewest.

I say as an individual that the ability to tie together ideas to move forward is a crucial sign of having internalized some skills and ideas. So I want to show my students I believe in this and I ask them twice a year to go through this process. I try to make sure that all my unit tests along the way include opportunities to make explicit the connection between current topics and past material. I hope that this helps some, but I still se a great deal of stress.

So, I ask for the wisdom of the internet community of teachers and administrators. Is this model of exam time just antiquated? Can it be modified in a meaningful way? What do you do at your schools?

What Do I find Interesting? – How Can I Help my Students Answer this Question?

One of the joys of my engagement in the interweb of math teacher bloggers (some of whom I stumbled across my self many of whom I have discovered through the MTBoS challenges) is that I feel my brain tickled and challenged. One of the most consistently challenging of the bloggers is Michael Pershan and he has recently been writing a series of posts about what he finds interesting. Kind of math-y in general, but also just musings. I used to think that success as a math teacher might be measured by running into a student after college and having him/her still be able to answer, say, a trig question about oblique triangles. Luckily, I grew out of that and have a different idea of what success might mean. I have a story from a former student (Chris S.) that I think is apropos here – and I apologize if it seems like I am patting myself on the back here, trust me when I say that there are many students who don’t see their experience with me the way that Chris did. For a few years, I was living in NJ a mere 35 minute express train ride into Penn Station. On days when I was off work but my wife was not, I would often head into Manhattan. Chris was working there at the time doing some high powered financial advising. He was one of the more brilliant kids I’ve ever taught and I have kept in touch with him since he graduate in 1994. So, one day we are having lunch and he is recalling a particularly thorny data analysis problem he had been wrestling with. Much of what he detailed was over my head, but it was great to sit and listen to him so passionately recalling a struggle. He said that his boss, let’s call him Ned (I don’t recall his actual name), helped him with a major breakthrough. He said that one morning – after wrestling with this problem for over a week on and off – he told Ned that he needed to take a long lunch and get out from under this problem. He came back a few hours later and Ned had made an important advance on the solving of this problem. Chris then said to me, “Jim, Ned kind of reminds me of you – he just asked some questions of the data that I did not think of asking and this allowed me to finally solve this problem.” I still get kind of chocked up thinking about this day. He did not say that he remembered a certain lesson or a success on his AP test. He did not say that he remembered having fun in my class, but I think he did. He did not say that he thought of me as a caring teacher, but I think our ongoing relationship says that he does think of me this way. No, what I took away from that conversation was that I challenged him by asking questions he did not think of. What I inferred (maybe this is just optimism on my part) is that he finds this to be a positive trait. He was speaking with admiration about his boss. Now, I know for sure that Chris is a smarter person than I ever have been and I know that many of my students fall under this category. But what I hope that I can convey is the value of questions. Many times in the classroom these are restricted to a mathematical context, but I want my students to develop an appreciation of a good question and develop the habit of asking good questions about the world around them. There is a quote I found in my reading years ago when I was a student and it is a quote I share with my students. I have not found the correct citation for it, so I apologize for not being able to give credit where it is due. Here is the quote:

Genuine enquiry is an important state for students to recognize and internalize as socially valid. Consequently it is an important state for teachers to enact. But it is difficult to enquire genuinely about the answer to problems or tasks which have well-known answers and have been used every year. However, it is possible to be genuinely interested in how students are thinking, in what they are attending to, in what they are stressing (and consequently ignoring). Thus it is almost always possible to ask genuine questions of students, to engage with them, and to display intelligent directed enquiry. For if students are never in the presence of genuine enquiry, but always in the presence of experts who know all the answers, then students are likely to form the impression that there is an enormous amount to know, and that experts already know it all, when what society wants (or claims to want) is that each individual learn to enquire, weigh up, to analyse, to conjecture, and to draw and justify conclusions.

QUICK UPDATE

One of the amazing librarians I work with found the citation for me. You find the whole text at http://www.math.jussieu.fr/~jarraud/colloque/mason.pdf

The author is John Mason