So, in my BC classes we are wrapping up our tour of integration techniques. It’s pretty easy when you are convinced that integration by parts is the strategy to use, or when you know you are supposed to use a trig substitution, etc. Little parcels are easy enough to deal with. Throw them all in a bag at once and choose? Much much more challenging. Yesterday, in our 40 minute classes, each of my two sections of BC Calc made it through two problems and I could not be more proud of them. They fought, they tossed out ideas, they stuck through some thorny algebra. They critiqued each other’s ideas. They questioned mine. I tried – I really did – to give them space and let it unfold. Other than one idea that I knew would lead to pain, I did my best to let them run the conversation. It was the kind of day that justifies – at least in my mind – our decision to have BC as the second year calc class. In a one year track these kids would not have days like this where they could just play with ideas without regard for the clock. The problem that was the real winner is below (if my cut and paste graphic works right)
Edit – Image pasting is not my strength right now. Sigh. The challenge at hand was to integrate the fraction dx/(x^(2/3) + 3x^(1/3) + 2)
One student in each class suggested completing the square and that was pretty thrilling. The first one even pushed a step or two through on a trig substitution involving secant. That’s where I intervened because I was pretty sure that this path would lead to pain. We looked at GeoGebra and tried to work backwards from its answer after we went down the partial fractions path. Man, what a good day and I was fortunate enough that one of my colleagues came to visit yesterday morning. She was their AB teacher last year so it’s possible that they stepped up their game for her. If that’s true, I’ll have to enlist her for future challenge days.
Our school had a day off on Friday (and a day off today as well) for a long fall weekend. We were asked by the powers that be to use Friday for professional development. I chose to drive a couple of hours in the morning to go visit another school. When I did my last job search in the early months of 2010 there were a number of schools that caught my eye and the school I visited on Friday was one of them. The chair there was remarkably kind and helpful in setting up a too short visit that morning. Since I had kid pick up duty that day AND we had agreed to house sit for some friends to look after their dog AND my boy had an ice skating birthday party to go to (there is a theme here about how life unfolds in the dardy household) I did not have quite the leisure I had hoped for. I arrived at 8:30 ish for a warm, quick chat with my host, I saw a Geometry class, then I saw a Precalculus class, then I saw an AP Stats class. A nice follow up chat and lunch with the chair, then I was off to home.
I always enjoy seeing classes – it is a part of my job as a chair that unfortunately gets buried under other tasks. It is fun to pick up tricks from other teachers. In this case, the geometry teacher had a lovely way to highlight parts of the parallel line with transversals problems that they were working with that morning. She had spools of different color tape that looked like athletic trainer tape. She pulled off two of one color to highlight which lines in the diagram were parallel to each other and a different color for the transversal. It was SO COOL to see this way of making the relevant information in the diagram just pop out to the kids. It was also fun to see her improvise. The kids were checking their work from the night before and were having disagreements about measures they had taken. Out the window went the lesson plan for the day and out came a class set of protractors so that they could practice with their measuring skills. The teacher confided in me that some of her attitude about this was strongly influenced by her husband who is a woodworker. In the AP Stats class I was privileged to watch someone who was a real, honest to goodness statistician before entering the classroom. As a stats novice myself, it was great to chat with her beforehand and to watch her in action. I think that she convinced me to try an activity that has been previously pretty intimidating to me. The precalc class was fun to watch as well as the kids were hanging in there working through some complex polynomial graphing ideas.
I know that I have a tendency to look at my world and see the potential for excellence in the people around me. I know that I focus at times on what is not quite right instead of celebrating what is right. A visit like this worked wonders for me on a number of fronts.
1. It’s always great to reach out to more people to bounce ideas off of
2. It’s fun to watch kids at work – especially when I have no preconceived notions of who they are or what they SHOULD be doing
3. It’s rewarding to talk to others who are working through some of the very same struggles. How do we accurately place test kids who are new to a school? How do we balance ambitions for kids with their abilities and previous track record of achievement? How do we find TIME in the school day/week/year for meaningful problem-solving while still serving an ever expanding curriculum? The chair I met with is thoughtful, experienced, and intelligent. The fact that she is struggling with these questions as well makes me feel better.
I’m proud of my school, our students, and my colleagues. I believe that we can all be better than we are but I want to try and focus on what we’re doing right and I think that this experience on Friday can help me with that.
A number of years ago I was fortunate enough to be able to attend the Ajna S Greer summer conference at the Phillips Exeter Academy. In addition to great mini courses, terrific connections, lovely food and drink, and just general all around wonderfulness there I was exposed to their Harkness table teaching. I had long struggled with the traditional rows and desks layout of my classroom and I remembered how I would utilize it as a way to hide as a student. Being relatively small in stature I was able to disappear into the classroom seating arrangement when I did not want to be engaged in class. As a teacher I want to avoid that. At my last school our desks were very moveable and I played around with a ‘pod’ system in my classes. The girls took to this system (it was a single gender school) and they took a certain amount of ‘pod pride’ in their group work. I played with group quizzes and I was feeling like I was moving forward as a teacher. I still found myself a bit frustrated by the lack of whole group interactions. I came to my current school in the fall of 2010 and inherited older desks that were not so flexible in their use. So, I prowled around campus looking for solutions and I found some old library tables that were sitting in a storage room on campus. My boss got our maintenance drew to help and over christmas break I removed all individual desks from my room replacing them with six rectangular tables. When my students returned they were taken aback by this new set up and by the new classroom behavior expectations I put before them. I simply asked them. I told them that it was understandable if I was the person who talked the most in class – not always desirable, but understandable – but my challenge to them was to make sure that my voice was not heard more than 50% of the time. I wanted them to talk to each other. I wanted them to toss out ideas and questions. I want them to interrupt me and to share their ideas. I want them NOT to interrupt each other when ideas are being voiced. In general, I wanted them to be active participants in the classroom not passive vessels waiting to be filled. I am the only math teacher in my school who has a room set up this way so when they get to me there is a learning curve. I encounter a number of different situations as they learn how to be a student in my class. There are social conversations occurring. More of them than there used to be. I wish that there were fewer of them, but it’s a price I’m willing to pay. There are students who are not sure enough of their ideas to share them with the class. So they try them out on their neighbors instead. I try hard to listen and pull those ideas out and toss them to the crowd. There are students who are distracted by their neighbors and want to just focus on me. I try to move people around so that this effect is minimized. The physical layout of my room means that some students have to turn around to see certain boards. I don’t know how to fix that and I am not sure it is a critical problem. Some students monopolize the class conversation. This used to happen anyway. I think that now I at least have the benefit of students making eye contact with each other. It is easier to listen to your classmate if you are not staring at the back of her head. Some students are quiet and more shy. I get that. I watched the TED Talk from Susan Cain about introverts. In fact, I shared it with my class and talked about the impact of the classroom set up. We had a great conversation about it. I know it’s not a perfect setup for everyone but I also hear back from my students that this setup has helped them put their own voice to their mathematical ideas. I have had students tell me that before my class they thought that their job as a math student was to know how to use certain formulas and know when to use them. After my class, I hope that they have a larger view of their job in the classroom. There is a learning curve at the beginning of each year and I hope that we are beyond it now as we head to week 7. I feel that I am more aware of my students and that I have a better sense of their level of understanding. I think that my students get better everyday at listening to a variety of voices in the classroom. I still spend too many days doing more than 50% of the talking in class but I think that I am still on that learning curve myself. I would not be so bold as to say that I am implementing Exeter’s Harkness style the way it is intended, but I think that I have found a way that makes sense inside my school.
So, we are almost done with our deep and quick tour of AB topics in my BC class. We use the Stewart text which has an interesting section at the end of each chapter. The section is called Problems Plus and I have been browsing through these sections for class examples. On Friday I picked a problem that looked pretty challenging. The set up is this – Imagine a square region with sides measuring two units. In the square a region is shaded. This region is the set of all points that are closer to the center of the square than to the nearest side. What is the area of this region? I did not try this problem first, I had confidence that we could work our way through it. In each of my two sections this was the second problem of the day. Each group dispensed with the first problem in about 5 minutes. Each class spent almost 40 minutes discussing/debating/arguing over this square problem. What thrilled me was that both classes (the small morning class of 8 and the large afternoon class of 18) stayed engaged offering ideas, questioning each other, thinking about circles, etc. We looked at GeoGebra to try and sketch some regions. We thought about the distance formula and circles since the kids were convinced that the region where the distance to the center and the side was equal would be somehow circular in nature. None of our ideas came to find a final solution. To me, this fact is SO tiny in comparison to the fact that they fought, they were engaged, and some of the afternoon kids stayed after to share new insights. I am so proud of this group for being willing to engage and not being at all angry or visibly annoyed when we did not come to a solution. I can’t wait until Monday to see what ideas they bring to the table.
So, I sort of pride myself on being the type of teacher who creates an environment in his classroom where conversations can flow. I have my kiddos at two large tables where they are elbow to elbow and talk regularly. Sometimes, of course, the conversation strays – bit there are often rich math conversations going on. I posted this quick story over at One Good Thing, but I want to share it here as well.
I presented my BC class (all in their second year of High School Calculus) with the equation of an ellipse centered at the origin and asked the following rather vague question – “Are there any two points on this curve where the lines tangent to the curve are perpendicular?” One girl, Chloe, immediately answered that the tangent ‘on top’ of the graph was horizontal and it would be perpendicular to the tangent on the ‘side of the graph’ which is vertical. I congratulated her and challenged the class to find some other more interesting points. A student asked what the slopes of these more interesting lines might be and then a boy, Sal, chimed in that any number you pick must be the slope of a line tangent to this ellipse. His argument was based on recognizing that between the two tangents that Chloe had mentioned the slopes range from 0 to positive infinity. In other quadrant the slope would range from 0 to negative infinity. If he had mentioned the intermediate value theorem I might have fainted on the spot from joy.
After I posted the story to One Good Thing I read Ben Blum-Smith’s most recent posting (http://researchinpractice.wordpress.com/2013/09/08/kids-summarizing/) and I now realize what an opportunity I missed by simply congratulating Sal instead of getting others to join in and complete the thought process. Read Ben’s post. You’ll be glad you did. I intend to try and incorporate this strategy into my daily practice.
We made the decision this year to start our precalculus classes (both honors and non-honors) in the study of trigonometry. This decision was made based on frustration with the traditional slow start of reviewing Algebra topics and based on the request of our physics teacher. So now kids can start off a little more productively if they are simultaneously enrolled in some level of physics and some level of precalculus. So, today I decided to try out an experiment with Desmos. I made a table of values of the average daily temperatures of my beloved former home (Gainesville, FL) and I both gave the students a physical table of values and displayed the plot of this table of values on on the board through Desmos. Their job, in their pods of 2, 3, or 4 students at a time was to match a function to this data. I was SO happy with the work they did and with the conclusions that they arrived at. Here is a link with the data and the various equations
Note that above I said (separately) that I was satisfied with their work AND their conclusions. I am trying so hard to make that distinction for my students. To talk about the process, the thinking that goes on. One group pulled out an iPad and called up their own Desmos app and kept tweaking their work. We had great conversations about how to identify the amplitude, how to deal with phase shifts, what should the period be, etc. I’m not crazy enough to think every day will go so well, but I sure am happy and optimistic right now.
We made the decision at our school to make AP Calculus BC a second-year course in high school Calculus so our kids in BC have all completed AP Calculus AB with some measure of success. I am pretty firm in believing that we are doing them a favor (and doing our teacher – right now it’s me) a favor as well. Our kids had been ‘succeeding’ in BC by the measure of the AP test, but they were exhausted and had no time for reflection. We probably have a bit TOO much time for reflection with it as a second-year course, but that’s the problem I’d rather have. So, we use Stewart’s Calculus text (I inherited it and I’m not in love with it but it’s more than acceptable) and he has these clever sections at the end of each chapter called Problems Plus. I spend the first four weeks or so reinforcing AB topics through the use of these problems. Our AB calendar doesn’t lend itself to too much of this so I feel like it is not simple repetition for the kids. We just finished chapter two on derivatives and I’ve been mixing in some AP FR problems for HW. I’m completely at a loss to explain the wide range of reactions from the kids and their range of performance on these problems. One of our more ambitious kids – a junior girl from China who earned a 5 and A’s all last year – was presenting an AP problem with a linear piecewise function. We were told that this was function f and a function g was defined to be its antiderivative. We were presented with a triangular area. She carefully explained how she found the line equations for each piece of the graph and then proceeded to anti differentiate them. I pointed out that her work was correct but perhaps it would be more efficient to simply calculate the area of the triangle bounded by the curve. She was not moved AT ALL by my argument. I’m torn between being happy that she knew how to convert into a function (I actually AM happy that she can do this) and troubled that she is so rigid in her approach to antidifferentiation. Rather than recognizing that it is also a geometric concept, she is locked into the function interpretation. I don’t want my students to spend so much extra time and energy on problems like this. I also don’t want them to think that I don’t respect their thinking. I don’t want them to think that this is all about doing it the way that Mr. Dardy wants them to do it. Figuring out how to respond in a genuinely positive way while also pointing out how much more efficient she can be was such a challenge. I need to work on this. So much work to do in this job, no matter how long I’ve been at it…
So, I have just started the Baker article in Harper’s and I’m already pretty annoyed. However, I’m more annoyed (I think) by the reality of the situation than I am by his argument about the situation. We are in day three of school here and one of my jobs as Dept Chair is to sort out math placements for new international and domestic students. I have had the following conversation (or some variant of it) at least three times in the past day and a half:
Student: Why am I in this class? I should be in a higher class, I’ve learned all of this already.
Mr Dardy: Well, your placement test indicated that this was the best combination of challenging you without putting you in a situation where you might fail.
Student: That test? that was unfair, I don’t remember all of that stuff.
So, my question (asked out of frustration) is this – Do other disciplines have the same plague of students not knowing things they claim to have learned already? More importantly, do they deal with some inherent assumption that this is okay to not know what you claim to know? I know I’m not asking anything novel here, but I needed a virtual place to vent a bit so I can continue to smile and deal with my young charges with good cheer.
So – as I anticipated, this problem was more challenging than the more typical direction where the number of people in the room is the given information. What completely knocked me out was that I saw two different diligent solution paths. One student convinced his neighbors that the formula to work with was x(x-1) where x is the number of people in the room. Nice thinking, not exactly right but a good start. Another group was diligently building a table of values. We plotted those values and it was pretty clear that a parabola was emerging. This was nice since the other formula was quadratic. However, the table of values did not match the formula. This led to a quick adjustment and we decided to solve the equation (1/2)(x)(x-1) = 253
I was told to multiply the 2 over to create x(x-1)= 506 and this is where class got exciting. The students, predictably, wanted to solve the quadratic. We had already concluded that we wanted numbers somewhere in the twenties based on the product. One of my students pointed out that we were looking at consecutive numbers whose product ends in a 6. The student patiently explained that it had to be 22 and 23 or 27 and 28. I LOVE this. The type of number sense at play here was so refreshing. First day of school a winner based on this exchange.
So, one of my favorite opening day problems for Algebra II is the handshake problem. You know, there are 18 of us in this room and if we were total strangers and went around and introduced ourselves to everyone in the room, then how many handshakes would occur are we go around the room meeting each other? I know, I know, this is a sort of pseudo-contexty type of problem but it always leads to interesting conversations. So, this year my lowest class is a Precalculus Honors class and I think I’ll start off by saying “There is a group of strangers in a room and they all go around and introduce themselves to everyone in the room. At the end of this process 253 handshakes have occurred. How many people are in the room?”
Is this any more interesting/challenging than the standard version? I’m not certain, but I think it is.