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.
At our school’s whole faculty meeting yesterday morning (we see our kids on Monday for the first time) we had an interesting PD session on listening/counseling/advising and I have some ideas rattling around the cave of my mind that I need to air out somehow.
We had an exercise where we were to pair up with someone that we do not know well. Our school has two campuses separated by 3 miles – so the pairing generally was one upper school member with a lower school counterpart. We were to then take turns talking about something important to us for 8 minutes and then switch speaker roles. When it was not our turn to speak we were to listen – not to ask questions, not to comment, just listen. After the activity the guest speaker – a therapist from the Stanley H King Counseling Institute – was gathering reflections and impressions from us about this exercise. One of my colleagues commented that he found it difficult to listen without asking questions and the therapist remarked that it is difficult to simply let another person’s thoughts go where they want to go rather than influence them to go where we want them to go.
This got me thinking about my questioning habits in the classroom. I have long thought that one of my strengths as a teacher is my willingness to trade telling for asking in many situations. After this conversation yesterday I am now wondering how much I need to try to trade asking for listening in classroom discussion situations. Am I just fooling myself into thinking that my asking is any less monopolizing an activity than telling is? I need to ponder this and I’d love to hear any reflections from the world outside my head.
So – I definitely do not want to come off as negative about the EdCamp experience and I don’t want to pile on at all, but I read a post this morning that did a great job of crystallizing some of my concerns. The post can be found over at The Space between the Numbers
I have enrolled in Jo Boaler’s online math course this summer as have about 20,000 of my closest friends. I was excited by the prospect of this class and I urged my middle school and upper school colleagues to enroll as well. I have one MS colleague and two US colleagues who took the bait and joined in. I have to say that at this point I am a bit disappointed by the course. Perhaps I was not really the intended audience since I have done a decent amount of Carol Dweck reading before the course. Much of what Boaler is presenting is through that lens. However, my main disappointments have to do with the problems created by the size of the audience. I submitted work over three weeks ago and I have not received any feedback yet. The assessment panel of the course indicates an estimated wait of three days for feedback and I am now well past that. I know that some work has been done to streamline the discussion forum but my forays into that area of the course have not been promising. Comments are not obviously tied to lessons and too often threads are one or two comments long with no sustained dialogue occurring at all. Perhaps I have just not been persistent enough (a little in joke for those who are engaged in this course!) to find the rich veins of conversation, but it’s been a disappointment to me overall. I will try not to bring that into my opening meetings this year as my colleagues may have a different take. I’m not sad I took the course, it has helped me get my thoughts focused as the new year comes zooming in, but I cannot say that I feel really enriched by the content so far. I also have to admit that this is my first foray into the MOOC world so perhaps the concerns I have about discussion forums and feedback cycles are simply a reality of this big, brave new world. I’d love to hear about the experience of others who may have chosen to join in as well.
I will always remember a conversation with my last division head who remarked that enacting change in a school is like steering the Queen Mary. A slow, laborious process. I understand that I am lucky to be in an independent school where our students do not have high stakes graduation tests looming over their heads (or, more truthfully, they just have different ones in the form of AP and SAT/ACT tests) so that my department has some real autonomy about curricular decisions. As long as we’re not setting our kids up for future failure we can eliminate some repetition and some downright unnecessary material from the curriculum. Christopher Danielson has written eloquently about this issue. Too often, we get hung up on curricular checklists convinced that the kids will suffer in their next class if we don’t adequately ‘cover’ some idea. I am trying to get my department to let go of that idea by trying to open a conversation between the team members about what they really need to have their students know in class X. I’d love to hear how others have conquered (or are simply tackling) this question.
So, a post over at Mrs. Reilly’s blog (http://mrsreillyblog.wordpress.com/2013/07/30/first-days/) this morning has me thinking about the locker problem again. I am trying to imagine a semi-elegant solution to this follow up question. “So, after all 1000 students have gone through and done their thing, how many total times was a locker door touched?”
I am curious as to whether my students will try to keep track from the point of view of adding how many times each door gets touched or by adding up the number of times that each student touches a door. I’m thinking tables and thinking that the greatest integer function will need to be invoked. Don’t have a clear idea myself yet, and I am certainly open to any suggestions.
Just rolled in after a two hour drive and my brain is still buzzing a bit. A couple of quick impressions with more thoughtful reflection to come:
- The STEM phrase (and related ones) means different things to different people/schools. I know that this is essentially true of most words, but it seems especially true here. I think it is vital for each school to clearly define for its constituencies what they mean when they are talking about STEM.
- It is clear to me that there are MANY educators who thirst for time to talk with each other about their practice and their ideas. We need to work at each of our institutions to make this more possible.
- The notion of collaboration (between students, between teachers, between departments) was at the heart of all of the talks I attended.
Oh yeah, I won two raffle items! Sweet.