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.
For as long as I can remember – even back when I was a student – the beginning of August signaled the beginning of dwelling on school. This year I get a two day headstart. Tomorrow I am off to visit an old friend in NJ on the way to EdCamp Steam. He has not only offered his home for Tuesday night (so I don’t have to leave home around 5 AM Weds) but he has offered to let me sit in on a PD session at his school. He’s the Upper School Head at his school so he has the right to do that…
I am excited about the EdCamp but really have no idea what to expect. I am just hoping to walk away jazzed about the new year. We are starting a STEM initiative at our school and I hope to come back with good ideas to share.
Speaking of sharing good ideas – I am scouring my brain to find ways to expand two of my favorite beginning of the year problems. I have been using the 1000 locker problem for years and I always find ti to be a great conversation starter. However, I have not found any particularly interesting extensions. I’ll poke around for some and I’ll certainly share if I find any. Another favorite is the question of how many squares are on a checkerboard. I have extended that to asking how many rectangles are on a checkerboard. I’ve always been pleased with that one.
So, this post is inspired by my most recent trip, my daughter, and by Christopher Danielson. We just returned from a trip to Pittsburgh to see my beloved Mets defeat the hometown Pirates. While on the road my impatient daughter is often asking how much farther we have to travel. Since I don’t imagine that she understands answers such as “28 more miles honey” (she will turn 4 on Thursday) I often answer by saying things like “only 20 more minutes, about the sam time as an episode of PowerPuff Girls”. So, when we were at the pool at a hotel on our trip we were playing a jumping game where she stood on the steps of the pool and jumped to my arms. One time I was too close for her tastes so she told me to back up. When I asked her how far away I should be, she said “Five minutes away.” I asked her, “Don’t you mean five feet away?” But she stood firm in her answer, she wanted me five minutes away from her. Christopher routinely writes about his children’s developing understandings of mathematics and, while I teach in the high school, it has made me more conscious of trying to get at what my students think that they understand about developing situations in the classroom. As a dad, these thoughts intrude as well. I am now debating my use of time as a marker for her, although I am certain that it is a more tangible way to answer her questions about how far away we are from our destination.