As I mentioned earlier in a post simply called My School we have a large and active international community at our school. Tonight, our dining hall is hosting a dinner of Chinese food prepared with the Lunar New Year (sometimes referred to as ‘Chinese New Year’) in mind. Today at our school assembly time our chaplain delivered a typically terrific talk, this time with Pete Seeger being the focus of his chat. When he finished – a little earlier than usual – he turned the stage over to one of our international seniors named Oliver. He mentioned that at this holiday time many of our students cannot be with their families to celebrate. He then introduced a video put together by another student who goes by the nickname Bobo. Bobo and Oliver had the idea of having the parents of our Chinese students (Bobo and Oliver are each Chinese but we certainly have many other countries represented here) send their holiday wishes to their children. We saw a five minute video of moms, dads, siblings, etc. all wishing their children a happy and safe new year. It was such a terrific gesture by these two young men and it had a great impact on the room. Many of the adults in the community were brought to tears by how sweet this was. Many of the messages were in Chinese and subtitles were presented for some of them. Our chaplain could not help but remark that the wishes of parents – be happy, do well in school, eat well – all transcend cultures.
What a great moment and a reminder of how terrific this community is.
By the way – remember, we have an opening in our math department. Think about it…
So, my afternoon crowd was not to be outdone by my morning crew. I slipped in a subtle reference early in the conversation with them so that they would not be inclined to simply introduce the phase shift idea. I wanted them to have a little practice untangling the mechanics involved in dealing with developing a Taylor series. They were very quick to recognize and agree that the coefficients were based on factorials so jumping from the 5th degree polynomial to the 7th degree was pretty easy for them. When I asked for the cosine they were confident about using even powers instead of odds and came to a conclusion pretty quickly. Where life got interesting was when I showed them Michael’s solution from the morning and discussed why i preferred the symmetry generated by an even powered series instead. I also discussed how Michael’s translation idea might give better results for approximating cos x with negative values of x. That’s when they stepped up and knocked me out. They suggested that we take the 6th degree polynomial approximation we had for cos x and do the following: phase shift by pi radians and reflect over the x axis. I am linking to a GeoGebra file that we created. If you want to dig into that file – here are the explanations of the functions.
a and b are self-explanatory
f is the 7th degree Taylor for sin x
g is the phase shift of this by pi/2 to approximate cos x
h is the 6th degree approximation of cos x
m is the crazy reflection/shift to move the cos x approximation backwards to another portion of the cosine curve.
Whew – what a day
We are starting our journey through the study of Taylor Polynomials today. I started with looking at y = sin x and asked them to find a ‘simpler’ function that behaved like sin x does around the origin. I sort of purposely asked this in a pretty vague way and we had a good chat about what I was asking for. One of my students offered up y = x as an answer. This gave us the opportunity to talk about the limit of sin x / x as x approaches 0. It also gave us the opportunity to talk about L’Hopital’s rule. A pretty good start in my mind. Then life got interesting. One student suggested a cubic function but I was able to get someone to urge an extension to a quadratic function that might match the sin x graph as the next step. I’m not sure what he (Michael) saw that made him jump to a cubic. He’s a really insightful student. So, I held that off and got to working on a quadratic. We agreed that the quadratic better agree with sin (0) and that the slopes should be the same. Someone suggested that the second derivatives should match as well. This resulted in a quadratic with a leading coefficient of 0. Not so good. It would have been easy for them to give up on this process, but Michael had already suggested the cubic. We had success in finding one and a GeoGebra graph confirmed that this worked over a larger region than the simpler linear function. We jumped into a fourth degree polynomial – again with failure due to a leading coefficient of 0. Here is where things really started getting promising. I asked why this was happening and a different student remembered something about even and odd symmetries. The precise language did not arrive right away, but we were able tp get that together as well. Pretty promising… A fifth degree polynomial was found and it graphed even better than the third degree. The students were getting a little tired of this process so I very quickly convinced them of the behavior of the 7th degree approximation. Michael (he was on fire this morning!) recognized the factorial pattern unfolding so we jumped ahead to the 9th degree polynomial. We were feeling pretty good about ourselves at this point. I asked them what function we might be interested in next and, luckily, I was told that cos x would be our next target. I told them that I would be quiet for the next few minutes while they worked this out for themselves. Normally, I am not at all interested in my students – especially ones at this level – simply mimicking my solution patterns. In this case, I thought that this new process was intimidating enough that they would just try to parrot my work. I was fine with that idea, this unit will take some time. However, my best laid plans were foiled. About a minute after I sat down dramatically Michael asked ‘Why don’t we just replace each x with x + pi/2?’ I was SO HAPPY, but i tried to hide that for a moment. Luckily, he spoke pretty quietly and his classmates were still working. I went back to GeoGebra and wrote a new function in his honor. Taking our last guess of h(x) which was our 9th degree polynomial and writing m(x) = h(x + pi/2) and I displayed this graph on top of the graph of cos x. It was a fantastic match but it did not have the symmetry that we had seen for the sin x approximations. The students who had plowed ahead with the polynomial model gave me their 9th degree solution and we looked at three graphs together. The cos x graph, the shifted sin x Taylor series and the cos x Taylor series. A really terrific conversation ensued. Today is what we call a T day where we have 50 minute classes. This felt like an enormously productive 50 minutes. I hope that the afternoon goes at least half as well.
The NCSSM conference ended around 1 today and my flight back home got cancelled due to snow. Luckily, a former student of mine lives nearby and he graciously has offered his place for me to sleep. We also had a great night out catching up.
I’ve been thinking more about @JustinAion’s quote from a couple of blog posts ago. He said that he only feels like he is teaching when he is answering student questions or going over examples. After the sessions here at NCSSM I am even more committed to fighting that urge – the one that Justin put his finger on. I want to be more invisible, give my students more room. Ask more than tell. I want to analyze their thinking more than their answers. I want to think deeply about their questions and misunderstandings. I hope to get better at this and I’ll report back as I try some new ideas out about how to do this.
A quick aside based on a conversation before one of the sessions. A teacher behind me was talking about his daughter Maya who had recently learned to spell her name reliably. He mentioned that she had also recently learned that her middle name was Sage and she decided that she’d rather be called Sage. She was still spelling her name the same way but she was announcing that she had written Sage now, instead of Maya. I have to imagine that the set of alphabet symbols that we would read as Maya simply read to her as her name. If her name is now Sage, then those symbols mean Sage. I was reminded of my own 4 year old who we call Mo. Over the past five months Mo’s ‘signature’ has evolved from OW to WO to OM to MO. However, whenever she sees something with her name on it she simply says it is hers. She seems to see no difference between them. I wonder how similar this is to some of our students writing down things like (x + y)^2 = x^2 + y^2 and not having any recognition that it does not match anything we’ve written.
A brief one tonight as I am solo dad while mrsdardy is off at a work event.
My AP Calculus BC class has a test tomorrow. Their last HW assignment – some review problems I put together was assigned for Monday night with the idea that it would help guide their studying and that today would be a day in class to discuss any concerns. Well, the class where I had the sprinting is similar to cramming conversation did not seem to take my hint at all. It was clear that plenty of cramming is planned for tonight. even asked the one girl directly about our conversation and she just sort of shrugged and said she’d be ‘sprinting’ tonight. I know that these sorts of habit are hard to change. I mean HARD to change.
So, tomorrow morning I teach a couple of classes then head to the airport. I am off to NCSSM for their Teaching Contemporary Mathematics conference. I went there about eight years ago and was wowed by Dan Teague, Floyd Bullard, and the whole atmosphere of the place. This time I am going to make a point of meeting Daren Starnes – the author of the Stats text I use. He’s also the Dept Chair at Lawrenceville and he’s been so helpful in setting up a professional visit for two of my colleagues to go there. They got snowed out Tuesday and have already rescheduled. It’s rewarding to start feeling connections outside of my building and it’ll be nice to meet him this weekend.
Oh yeah, while I’m out on Friday my Calc BC kids will work on Problem Set 3
Okay – so I’m thinking out loud here. Hoping some wisdom comes from this exercise and/or from brilliant comments by my dear readers.
Working with my outgoing Calc BC group and I comment to one of my students that it’s a tough day for him. He’s on our swim team and they had a 6 AM practice Tuesday morning and a meet that afternoon. One of the other students – a member of our field hockey team – says that her team never ran sprints the day before a big meet. Now, it’s important to understand that our field hockey team has won’t he state championship three of the past four years. This student was a member her whole high school career. I take this opportunity and I ask her if she thinks that this strategy (don’t stress out your body the day before an important match) might be carried over to another realm. I am greeted by a quizzical look and I say ‘Maybe you should not cram the night before a big test.’ Another quizzical look. She asks if I am advocating not studying. I say that the daily diligence of regular work and studying is comparable to daily hard practice in field hockey. Then, relax a bit before an important match (or test) and maybe this is a formula for success. I don’t think that many of my students saw that as a winning strategy.
I just observed a lovely Precalculus class taught by one of my colleagues. The class was working on a variety of word problems – coins, movie tickets, area/perimeter, etc. My colleague is a remarkably calm, zen-like fellow. He sat in a student chair the whole time (sort of invisible!) and asked one student at a time to come to the board. The rest of the class was attentive, offering help to their colleague and generally being cooperative and positive. The teacher kept asking nudging questions of the student at the board. “What do we want to find here?” “How can we relate the number of coins and the value of the coins?” etc. Being an observer in the class (and not stressing out about HOW to do the problems) I saw that my colleague was modeling for his students a lovely strategy for tackling these problems. If each student could play that conversation back in their head as they struggled with any problem, then they would see much more progress. They might still make mistakes, but they’d have a sound strategy for success. What troubled me – and I spoke with the teacher about it the next morning – is the fact that I KNOW that some of them will not ask themselves those questions. They won’t take his advice for attacking these problems to heart. I am not saying that all of our students need to mimc our behavior. What I am saying is that students who struggle, ought to feel that it is a lifeline that is being offered here. When I asked him about this the next morning, I told him that I was impressed by his careful teaching and modeling. His response was something along the lines of ‘I am a big believer in teaching. I just think it works better when learning happens.’ I really don’t think he was being mean or cynical.
This morning, I awoke to another terrific blog post by @JustinAion over at his blog Relearning to Teach. If you are not familiar with his work, you should change that and visit him. Pay attention to his tweets as well. Life will be better. He closed out his post today with a powerful quote – “Even with everything I’ve seen, done and learned, even with all of the conversations I’ve had with other teachers, I still only feel as though I’m “teaching” when I’m answering student questions or going over examples.
I wish I could scrub that feeling.”
I think that I’ll walk away from my computer now and let these conversations and this quote marinate a bit. I know I have some questions, but I am not sure that I can ask them accurately enough yet.
I’ve been VERY pleased this week with my classes and with carrying out my new year’s resolution about shifting my classes. In my AP Stats class we worked on a fun project from the Mathalicious crew examining Simpson’s Paradox. The kids were in small groups working pretty well together. They did not need me to talk much (remember – it’s important for teachers to be able to become invisible!) and I was able to listen in on some pretty great conversations. We looked at three years of batting information about Derek Jeter and David Justice and saw that Justice had a higher batting average each of the three years but a lower cumulative percentage for those three years. The activity starts off with a fun parlor game. I had six groups in my room. Three of them received year-by-year data and the other three received cumulative data. I asked them to conclude who was the better batter and the predictable vote of 3 to 3 happened in one class, the other class went 4 – 2 for reasons I could not decode. Then we worked our way in to a conversation about school admissions and looked for evidence of discrimination. It’s a great activity that they designed. We also watched a pretty interesting TED talk by Dan Ariely. It’s been a good couple of days and we close out the week with another Mathalicious activity.
I am typing this while my quiet morning Calculus BC class works on the second of my in-class problem sets. Again, they are silently working and I have decided not to fight against this as long as they are willing to share when we look back at the work.
Next week it is back to the text for my AP Stats class, but I feel that this week has been an important breather for them AND an important reminder that they don’t need to hear my voice all the time. they’ve been doing fine listening and talking to each other.