So we are starting our final push for AP review in both my courses now. I teach two sections each day of AP Stats and two sections each day of AP Calculus BC. Yesterday we had our last Stats test for the text and today I gave them a complete released multiple choice section. I thought it would be more helpful to them (and to me!) if I sat quietly and listened and worked while they worked on these questions. It’s probably helpful to know that I have my class set up in two large tables that seat ten students each. They are elbow to elbow and they can all face their peers directly. They don’t need to stare at the back of people’s heads. I encouraged them to scour their own brains. to pick the brains of their neighbors, to prowl through their books and notes, and to air out their ideas and questions. Now, when I was a senior our AP Calculus teacher, the great Barry Felps, rarely ever spoke for more than 15 minutes a day. He’d field a question, maybe two, from the most recent homework, he’d introduce a new idea or work an example to lead us on our path. Some days he’d really work the boards but most days he said very little. He told us he had work to do and so did we. We’d huddle up in groups and work. I LOVED it and I keep thinking that my students will love that freedom as well. Well, it doesn’t seem to always work this way. I just read a great post earlier today called Can You Just Tell me What to Do? and, although he is addressing a different classroom environmental concern, I feel that some of my students probably want to say something like this to me. I know it’s late in the year and I probably cannot make major strides in changing this, but I REALLY want to be more helpful in establishing a classroom structure where we are comfortable exchanging ideas with each other. As I have written before, one of my Calculus classes tends to be terrific at this. One of them is very quiet by nature. I get that, and it’s a small group so I don’t push a great deal on them. However, my two stats classes are each big (by our standards) and I just have not been able to create a space where they seem comfortable having the kinds of rich conversations that I would love to hear. When I am guiding the conversation, I sometimes can get some really great chatter going. Those are fun days and I long for more of them. However, when I sit down and shut up, so do they. I’ll hear a few pockets of chatter among neighbors but nothing like the heated exchange of ideas and opinions that I dream of. So, my question to my dear readers is this – What strategies have you found to be effective in helping to create a culture where the students see it as their job to share ideas?
I’m looking forward to adding to my bag of tricks.
We just finished our course material in our AP Stats class on Friday. We are using the delightful text by Starnes, Yates, and Moore – The Practice of Statistics, 4th Ed. and the last two sections of the course focus on linear regression and transformation of data. This course, for many of my students, has been a relatively algebra-free zone. In this last chapter when we are talking about transforming data, there is NO way around trying to remember some algebra and precalculus ideas. When we looked at some example scatter plots and talked about what shape it looked like to them, there was a bit of a gulf in terms of confidence and comfort in my students. Some of this is fatigue at the end of the year but some of it is an indication of the fact that many students are willing to let some of these facts just kind of disappear. I know that my students have worked through graphing functions of the form y = 1/x , y = 1/x^2, and y = ln x. All of these functions were referenced in class the last two days. On Thursday we spent 40 minutes on one data set that ended up being a very close fit to y = k / x and we had some real transformation work to do to find the missing k value. I was really pleased with the patience and attention of each of my two sections. On Friday, I had these notes prepared and big hopes to make it through the problems on the note sheet. The setting of the problem (tossing M & M’s on to a table and eating only those with the M showing) made it pretty clear that some sort of exponential function was at play. In fact, in the discussion of the data set we touched on the idea that the proportion of remaining candies at each turn should be about 0.5 of the previous value. The number of M & M’s remaining after each round of this set up was 30, then 13, then 10, then 3, then 2, then 1, then 0. I was pleased that in each class a student immediately asked whether we could find the original amount in the bag. They’re thinking like statisticians! However, they are not completely comfortable thinking like precalculus students although they all have been already. Quick conversation led to thinking about a half-life formula but I really wanted to push them in the direction of trying to find a linear model for the data somehow. Playing with the data entered in the TI was slower than I wanted it to be but resulted in some great conversations. We wanted to think about logs since we were thinking about exponents. We debated whether to take the log of the # of candies or to raise e to the # of tosses involved. We tried the exponent first and did not like the looks of the scatter plot much. We tried to take the natural log of the # of candies and got a dire warning about domains. I thought that this might trip people up but in each class I was quickly reminded that 0 is a bad input for the log function. One student even answered about WHY that’s a problem, not just THAT it’s a problem. So, we tossed out the data point with the zero output. We looked at scatter plots of both and decided we liked it better when we took the log. Some kids seemed suspicious of the log idea but were convinced that the natural log was okay after seeing the scatter plot on the TI. Each class asked for a linear regression on this new scatter and they were impressed when the correlation coefficient was -0.99 and the linear regression equation was y = 4. 059 – 0.681x Here is where each class got interesting and why I think this was worth blogging about tonight. I anticipated that someone in each class would tell me to use this to figure out an estimate of the original amount. I was prepared to remind them that the y here is really ln y and we can solve for the ‘real’ y. What happened instead in each class is that someone recognized that the slope was familiar. Now, I’ve been teaching longer than my students have been alive. I recognize and remember certain helpful numbers and I knew that the slope needed to be related to the natural log of 2 since this is a half-life problem. What surprised me was that each class contained a student who knew this. I excitedly congratulated the student in each case for recognizing that and talked my class through why this was so. But as I pounced on this recall with my complimentary response, I noticed that certain students looked dispirited. I made a point of backing up now. I reminded everyone that it was a great thing to be able to recall this kind of number and I tried to impress upon them the power of noticing these connections. But I tried to make sure that they understood that I would never set up a problem where they needed to make this sort of jump. It was interesting to think about this. I want to reward (with enthusiasm) cleverness and the ability to make connections. I want to celebrate this kind of create and thoughtful analysis. However, I do NOT want to create a stressful environment where the majority of my students are wondering whether this is what is expected of them. I tried to patiently point out that I was thrilled and surprised by this recognition. I am happy that I have made it to the point where I am able to feel that stress in the class, where I can see the almost visible sighs of some of my students as they recognize that some of their peers can do things that they don’t del that they are capable of doing. I want to think that I am creating an environment where students feel that they are safe in making guesses publicly and eel safe in not being able to understand where those guesses come from sometimes. When I am the one making this kind of guess it is easy for my students to raise their hand and press me about why I made the connection I made. When one of their peers is the one making a creative connection, I think it is a little more intimidating. I hope that I reassured the vast majority of my students that knowing the fact that ln 2 is approximately 0.69 is a nice thing to know but not a crucial thing to know.
Been away for a while for a number of reasons.
I just read an article on slate.com the really got me thinking about what learning looks like and, therefore, what teaching means in this context. Read a great quote sometime ago that basically said teaching does not exist unless learning has happened. This is quite a challenge for us, obviously.
I shared the article with our AP Psch teacher and he said it was a valuable read and that he would share it in the future with his students. I think it’s worth a read, but if you don’t want to follow the link the article discusses a famous memory study subject who suffered damage to his hippocampus. This caused amnesia to set in but over the course of his life he was still able to form new memories of a certain sort. Here, I think is the interesting quote
After the motorcycle accident, K.C. lost most of his past memories and could make almost no new memories. But a neuroscientist named Endel Tulving began studying K.C., and he determined that K.C. could remember certain things from his past life just fine. Oddly, though, everything K.C. remembered fell within one restricted category: It was all stuff you could look up in reference books, like the difference between stalactites and stalagmites or between spares and strikes in bowling. Tulving called these bare facts “semantic memories,” memories devoid of all context and emotion.
I immediately thought of my AP Stats students who are always asked to report conclusions in context, but I also thought of my Calculus students. Both of these groups of students have a deep reserve of the qualities that usually mark a student as a good student. However, too often I have conversations where it is clear that much of what they have displayed as learning in many classes might not go much beyond the sort of semantic memories referred to in the pull out quote. Skill such as setting a derivative equal to zero when solving optimization problems, or running a two sample t test rather than a z test are often reduced simply to factual memory with no conceptual anchor. In stats when we ask about rejecting or failing to reject a hypothesis based on a reported, or calculated p value, it feels like a particular student should either ALWAYS get this decision right or ALWAYS get it wrong based on a conceptual idea about what the p value says. However, I have seen too many instances where this decision seems to boil down to not much more than a coin toss as the student tries to remember a rule. If the p value has a meaning related to probability, then the answer should be clear and consistent. It feels to me that the biggest challenge in teaching these days is to figure out how to help my students slow down and think. Really think about the ideas that they are working with. Too often they have been rewarded with good grades without reflecting on what they’ve learned and how it applies to anything. This sounds (and kind of feels) like a criticism of my students and my colleagues. I don’t intend it that way. I intend this as a question for me and my colleagues (both in my building and around the world) and my students to consider. How can we construct our classes in a way that helps to develop understanding for our students in a more meaningful, more permanent way? I certainly don’t pretend to know the answers. I know that the way I run my class works for some. It makes other crazy. Two super quick anecdotes, then I’m off to pick up my little girl.
- This year when I was reading my teacher/course evaluations that the students fill out I ran across a great written remark. One of the questions asks whether the instructor challenges the student to think critically about the subject matter. This student in question marked that he agreed with the statement and then wrote ‘TOO MUCH THINKING’ I hope that this was meant in a good natured way, but I DO know that I wear some of my students out with my questioning. They often ask me to just tell them HOW to solve the problem.
- Last year when we were wrapping up Calc BC and working in class on review material for the AP test two students were talking. They did not know I was close enough to hear (or they did not care) and one said to the other ‘last year I knew how to solve these but I had no idea why it worked.’
Here’s to the never-ending struggle to make this all meaningful.