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Professional Growth in a Connected Age

I’ve been teaching for a long time now. It makes me feel old when I realize I am in my 27th year in the classroom now. I joke with my students that I have been teaching longer than any of them have been alive. When I started teaching the predominant models of professional development were the inservice days at school where the school administrators decided how we needed to grow, the weekend workshops or summer workshops that I would scramble to find funding for, or the one or two day workshops that would cause me to miss school. It’s a different world now. I know I’m preaching to the choir if you are even reading this but this world of twitter, of blogs (both writing them AND reading them), of online simulcast workshops, or improv EdCamps, the list goes on. In this day in teaching I am fully convinced that if you want feedback and you want connections to help you think about your craft and to expand your toolbox – if you really want it – there is an ocean of resources at your fingertips. Literally (since I’m typing this right now!) at your fingertips. Not all of it fits everyone. I know that I am still wrestling with the timing and pace of my twitter feed, but I think I’m getting better at it and I KNOW I’m growing as a result of it. I spent a long time reading blogs, then commenting on blogs before I felt confident enough to launch my own. I have two kids at home so I know how tight time can be, but I also know that the past two Saturdays (that I blogged about separately here and here ) where I spent a combined 16 hours out of the house were worth the time and effort. Luckily Mrs. Dardy is kind and flexible and supportive of this pursuit.

I’ve been thinking quite a bit about this, about how different my life as a teacher is in the past few years. I’ve been in regular communication with one of my former colleagues, Gayle Allen. Gayle (@GAllenTC) hired me seven years ago when my family left Florida and we landed in New jersey for a while. Gayle is a remarkable, energetic thinker and was a great boss. She and I have been engaged in a long conversation about professional growth and one of the results of this conversation is an article that got posted today over at a website called Getting Smart. I know that I am not unique in this journey, but I also know that there are still many of our colleagues who have not taken this plunge. Some because they are not interested in doing so, some because they don’t know where to start. I’m pleased to be able to give shout outs of thanks to Dan Meyer (@ddmeyer) and to Sam Shah (@samjshah) through that article and I’m pleased to be connected again (even if we are nearly 3000 miles from each other) with Gayle.

This summer will see a trip to OK to take part in TwitterMathCamp. This would not have happened if Tina Cardone (@crstn85) had not reached out to me and asked me to join in on the fun. This summer will see me finish an in-house Geometry text for our students. This project would never have happened without the encouragement and advice of Jennifer Silverman (@jensilvermath). This summer will see me work on plans to help a brand new teacher in our high school take the leap from teaching Algebra I in the middle school to teaching Honors Precalculus for the first time. All of these experiences will help me grow as a professional. 27 years at it now and I feel like I still have an awful lot to learn. I hope to be smarter this time next week about this craft than I am right now.

Making Connections

We normally have very few school days between AP tests and graduation. This year we have 8 days left (counting today) so I am trying to have a little exploratory fun with my AP Calculus BC kids. They’ve worked hard on Calculus for two years now and I know that there is plenty more Calculus out there. Instead, though, I’ve chosen to take them on a little tour of some topics that they normally would not see in their high school days. Today’s topic was different number bases (the link is to my classroom document for today) and we had some fun with a series of base 8 addition and multiplication facts. I presented them with a  picture of Lisa Simpson as a clue and one of my students noticed that she only has 8 fingers. I use this as my motivation for this conversation. So, I had this nice little handout prepared that I hoped would guide us through a fun conversation. What I did not anticipate was a terrific question that came up. Let me set it up. We visited a website that converts base ten numbers to base 6 while you type. We had fun playing with it typing in things like the year of my birth and a few other nuggets. I asked them whether a base 6 representation of a number would always be longer than the base 10 representation and we had a nice chat about that. Then a student suggested that I type in a decimal so I typed 9.2 and the website did not like this. It would have been very easy to just shrug it off. My kiddos did not. They pushed me a bit and someone suggested that I write the first few negative exponents of 6. One student wisely suggested that I work on the assumption that the coefficients would all be 1 since 1/5 is SO close to 1/6. Here’s where it got fun. Another student said something along these lines – ‘Isn’t this just going to be an infinite series?’ WOW! I was so so so pleased with this. Connecting to Taylor’s and other infinite series in the face of a (relatively) harmless decimal? So proud I was.

We tried this conversion and another student seemed suspicious of the assumption that every coefficient would be 1 (or 0) so I reverted to binary. Now our task was to convert 9.2 into a binary number. The whole number part of 1001 was easy to sell. Now, we had to convert 1/5 into a series of decreasing powers of 2 (or increasingly negative powers of 2). Well, there was quite a bit of computation involved, some nice guess and check strategy was employed, the TI calculator function of turning decimals into fractions was helpful and in the end we discovered (and were able to verify) that 9.2 in base 10 is 1001.0011001100110011…

I have made a few mentions recently of my battery being recharged. It was awfully nice to see that some of my students still have some juice left in their batteries as well.

Charging My Batteries

Two Saturdays in a row now have been spent with some pretty amazing educators. Yesterday I was one of the attendees at EdCamp NEPA. One of the organizers, Mike Soskil (@msoskil) is someone I ran across on twitter some time ago. We had made plans to meet and lunch together at a tech conference but I had a minor car accident that morning so our lunch never happened. I finally met him yesterday and he was as terrific in person as he is online. He was full of energy and enthusiasm and it spread through the room. The vast majority of the folks there were first time EdCampers. I was a second timer so I felt like an old pro in this group. I’m guessing most of you reading this are familiar with the EdCamp model. You arrive to face a big blank session board and write up anything you want to talk about or listen to and people vote with their feet. Mike asked me to consider running a geogebra session and it did not take much to talk me into doing it. So I ran a session I called Visualizing Mathematics – Exploring with GeoGebra and Desmos. I showed off the work my students did in exploring the average daily temperatures in Gainesville, FL on desmos and we got into a great conversation about the power of this kind of visualization. I shared the story that all of my students missed some of the data in the same way and I got to make my standing joke about how long summer lasts in FL. One of my students asked if we would be more accurate with a city farther from the equator. Another student suggested that being closer to the equator would result in more symmetry. A conversation like this would never have happened without this visual support. I also showed off a geogebra file I created to model Taylor Series expansion as well as one of the files that the great jennifer Silverman (@jensilvermath) shared last week at our workshop. This one was called Quadratic Palooza. I had a small group in my room but they were engaged and asked me some great questions. We had a lively conversation about the power of this kind of visualization and how it enables students to ponder and ask questions that they likely would not have thought of before.

I also sat in on a couple of great sessions. The last one of the day was called Love It / Hate It. The moderator would post a statement about some school related policy/issue and we were to move to parts of the room based on whether we loved it/were on the fence / or hated it. We were to discuss with our group to construct an argument to have with our colleagues in the room.

Some of the sessions had participants taking notes together on google docs and here is the link to the schedule page

 

At a time of year when I am usually dragging (we have 10 class days left) I find my self with my batteries feeling recharged. Instead of feeling a bit sluggish, I am feeling perky and peppy, Love it.

 

A Note of Thanks

So, yesterday I had the privilege of spending about 8 hours working. I know, working on a Saturday does not necessarily sound like a privilege. But it was. I had the pleasure of working side by side with three of my colleagues who work in the same building as I do day to day. Mary and Mary and Kathy all agreed to take a Saturday away from their families and friends and join us in learning to use GeoGebra. I cannot feel them (or you) how much I appreciate their willingness to do so. I also had the pleasure of meeting three people from the Philadelphia area who all agreed to give up a Saturday AND to travel two hours to do so. Ed, and Whitney and Andy all took a chance. They heard about our workshop through a flyer that was distributed on a Philadelphia area teachers email list (thanks Ruth!) and, knowing little to nothing about me, my school, or our presenter, they chose to commit a Saturday in May to come and learn with us. I sure hope to continue hearing from them and to build some real community with them even if they are two hours away. I finally had the pleasure of meeting Justin in person! I’ve been reading his blog (and commenting there) and communicating with him on twitter. It’s been a pleasure feeling like we are building a relationship and it was a total treat to finally meet him in person. He drove from the Pittsburgh area to his mom’s house on Friday. Drove to my school and back to his mom’s house on Saturday and today he is driving back to Pittsburgh. He was full of energy, joy, and ideas yesterday and he helped make the day for me. What a treat! Lastly, I finally got to meet Jen in person. Jen has been completely generous of her time and her knowledge. I”ve been picking her brain via emails and google chats. I’ve been stealing ideas from her and she is the main reason I’m brave enough to tackle a curriculum project this summer. I’ll be writing an iBook for our Geometry course next year and I could not have conceived of doing this without her inspiration. She and her husband Charlie drove down from Connecticut and Jen was our leader in this exploration. We made some fantastic discoveries with each other (and through the help of remote twitter colleagues), we wondered and played. All of us were tired at the end and our brains hurt. That’s a great feeling, isn’t it? Community can mean many things and yesterday I felt that my community exploded beyond the walls of my school.

When My Students are More Clever than I am

This happens often enough to keep me excited in my job. I love the feeling when I learn from my students. Nearly everyday I learn important things about people, about human interactions, about kindness and community. What happened today was a great example of when I learn some math from my kiddos. I blogged yesterday about helping a boy with a L’Hopital’s Rule assignment. What I did not mention was that there was a problem I could not solve. I’m not good enough with typesetting here on wordpress so I’ll do my best here. We were trying to evaluate the right hand limit of (x – 1) * tan (pi*x/2) as x approaches 1. Two clues made me think of L’Hopital. The first is that the student told me they were working on L’Hopital’s Rule. the second was the indeterminate form of 0 *  negative infinity.

We looked at a graph to convince ourselves that there is a limit and then we tried to make this a quotient to fit the rule. My instinct with trig functions is to avoid the cotangent, cosecant, and secant functions simply because I feel more comfortable visualizing the basic cosine, since, and tangent. So I made the product a quotient by dividing the tangent expression by 1/(x-1) creating an indeterminate form of negative infinity divided by positive infinity. A quick ratio of derivatives yielded something even word and the second time around was even scarier. We walked away from the problem – in part because I was trying to do three other things at the same time and in part because he was exhausted from thinning his way through the other examples. I asked one of my best BC students to consider it overnight and had some hope that he would. 

This morning at 8 I posed this question to my quiet BC class of 7 and they pounced on it. One student advised that I rewrite the tangent function as a quotient of sine divided by cosine so that L’Hopital is immediately satisfied. Another advised that I rewrite tangent as a cotangent so that I have a ratio as well. In each case, my students saw a way to rewrite a product in terms of 0 / 0 rather than my way of infinity / infinity. Neither format is lovely but we all seemed to agree that 0 / 0 seems less scary. One round of derivatives on each idea led to the conclusion that the limit was – 2 / pi. Geogebra agreed.

What strikes me is that, despite me showing an undesirable approach AND me asking them to recall something from the past, my morning class was perfectly willing to be flexible and to TRY something. Two sound ideas in about 90 seconds and we were off. I was so delighted to start my way this day.

Why So Different?

I live in a dorm on our campus and we have about 80 boys in our building. Tonight one of them asked me to help him study for a test tomorrow in his Precalc class on conic sections. Earlier in the day at school a boy from Calculus Honors was working through L’Hopital’s Rule and struggling a bit. He was in my room working during one of my free periods and I helped him out a bit as I was running in and out of the class. In each case, I found myself really concentrating on ‘how do I do this?’ questions with these guys. Normally, I think of myself as a teacher who really focuses on process. I ask my students many questions and I try to emphasize ideas, connections, principles at work. But I find that when I am in this sort of tutoring type of environment – especially when I am working with students from other classes, not my students – then I switch to a much more functional, how do you solve this problem approach. Don’t know why I do that and I think I’m unhappy about it. I just noticed this in a very obvious way today and I want to toss that observation out to the world. Do you find yourself being a different teacher in these situations? Is there a good reason for it?

Creating a Culture of Sharing Ideas

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.

Managing Expectations

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.

Thinking about Learning (again…)

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.

Observation of Student Behavior

As part of my ongoing commitment to taking one day per week away from the Calculus curriculum, I spent yesterday playing the game of set and 2048 with my students in Calc BC yesterday. My afternoon class was fully engaged in set offering different answers and we found another site which gives you more than one game of set per day. The I opened up 2048. I became aware of this game int he past week due to constant twitter references. I played it some Wednesday night and shared it yesterday. Well, for about ten minutes or so the entire class was engaged tossing out advice and arguing moves. Then the class started to get more and more quiet. What happened was that my students started pulling out their phones and playing the game for themselves. I like the fact that they were interested enough to make sure that they had the game for themselves. I was disappointed that what felt like a great community conversation devolved into individual focus and lack of communication.

I mentioned this and two students told interesting stories. One girl told me that she and her friends recently received free dessert at a restaurant because the server (or maybe the manager) commented on the fact that none of her friends spent dinner on their phone, they were engaged with each other. A boy told me that his friends have a standing challenge sometimes when they dine. Everyone puts their phone in the middle of the table and if someone breaks down and picks up their phone they get to pick up the tab.

It’s interesting that each of these stories was told in a way that made me feel that the student was aware that their technology sometimes gets in the way of interactions. I wonder how much saying that out loud affects their behavior?