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?

Exciting Opportunity

So, one of the benefits of creating a virtual presence has been that I have all sorts of new friends that I have never met. I look forward to thoughtful exchanges on my blog and on theirs, I chime in every once in a while to the torrent of information that is twitter and I am happy that I’ll be able to meet a bunch of these folks at twittermathcamp 2014 in OK this summer. However, another opportunity to actually meet some of the army of talented math folks on the internet has reared its head. The amazing Jen Silverman (@jensilvermath on twitter and at http://www.jensilvermath.com on the web) will be traveling to my school in Kingston, PA to host a one day Geogebra workshop on Saturday, May 3. Here are some reasons you should think about attending:

  1. Jen does amazing work on GeoGebra, she is sort of a GeoGebra Jedi Master. See this page for evidence.
  2. We are hoping to have a manageable crowd of about 12 – 15 folks here. Enough to share ideas but not enough to get in the way of some direct instruction when you need it.
  3. I’m working on taking care of lunch for everyone – so that is a definite plus.
  4. Oh yeah – it’s free!!!


Jen created a lovely flier for this event. If I was smarter about managing my blog I would display it below, but you can click the link to see the document.

I hope that many – if not all – of my colleagues from our middle school and high school can join us and I am reaching out to anyone within a reasonable drive of NE PA to come and join us for a day  of learning and sharing.


Different Perspectives

A quick reflection here before I wake up my kiddos.

Yesterday in BC Calculus we had onto of our weekly problem days where we (mostly) put aside our current Calculus work and look at interesting problems that may or may not involve any Calculus at all. Here is problem #1 from yesterday (a problem I borrowed from @bretbenesh.

A mountain climber is about to climb a mountain. She starts at 8 am and reaches the summit at noon. She sleeps at the top of the mountain that night. The next morning, she leaves the summit at 8 am and descends using the same route she did the day before, reaching the bottom at noon. Why do you know that there is a time between 8 am and noon at which she was at exactly the same spot on the mountain on both days? We should not assume anything about her speed on either leg of the trip.


One of the things I enjoy most in teaching is seeing/hearing different ways of attacking a problem. When I read this problem I immediately sketched a height v time graph with the base of the mountain and 8 AM as the origin and the top of the mountain, noon as some arbitrary point in the first quadrant. A wiggly sketch connected the points. Day two has a y-intercept of top of mountain, 8 AM and an x-intercept of bottom of mountain, noon. No matter how I connect these points the sketch intersects my other sketch and I see the reason why. I’m surprised by this discovery, but I see it. In each of my 2 BC classes yesterday there was one student who saw through this problem and explained it away so quickly that I was wowed. In each case the student immediately switched to thinking of 2 people rather than one. If one person starts at the top at 8 am and walks down while the other starts at the bottom and walks up then they must pass each other at some point! Simple, clean, elegant. It’s fun to learn from your students, isn’t it?

That Elusive A – ha Moment

On Monday we returned from our two week spring break and we finally took the plunge into Power Series in our BC classes. Oh, by the way, we were looking at snow in our area on the weather forecasts. Great first day back after spring break!

So, on Monday and Tuesday we were dealing with defining Power Series’ and looking at the radius of convergence and the interval of convergence for these series’. My students seemed to be dealing with these problems pretty well. Some number of weeks ago – I cannot even remember right now – I introduced this last full chapter of our text by talking about our ultimate goal of developing Taylor series approximations and I used the function f(x) = sinx as my example weeks ago. I convinced my students that we could create a polynomial the behaves like sinx as long as we were willing to be patient enough. I started off (again, this was weeks ago!) with an approximation of sin(0.1) using geogebra and talking them through the idea that we wanted (more accurately, I wanted) to create a polynomial called P(x) that agreed with f(x) at x = 0, and whose first, second, and third derivatives all agreed with those of f(x) at x = 0. We chose x = 0 for relatively obvious reasons and since they had never seen this argument before they were willing to go along for the ride. So, we finally get to the point now where my students can follow along in the logic rather than simply watch and/or write down notes. They come to class yesterday and I tell them that in our 40 minute class I hope that we can finish 2 problems. This creates some visible unease as the idea of 2 problems each taking 20 minutes generates some snarky remarks about how hard this is going to be. What follows is a summary of the conversation with my second BC class of the day – my much more vocal and active group of the two.

Problem #1 – Estimate, correct to three decimal places, the value of sin(0.1) without using your calculator. I start a conversation about what we might be able to know about this value. We pretty quickly agree that it is positive and small. In my morning class I had a great estimate in degrees of what 0.1 radians might look like and I hope to prod the conversation in that direction. I start by asking what a logical upper bound for the estimate might be and I hope to hear someone say 1/2 since that is the smallest exact sine value they know in the unit circle. Instead, Jon tells me that it has to be less than 0.1 which is true and much more accurate. I ask him why this must be so while a number of his classmates are generating their own guesses. His neighbors are in a debate about why 1/2 is an upper bound for reasons that hover around the unit circle. When I question Jon he tells me that the function has a slop of 1 at the origin and that this slope decreases as x increases, so therefore when x increases by 0.1 y will increase by less than that. Wow. I was SO happy to hear this reasoning and I wanted to make sure that the rest of the class heard it as well. I should have dusted off Ben Blum-Smith’s idea of having another student try to restate but I honestly was not sure how many kids had even heard him. I was standing near him and he was speaking to me while his classmates were involved in conversations with each other. So, I took over and restated his point. I then pushed a bit and asked the class why Jon knew that the slope of f(x) = sinx was 1 when x was zero. Here, my mind was anticipating and hoping that someone would mention the limit of sinx / x as x approaches 0. I might have had to take a break at that point to calm my heart down. Instead I got another terrific answer – we know the derivative of sinx is cosx and we know that cos(0) = 1. I asked a student why we were suddenly talking about derivatives when Jon discussed slope and I was calmly told that the derivative IS the slope and we were ready to march on. The procedure for setting up the system of equations is tedious and time consuming and as I started the problem a number of students were rifling through their notes and found the example we did weeks ago when we generated a third degree polynomial to match up with f(x) = sinx. I was again delighted that they (a) remembered we had done this and (b) could find it so quickly in their notes. So we get the function we want and now substitute x = 0.1 into the polynomial. We have the fraction 599/6000 at this point and Jon is pretty pleased. We see that it is less than 0.1 but just barely. I remind them that the directions asked for an answer in decimals without their calculator so we dust off some long division skills and get to 0.0998. I ask a student to pull out his calculator and give me the four decimal answer that his calculator has for sin(0.1) when he recites the exact same decimals I can see some noticeable smiles on my students’ faces. They are pretty impressed. We are almost there, I can feel it.

Problem #2 asks for a four decimal approximation (I correct myself midstream because of the first problem and what I remember of our morning work) for ln (0.9), again without their calculator. So this problem has a different wrinkle. I have not yet introduced formal notation from their text regarding these series, so they don’t know about the center of convergence yet and we are not assigning the mystery, powerful a t this yet. I’m using the phrase ‘we are concentrating on x = ___’ and we want the blank to be a value close to our target but one where we can easily compute and exact value if we need to. We all agree pretty quickly that x = 1 is where we should concentrate and that ln (0.9) will be negative and small. I’m happy that I have enough discipline now to weave in this kind of ‘what do we know, what do we wonder, what can we guess’ kind of conversation into class regularly now. All this twitter and blog PD is taking hold!

So, we go through the tedious process AGAIN of matching a power series out to the third degree so that P(1) = f(1) [where f(x) is now lnx], P'(1) = f'(1), P”(1) = f”(1) and, finally, P”'(1) = f”'(1). However, we have an interesting decision to make here. For the first problem, with x = 0 as our focus, we all agreed that P(x) = a + bx + cx^2 + dx^3. With x = 1 as our focus now, we were a little anxious about this model. Students quickly offered two solution ideas – replace each x with an x + 1 or replace each x with an x – 1. I have to say I was pretty thrilled with how this conversation was unfolding. Agreement on x – 1 was reached. When I was asked why, I responded with the following two questions – (a) What is the simplest equation of a parabola with its vertex at the origin? (b) What is the simplest equation of a parabola with its vertex at the point (1,0)? Everyone seemed okay at this point. We get our polynomial and evaluate it at x = 0.9 and we arrived at the fraction -79/750. When I did the long division we arrived at -0.1053 and, once again, someone’s calculator matched this exactly. A wave of smiles and nods went around the room. Those elusive moments when you can actually see a group of people lock in on an idea are so exhilarating. It was so much fun to see this group of students attentive and engaged, not intimidated by two problems that each took about twenty minutes each. This class is my last class of the day and I ended the day in a very positive mood as a result of this conversation.


PS – Another problem day today. Here is my newest problem set. I borrowed problem #1 from @bretbenesh who was clear in explaining that he borrowed them from all over. Problem #2 is an old favorite and problem #3 is from a recent math league competition. 

Fishing for Ideas

When we return on Monday from our two week spring break, my AP Calculus BC kids will be finishing up their version of their Calc text with the final push of study on infinite series. We’ll be gearing up for our tour of Taylor and Maclaurin techniques. I want to design a final unit to tie together some loose ends from their trig days and formalize their knowledge of vectors. I feel that I can teach the required AP vector techniques in about 2 days but I want to craft something a little larger. I’d like to try and frame this by reviewing important trig highlights first. Our kids do not see DeMoivre’s Theorem in their precalc try unit and I don’t want to send them off to college without that tool. I am dreaming of a way to wrap all of this up in a nice, tidy bow. Trig/complex numbers/vectors as a meaningful and lively final unit. I have about two weeks before I would be starting this off in class and I would appreciate any clever ideas/links/words of encouragement/etc. that I can gather from the collected wisdom of my virtual colleagues.

Why Do I Blog?

Once again, Dan Meyer has me thinking. This time the blame can be passed along to Michael Fenton. Michael raised a question on twitter.  His question to Dan was

Could you have written a list of 5 reasons you blog 5 years ago? And a list of 5 reasons you blog now? Lists match? What’s changed?

As usual, Dan’s page is generating some great responses. You can jump to that page here.

As I write, I am on my first day of spring break. It is a quiet, beautiful morning on my mom’s back porch with everyone else sleeping. Let’s see if I can make sense of this question.

I am a relative newbie to this blogging business. My first post was in the first week of July 2013. This is my 58th post. I started reading blogs with some regularity and subscribing to them a few years ago. The first one I subscribed to for regular delivery was either Dan’s blog or Sam Shah’s. My memory is not sharp enough to recall which was first ( Edit – A little research shows that I subscribed to Sam’s blog in Oct 2010 shortly after relocating to NE PA). I was living in NJ where I moved after leaving FLA. While I was in FLA I was enrolled in a program to earn a doctorate in education. I was taking night and weekend classes while teaching full-time and raising a little boy. I even took a year off from the classroom to be a grad assistant and full-time student. I worked on a super cool research project looking at arts integration in schools and was feeling super energized by the reading I was doing, by my classmates, and by my professors. We headed off to NJ where I was going to carry out my dissertation research (where I was lucky enough to work with this guy as one of my research participants) and I worked with a really inspiring Associate Head/Director of Studies and some other good colleagues there. But, once I was back in the classroom walls, I started realizing how much I missed the level of conversation from my graduate classes. As most of you reading this know, we spend TOO much of our time in our four walls closed off to our colleagues. As many of you have probably experienced, there is not much in the way of structured time during the day where we can have meaningful conversations about what goes on in those walls. As a math teacher, I get too much semi-snarky jokes when I try to talk about my class at the lunch table or in the faculty lounge. There is something about the pace and structure of our days that seems to work against built-in reflection time. So, I found that time and space out on the internet. For a few years I read blogs and occasionally commented on them. I finally took the plunge last July and started my own. In the fall when the MTBoS blogging initiative kicked in for its second year, I was all in. I even took the plunge into twitter based on one of the challenges presented there. That must have been in October. Since then I have sent out almost 1300 tweets. All but about 50 of them have been to people I know only through my math experiences or through their blogs and tweets. The majority of them have been sent to people I’ve never met. Through these experiences, I have had the pleasure of being invited to my first EdCamp – with another on the way. I have been invited by the amazing Tina Cardone to take part in a workshop presentation at this summer’s Twitter Math Camp, I have received tweets as answers to questions from Ketih Devlin and Steven Strogatz, I have shared amazing lessons from people I’ve never met, I have pestered my colleagues with emails and document attachments that I have gathered from the web. I spend the first 45 minutes or so of each morning (after I feed the cats and start some coffee) reading my email alerts from the previous night, scanning what might have happened on twitter after I fell asleep, and checking my wordpress reader. I go to school every morning thinking about something that I might not have thought of on my own. I look forward to this quiet time in the morning before my family wakes up as a time to wake my brain up and recharge it. It’s a way to improve the lives of my students and to help ensure that I don’t feel stale and bored. I am in my 27th year in the classroom and I feel as energized about it as ever and I think that much of the credit belongs to a world of people I’ve never met.

So, I realize that most of what i have just said explains why I read blogs and prowl twitter. Why do I write my blog? A much simpler answer. I think that there are two reasons.

  1. I want to give back – at least a little – to this rich world of ideas.
  2. I want feedback on my ideas as they develop.

Not profound, but it feels good to chime in. Now, I’m off to breakfast.