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.
Author: mrdardy
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?
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:
- Jen does amazing work on GeoGebra, she is sort of a GeoGebra Jedi Master. See this page for evidence.
- 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.
- I’m working on taking care of lunch for everyone – so that is a definite plus.
- 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.