Monthly Archives: October 2014

Wrestling with the Modern World

Sometimes I am convinced that the universe is sending me important messages to sort out. I am not sure if I am always up to the task of making sense of these meanings. In my last post I was wondering aloud about how to incorporate technology into my assessments in a way that made sense. I asked my Calc BC kids to wrestle with a tough problem about circles. The problem made much more sense (to me, at least) when I graphed it using GeoGebra. It allowed me to lock in on a region of reasonable solutions. I asked if anyone out there has logical ways to incorporate this newer technology during assessments. For years my students have come armed with TI calculators. Sometimes they know how to unlock its powers, sometimes they do not. Somehow, the world of GeoGebra and Desmos (and Wolfram Alpha, and and and) seems more dangerous or intimidating to open up to classroom assessments. I worry about how to evaluate my students’ progress when I do not know where/how they found answers. So, that’s one part of what is in my head now. I have struggled with cell phone presence in my school. A little background might help explain where I am. Eight years ago when we moved north I became involved with our local Unitarian Universalist church and I volunteered as a youth group counselor. I attended a number of weekend ‘Cons’ with our youth. One of the persistent messages at these events was that this was an intentional community that was being created for the weekend. The youth were urged to be present to each other and to the event. They were expected to put electronics away for the weekend and they were asked not to engage in public displays of affection. For the most part, they bought into these requests and the energy was palpable. Kids were engaged with each other, they were talking, singing, laughing. It was a fantastic, but exhausting, weekend environment. Just last week I visited a school and sat in on four classes and two assemblies while I was there and did not see one student (or one faculty member, by the way) staring at a screen in their palm or in their lap. Kids were present to each other, to their classes, and to their assembly speakers. I found it refreshing. In my school there is a gathering area right outside my classroom window and I often see two or three kids on one bench all staring at their phones. I know that this is my bias (maybe this bias belongs to others as well!) but I find this dispiriting. In my class, I tend to stand near the door to greet people as they come in and some of them are trudging through the halls staring in their hands and barely aware of those around them. I used to have to spend time getting my classes to quiet down at the beginning of class because they’d be talking to each other as they sat down. Not so much anymore. Again – I know that this is my bias here, but I find this a bit depressing. I try to utilize the language from my UU experiences and since I teach in an independent school I CAN invoke the idea that there is a choice made in being at our school. The reality though is that this choice is often the choice of parents and not my students. At the youth group it was much more a matter of choice by the youth engaged. So, after my school visit I was feeling that my bias was being confirmed and supported by the environment of the school I visited. Then my brains was rocked yesterday by Justin Aion. Justin blogs over at http://relearningtoteach.blogspot.com and his posts (nearly daily ones!) are a treat. I have also had the pleasure and privilege of getting to know him in person here at a workshop we hosted (run by the wonderful Jennifer Silverman) and at twittermathcamp this summer. He is as delightful in person as he is through his blog. Yesterday Justin wrote a pretty moving post (you can find it here) about cell phones and I want to try to address his points as a way to help me clarify my own mixed feelings. His final point is the most important (by the way – read his whole post, don’t just take my highlights!):

If the answers to my tests can be looked up on Google, are they really worth asking in the first place?

I want my students to be creating, to be evaluating, to be synthesizing information.  I want them forming opinions and interpreting answers.  It would be great if they could determine the circumference of a circle from it’s diameter.

It would be better if they could tell me which of the given answers is the most reasonable estimate.

A smart phone can’t make judgement calls.  They can’t interpret answers.

If a smart phone can answer my test questions, I’m asking the wrong questions.

I agree 100% with these sentiments. When I first visited my current school I saw a chapel presentation that completely won me over. It was one of the 4 or 5 major reasons why I am here. Our  Reverend addressed these ideas and won me over. I do not think that this is the real reason why I worry about cell phones or other connectivity issues on assessments or in my class. Justin writes passionately about students doing what he wants (needs?) them to do while still being connected electronically through their phones or their headphones. What troubles me is a persistent belief that I have that we all benefit when everyone is engaged in class. The student who is doing solid math while wearing headphones is depriving their classmates of a strong voice and they are depriving themselves of the opportunity to explain their own thinking or to hear the thoughts of their classmates. I believe SO strongly that learning ought to be social and interactive. Maybe I am just inflating any logical concerns about relating to each other but that is where my heart and my head are right now. I don’t know how to balance what I want, what my students want, what I believe is best for the group as a whole, and the needs of the individuals. I know that there is a sweet spot there and that it almost certainly varies by class – hell, even by time of day.

I have asked my students to have their phones on their desks this year. We know that they are in the classroom and I don’t want surreptitious use in their laps. I ask them to look up stuff, I recognize that some of them use their phone as a rudimentary calculator. I don’t pretend that these don’t exist and I want to encourage honesty and openness about their presence in the classroom. Some students have complied while others have not. I speak patiently (but consistently) with those who keep them in their laps and text friends during class.

I know that I want my students to interact and I believe that they do less of it when they are plugged in to their phone or their headphones. I want students to research and solve challenging problems and I know that they do less of that when they are not connected to the internet through their phones or tablets or laptops. I chaired a committee at our school that helped develop a 1 – 1 program in our middle school. That program should soon bubble up to our high school. I believe in technology. I do, I think it improves learning and depend understanding. I am jealous of my students when I get to display complex ideas with Desmos or GeoGebra because I am old and did not even have rudimentary graphing technology available when I was trying to learn trig and calculus. I cannot tell if my visceral reactions to cell phones is at all logical and I am trying to sort that out. Justin – thanks for making me think and making me uncomfortable. Anyone else out there reading this – please poke at me through comments or through twitter (I am @mrdardy) I want to sort through these conflicts. I want to create an environment that is meaningful for my students AND for me. I sometimes feel like the grumpy old man yelling at kids on the lawn (even though I don’t have my own yard!) even though I don’t want to believe that is me.

sigh… This stuff is hard.

Catching Up and Looking for Some Ideas

There are a few ideas/questions banging around in my brain. No school tomorrow here so I can relax a little more than usual on a Sunday night. I’ll try to be coherent and I hope to get some feedback here or through twitter (where I can be found as @mrdardy)

On Friday our school had the day off and we have been encouraged to use this as a professional development day by our administration. I chose to travel a few hours to visit a school where an old friend is working. The school does some interesting work in the STEM arena and they balance an IB program as well as AP expectations. I gathered some ideas that I will be bouncing off of my colleagues and administrators, but more importantly I just felt energized. I walked away excited to have made some new contacts, happy about many of the things we do at our school so well, interested in figuring out how to develop cultural pieces to support some ideas that work there, and filled with some ammo to talk about the need for schedule changes at my school. As a young teacher I never visited another school. I have long had the habit of visiting other classes at my school and I never feel like I do that enough. It has only been since I moved north 8 years ago that I started making the effort to visit other schools and I cannot recommend this enough. Where I live I pretty much have to drive two hours or so and I have done that the past two years. Every time I have reached out to another school I have received nothing but positive responses and a generous  expense of energy in making the visit happen. I also want to take this space and time to extend an invitation to anyone who wants to come and see our school in action.

On Thursday my AP Stats classes had a group quiz. I stood at the door with playing cards in my hand. Students took one (blind) from me and were randomly assigned to groups. Each class had four groups and each group had a different quiz. There was one question in common to all quizzes but otherwise they each had five different questions. It was SO much fun to listen in as they wrestled with these questions and as they explained ideas to each other. There were some healthy debates but it never got tense or unpleasant. Our school has a very international flavor and I was especially pleased to hear the voices of my international students in these conversations. So much of the material in this course is based on careful reading and vocabulary and I sometimes worry about whether this gets in the way of these students accurately showing me what they know. Have not graded them yet – that is tomorrow morning’s task – but I fully expect them to shine.

On Thursday my AP Calc BC class took a test on integration techniques. The last question on the test was this – Divide a pizza of 14 inch radius into three equal portions with two parallel lines. Most of my students wisely chose vertical lines. Two chose lines in the form y = mx + b, a bad choice. I went into this intending to give full credit even without a numerical value for the line equations. Setting up the integral appropriately is where the calculus is in my mind. Here is what I find myself wishing after this test and after looking at their work – I wish that they had access to desmos or geogebra while they were taking this test. I wish that they had something much more powerful than their TI calculators to visualize this, to try out ideas, to narrow down where the solution needs to be. I had to struggle through some ugly algebra and some calculus that should have been cleaner and more obvious. I’m impressed by the patience and perseverance I saw but I am frustrated since I know that better tools can help them work smarter on a problem like this one. How many of you out there have a setup where your students have access to these tools on assessments? Am I overthinking this by worrying about internet access during a test? Should I just trust that reasonably written questions can allow them to show me what they know and allow me to judge my students’ progress? I’m thinking hard about this and I would love some ideas.

Great Follow Up Day

On Friday I wrote about a pretty terrific conversation that came up at the end of our BC class. We had tackled a particularly gruesome integral – (tan x)^5 and I had done so by repeated patience substitution and I chose to let u = sec x and look for combinations of sec x tan x as the du piece. One of my students stayed after and showed me his work where he shoe u = tan x and du = (sec x)^2 He was frustrated and told me that he spent about a half an hour trying to figure out why his answer was ‘wrong’. So, I typed up my solution and had it on one side of a page. (This document and the graphs I created are all linked on my last post from Friday.) I typed up the solution my student arrived at and placed it on the back of the page. I had my students working in teams so they each had my solution and their classmates’ solution in front of them simultaneously. I asked them to examine each of them and explain why they did not agree. I heard some pretty good conversations, most of them simply concentrating on making sure that they even followed each of the solutions. We had talked about it on Friday so it was good to hear them reflecting clearly on that experience. After a couple of minutes, one of my students announced that he proved that both solutions worked. I played dumb and asked what he was talking about. He explained very calmly that since one answer was based on even powers of tangent and one was based on even powers of secant, we could show that they were nearly equal. They seemed to differ only by a constant. I then showed them the Desmos graph I created and the GeoGebra graph I created. Both programs were happy with my solution and with my student’s solution. Neither program was happy with the difference between them. But I showed them that every x input we could guess at in the difference function yielded either an undefined answer or an answer of -0.75

I used this conversation with a number of goals in mind. I want them to get in the habit of talking to each other. I want them to see that there is not just ONE way to do math problems – especially ones as sophisticated as the ones we talk about in BC Calculus. I want them to think about graphs. I want them to utilize resources such as Wolfram Alpha, GeoGebra, and Desmos. I want them to notice and wonder about relationships. They are not yet where I want them to be in these terms, but the more often I remind them and the more often I model this behavior, then the more likely they are to adopt these behaviors.

If I did not believe this, I might not have the energy to keep on keeping on in this job. But I do believe it and I do keep on keeping on.

Thank you world of math resources for my students! Thank you world of recourses for me!!

We Broke Graphing Technology Again – A Success Story

In AP Calculus BC we are doing some pretty unexciting stuff right now – techniques of integration. The problems are (sort of) fun little algebraic puzzles but I find little room for conceptual conversations. Maybe I am just missing something obvious. But today was a bit of a revelation and I wish I knew better how to try and insert equations to tell the story. I’ll just have to use some tortured syntax to get my point across. I put up three pairs of integrals and told them that one in each pair was something they knew how to do before they met me (our school does BC as a second-year calculus course) while the second was one they needed my help with. I had an integration by parts example side by side with a boring old u substitution (the integrands were x cos(x^2)  versus x cos x) and they knew which one they COULD do and we talked through integration by parts. I had a partial fraction problem side by side with a  natural log problem (the integrands were (x – 2)/(x^2 – 4x + 3) versus (x + 1)/(x^2 – 4x + 3)) and again they knew the difference and we talked about partial fractions. I had a trig substitution problem against a boring old square root (this time it was sqrt (9 – x) versus sqrt (9 – x^2)) Then someone asked me a HW problem. They were asked to integrate the fifth power of tangent x. I took off writing and trying to get buy in at each of the many steps. I told them at the end that they knew each of the steps they just did not know which direction to move. I assured them that this was a process they would master with a bit of practice. As I was working, I made the decision to substitute for sec x and set up the answer in terms of that function. A student asked me why he could not use tangent to substitute. I did not have a bunch of time left so I asked him to hold his thought and talk to me at the end. He did. As a result, I made a document we’ll examine as a class on Monday comparing his solution and mine. You can grab that here I went through with math type to show his solution and mine. I’ll leave it to the students to determine why they look different and I hope they come to the conclusion that they are NOT different. To help push the conversation I created a Desmos graph and a GeoGebra graph to show my function (called d(x) in each case) and my students function (called j(x)) in each case, I will erase the f(x) that you can see by following these links because I don’t want to give the game away immediately. What troubled me was that each program dealt with my function and my student’s function just fine. When I combined them the graphing technology broke. I tweeted out to @desmos and received – as usual – a quick and helpful reply. In this case, the reply was simply ‘Thanks for sharing. This will help us make better graphs for the future.’ This is the second time this year that we have found a little glitch and I could not be more pleased with the response I have gotten each time. It is such a great way to emphasize to my students what a connected world we’re living in and how they can reach out and find help. My student said he spent a half an hour trying to figure out why his answer was ‘wrong’ since it disagreed with his text’s answer. I hope after Monday that he will begin to internalize the idea that he can check his answers in pretty powerful ways. Ways that I did not dream of when I was learning this stuff in 1982. What a fun fun experience seeing his work and getting the reply I did from Desmos. Add in the fact that I get a date with my wife at a local farm to table restaurant and the day could not get much better.