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I am kind of embarrassed that I forgot one of the best highlights of the TMC17 conference. A while ago I received a tweet from John Golden (@mathhombre) asking if we could have a video chat about calculus. He was putting together an idea about a resource for his calculus students and wanted a variety of perspectives. Well, after a series of attempts we finally settled on a group chat on Saturday night. It was pretty loud everywhere on the lobby level so I offered my room as a quiet refuge. I had the joy of chatting about calculus with John, Jasmine Walker (@jaz_math), Edmund Harriss (@gelada), and David Butler (@DavidKButlerUofA) You can find our conversation here

 

I was SO flattered to be asked to do this and it was such a blast to chat with these four lovely and brilliant people. I told John on Sunday that I was jealous of his students. My apologies for having this wonderful experience slip my mind when I posted earlier today.

At our church this past Sunday one of the members of the congregation gave a thoughtful sermon about what it means to keep the sabbath. At least that was the primary framework of her conversation. Much of the time was spent talking about learning to take care of herself and what that looks like. Is it listening to a sermon at church? Is it staying home to garden instead? Is it taking a long walk? Is it listening to an inspiring TED talk? Naturally, there is no universal answer for this, but it sure got my mind spinning thinking about what it means for me to take care of myself.

I have engaged in a number of twitter exchanges recently sharing podcast tips. I have become hooked on a number of them and I listen while walking/jogging outside. I listen while I am on my Airdyne in the basement or on the treadmill in the basement. I was listening to Marc Maron and Randy Newman talking to each other just half an hour ago while eating my breakfast here in the airport (I am heading to TMC17 today!) Last month I was visiting my last hometown in New jersey and I went for a long stroll in a park where my wife and I used to walk as a pit stop on the way to pick up our son from day care. At the urging of an old friend – again, through a twitter conversation – I unplugged for that walk. For about 30 minutes I was strolling through this park, remembering cool fall days walking with my wife, listening to the sounds of the park and the neighborhood. It felt energizing. However, I have to admit I have not unplugged like that for another walk or run since. As energizing as that silence felt, I also recognize that I draw a great deal of energy from taking in ideas/content/entertainment through my podcasts. I tend to have music on in my house most of the time I am there. My wife and kids bought me a hammock for Father’s Day. I always bring a book with me to the hammock. I wonder (worry?) if I am just hiding from silence and from being with myself this way. I justify it by recognizing how much I enjoy being tapped into a number of conversation. By recognizing the joy I find when something I hear about that seems brand new suddenly starts popping up all around me. I am excited that I have been spending more time and energy listening to new music again due to my summer DJ gig (which I hope will turn into a fall one as well!) However, I also worry that this is making my time and mind feel even more crowded. I worry that I should put that aside and be quiet. I worry that I should be goofing around with my daughter at home more often instead of curling up with a book while she plays in her room or watches a show on TV. I justify this by thinking that I am ‘taking care of me’ by indulging in books, music, podcasts, exercise so that I can be better at helping others – wife, kiddos, students in the fall, etc.

I am in the last few weeks of summer here and I have taken on a teaching overload for the upcoming year. I’ll be teaching five classes with four different preps. This is on top of being a department chair. I think that the looming concern about what this will feel like has also made this past Sunday’s sermon more meaningful. I mentioned earlier that I know that there are no universal answers to this question, what I am worried about is that I am not clear about what the answer is in my particular situation.

I work better when I set specific goals. Last fall I was waking up early three or four times a week and going for long walks before coming home to wake up my wife for coffee in the morning. I think that I want to commit this fall to picking one day each week, Monday feels like a good choice, to making sure that I go out on this walk with no earbuds. Take a long walk or jog with the silence of an early morning in my ear. Keep my mind clear thinking about what the upcoming week holds. By putting this in writing, I am convinced that I am more likely to carry through with this plan. At the very least, I will feel vaguely guilty or embarrassed if I cannot carry through on this commitment. Not a BIG game changer, but at least this feels like a start. I will check back in on this after the school year is in gear.

As always, feel free to join in the conversation through comments here or by poking at me over on twitter @mrdardy

 

Yesterday was graduation day here at my school. I am pretty sure that this was my 31st high school graduation ceremony – mine and 30 years as a teacher. I think that I did not attend my little brother’s graduation for some reason or other. At least, I do not remember it if I was there.

There are always waves of joy/sadness/pride/regret that run through me on graduation days. I saw some alums and had lovely conversations with them that made me happy. One joked that his Calc 3 class at McGill was easier than his BC Calc class with me. I think that this is probably a compliment in the end. Some students went out their way to find me to express gratitude while others certainly showed no inclination that I was on their list of people that they wanted to talk to on graduation day. Every year ends with the good feeling that there are students who I have made connections with in or out of the classroom. Young people who appreciate that I was part of their lives here. Every year also ends with the disappointment that there are some students I was not able to connect with. Students who were frustrated by my class, did not connect with my goals or my classroom strategies. Students who will not remember me fondly – if/when they remember me at all. This is both a cause for sadness/frustration and motivation to recharge soon when I think about next year and plan for how to reach a broader set of students where they are.

I had a conversation with a colleague recently that made me reflect on graduation feelings and helped me make sense of them. My family moved in the middle of August 2016 from the boys’ dorm where we lived for six years and into a house on campus that the school owns. Since school was already looming when we moved, we did very little in the way of yard work to make the place feel like our own. This past week, my wife and I were able to spend a notable amount of time working outside and trying to make the place feel like ours. I was talking to a colleague at brunch and mentioned that I felt satisfied about the work we had done that morning. While I do not find any zen-like sense of peace and serenity while doing yard work, I do find a sense of satisfaction in looking back after two hours of work and seeing a recognizable change in our flower bed. When talking at brunch about this I contrasted the work in our flower bed with the work we do int he classroom. It feels pretty rare that we see noticeable change in just an hour or two in the classroom. The sense of satisfaction and pride I felt on graduation day when reflecting on the successes I have had is certainly deeper than my satisfaction about the flower bed, but it takes a great deal more patience to get to that graduation day feeling.

 

Apologies to anyone who tried to follow the links on my last post. I am still learning the ins and outs of google forms. I have gone back and corrected the problem, so you should be able to view the question form if you are interested in doing so.

In my last post I was reflecting on some of the important differences between students based, in part, on their age and experience. Thinking about that since the post, I also realize two other important differences between my Geometry classroom and my AP Calculus BC classroom. In our school, Geometry is the last class in our curriculum where there is not a distinction available for Honors credit. Starting in Algebra II, kids get sorted out and those students who don’t see math as ‘their thing’ or simply want to back off a bit in my subject area can. This creates rooms, in both the honors track and the non-honors track, where there is more homogeneity in interest level. In my Geometry class there is a wide divide in interest/background/ability/age in the same classroom. In my Calculus class there is a more level playing field. I think that this goes a long way to explaining some of the data I received this week. The second major difference is that, due in part to the fact that new students enter our school at every grade level, there is a more noticeable age difference in my Geometry class than in either of my other classes. I have students from grades 9 – 12 in Geometry. In my other classes I have only juniors and seniors. I think that this leads to a real difference in the social environment in these classes.

Our school asks each teacher to administer course evaluation forms to all students. The format that the school developed asks many questions, almost all of them Likert scale questions with space included for short answer explanations. I appreciate the emphasis our school places on seeking student input but I have developed the feeling that too many students just glide down the page circling essentially the same answer to questions and they are reluctant to write much down. Some have stated that they are concerned that their teachers recognize their handwriting, but I suspect that most just aren’t that terribly invested in the process. We spoke about this extensively in our last department meeting and one of the conclusions we reached was that we will administer some form of course evaluation at the end of each of our trimesters next year. After all, if the goal of the feedback is to improve the students’ experience, then telling me what to change in May does not have much weight to students who are leaving my classroom in another week. I am kind of embarrassed that I have not come to this conclusion by myself, but at least I am learning, right? I did do two things differently this year. I wrote my own surveys for each of my three classes and I administered them electronically in the hopes that I would get a little more detail from my students. If you are interested, you can see my surveys here for Geometry , here for my Discrete Math elective , and here for my AP Calculus BC class . There are not many differences between them, but I did tailor a bit for differences in the classes.

Here are screenshots of the pie-charts generated on the Google forms. First the response of the Geometry students to the group seating decisions I made this year.

Here is the response of the Discrete Math Students. One of my sections was small enough that we stayed in one group together all year. The other section had rotating groups for two of the three terms of the year.

 

Finally, here is the image for my Calculus team

 

I think that there are some interesting things happening here. I had the chance to talk to my Discrete class and they were willing to share some interesting insights. I think that the comfort level with rotating groups is closely tied to a combination of comfort level with the material and with each other. My BC kids all came from AB the year before and many of them were in the same AB sections. They know each other and they are confident with math. My Geometry kids are a wide blend of ages (grades 9 – 12) and backgrounds (a good number are new to our school this year) so there is not as much cohesion. My Discrete kids come from all over the place. Some just finished Algebra II, some had a year of Precalculus. Some had part of a year of Precalculus before switching over. Some are brand new to our school. There is a good degree of camaraderie in the classroom, but there is not a consistent feeling that everyone is on the same page. This is something I need to be more aware of and a piece of classroom culture that I think I can help improve next year. During my conversation with them yesterday, we focused on two topics. The first was group assessment – I have some group quizzes and we have a group final each term. We had all eleven people working together and they seemed to appreciate that, but felt that I needed to trim the number of questions since debate/discussion took some time. Duly noted. They also seemed to largely feel that rotations are okay, but maybe they should change after more than a week. I am thinking that they may change at the beginning of each new chapter. With our new schedule at school next year, this might work well.

I feel good looking back at this year. I took the plunge and moved from static seating in small groups to dynamic seating that created broader networks of communication among my students. I personalized the feedback I ask for and I feel that my students took these questions seriously and shared some remarks with me in a pretty honest way. I have a lot to think about this summer (as always!) but I feel that I am moving toward becoming the teacher I want to be.

Yesterday morning as my daughter, dubbed Lil’ Dardy by Christoper Danielson, and I were walking to the dining hall we had a delightful conversation. I shared it with a couple of folks who urged me to write it down to remember it. I feel that this platform is probably the most permanent one I have access to, so here goes.

We have two cats, one is named Olympic and one is named Titanic. Here is a picture of them.

The black one is Titanic and he is the subject of our conversation.

We have a neighbor cat across the street who closely resembles Titanic. The neighbor cat, my daughter has dubbed this one Mr. Whiskers due to his long white whiskers, was sitting on his porch Monday morning. Lil Dardy says that she likes Mr. Whiskers because he makes her think of what Titanic will look like. She said he looks like Titanic five years from now. I proposed that, perhaps, this was a Titanic from the future who traveled in a time machine to look after his younger self. Lil Dardy responds by telling me that scientists are working hard on building a time machine. It would be great, she informs me, for kids who don’t like school. They can just skate past their school days in the time machine. She then gets serious and says ‘Dad, I’m sorry but I think I like science more than math.’ I assure her that this is not an insult and I follow by asking why it is she likes science so much. Her quote was pretty great. ‘In science you think of something and then try to make it true.’

Pretty great conversation to start my day.

A while ago I wrote a post over at onegoodthingteach.wordpress.com (link here ) about being proud of my students on a day I was away. I have been engaged with a local leadership group this year that has had me away far more than I prefer to be. I have routinely received positive reports from my colleagues about how my students handle their responsibilities while I am away and I have always tried to share these comments with my students.

I was reminded of the importance (and joy) of this recently by two events. My blog post got a belated reply from a fellow named Josh. He linked to a post he had written on this idea. This was an important reminder not to take it for granted that my students know how much I appreciate being able to leave and knowing that some good math might still occur. The second reason is linked to my leadership program. The group in our area works with business leaders, teachers, college students, and high school students. I was asked to host one of the college students recently. The young woman who was my guest is a math major in the education program at her college. She sat in on three of my classes that day. Unfortunately, she had to leave for her class before my Geometry group met. The feedback I got from her was wonderful. She remarked on the conversations that my students were having and on the level of ideas that they were willing to wrestle with. I was SO pleased not only to hear kind words, but specifically to hear her compliment the discourse in my classroom as this is a big focus of mine. The best part though, was being able to share the remarks with my students the next day. I think that they just shrug it off a bit when my colleagues say nice things, like, maybe, they are just supposed to be nice. However, there was a more tangible reaction when the words of kindness came from a stranger, especially one who is studying math in college.

The fact that these two events happened in the same week was pretty awesome for my flagging energy level and it was a reminder of just how fortunate I am.

I’ll also be posting this reflection over at onegoodthingteach.wordpress.com

If you are not a regular over there, you should think about subscribing.

One of the ideas that has been pinging around my brain recently is that the order of questions on a quiz or a test has a pretty large, but unmeasurable, effect on student performance. Not performance of the whole group, mind you. I am thinking on the granular, individual level. I am now reading Thinking Fast and Slow by Daniel Kahneman (it’s pretty fantastic so far) and he just described an experiment that got me thinking (quickly!) and I put the book down to bang out this quick post.

He describes a survey done of some students where one group of students saw these two questions in order.

• How happy are you these days?
• How many dates did you have last month?

When presented in this order, researchers saw essentially zero correlation between the answers to the questions. Kahneman concludes that dating is not the measurement by which students assess their own happiness. However, when the questions were reversed there was a remarkable correlation. He concludes that the first question gets students focused on that particular aspect of their lives and colors how they view the question about their own happiness.

Does something similar happen to many of our students? If an early question or two seems comfortable and familiar, does this build a sense of confidence and help lead to better results? I often find myself moving at least one of the questions that I anticipate to be a challenge toward the front of the test. My reasoning is that I want them to engage these richer questions while they still have more energy and more time to wrestle with the ideas built-in. After reading this brief description by Kahneman I am now doubting myself. I wonder if I would see better performance from some of my students if I intentionally arrange the test so as to build confidence and, perhaps, give some more built-in clues for later on in the assessment. The real problem with this type of thinking, and the type of thinking I have been more traditionally doing, is that my sense of which problems might pose a challenge do not always correlate to my students’ point of view. I think I want to have a conversation about this with my classes. My three subjects this year have such different sets of students that I suspect I will get pretty different feedback on this issue. That might serve me, and my students, well.

Any thoughts? Drop me a line here or over on twitter where I am @mrdardy

 

Some of this blog post is kind of embarrassing to write, it regards an idea that I have worked with for years and I feel I should have had a better sense of it than I do. I’ll start with a conversation I had with my wizards in BC Calculus recently.

We are discussing convergence tests for series and one of the standard tests is the integral test. As Stewart presents it, at least how mrdardy presents Stewart’s presentation, we discuss an infinite series and if the integral of the function has a finite answer, then the series itself converges. Formally, what my students were told is this:

Suppose f is a continuous, positive, decreasing function on the interval [1,∞) and let the sequence a(n) = f(n). Then the infinite series ∑a(n) is convergent if and only if the improper infinite integral ∫f(x)dx is convergent. If the integral is convergent, then the series is convergent. If the integral is divergent, then the series is divergent.

 

Pretty clean and clear for a calculus statement, right? We had a nice discussion about the difference between looking at the integral which is a continuous summation versus the value of the series which only sums the natural number inputs. A question arose about how to compare the value of the integral with the value of the series. We decided, I really leaned them in this direction by thinking out loud, that the integral sum would be greater than the sum of the series since the integral also added “all that stuff between natural number inputs”. We all seemed happy with that conclusion until yesterday. My LaTex skills are lacking, so I will try this as best I can without that tool. Apologies in advance.

The series we were looking at was the infinite series of 4/(n^2+n). They quickly recognized that this could be rewritten using the method of partial fractions. We rewrote it and, by the magic of the telescoping series, we saw that the sum was 4. All good, right? Well, someone asked me to remind them how we could use the integral test to confirm convergence. We integrated the function using partial fractions and arrived at a sum of 4 ln2. Nice, the integral converged as well. Not so nice, the integral is less than 4. This violated what felt like a nice conceptualization that we had talked through a few days earlier. I sent out a quick call to twitter about why the integral test, that I had convinced myself of providing a ceiling for the sum approximation, would do this. I got a gentle reply from @dandersod reminding me that the integral is the floor for the approximation. I am pretty sure that I have known this at some point, but had constructed such a compelling Riemann approximation argument with my students that I was stumped.  I dismissed them for lunch.

One of my advisees dropped by for a quick chat. He is one of our three Differential Equations students this year. He took a look at the problem and I tried to convince him of my mistaken impression that the integral gathers up all this other area and should be an overestimate. He paused, thinking about the problem and then had a great ‘A-ha’ moment. He said, isn’t the series just the left hand Riemann approximation? Since the function is decreasing (otherwise, it wouldn’t converge!) this left-hand approximation will necessarily be an overestimate. Nice, clear reasoning. The kind of reasoning I should have had at my command when talking with my class a week or so ago. Sigh.

My Calculus kiddos are taking a test today. I get to apologize AND praise one of their colleagues on Wednesday when we revisit this ‘mystery’.