Thinking About Speed and Time

On first glance, the title of this post has me thinking about my Calculus classes, but that is not the speed and time angle that is on my mind this morning. Yesterday, I finished listening to the newest episode of Malcolm Gladwell’s Revisionist History podcast. The episode (found here) is called Puzzle Rush which is the name of a variant of chess. In the episode Gladwell raises some interesting questions regarding chess, the LSAT, and various places in our society where it seems that speed is valued more than deep thought. He keeps referring to the hare and the tortoise and wonders when the hares got to make the rules. This pod has me thinking about my assessment practice. As I often do, I am going to use this space to think out loud and I am going to hope for the usual outpouring of wisdom here and on twitter to help me work through my questions/concerns.

Earlier this year, some colleagues were having a grumpy conversation about the kids these days. You know, the usual grumpy late winter talk about what is wrong with kids. A totally natural conversation that happens at some point every year. Not a criticism here. However, I did push back a bit and I said that while my current Calc BC kids would be dismayed by my Calc BC tests from 20 years ago, my kids from 20 years ago would also be dismayed by my tests from today. I am pretty convinced that my students today are being asked for deeper analysis of why the math they have learned works the way it does and they are asked to make more predictions and asked to tie together information more deeply. I am also pretty convinced that they are slower in their calculations and in their algebraic manipulations. If my students from today tried to complete in 50 minutes a test I wrote more than a decade ago, many would flounder. If my students from ten years ago tried to complete a test I wrote this year, many would be flustered by the open nature of some of the questions. In general, I think that the thinking I am asking for now is more important. If I still thought that the old ways were more important, I would not have evolved in my assessment practice in the direction I have moved. Where Gladwell has me questioning myself is that there is still a distinct flavor of speed that comes into play. I have a number of students who are still furiously writing when I give them a three minute warning. They are still furiously writing when I give them a one minute warning. Heck, they are still writing as students are passing from class to class in the hallways and I have to bark at them a bit to give up their work. I am somewhat convinced that this might be true no matter how much I shorten the tests. I also admit, not proudly, that I am a little uncomfortable with the idea of a 50 minute class test only taking 20 minutes for some of my best students. I do not believe that speed is the best judge of talent, I know better. But I also suspect that speed is an ingredient in success in many endeavors. What I am wrestling with in the wake of Gladwell’s pod is how do I strike a balance here. I keep flashing back to an essay I read years ago by Dan Kennedy in which he advises ‘Value what you assess and assess what you value.’ I think that there is a very real part of me that values some level of automaticity. Maybe I am being shallow here, but it feels like my best students, the ones who have really mastered ideas, can do so quickly. Maybe I am just fooled into thinking that they are my best because they move quickly? I can keep rambling with this internal monologue, but I won’t bore you this way. I will just jump to some questions that I have for you, dear reader, and I hope to get a nice conversation going in the comments here or over on the twitters where I am still @mrdardy

  1. How do you estimate the time needed for your students to complete a task in class? I have 50 minute classes (mostly) for testing. I generally work on the idea that I should be able to carefully write out my solutions in about 15 minutes. No real science behind this, just accumulated experience.
  2. When writing a test where I am pretty sure that there is one especially challenging (I usually call them interesting!) question, I try to place that one near the front half of the test. Students can, of course, skip around but most just plow through. I want the problem requiring the most thought to be placed where there is still some time for that thought to occur.
  3. When students finish their test, they are dismissed. Is this smart? How do you approach this?
  4. Our schedule, like many of yours I would guess, does not really encourage flexibility with students who might want that simple two to three extra minutes to wrap up work. I have students coming in for their class and I want to respect their time. My students are on their way to their next class and I do not want to interfere with that time. I am uncomfortable, for a number of reasons, with the idea of having them just come back to wrap up later. Any comments/ideas/hacks that have worked within these pretty common scheduling restrictions?

As always, thanks in advance for any wisdom. I am looking forward to a good conversation that will benefit me and my students.

TMC Reflections, Part Three

In this post I want to concentrate on a couple of the afternoon sessions I attended. The TMC program (you can find it here) was filled with so many interesting opportunities that I kind of agonized over some of the choices. One that I knew I would attend was the session run by Danielle Racer (@0mod3) discussing her experiences in implementing an Exeter-style problem based approach to Geometry this past year. Danielle and one of her colleagues (Miriam Singer who is @MSinger216) came back from the Exeter summer math program (it is called the Ajna Greer Conference and if you have never been, I suggest that you try to change that!) all fired up and ready to reinvent their Honors Geometry course. Danielle spoke eloquently about their experiences and shared out some important resources. We had a great conversation in the session about the benefits and struggles of problem based curriculum. This conversation tied in to another session I saw as well as some thoughts and conversations I have been having for years. First, the afternoon session that I think linked in here. Chris Robinson (@Isomorphic2CRob) and Jonathan Osters (@callmejosters) are colleagues from the Blake School in Minneapolis.  Chris and Jonathan spoke about a shift in their assessment policy that centered around skills based quizzes using and SBG model and tests that were more open to novel problem solving. I am simplifying a bit here for the sake of making sense of my own thoughts. I thought that their presentation was thoughtful and it generated great conversation in the room. Perhaps we (especially I) spoke out more than Chris and Jonathan anticipated and we ran out of time. Another sign of a good presentation, I would say. When there is more enthusiasm and participation than you thought you’d get, it probably means that you are tapping in to important conversations AND you have created a space that feels safe and open.

These two sessions had me thinking about some important conversations we have been having at our school and I am totally interested in hearing any feedback. The first conversation I remembered was with a student who had transferred to our school as a senior and was in my AP Calculus AB class. She was complaining about my homework assignments which were a mix of some text problems and some problem sets I wrote. She said in class, ‘You seem to think that AP means All Problems.’ A little probing revealed that she saw a difference between exercises and problems. A brief, but meaningful, description I remember reading is that when you know what to do when you read the assignment then it is an exercise. If you read it and you don’t know what to do, then it is a problem (in more meanings than one, I’d say). The next conversation I recalled was with a colleague who has now retired from math teaching. We were talking about homework and the struggles with having students persevere through challenging assignments. He also used this language making distinctions between exercises and problems and he suggested that HW assignments should have exercises and problems should be discussed in class when everyone was working together. He felt that the struggle and frustration of problems when you are on your own would be discouraging to too many students and would likely lead to less effort toward completion on HW. A similar conversation came up with another former colleague who was frustrated with some of the problem sets I had written for our Geometry course. She did not want to send her kids home with HW that they would not be able to complete successfully. I recognized that this was coming from a fundamentally good place. She did not want her students to feel frustrated and unsuccessful. However, I firmly believe that real growth, real learning, and real satisfaction are all related to overcoming obstacles. I have witnessed this recently with my Lil’ Dardy who just became a full fledged bike rider this summer. I heard it from my boy, my not so Lil’ Dardy, who made the following observation recently, ‘You know, I find that I like video games much better if they are hard at first. Why do you think that is, dad?’

I know that we can anecdote each other to death on these issues and I also know that there is not ONE RIGHT WAY to do this. But I am in the process of trying to make coherent sense out of my inherent biases toward problem based learning. I want to have deep and meaningful conversations with students, with their parents, with my colleagues, and with my administration about how to approach this balance and about what a math class should look like and feel like in our school. While I have been writing this I was also engaging in a meaningful twitter chat about some of this with the incomparable Lisa Henry (@lmhenry9) and with one of my new favorite people Joel Bezaire (@joelbezaire) so I know I am not the only one struggling with these questions. Please hit me up on twitter (@mrdardy) or start a raging conversation in my comments section sharing your successes/failures/theories about how to strike a balance between exercises and problems between challenging students while making them feel safe and successful and between running your own classroom with your own standard and fitting in with a team at your school. These are all big questions and I wrestle with them all the time. I want to thank Danielle, Chris, and Jonathan for sparking them up in my mind again and for creating lovely spaces for conversations in their afternoon sessions.


Coming soon will be my last entry in this series where I think out loud about the amazing keynote delivered by Tracy Zager (@TracyZager)

TMC Reflections, Part Two

This was my third year in a row attending TMC and for the past two years I was co-moderating a morning session. (Thanks to both Tina Cardon (@crstn85) and Lisa Bejarano (@lisabej_manitou) for working with me the past two years!) While I enjoyed each of those experiences immensely, I must say that this TMC felt a little less stressful for me. There were a number of appealing sessions and two in particular jumped out to me. I was torn between Henri Picciotto’s (@hpicciotto) morning session called Advanced Transformations and the session run by Matt Baker (@stoodle) and Chris Luzniak (@pispeak) called Talk Less, Smile More: Getting Students to Discuss and Debate Math. I chose the latter and it was a pretty terrific way to spend six hours over the three days of the conference.

A little background into why I made this choice. Nine years ago when I moved north I made a commitment to blowing up the traditional rows/columns seating arrangement in my classroom. I had three years of small, moveable ‘pods’ of desks at my last school. Here, I had four years with two large conference style tables before asking for desks and now I am back to smaller pods. I have been explicit with my students about my expectation that they be active participants in the classroom thinking process. I think, for the most part, that I have managed this reasonably well and have generated interesting conversations in class. I believe that my students gain some important skills in being able to think out loud and I am certain that they all benefit from hearing so many voices. What I know that I do not do well enough is to decentralize myself in the classroom. Too often fantastic conversations from small pods gets directed to me instead of to the rest of the class. The students use me and their sounding board and as their speaker and I want to learn how to get out of the way ore often and figure out how to elevate small group conversations to the space of the entire classroom. The course description seemed to match this goal.

It was an extremely popular session and we were kind of crammed on top of each other in our classroom, but it helped to develop an easy, comfortable rapport in the room right away. So, my big takeaways are as follows:

  • have to figure out some strategy for randomizing groupings somehow. I want to balance what the research says with the norms of my school. I also have to contend with my weakness in bookkeeping. Not ever having a seating chart works well with my lack of attention to this sort of detail. Conversations in this morning session and vigorous twitter conversations have me convinced I need to do something. The big debate in my mind now is how often to shuffle the pod memberships.
  • One remark on twitter today really has me thinking. In debating randomizing seeing every day versus once per week, Anna Blinstein (@borschtwithanna) observes that daily regrouping seems to focus attention on mathematics conversation while weekly regrouping seems to focus attention on classroom discussion norms. I am inclined to think that weekly regrouping will work best with my student body and with their previous experiences. I want to foster some familiarity and comfort in small group conversations and I think that daily switching might make that challenging. I am open to being convinced otherwise.
  • I am inclined to ask my boss to have my desk removed from class so that there is no longer any centralized seat of power of any sort. I think that it would go a long way to creating the classroom culture I want if students came into class and everyone had the same desk.
  • I need to get in the habit of sitting down while a student is talking and have that student stand to make sure that attention is directed to the person sharing their ideas/questions rather than being directed at me to see my terrible poker face in action.
  • I have three large walls of chalkboards. I need my students up and at them regularly. I think that this might look different in my three very different classes that I teach, but this needs to happen.
  • I need to be careful and consistent about the use of language from me and from my students. Chris strongly advocated formal language from the world of debate where students make claims and support them with warrants. This feels like it would work particularly well in Geometry this year.


I need to be clear that some of these remarks/reactions are directly prompted by the helpful session that Matt and Chris ran but some of these are older ideas that have been clanging around in my brain. My reactions were given shape by the meaningful conversations we had together in this morning session.

More Thoughts About ‘Helpfulness’

Tomorrow three of my four classes will be taking unit tests. I have always devoted the class day before a test to review. Over the past 5 – 7 years I have become more and more insistent that a review day should be a day where I am here to answer some questions that students come to class with and to help facilitate some meaningful conversation between my students. What many students seem to believe is that review day before a test is simply a time for me to tell them exactly what will be on the test. I always come to class on these days with some prepared questions in my back pocket and I always dream that those questions will stay there. That is not often the case, and it certainly was not the case today.

My Geometry class, the one I’ve been SO proud of recently, was in pretty good shape. We looked at our last HW together, they had some good questions about that but they could not really generate too many meaningful questions of their own. I displayed the review questions I had prepared and they perked up and were terrific in joining in the conversation. I just came away wishing that the class had been more about them and what was on their mind. In retrospect, perhaps it was exactly about what was on their mind. They are concerned about what I am interested in right now so that they can glean some important clues about preparing for tomorrow’s test. Sigh…

My two AP Statistics classes are in a different place emotionally than my Geometry class is. They are almost all seniors and the energy level that they brought back from winter break is distinctly different than the energy level I see in my Geometry students. I gave them class time yesterday to work on their own or with their neighbor on the review exercises at the end of their most recent chapter and my observation is that there were relatively small pockets of productive conversations. However, there were also quite a few incidents of aimless chatter, obsessive checking of their phones, silly debates, and general non-statistical conversations.

So, I feel that I am asking myself the same question I asked myself on these pages just a couple of days ago. How can I be less helpful in the standard sort of hand-holding way that my students want me to be while actually being helpful to them in modeling smart behavior about how to work, how to be metacognitive, how to be reflective, and how to be more self-aware. Trying to recall who I was when I was in high school is probably not the best exercise in answering these questions. I was a different person then than I am now. I am remembering through a distinctly tinted memory lens and I am not teaching four classes of teenage Mr. Dardys.

Gotta keep thinking and keep pushing.

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.