The Decisions We Make…

I have two sections of Discrete Math this year, one in the morning and one right after lunch. During the fall term, each of these sections had 7 students. We all sat at a single group of desks together and had some great conversations. A number of the students have spoken to me about how much they enjoy this atmosphere. It does not work for everyone of course, some students prefer not to have the expectation of participation, they would prefer to quietly observe and have more time to think before speaking. Our school is on a trimester schedule and this Discrete course is set up as a trimester course where students can move in or out and not have the demands of previous knowledge from this course. So, I have done some thinking about how to make this course modular. One of my sections expanded from 7 students to 16 this term and we are in the process of figuring each other out and how this new group will mesh. One of the students who has been in the class the whole year commented that class seems more quiet this past week. Interesting that more than doubling the size of the class has resulted in a quieter atmosphere…

All of the above is just to sort of set up what our week together was. We started a probability unit this week and so far all of our energy has been spent on counting techniques. When does or matter? When does and matter? What is the difference between them? What is the deal with that ! function anyways? When can we tell whether replacement matters? These are the kinds of conversations we have been having and I have had in-class activities for us to work on together while I have been asking them to do some reading and some HW on their own outside of class. The great Wendy Menard (@wmukluk) shared some fantastic resources that one of her colleagues shared. She was also kind enough to spend some time on the phone last weekend to serve as a sounding board. One of the decisions I made was not to spend much time emphasizing notation together in class. For example, our text explains permutation notation pretty cleanly, points out that our calculator writes 10P4 while you might also see P(10,4). It clearly shows that this calculation is 10!/(10-4)! while also introducing this notation in more general form of P(n,r). In class we had a number of examples of drawing some subset of members from a group, so I thought that the text’s approach and our class approach would support each other. I also figured that any students flummoxed by the text notation would ask me in class what the deal was. So, the first HW question on Wednesday night was this in fact – Which is equivalent to P(10,4), 10!/4! or 10!/6! ? We had a quiz scheduled for Friday and on one of the questions I gave the students the numerical value of P(26,3) and asked for an explanation of how to get that answer. On Thursday I had a couple of review problems thrown together from the textbook author’s supplemental test bank. I planned on starting class by fielding any HW questions then turning them loose to work on the review problems. In my morning class I projected their HW from Wednesday night and had the first HW question on the board. Not one student knew what P(10,4) meant. They asked whether that was a point on the plane. I have to assume that they did not do the reading or the HW on their own. I quickly untangled the notation, pointed out how it matched some other conversations we had and then gave them their review sheets to work on. That was my morning class of 7 students. After lunch I had the book projected with the first HW question. Not one student in that class knew what P(10,4) meant. I decided to remark on the importance of doing the reading and the HW and then just gave them their review sheets and sat down.

One student came by on Friday morning during my free time to ask me a question about the notation and she remarked that it was clear I was disappointed (annoyed?) that no one had done the HW. She wanted to make sure she understood that notation. Each class on Friday began with me answering questions before the quiz and I do not recall anyone in either class directly asking me to revisit the P(n,r) notation at all. I have not graded the quizzes yet but I know that there were a number of students in my second class that either left the question about P(26,3) blank or simply wrote something to stumble into extra credit. A number called me over to ask about it and I said this was something they needed to know. I do not remember my morning class as clearly, they may have been in a similar boat.

So, as I think about this I realize that I made two very different decisions with my two groups of students and I am not happy about either of them. In one class I came to their rescue and explained something that they clearly could have come to terms with – in some way – on their own. In the other class I let my annoyance take over and I did not address the question at hand. I also realize that my students, especially those in my second class had two decisions to make. On Thursday night, after seeing my disappointment/frustration they could have gone back to their reading and either understood it themselves or they could have checked in with me during review on Friday. It is clear that a number of them did not do that. So I am faced with yet another decision when I grade what are likely to be disappointing papers. I feel that I want to get across a pretty clear message about responsibility but I also need to recognize my responsibility here. It is reasonable, I think, to see my role as someone who expands the conversation from the text, not as someone here to simply recite what the text already explains. But I also recognize that I have 9 students who are new to our class and all of them are new to me as a teacher. If they are used to teachers making sure that every question in their text is also addressed in class then my idea about my role might be a bit of a shock and I did not spend much time together on Monday explaining this about myself. However, I also have 14 students who were with me all fall and it is pretty clear that none (or very few) of them did the reading and the HW either.

I am not happy with myself that I let my annoyance get in the way of clear thinking. I am also not happy that I was not more clear with my morning class about my disappointment that none of them had done what I asked. I am not happy that so many students did not do the reading or the HW. I AM happy that I had a student come by and clarify the question for herself while also recognizing that she should have done so on Wednesday night. I feel that including the question when I compute the grade will likely have a pretty significant impact on many grades as it was one of four questions on the assignment. I also feel that it is a reasonable question to ask, but it relies on notation that I did not explicitly present.

I have been reading a number of the DITLife blog posts and there is a constant reminder about the number of decisions that we make on the fly everyday. These are complicated decisions and I know that I hope that I make them clearly. Here is a case where I think I was probably not as clear thinking as I should have been and I will likely need to make a decision about grading that will, luckily, not have to made on the fly. I have a bunch of new students who are only one week into their experience with me. I want it to be a good experience where they grow as scholars. I need to think carefully about how I respond to this disappointment – in my own behavior AND in the decisions they made.

Assessment Is On My Mind

Our school is making a pretty big change next year. We are moving from a static schedule where we have seven classes that meet in the same order every single day. Our class lengths are either 40, 45, or 50 minutes depending on the length of our assemblies and our ending time each day. Next year we will be on a rotating schedule in 7 day blocks. Each class will meet four times for 50 minutes and once for 90 minutes over that interval. Five classes will meet every day, four of them will be 50 minutes long and one will be 90 minutes long. As a result of this upcoming change we are being asked to do some deep reflection about our practice and our curricular choices. As department chair I have been encouraging my department to think deeply about trimming or eliminating items from our curriculum to make time to think and have more open ended (and open middle!) problems. This schedule change will really push us to do this and I suspect I’ll be thinking out loud on this space as we move through this process. Where my mind is tonight is on the subject of assessment. I was asked to facilitate a group this morning to talk about our assessment habits and goals. Years ago, I moved to a school that had a rotating schedule with a long block. At the time I was not the same teacher I am now (at least I don’t think I was) and I don’t remember being all that thoughtful about the impact on my practice. The fact that I had moved to a new school and just had to adapt probably diminished any sense of dramatic change that I might have felt. Around here right now we are having some pretty valuable conversations and I was part of one this morning.

My takeaways from the meeting are:

  • I want to have more frequent, but shorter, opportunities to check in on my students’ learning. Ideally, some of these would not have any letter or grade attached at all but simply serve as formative checks of their developing understanding of the material at hand.
  • I think that the old model of a 45 – 50 minute timed sit down assessment where everybody takes the same (or VERY similar) tests needs to shrink, at least in my discipline. Our time together will be too precious to take up too much of it silently hunched over a piece of paper with problems, no matter how creative the problems are. We have 154 class days this year that are not devoted to term exams. Meeting 5 out of every 7 days changes that to 110 days. Granted, these 110 days will be a minimum of 50 minutes each (our current longest class) so contact time will feel different. But we are still only going to see our students for 110 days and we need to make those days count.
  • I want to expand my palette. My students’ grades are almost entirely dependent on times tests and quizzes. I know that there are other smart ways to do this, I want to learn more and I want to grow my toolbox.
  • I have been sneaking in group quizzes lately given that my class is set up on pods. I want more of this. I experimented with my Discrete Math elective in the fall. We have a stretch of days leading up to our term exams called test priority days. On these days we are limited by department for who can have assessments so that our students do not have these pile up on them. Each department has two of these days. I chose to have a group test on the first day where everyone chipped in ideas (obviously some students were more vocal than others) and they all got copies of my feedback on their test. Four class days later, after one new topic was introduced, each student had an individual test. My best students performed about the way that they always had, I wasn’t too worried about them. What pleased me was that some of my students who had struggled during the term clearly benefited from the group work not only in improving their grade by the addition of this group test score, but they performed at a much higher level on the individual test. The combination of the work with their peers and the ability to study from that work and from my feedback all seemed to work well together. I want to capitalize on this and try more ideas like this moving forward.


So, the reason I am thinking out loud here about this is the reason I always cite. I would love to hear from you, dear readers. Please share successes and failures you have experienced as you expand your assessment toolbox. I want to hear how you wisely spend time with your students on extended classes like the 90 minutes we will have periodically next year. I want to hear what pitfalls to avoid that we might not even be anticipating. In general, I just want to continue learning from all of you!

As always, feel free to comment here or to engage me on twitter where I am @mrdardy


Thanks in advance

Hands-On Geometry

I’ve been at this high school math gig for a good long while now but I periodically have to remind myself of a couple of important facts. The most important one is that not everybody’s mind works like mine. Just because I like a certain way of thinking, or dislike a certain way of learning, I should not assume all my students will agree. In fact, I can be pretty certain that all of my students will not agree, there’s too many individuals for that to work.

When I studied Geometry I did not like physical drawings and constructions. In part because I am a bit inept when it comes to controlling something like a compass, but also because getting my hands engaged does not seem to fire too many of my neurons. So, when I wrote my Geometry book a couple of years ago I did not include much in the way of hands-on manipulations. The past couple of years of working through the text with our students has pointed out the weakness of this approach. So, I put my head together with one of my talented colleagues to try and make an activity that would trigger some neurons for those students who come to life when they get their hands busy. I had been using a pretty cool activity I ran across from Jennifer Silverman but I made pretty flimsy paper copies to work with on a pipe building activity where kids had to manipulate bent angle joints with different pipe lengths. It’s a great activity but using simple paper copies dragged the activity down. We invested in some packs of AngLegs this year and my colleague wrote a pretty cool activity modeled off of our pipe building activity. You can find his document here.

I was impressed as each of the seat groups in my class played with the AngLegs making some discoveries about combinations that worked and those that would not. We discussed, without naming it yet, the triangle inequality theorem to explain why some combos did not work. But the real fun, and the clever heart of my colleague’s activity, was when I asked one student from each group to come to the front of the room. When they left their group the remaining group members were given the following task – I slightly modified the original document on the fly – I asked them to make and measure a triangle. Find six measures, the three side lengths and the three angles. They then put the triangle away where it could not be seen. I sent the volunteers back and their teammate gave them three pieces of information. I left it to each group to decide what information to share. Once given three clues the volunteer student needed to manipulate the AngLegs to copy the triangle described. What ensued was a terrific conversation about what information is necessary to guarantee that I have to make the same triangle. We used this as a launching pad to discuss congruence theorems for triangles. I have some great links in the text to some wonderful GeoGebra activities up on the GeoGebraTube site but I know that many of my students do not do these explorations.  I also know that some just need to get their hands dirty, so to speak. Some kids were able to recreate the triangle but admitted that it was a bit of luck. Some stumbled upon the ambiguous case of the Law of Sines without being told that this is what happened. Some realized that they had no choice but to create the correct triangle.

I was really pleased by the level of engagement and I am now thinking about ways to use the AngLeg sets again soon when we start talking about side and angle bisectors. I want to have tables create and draw their own triangles before we stumble into discoveries about concurrence of these bisectors. This will feel, I hope, a little more authentic than me just giving them a prescribed triangle which may feel a bit like I am just luring them into some pre-prepared trap. I think that this activity we ran benefited my students and we have referred to it on a number of occasions already. The grouping of three or four students together at a time helps and allowing them to get their hands busy has helped. Looking forward to loosening up a bit more and letting my students be more tactile in their approach to Geometry. I’ll still show them the GeoGebra and introduce them to Euclid the Game  but I need to remind myself that they are not a bunch of mini Dardys in the room.

Questions about Questioning

I feel I am long overdue to write this blog post. In part, this is due to, you know, life getting in the way. In part it is because I have about three posts swirling around in my head right now. Next week our students are taking term finals so I will have a little more unstructured time and I may finally get around to writing more. That is, if I get around to writing plans for the short stint between thanksgiving and winter holidays.

Today, I am going to try and make sense of a fantastic post by Mark Chubb (@MarkChubb3) that can be found here. In the post (which you DEFINITELY should read) Mark raises important questions about the questions we ask our students AND the purpose, the goal, of those questions. I often tell a story about a student who graduated back in 1993 named Ashley. I had the privilege of teaching Ashley for four years in a row up through Calculus BC. The week before her AP exam I asked her how she was feeling. She told me that she was not worried at all because she knew that if she got stuck on a problem she would hear my voice in her head asking her what that problem reminded her of or what have we done in the past when we have seen this. I was flattered that she had internalized some of the strategies we had worked on together and I felt good that she felt comfort in my leading questions that I had been asking her over the years. She was also a tremendous student who was in a group of talented kids who pushed each other over that four year span. Since then, however, I have begun to question myself about the sort of questions I pose. I still believe that most of my students would be able to effectively work through problems they are presented if they can have an internal monologue that is similar to the conversations we have as a group. What I worry about is whether my guided questions are taking away their agency, their ability to discern what they think is important in a problem. I made it through high school math pretty successfully and I have confidence that I can guide students through this journey. But posts like Mark’s, and conversations I have had through this blog, in conferences, through twitter all push me in the direction of making my voice less central in my class. I have taken great strides in this direction in the past few years, but I still feel that I talk too much, that I initiate conversations and lines of questioning too often. That I impose my sensibilities about what to notice and what to wonder about on my students. The trouble is that many of them are happy to have me, and their other teachers, take on this burden. It is easier, it feels more stable and safe to hear the expert in the room direct the conversation. I know that this is not the best strategy but I too often fall into this trap.

I am going to lift a portion of Mark’s post here to draw attention to the central question about questions that I think he was trying to raise.


Funneling vs. Focusing Questions

As part of my own learning, I have really started to notice the types of questions I ask.  There is a really big difference here between funneling and focusingquestions:


Think about this from the students’ perspective.  What happens when we start to question them?

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After reading this and playing back a number of classes in my head, classes that I was really proud of, classes where I felt that my students had made some major breakthroughs, I realized that I do FAR too much funneling and not enough focusing. An easy excuse is that my students struggle in metacognitive processes, so it is more painful and time-consuming to do this. And, just like my students want the more comforting path of me telling them what is important, I am prone to take the comforting role of the guiding questioner. But my students are not going to get better at monitoring and understanding their own thinking this way. They are not going to take ownership of what they feel is important in a problem this way. They can get to be better mathematicians and be more successful at high school math, but I fear that I am training students to be mini mrdardys instead of being better high school mathematicians as themselves.


Our school is moving to a new schedule model next year. We will be on a seven day cycle where in each rotation we meet four times for 50 minutes and one time for 90 minutes. This will force us to re-examine how we run our classes, how we will value and plan for our time together with students. There are many layers of what we will have to examine but for myself, I think that I will be going back to Mark’s post over and over, I’ll be looking at a few classes that I have had videotaped and I will be working out how to hand ownership of ideas to my students. I will be working on how to make sure that the classroom and the class time we spend together is not so dependent on my point of view and my insights into problems. I want my students to leave my class better high school mathematicians, that is absolutely true. But I want them to be better models of themselves as high school mathematicians, not imitations of me. As Ben Folds sang once, I do the best imitation of myself. I don’t need my students to be imitations of me.


Linear Functions

Some of my Geometry students are wrestling with being able to accurately write linear functions given information about points and slopes. I am struggling with how to help them overcome this and I have been doing quite a bit of thinking about how we teach this and what kind of sense it might make to my students. I know that I have these fundamental ideas in my head – an equation is a relationship between the variables stated in the equation (it tells me how to turn an x into a y or vice versa) AND the graph of an equation is the set of points that makes the equation true. I know that I say these things and I am fairly certain that previous teachers have said similar things. I know that my students have repeated some of these things and they can (periodically) carry out these operations. Where the mystery lies for me is why this skill can only be inconsistently displayed and I have a couple of thoughts. I am interested in any wisdom you have about this question.

Almost every single one of my students prefers the slope-intercept equation of a line to any other form. Partially because of my history as a Calculus teacher and partially because I favor more direct problem-solving approaches, I am an advocate for the point-slope equation. I consider it a minor success that most of my students answered the first question on their recent quiz in this form. Here is the question: Find an equation of the line that passes through the points (3 , 1) and (5 , 4). Now, I am careful to ask for an equation rather than the equation but I do not know how much of an impression this might leave on any of my students. Almost every Geometry student answered this correctly and most left it in point-slope form. I was pleased. The next question was this one: Find the coordinates of the following points on the line you found in problem #1.

  • The x-intercept.
  • The point with an x-coordinate of 1.
  • The point with a y-coordinate of 1.


Here is where things started to fall apart for many of my students. I have been thinking about the mistakes I see in class and on assessments and it occurs to me that there might be a fundamental problem that I do not know how to solve. When a student wants to write the first problem in slope-intercept form I instruct them to first find the slope, then replace the x and y in y = mx + b with coordinates of either given point to find the b value. I tell them that this way is harder, but many want to hold on to that equation form. If they want to approach the first problem with the point-slope form I tell them to first calculate slope then replace x1 and y1 in the equation    y – y1=m(x-x1)  with the coordinates of one of the points they know while leaving the x and the y alone. I am embarrassed that this inconsistency has never jumped out to me before, but why is it that in one equation we leave the x and y while in the other we replace the x and the y with coordinates of a given point?!???!? I have to imagine that some of my students are absolutely baffled by this inconsistency. I wish that they could verbalize that sense of confusion, but I just now figured it out for myself, so why should they be able to lock in on this? So, dear readers I implore you – help me figure out a better way, a more logically consistent way, that I can help direct my students. This is not an intellectual task that is beyond any of them, but I have to guess that a handful of them are so tired of being asked to do this and have sort of given up on the idea that they will ever master this concept. It is way too easy to just write it off and hope it will go away. I do not want this to be their reaction and I want to see them reliably be able to answer these questions.

Busy Days

My last post was about a professional development conference I attended and presented at. Last week I went to another and presented again! In between, we had grandparents’ day at our school, so there are a couple of things I want to share today.

Last Friday I attended the Biennial Conference of the Pennsylvania Association of Independent Schools. It was hosted at the lovely campus of The Episcopal Academy in a Philadelphia suburb. I attended two sessions and presented my MTBoS love song at a third session of the day. When I presented at ECET^2NJPA I had a small, but engaged, crowd. At PAIS I was fortunate enough to have a full room with some people sitting on the counters. We had a lively conversation and one person in particular had some great questions. Recently, NCTM issued an editorial statement about the importance of curricular coherence. It can be read, in part, as a warning against using open source curriculum without deep and careful thought about how it all fits with what you are trying to accomplish in your classroom. My presentation focuses on my journey through the MTBoS and how the resources shared by our community helped inspire me to take on the task of writing a text for our school. It is a text that I hope represents some important values in our department. The text challenge I tackled was for our Geometry course and I have to admit that I felt a certain amount of freedom that I might not have if I was writing for Algebra I, Algebra II, or Precalculus. Those courses that are more in a direct vertical relationship with each other feel like they bear more weight in terms of coherence with each other. There is also more of a feeling that these courses depend directly on each other. I mention this because of a great question that came my way about this. One of the members of the conversation directly asked me about the decisions I made regarding the course content and I had to admit that I probably would have been more intimidated if I had tackled one of these more ‘core’ courses in the high school curricular stream. I felt that we had a really good conversation in the room about incorporating different activities into the classroom. Since the audience were all members of independent schools they probably have a little more flexibility than many of our public school friends have in terms of deciding what resources to incorporate into their classrooms. I have now made this presentation three times for three different organizations and I will probably put it to sleep now, but I am glad that I have had the opportunity to engage in this conversation and to spread the word about the deep well of resources that is the MTBoS.


Last Wednesday, two days before the conference, our school hosted our annual grandparents’ day. I have often been teaching Calculus in the afternoon and this rarely brings many grandparents into the room. This year I end my day with my Geometry class and we had about a dozen guests in class. I found a lovely activity at the Nrich site and my students and our guests had a terrific conversation tying together Cartesian coordinate plane ideas, transformations by vectors, and the idea of being able to project ahead in a sequence. I have been including problems from the Visual Patterns website and I think that Wednesday’s activity might have been a bit of a breakthrough. The conversation we had, and the inclination to want to show off for our guests, was more lively and engaging than I anticipated and on last Thursday’s test I saw better performance on the pattern recognition problem than I had previously seen. I cannot recommend Nrich too strongly. There is a wealth of great problems there and I am using another one for this Friday’s parents’ day visit when I expect to have another crowded room. One of the grandparents was here from California to visit her grandson and she stayed after class to chat and ask for a photo with me and her grandson joining her. I was flattered by her words and by the fact that she wanted to have this memory.


My ECET^2 Experience

This past weekend I traveled to Ewing Township in New Jersey to attend the third annual ECET^2NJPA conference. A bit of alphabet soup, this name. Let’s dissect it – The ECET^2 is for Elevating and Celebrating Effective Teaching and Teachers. The NJPA refers to New Jersey and Pennsylvania. I found out about this, I think, through a tweet where we were asked to nominate ourselves or somebody else. Since I was not sure of what the conference would be like, I did not nominate one of my department members (but I will for sure next year!) and I nominated myself. I also pitched a conference session. I modified one that I had presented in summer 2015 at a Pennsylvania Council of Teachers of Mathematics conference. I was flattered to have my proposal accepted, especially since we were told at the conference that about 200 proposals were submitted about 46 were accepted. It turned out that the timing of the conference was terrible for my family. My wife works at a nearby college and they were celebrated their homecoming weekend. A time of much stress and many hours for my wife. We also had committed months ago to traveling to see Brian Wilson perform The Beach Boys’ album Pet Sounds about an hour and a half from our home. So, I left around 5 AM Saturday morning and we had a family friend come and watch our 7 year old all day while I was gone and my wife was working. I drove home Sunday afternoon – essentially right past the town where the concert was – to get the family and turn around to the show. Long story short – both trips were WELL worth it. I won’t use this space to expound on the wonders of Brian Wilson, but I will use it to talk about my tremendous experience at ECET^2NJPA.

First things first – I have mostly been going to math professional events the past few years. I attended an EdCamp last year but in the recent past I have gone to 3 twittermathcamps and the aforementioned PCTM. This year I am going to NCTM in Philadelphia for a day, so this event with educators from different fields – administrators, elementary school teachers, special ed folks – seemed like it would be refreshing. I certainly ended up gravitating toward some math peeps, but it was great to be immersed in wider conversations. I also made a commitment to myself to try and go to some sessions that were not particularly math-y. This commitment is tied to the fact that I recently joined a local leadership program being run for educators. One of the goals of this program is to try and develop a school improvement project and to discuss aspects of leadership ranging from department to school to district. So, I am trying to broaden my horizon a bit and immerse myself in school conversations outside of math curriculum, and pedagogical techniques which is where my heart and mind have been living for some time now.

I will start off with my only complaint about the whole weekend. I am a bit of a holdout when it comes to phone technology. I stand out like a sore thumb at twittermathcamp because I am the only one who does not have sore thumbs from texting. I do not have a smartphone and the devices I brought with me (an iPad and a MacBook Pro) both belong to my school so I am reluctant to load much in the way of applications on them unless they are school related (with a few indulgent exceptions like Spotify). So, when I arrived at the conference there was no printout of the sessions being offered or indications of where they were located. We were all encouraged to have an app called Whovo to navigate this. Submissions of session evaluations were also to be done this way. Again, I recognize that I am a holdout, but it felt odd that I had to make a special request to see to agenda for all the sessions. To the staff’s credit, they printed one up for me right away. So, that is the end of my complaining. Now, on to the praise!

The first session I attended on Saturday morning was focused on effective feedback and was presented by Dr. Stefani Hite and Dr. Christine Miles. We discussed some ideas I was already familiar with but the real takeaway for me was a phrase they used that caught my attention. They talked about what they called ‘Feed Forward’ this is feedback or information to share with students about their work that allows them to move forward to grow. I feel that I do a good job of getting work back quickly and discussing issues with problems together in class. But I realize that most of my feedback in this format is looking backwards over what went wrong, not looking ahead to help prevent future problems or prevent the same problem from arising again. I hope that I can wrap my head around this and give my students constructive feedforward to help them grow.

The second session I attended was run by Baruti Kafele a former principal (in fact he exists in our virtual worlds as and @principalkafele) His session was called Critical Questions for Inspiring Classroom Excellence. He challenged us to answer the following four questions:

  • What is my classroom identity? (Who am I?)
  • What is my classroom mission? (What are you about?) – He called this our what
  • What is my classroom purpose? (Why do I do this?) – He called this our why
  • What is my classroom vision? (Where am I going?)

One of the debates we often have concerns dress code and why we have one (versus why we say we have one – these are rarely the same thing) and he had an interesting story to share that is causing me to think a bit about my stance on this question. He talked about people who wear uniforms. If you see a fireman without his uniform you have no idea who he is and no expectations about him. If you see a chef without her uniform you have no idea who she is and no expectations about her. However, when they are in their uniform you have a set of expectations about how they will perform, about who they are. I am wrestling with where this will go in my mind, but I know that I found it to be a striking conversation. He followed this by telling a story about how his children have expectations of him as dad, his wife has expectations of him as Baruti, we in the room have expectations of him as Principal Kafele. I do want to feel different when I am dad, husband, son, brother, friend, Mr. Doherty…


The next session was a wonderful one run by Manan Shah (@shahlock) and his wife Meredith Valentine. Mann is a college math professor (among other things) and Meredith is a second grade teacher. I was drawn to their session for a number of reasons. I was excited to finally meet Manan in person after interacting with him on twitter for some time. My daughter is in second grade so I wanted to hear some second grade stories, and it was time for some math. They discussed some fantastic math games and strategies for sneaking in some high level math ideas with little ones without burdening them with formal notation and imposing formulas. I am going to share at least one of their ideas with my daughter’s teacher. They talked about having two classmates skip count while walking (or even skipping!) together. Imagine this, one person is walking and counts off every second step out loud because two is her number. She goes step, TWO, step, FOUR, step, SIX, … Her partner has the number three so he goes step, step, THREE, step, step, SIX, step, step, NINE,… What a fun activity to plant ideas about common factors, about least common multiples, about a number of number patterns. What was great was that they never used this formal language, just let kids notice things and ask questions.

Session four was when I was presenting my love song to the MTBoS called Escaping the Tyranny of the Textbook. The theme of the weekend was ‘The Power in the Room’ and I felt that my message of self-sufficiency and the power of the resources of educators on the web sharing resources really fit in. I’m glad that the organizing committee felt the same way.

The last session was by Steve Weber (@curriculumblog) and it was called Building a Culture of Learning. I’m glad I was there and I was moved by Steve and the conversations in the room. I’m following him now on twitter and I expect to get some great nuggets from that.

We also had a series of speakers between sessions and had plenty of time to share ideas and stories not only at the conference, which was hosted on the campus of The College of New Jersey, but there was also a lively get together in the hotel downstairs on Saturday night.

I want to make sure that include a couple of important references and thank you notes here as I sign off. The lead organizer for the event was Barry Saide (@barrykid1) but he will be quick to pass off any credit and share it with the energetic team of coordinators and volunteers. I owe a thanks to the Gates Foundation who help underwrite these events. I arrived Saturday morning around 7:30 AM and left Sunday around 1:30 PM. In between I was treated to five nice meals, snacks in between, and a free room at a nice hotel nearby. Can’t beat that cost! Anyone interested in this organization can start out by checking out the website of the local event –

Special thanks also to Manan Shah (@shahlock) who I have been interacting with on twitter for some time now but until this past weekend he was a virtual friend. He’s a for-real flesh and blood friend now and it was delightful chatting with him at the conference and at the hotel.


Experimenting with Visible Random Groupings

In some ways I think that I am intellectually adventurous, that I am willing to try something new in my classroom. In other ways I struggle with change. I try to make myself feel better about this by reminding myself that we all struggle with this in varying degrees.

This past summer – my third lucky summer at Twitter Math Camp – I finally committed to trying visible random grouping (to be referred to as VRG for the rest of the post) for this academic year. A little background here.

When I moved up north in the fall of 2007 I made a commitment to not have my students in rows and columns. I no longer felt comfortable with most of my students looking at the back of other students’ heads. So, I rearranged the seats at my old school in ‘pods’ of three or four desks. However, I always let students pick where they wanted to sit. As most of us know, even if WE don’t assign seats, the students essentially do this themselves. I comforted myself by thinking about the camaraderie I saw, by listening in on the lively conversations that did not happen when my students sat as if they were in a matrix, and by the fact that I know that I would have preferred life this way as a student. When I moved to my new school I had two long conference style tables so I had two largish groups of students working with each other. Two years ago I ditched the conference table and went back to pods.

Over the past three summers I have heard more and more conversations about the power of rearranging the students, about shaking them out of these simpler comfort zones and encouraging everyone to be comfortable sharing ideas with everyone else in class. Alex Overwijk (@AlexOverwijk) has been an especially articulate proponent. So, this summer I learned about a pretty cool website ( where you can build a roster for a class and anytime you want this program will randomize your class. In groups of 3 or 4 or 5, by the number of ‘teams’ that you want, etc. It creates a cool visual that you can project and the kids get rearranged instead of staying in their friendly neighborhood comfort zone. I committed to trying this for a number of reasons, the primary one being my experience the past few years in Geometry. I had been teaching mostly AP and upper level honors classes and these students mostly knew each other for awhile and they were comfortable sharing ideas and debating/challenging each other at times. Not true of my Geometry class. Even last year’s class which was outgoing, chatty, and engaged. They did a great job in their pods discussing ideas but did not do a good job projecting those ideas out to the class. They always wanted to filter ideas through me and, over the course of the year, inevitably fell into some ruts about who took command when I asked them to work together.

Now, enter VRG. The strongest proponents discuss doing this every single day to continually shake things up. I got a little scared of this because I really value the sense of camaraderie that I have seen developing over the years, so I came to what seems like a nice compromise. On the first day of each week, I shuffle the class. I am now ending the fourth week of the school year and I have some observations I want to share. I am particularly motivated to do so by a twitter chat this morning.

There is sound research in the field about VRG and its effects. This research suggests that the positive effects of this practice are most clearly seen when this happens every day. I do not want to discount this and I do not want to feel like a contrarian. What I want for my classroom is for my scholars to not only know everyone else and hear the ideas of their peers, but I want them to be in a zone that feels comfortable and safe. My prejudice is that this zone is more likely to happen if I have some time to get used to my new teammates. What I have seen in four weeks can be summarized as follows (and I will make separate remarks for my AP Calculus BC group and my Geometry group)

  1. In BC Calculus I have also been incorporating whiteboards that the pods write on together. The combination of whiteboarding (and presenting the ideas of the pod) out to the class along with VRG has been pretty spectacular. Again, these are kids that know each other well, but I have been seeing active conversation across table groups to former teammates that is lively. I can step out of the way and let them bounce ideas around as I wrote about yesterday.
  2. In Geometry we have not done as much whiteboarding, I want to improve on this. What I have seen is students talking to people they did not choose. I see them making guesses to/with their neighbor. I have seen students more willing to stand up and talk. I have heard some lively discussion between students and I know it is not just with their good buddies unless they all magically happen to love each other.
  3. I have been able, in both classes, to call on a wider variety of people because even the shy/underconfident/nervous kid has someone in their group whose ideas they can paraphrase. In the past few years I felt that there was more of a posture of looking to one person in each pod to be the spokesperson. I see less of that now.


When I tweeted out my happiness about weekly VRG I was promptly congratulated AND reminded that this would be even better if it was done daily. I may get there, but I kind of feel that this is my 10% moment. That place where I am making a change I know is for the better but I am limiting myself in my own discomfort a bit so that I can still feel sane and effective in other arenas.

More Creative Problem Solving



The problem above came across my twitter feed this morning courtesy of John Joy (@johnjoy1966) along with the suggestion that this was a problem from a trig unit. John also questioned who this problem would be appropriate for. I told him I would feed it to my wizards in AP Calculus BC. I also had a class coming in right after John posted it so I did not see any of the conversation – this way I could present it to my class with no prejudice about what to say. When they came in the next period I had the problem from the tweet up on the screen with no other support. I simply said that this seemed like an interesting problem and I had not had time to try it myself. I handed out the whiteboards to each desk group – this was their suggestion! – and I got out of the way. I heard them talk about the function being odd so that they knew f (-3) right away. One group found f (6) by imagining it as f (3 + 3). This meant, of course that they also knew f (-6). Progress, right? But nothing about knowing the values of f (1), f (2), or f(3). I asked if they wanted to hear a hint and three students quickly waived off that notion. After another couple of minutes I went to the board and started writing what we seemed to know about the function.   I wrote f (3) = f(1 + 2) and wrote out what the definition of the function suggested. We got an ugly expression for comparing f(1), f(2), and f(3) to each other but it was not promising. I was itching to give them a hint but they were holding me off. One of my students – thinking out loud – wondered if this might be a periodic function based on the values we knew on the board. Another group suggested that it might be a sine function. I hopped at this – another example of how I need to work on developing a poker face of some sort. The group backed up a bit and suggested that they were kind of joking, but I buoyed them up by reminding them of the periodicity suggestion. I finally gave them a vague clue – one too vague to have helped them at all. One of my students during a class warm up a few days before had the back of his book open to a series of formulas and review facts from their study of trig. I reminded the class that I complemented him on that and I pointed out the similarity of the given function to a trig identity involving the tangent function. The kids were a bit flustered claiming that no one remembers these formulas but they sealed the deal right away once they had this fact in hand.

So, what did I learn from them today?

  1. I need to work on my poker face.
  2. I need to stop giving clues, they are too good to need them.
  3. This group of students is super persistent and creative.
  4. The small desk groupings AND the randomization every Monday seems to be working.
  5. The whiteboards give them space to ‘think out loud’ and effectively share ideas.


Man, a terrific day in Calculus thanks to my wizards and my virtual friends who prod my brain with their great problems.


Today was a pretty blah day until my last period class. My first three classes all had assessments so I had no fun conversations and I watched work pile up. As I came in to my last class of the day – my Geometry class – one of my Geometry teammates was waiting in my room to share that his students had been making some great strides in GeoGebra. He told me that a number of his students were really beginning to dig into what GeoGebra could do for them, especially now that we are talking about transformations. I used Geogebra extensively when writing my text and I borrowed from resources around the web for activities. One of them was an activity called A-Maze-Ing Vectors which had been created by the amazing Jennifer Silverman (@jensilvermath) and we used that activity the past two years. My teammate who had been waiting to share his good news had asked me this past summer about modifying this activity. We had had trouble completing the activity in one day and it did not take up enough for two solid days. He also had an idea about combining vector transformations on objects more complex than points. He created a pretty wonderful adaptation of the activity (you can find it here) and my students worked through it yesterday. I opened class today by projecting the last page on my AppleTV where we had to navigate a triangle through a maze and I invited a student to come up and draw on the TV (with a dry erase marker, don’t worry!) and I cannot tell you how great the conversation was in class. I sat down – a commitment of mine based on my #TalkLessAM session at TMC16 – and just watched the fireworks unfold. Kids were challenging each other, going up to the TV to draw their ideas, debating distances, talking about slope, worrying about vertices colliding with walls and discussing the option of rotating the triangle as it moved. I was SO thrilled with the engagement and the level of conversation. I credit this to a number of factors. The original activity was terrific and my colleague’s rewriting of it is creative and concise. Kids like drawing on a TV – it feels naughty or something. I sat down and got out of the way. Kids had worked this through the day before in their table groups and were invested in both supporting their teammates and making sure that their memory and their perspective was clearly heard. They were supportive of each other and slightly defensive if someone else had a different approach. After a pretty uneventful day at the end of the week it would have been easy to just limp tot he end of the day, but these kids brought each other to the finish line for the week sprinting. I am optimistic that we can pick up with a similar level of energy on Monday.