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.

Scouring for Resources

In a couple of weeks I will be presenting a session at the ECET2NJPA conference. I’m pretty thrilled about this opportunity to network with other educators and to sing my love song to the MTBoS community. The presentation I pitched is called Escaping the Tyranny of the Textbook. I presented a version of this previously at a summer conference for the Pennsylvania Council of Teachers of Mathematics. The focus of my presentation is to help build a network of people to support each other in the creation (and curation) of meaningful classroom experiences without having to rely on textbook publishers to always be the resource of activities, worksheets, practice problem sets, etc. Those of us who are engaged in the MTBoS know that there are an ocean of resources available to us – blogs, twitter feeds, the MTBoS Search Engine , various virtual file cabinets, etc. I incorporated a bunch of these links into my presentation to PCTM because I had a pretty targeted audience. I expect a wider range of classroom interests in my upcoming presentation, so I want to broaden the scope of the links and resources I highlight. This is where you, my dear readers, come in. Please share with me here in the comments or over on twitter (remember – there I am @mrdardy) any rich resources for social studies teachers, or English teachers, or World Language teachers, or elementary ed folks, etc etc etc


Thanks in advance for any wisdom!

Questioning Myself

This is going to be a super quick post and I would LOVE some feedback here and/or over on the twitter (where you can find me @mrdardy)

I am one of three teachers of Geometry at my small school and I am also the chair of our department. I feel that the three of us ought to have pretty similar policies to make life feel a little more unified and fair to our students. I have opted in the past to keep most of Geometry calculator free. I feel that this is one of the last opportunities to try and help firm up some number sense and some self reliance with minor calculations. I also encourage my kiddos to leave answers like the square root of 160 as a final answer or even something like ‘the sum of the first 98 natural numbers would be the number of handshakes’ to help offset any anxiety about calculations beating my students down. I also place very little emphasis, pointwise, on arithmetic mistakes. One of my colleagues pretty vigorously disagrees and feels that having a calculator by their side eases pressure, and is simply a more realistic way to approach life for her students. I find myself questioning my decision here since I do not restrict calculator usage in general in my other classes. I do, however, worry a bit about all of this since I am hearing pretty consistently from recent alums that they head off to college and are not permitted to use calculators in their freshman classes. My recent Calculus students report this very consistently, they take freshman Calculus at college without a calculator. I know that there are all sorts of reasonable arguments that we should not make high school decisions based on college realities. I also know that I am hearing back from a small group of students.

So, I guess all of this rambling is really about one thing – Give me advice! Let me know how you approach this question. Does it depend on the level of the class? Is it a departmental decision? A school or district policy? Am I simply holding on to some quaint idea that mental arithmetic really matters? I fear that I am not being coherent or consistent in how I think about this issue. HELP!

Some Class Silliness

One of my pet peeves is when students use acronyms to mask knowledge that they already have, thinking that this somehow makes their life easier. I cannot count the number of times I have heard Precalculus students chanting about ‘All Students Take Calculus’ as a way to remember what trig functions are positive in different quadrants. This, despite the fact that they have known for years that above the x-axis is positive for y and to the right of the y-axis is positive for x. Clearly (to me,at least) this does not add to their knowledge, it just adds a filter that obscures relationships in their knowledge. So, I tell them a story to express my disdain for these kind of meaningless mnemonic devices.

Years ago I had a hilarious student named Jeanine in one of my classes. One day, when walking to lunch, Jeanine calls out to me and says ‘Hey, Mr. Dardy I have a mnemonic to remember how to spell my name!’ She tells me that she just needs to remember








I love this story for a number of reasons. More than 20 years after this happened this still makes me giggle and makes me think of a wonderful former student. It is a true story and my students instantly see how silly it is to have this sort of memory device. I do not know if this has resulted in enough of them abandoning mnemonics but it is a fun story to tell.

This year, I told the story to a class with a student named Genevieve. The class took on the challenge of trying to develop a mnemonic for her name. Here is their result:











This makes me all kinds of happy.

Beginning of the Year Updates

A brief post here as our labor day winds down. As with most boarding schools, we actually have classes on labor day.

Yesterday I wrote about the clever solution that one of my Calculus students presented and, as I guessed, he did make a clear decision about when a power of x can be negative versus when powers of x cannot be. I had him present his solution to the class to start things off today. In my Calculus and my Geometry classes I am using to generate random groups at the beginning of each week. I am not brave enough yet to randomize groups every day, I feel like it is important to me to have some comfort (even if it is just a few days at a time) within my small groups. I have also (finally) bought some whiteboards and have one at each table group. It has worked fantastically well in Calculus. The kids have been talking vigorously, they have been enthusiastic about sharing their work out to the whole class. One of my goals coming in to the year was to increase student voice – especially in whole class conversations – and so far I have accomplished that in Calculus. In Geometry I am also shuffling groups at the beginning of each week. They have been better at talking in their groups rather than projecting out. This is not surprising to me. They are younger students and generally not quite as confident as the Calc BC kiddos. However, last Friday I was pretty insistent about having students stand and say what was on their mind and to say what their questions were. I made a big show of sitting down and asking students to stand so that everyone could focus their attention. We had multiple solutions to a problem offered and a great question from one of the students about a particular solution. I think that as long as I can be consistent, and insistent, about stepping aside and having students take the lead in the conversations then I think I can make some progress with these students and help set the table down the line for our department in having a student body that sees participation as a central part of their job.

My other two classes are each super small right now at 5 students each. They will each grow a bit but the whole idea of random grouping does not work with groups this small. We all sit together at a conference table. These are the sections of our senior-level elective called Discrete Math. We are having some good conversations about voting and ballot strategies. I was delighted to have one of my students tell me how excited she is that a number of her classes are all touching on the same ideas. She is in an AP Government class and I love the idea that students see that ideas can work across course departments.



Curious Creativity

A quick post here reflecting on a great solution presented by one of my AP Calculus BC kiddos. They had their first assessment on Friday. At this point, we are doing a quick and deep review of last year’s work. In our school, BC students have already completed AP Calculus AB and we spend this year digging deep and moving into the BC only topics. So, one of the questions I posed was in two parts:

  1. For what value(s) of x is x^10< x^6
  2. For what value(s) of x is x^7 < x^3


This, by the way, was a non-calculator assessment. I will be writing soon about my wavering on this issue. One of my students presented the following work:


  1. x^10 < x^6 becomes x^4 < 1 and this is true whenever |x| < 1 (other than x = 0) so the intervals are (-1, 0) and (0, 1)
  2. x^7 < x^3 becomes x^7 – x^3 < 0 which becomes x^3 (x^4 – 1) < 0 since x^3 < 0 for all x < 0 we need x^4 – 1 to be positive. This is true when 1 < |x|. So, the overlap here is x < – 1. If x^4 – 1 is negative while x^3 is positive, then 0 < x < 1.


What knocked me out here was that he divided in one case (when it was safe with even powers) while he subtracted in the other case with odd powers. Now, I have not had the opportunity to ask him about this yet, but I have to imagine that this was not just luck. I think he had some instinct, and I want to gauge how conscious this instinct is, that there is a problem with dividing by x^3 which can, of course be negative. I get to labor on labor day, we have classes here on this holiday, so I will quiz him a bit about this. I’ll report back.


Thanks to Sam Shah for catching a mistake in my earlier version of this post!

Long Overdue Thanks

I will be cross-posting this over at the One Good Thing site as well.


I graduated high school in 1982 and I just started my 30th year as a high school math teacher. During my time in school there were three particular teachers that had a huge positive impact on me. My junior and senior English teacher, Mrs. Myra Schwerdt; my junior Honors Introduction to Analysis teacher, Mrs. Sally Giles; my senior AP Calculus BC teacher, Mr. Barry Felps.

I was fortunate enough to have run into Mr. Felps about 8 or 10 years into my career at a workshop. I graduated high school in Jacksonville, FL and I went to college and started my teaching career in nearby Gainesville, FL. My graduate advisor at the University of Florida was also Mr. Felps’ advisor when he had been in school. This made me feel that there was some sort of deep connection there and, in an odd way, reaffirmed my decision to think about teaching. Anyway, I was able to see Mr. Felps in person and thank him for his influence. I hope that this meant something to him.

I never saw either Mrs. Schwerdt or Mrs. Giles again after graduating. I fear that Mrs. Schwerdt may no longer be around (this was a long time ago that I graduated!) but recently a friend and classmate sent me a link to a profile of Mrs. Giles. She had changed careers at some point and was working for a local agency in Jacksonville called Cathedral Arts Project. A local paper wrote a feature on her career there as she was preparing to retire. I reached out to the communications director at that program and she shared Mrs. Giles’ contact information with me. I just finished writing an email to Mrs. Giles thanking her for what she did for me and letting her know that she is a major reason that I chose to do what I do.

I know that it means a great deal to me when former students reach out to thank me or simply to share some story as a way to keep in touch. I have no real way of knowing whether Mrs. Giles will remember much about me as it has been over 30 years now since I was in her class. I also know that she no longer teaches, so this note will not serve as a pick me up on a tough teaching day. But I also know that I fell MUCH better having written this note and I hope that in some way it brightens her day.