The Language That We Use

I recently engaged in a spirited discussion prompted by Patrick Honner (@mrhonner) on twitter and on his blog. The original post that started this whole discussion can be found here and it is well worth your time. Engaging comments there an on the twitters and a friendly suggestion by Patrick himself has me writing here, thinking out loud.  To set the stage for this post, an image from Patrick’s post is important.

Screen Shot 2016-07-31 at 8.34.15 PM

A quick glance at this certainly suggests that these are congruent figures until you look more carefully at how the question is worded. This is a pretty classic example of the kind of question that makes students think that test writers are gaming the system to catch them in a mistake. We are looking at two figures that are equivalent to each other. A rigid transformation maps one onto the other. However, that mapping does not map them in the order suggested. A classic mistake that I lost points for as a student and one that, sadly, I admit that I have probably deducted points for when grading. The debate on the blog and on twitter raised some really challenging questions about our goals with this type of specificity. Yes, mathematics is a precise language and precision is a powerful habit to try to help develop. However, I keep thinking about my fun Geometry class from last year. When we were discussing how to determine whether  a triangle with given side lengths was acute, right, or obtuse we worked out a strategy where we assumed that the Pythagorean Theorem would hold and we decided what the consequence was when it did not. This led to my students saying things like this; “If the hypotenuse is bigger than we thought it would be, then the triangle is obtuse.” Now, I know that the largest side of an obtuse triangle is not called the hypotenuse. When pressed on the issue I suspect that almost all of my students knew this as well. Optimistically, I want to say that they know this as well, but is is early August… My concern here is that I was letting them down by letting them be a bit lazy with their language. What I did at the time was to gently remind them that hypotenuse was not the best word to use there but I understood what they meant when they said it. Should I have made a bigger deal about this at the time? Was I being understanding and flexible? Was I being undisciplined and imprecise? I suspect that there is a decent amount of both of these in my actions and I have to admit that I did not think too deeply about it at the time. In the wake of the conversation that Patrick moderated, I am thinking deeply about it. It is also early August (again, I not this) and it is the time of year that my brain reflexively starts dwelling on teaching again. I am also thinking about a distinction that I got dinged for as a student but this time it is one that I do not ding my students for. I remember losing points in proofs if I jumped from saying that if two segments were each the same length then they are congruent. This is, obviously, true but I was expected to take a pit stop by making two statements along the way instead of jumping straight to congruence. I know that equivalence of measure and congruence of segments (or the same argument with angles) are slightly different meanings. A nice explanation is here at the Math Forum. But I feel pretty strongly that my 9th and 10th grade Geometry students are not tuned in to the subtle differences and I think I am prepared to defend my point of view that they do not need to be. I want my students to be able to think out loud and I DO want them to be careful and precise in their use of language but I do not want them to think that this is some sort of ‘gotcha’ game where I am looking for mistakes and looking for reasons to penalize them.

I am thankful to Patrick for getting this conversation started and for gently nudging me to try and work out my thoughts more thoroughly on this issue. I am interested in hearing from other teachers – particularly Geometry teachers – on how they try to navigate these conversations. How precise should our high school students, especially freshmen and sophomores, be when discussing these issues?

As always, feel free to jump in on the comments section or reach out to me through twitter where I am @mrdardy


This is also posted over at my new site and this may be the last one that I cross post. Please visit me over at


Brief Thoughts About Classroom Closing Strategies

I am still processing the keynote speech that Tracy Zager (@TracyZager) gave at TMC16 and I hope to write a coherent blog post about it soon. One thought that is on my mind because of her is that I need to make a serious commitment to thinking more deeply about how my classes end each day. FAR too often they end with me, or a student, noticing that time is up (we do not have beginning or ending bells for our classes) and everybody packing up quickly and scurrying off to their next class. This has to stop. It is not fair to my students AND it undermines any habits of thoughtful reflection that I claim to be important for me and for my students.

At my last school I had an alarm clock app on my laptop that was relatively easy to manage. I programmed it every morning to sound with 3 minutes left in class. I had it linked to a portion of Steve Reich’s Music for 18 Musicians on my iTunes library. What happened at this point was that the music faded in and increased in volume for about 30 seconds. This was our cue to start wrapping up class. My students got in the habit of using this time to slowly pack up their page while we tried to have a conversation about what went on in class that day. When I changed schools I changed laptops and I did not find a similarly friendly app. I also was not thrilled with how those conversations went so I sort of abandoned this idea. Looking back, I am pretty disappointed in myself about this. Whether or not I can find an appropriate timer app (I could, of course, just use something as direct as a kitchen timer) I certainly did not need to abandon this idea. We have been having conversations at our school about our schedule and about the busyness of each day for our students. Our classes are in three different building on our campus and there are many times during the day where the five minute passing time between classes is a stretch for some of our students. If they simply dash from one class to the next with no structured opportunity to pause and reflect, then we teachers should not be surprised when there is less than ideal recall about recent conversations and activities in class. I do not want to obsess about each minute we spend together and get defensive about the idea that ‘class time’ ends before the actual end of the scheduled class. Looking back at the habit I tried to instill at my last school, I think I was coming from a good place. I often framed these two to three minute chats just in the light of ‘What do we know now that we did not know 40 minutes ago?’ and I suspect that this is a decent place to start. However, as with most classroom practices, if I am not diligent and thoughtful about its implementation, I suspect that students will simply see my class as one that ‘ends’ three minutes sooner than it is scheduled for. I also think that they would be largely appreciative of this extra time to breathe, to dash to the bathroom, or to simply stroll to their next class at a more leisurely pace. Honestly, this little advantage in and of itself would not be a bad thing. But I have larger goals than that. I want my students to start to get in the habit of reflecting on our time spent together, to think about how we have grown as a group in our time together, to pause and reflect on an important problem we discussed, a surprising result we found, or a challenge that still lies before us. If I have any real hope of this happening, I need to be a disciplined and structured role model for this habit and, I think I need to be transparent about this. I want to discuss this as a goal at the beginning of the year and I want to refer back to this conversation. Recently I shared a document I created called How to Succeed in Geometry. You can find it here. It is a draft in progress and I intend to create similar docs for my other courses. I need to add in some more description at the beginning about how we will conduct discussions in class (thanks to the great TMC16 morning session run by Matt Baker (@stoodle) and Chris Luzniak (@pispeak)), how we will try to end our time together on most days, and how we are going to commit to paying attention to each other and not just to what old Mr. Dardy has to say on any given day.

Please help me flesh out and improve my ideas about classroom closing strategies by sharing your questions, comments, stories of success in this area either here in the comments or over on the twitters where I am @mrdardy



Note: This is cross posted from my new site. You can find me at

Soon, I will only post from there. I hope you’ll migrate over there with me.

I’m Moving!

Both in real life and in my virtual life, I will be living in new spaces. Sometime in the next month I will move out of the dorm where I have been for the past six years. I’ll still be at the same school and still on campus, but with more privacy and less responsibility.

On the virtual front, I am moving from this comfortable place over to a new address


I’ll be tweeting out links for new posts from there. Apologies in advance for any inconvenience in managing blog readers or stored websites to visit.



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.

TMC16 Reflections, Part One

I’ve been trying to sort out my thoughts from the past week in Minneapolis and I have found that one of the best ways for me to do this is to sit and type them out. I am thinking that I may partition these reflections into three or four parts over the next day or two so that the ideas I am wrestling with will feel more bite-sized to me. Lil’ Dardy just had a terrible dental appt this morning so I am home with her all day. This will give me some writing time as she just naps away her pain and discomfort.


First, I want to concentrate on a small roundtable discussion section that I had proposed. I called it Building our own MTBoS at Home. A little background helps. I work in a small independent, day and boarding, PK – PG, co-ed school. I have five full-time colleagues in my department in our high school. Our other campus is three miles away and that is where my two children attend school. There are few other independent schools in my area and I have not found a way to connect logically with the public schools in my region. When I proposed this session I was hoping to crowd source some wisdom. I LOVe the online community I have tapped into and I suspect that if you are reading this that you do too. I also know that as valuable as you all are as an online resource, it is even better when I can sit down face-to-face to share ideas and energy. That is one of the beauties of the Twittermathcamp (TMC) experience. So, I was hoping to gather some ideas about how to build outreach so that we can find some of the same sustenance that comes from TMC more regularly in our home areas.


One of the GREAT problems posed by attending TMC is that every session slot has multiple promising events occurring. I was happy to have five energetic folks come to my session. I know that Sam Shah (@samjshah) and Tina Cardone (@crstn85) had a session with a  similar theme happening the next day. I look forward to picking their brains to see what came out of their session. A couple of the folks in my room where newbies to the TMC experience and it was great to hear what was on their mind. Our speaker at the Desmos pre-conference challenged us to think of evangelist as part of our job title, so that was on my mind all weekend. As we chatted in my session this idea kept coming up. Glenn Waddell (@gwaddellnvhs) spoke eloquently about his journey building community in Nevada. The phrase that came to my mind listening to him was ‘death by a thousand paper cuts.’ He spoke of sending out emails with links to administrators and other teachers every Monday. Simple, short links with a friendly message along the lines of ‘I saw this in my feed and thought it might be helpful.’ Every Monday – this is the part that really resonated with me. Be persistent, be consistent, be short and to the point. There is a local group that runs a math competition in the spring here – usually during our spring break, unfortunately – and I want to reach out to them. I want to find the email address of math teachers at my local schools. My goal this year is to build a couple of email group with addresses of these folks and reach out and share on a regular basis. Currently, I have been in the habit of emailing (or tweeting) links to colleagues – both those in my building and my online community – whenever I see something interesting. I think that I am going to adopt Glenn’s idea and make it sort of a weekly roundup. Perhaps I will use this space as the forum for my online team to share out ideas I have gathered or developed in addition to sharing out my classroom experiences. The other big idea I took away from Glenn was that he arranged a sort of happy hour meeting with some teachers in his area and, through the help of some grant money was able to provide some appetizers. He said that he also shared out some ideas regarding improving personal efficiency through some nice applications in addition to discussing class ideas. So he summarized by saying that he was able to provide a space that age each teacher three things to takeaway – (1) Some free food; (2) Something to improve their own personal life; and (3) Something to improve their own classroom.


I am pretty confident that I have a model to emulate and I hope to be able to start small with a meeting of local math teachers so that we can start building a support group for each other here in NE PA.


I want to thank all of those who came and I am pretty sure that I got all the names correct. I apologize if I missed someone in my scattered notes or if I got your name wrong. In the room was Kathryn Ramberg (@KathrynRamberg), Chris Robinson (@Isomorphic2CRob), Stephen Weimar (@sweimar), and Mary Langmyer(@mlangmyer)

Please reach out to any of these folks to improve your own community or to continue this conversation of how to enrich our local spaces the way we have enriched the online community that continues to grow. As always, also feel free to poke at me through the twitters where you can find me @mrdardy


A String of Good News

Our school year ends early, we graduate the day before Memorial Day here. So, I have had some time to unwind AND to look ahead to next year. I’ve been thinking about my new Discrete Math text, problem sets for my AP Calculus BC class (thanks to inspiration from Lisa Winer (@lisaqt314)), and my upcoming trip to TMC where I will be hosting a brief session to discuss how to develop communities similar to our MTBoS back at home.

Recently, I received not one, not two, but THREE pieces of good news that has happily distracted me a bit from thinking about the fall.

Last summer I led a session at the Pennsylvania Teachers of Mathematics summer conference. I gave it the dramatic title Escaping the Tyranny of the Textbook and it is essentially my love note to the MTBoS community. The goal of the presentation was to have any participants in the session leave the room feeling empowered to write their own curriculum or to learn better how to crowdsource curriculum that is tailored for their classrooms. I was pretty happy with it but I know it needs to be punched up. What better motivation to improve something than to put yourself in a public position where you need to be up in front of people all over again? So, I sent proposals to two upcoming conferences and I learned in the past few weeks that I was accepted to both of them! I will be presenting at the fall conference of the Pennsylvania Association of Independent Schools in October and I will be presenting at ECET2NJPA in September. I am flattered that my proposal was approved by each of these conferences and I am excited to meet some new folks to expand my circle of colleagues even more.

Our school started a STEM initiative shortly after I arrived here in 2010. The first director of the program has decided to step down in large part due to other responsibilities that she has since taken on. She has put the program on firm footing and when he school announced this opening they committed to having a director and two associate directors. I received the great news last week that I will be one of the associate directors of the program. All of our freshman take a STEM class that was designed by the program director and some of our students. They created some lovely iBooks that are still works in progress and that I feel a kinship to since I created our text for Geometry in a similar fashion. We have been hosting guest speakers, alumni, panels of regional experts to discuss items of interest. It’s an exciting program and I am looking forward to being part of the team for the next year. If any of you have advice regarding possible directions for STEM programming, please share here in the comments or over at twitter where I am found @mrdardy