TMC14 – A Newbie’s Reflections Part 2

I want to touch on two more aspects of the time at TMC14 as well as share a story about my travels home. Since I’m still processing the week and hope to be able to write something that has some meaning, I’ll write about the person I sat next to from Tulsa to Detroit. There was a group of teachers who were on this plane, but they were not a subgroup of the TMC team. There was one other TMCer on my plane. No, these teachers were on their way to Michigan to conduct a teacher training workshop. The man sitting next to me has been in the business about as long as I have been (hint: it’s a loooong time) and he told me he has been working with the same training group since 1991. I asked him how much this training job has changed during that time and he kind of paused. I shared with him a few observations I have about ways in which my students are different than they were twenty years ago. He listened carefully – I don’t think I was making any revolutionary points at all. I spoke about how I have been more conscious about the need to break class up and to lecture less than I used to. I talked about how I work to make sure that there is time for reflection built in to my classes in a way that I was not always conscious of before. I was kind of excited that I might have another interesting conversation about pedagogy with a stranger as I had been having the past few days. He calmly told me that the professional training that they do is the same as it has been the whole time he has worked there. He said this with no hint that it should have changed. I don’t want to sound instantly judgmental here, I honestly don’t. I looked at the website for the company he works for outside the classroom. They look like they are caring, concerned teachers who are trying to help others. But I cannot shake the idea that there is something inherently troubling about being in our business, working with young people in such a time of change and not changing a thing about what you do. 

I think that this is at the heart of what excites me about feeling a part of a community of teachers – those who came to TMC14, those who blog and question, those who engage in twitter chats and debates, etc.

There was an exchange this morning on twitter and part of the conversation centered around trying to be an ‘All-Star Teacher’. I think that everyone at TMC14 – all 150 of them were there to try to become an ‘All-Star Teacher’ and to help each other in that goal. I also believe that the teacher I spoke to on the plane is interested in helping others become ‘All-Star Teachers’ as well.


It’s just a matter of how it is we get there. I feel pretty sure that sharing the way we do in our community is the right path.

TMC14 – A Newbie’s Reflections Part 1

I am currently sitting quietly in the Detroit airport and I’ll be here all night. Weather foiled my return home today after 4 pretty terrific days at twittermathcamp2014

I have quite a bit that I want to sort out about this event and I’ll do so over the next few days in a couple of parts.


A number of people have already written and many more will about their experiences. There will be posts talking about the transformative effects of this working/learning/playing experience if history is any guide. A number of thoughtful posts have already been made. A couple of them are really touching and reflect insecurity (see here by @MrKent800, see here by @lmhenry9). Glancing at my neglected digg reader I see that there are posts waiting to be read from @pamjwilson over at her blog The Radical Rational, @algebrasfriend over at her blog and others as well. My tweet deck is hopping and I am paying closer attention to much of it because I can place some human names and faces to the tweets and to the quick chatter between folks. I’m going to try and capture my days in Jenks, OK and I hope I am fairly thorough in recognizing and thanking some folks who have been pretty important in this experience. Before I embark, it’s important – at least to me – to note that before arriving in the Tulsa airport Weds shortly before noon that I have physically met only one person who will be at the conference. That person is the charming @JustinAion who wrote a daily blog (honestly, he wrote every school day!!!) over here. I’m pretty outgoing when I am comfortable in a situation. I was worried about whether I’d hit that comfort zone at all in my time in OK on this trip.

In advance of the trip I, and many others, had filled in information in a google doc about our arrival and departure plans and folks started volunteering to take people from the airport to the hotel. Pam (@pamjwilson who blogs here) volunteered to take me to the hotel in her rental car. We had a lovely chat at the airport waiting for our car, ran into another group who we followed to the hotel. This trip was a bit of an adventure as our ‘guides’ made two quick reversals during the trip. We got to the hotel a bit early before rooms were supposed to be available, but we lucked out and got into our rooms pretty quickly. Hungry from our travels Pam and I ate lunch with two of the people from the car we followed. I’m disappointed in myself that I do not remember their names.

A nice nap back at the hotel before heading to a gathering room in the back of the hotel where people were chatting, snacking, playing games, etc. The name tags that we would wear the rest of the week were at the host school so we were introducing ourselves mostly just by first names.  Now, remember that I have never met any of these folks in person – other than Justin who brought piles of games to share so I’m feeling a bit overwhelmed here. I chat for awhile and enjoy myself. I briefly meet Tina (@crstn85 who blogs here) – this was a major goal of mine for this first night. Tina is the reason I came to TMC. She invited me to help her organize and run the Precalculus morning session. She and I had exchanged some comments and emails about her amazing Nix The Tricks project and I was (and am) totally flattered that she asked me to join in. We chatted very briefly but I felt better actually meeting her before our work commenced on Thursday morning. After bouncing around a bit I retire kind of early. I could not help feeling a little bit like an outsider. It’s a feeling that is hard for me to shake during the week. This is mostly on me, by the way. All week long folks were welcoming and friendly, but being brand new to this community in the physical sense and being relatively new to the community in the virtual world as well, it was hard for me not to notice the deep bonds between many of the folks here.

Sleep calls to me now. Although it’ll be sleep on the floor in the Detroit airport, I need to succumb. More thoughts soon on the week’s activities – both in the context of the day’s formal activities as well as on the informal social life of the twittermathcamp experience.

Baseball and my Math Classes

I’ve been sitting on this post for a few days trying to find the time to organize my thoughts. This post is motivated by this recent post by Michael Pershan over at Rational Expressions, by my son’s recent little league season, and by multiple recent conversations with a good friend and colleague who is our drama director. I hope that I can make it all make some sense.

Michael talks about what feedback looks like and sounds like in different contexts. He offers these two examples of how a baseball coach might talk to a player who is struggling with his hitting. 

“Tommy, you haven’t been hitting as well as you could’ve lately, amiright?”


“Each time you swung and missed, you raised your head as you swung so you didn’t really have your eye on the ball. On the one you hit hard, you kept your head down and saw the ball.”

I think that any of us (teachers, coaches, parents, people who have lived for a little while…) would agree that the second remark is more helpful. [This reminds me of a former student of mine who was a basketball player. At halftime of a game where his team was being routed his coach was blistering the team questioning their will to win etc. He raised his hand and said 'Could you tell us what we're doing wrong and how to correct it?' Unfortunately, his fired up coach did not take kindly to this interaction.] After I read this – and, again, I’d urge you to read Michael’s whole thoughtful post – I started thinking about the language I use when speaking to my students and when writing to them on their work. I also thought about my son’s recent experiences.


I watched my son struggle through his inaugural little league season with good cheer. He just turned 11 last month and had never expressed interest in baseball – much to my chagrin as it is my favorite sport. This spring he announced that he wanted to join the local little league. Luckily, they have a level for kids aged 7 – 10 with a variety of experience levels and they do a nice job balancing the teams. He was on a team with fantastic, supportive coaches. They were consistent, they were enthusiastic, they were patient. When a player made a mistake they pointed it out to him/her right away but they focused on what should have been done rather than simply criticize the mistake. They regularly referred back to practices and reminded the players how they were taught to play. I have seen my son give up on tasks at home and tasks from school in the face of frustration. Despite only getting three hits all season (but two of them resulted in RBIs!) I never saw him frustrated and he never gave up. I think that a big part of the reason why is that he was part of a team and felt that he was responsible to the team and his teammates. The fact that the coaches helped create an atmosphere of positive energy was a huge factor. The fact that the team won all but two of their games made a big difference as well.


I have had a number of conversations with my friend Jason who is a drama and English teacher at my school. We have talked about how different perceptions are at school about certain students. I think that all of us who have taught for some time are familiar with the fact that some kids ‘click’ better with certain teachers or subjects. But what we have talked about goes beyond that. I have noticed that there will be students who routinely struggle with follow through and commitment in their academic classes but will be praised for their persistence and determination by coaches and/or drama or music teachers. There are students who won’t do homework, they’ll bomb some tests due to lack of preparation, they might skip a class or an assessment but some of these same kids (generally) do NOT miss practices, they learn their lines or their parts for the symphonies, they stay extra time for practices or they regularly get there early. I know that there are all sorts of factors at play here. But I think that the two biggest ones are these: (1) Few kids CHOOSE to talk Algebra II or Chemistry or US History. They are required to take these (and others, these are just some examples) classes. They have made a conscious choice to play basketball or lacrosse or to play cello in the symphony or to try out for a role in the school play. It’s easier to be committed to something you choose to do. (2) In all of these activities a failure to carry out your job has a direct impact on the chance for others to succeed. If I failed an Algebra II test in school no one else was going to suffer, it was just my failure. If I run the wrong route for a pass play in football the entire team might suffer the consequences if there is an interception. If I flub my lines on stage it has an impact on everyone’s performance. If I don’t learn my part in the string ensemble the performance suffers for the players AND for the audience. 

I think that this second aspect is the most important one. This is where I finally try to get to my big point in thinking about this blog post and sharing it out to the world.

How do we create a team atmosphere in our classrooms? How can we encourage our students to feel responsible to each other and pull all together in the same direction? At my last school I had furniture that was easily manipulable and had my room set up in ‘pods’ of three or four desks put together. My students there took great pride in their pod performing well on assessments. I had a few ‘pod’ quizzes and they worked hard together on those. My current classroom is set up in two long tables that seat ten students each. The students interact,they share ideas and when I give them time to work together they do so well. But this does not create any sense of accountability to each other. Perhaps I’m dreaming and hoping for something that just doesn’t happen inside the classroom walls. I don’t think that I am imagining a grading system where their grades depend on each other in any way (although Dan Kennedy wrote a really important essay where he proposes something slightly along those lines as part of a much more complex argument about assessment) since I know how delicate the relationship between learning and grading can be. I look to the hive mind of the internet for inspiration and suggestions. How have you been able to foster an environment in your classroom where students support each other’s learning and where students feel that they are part of a team where their contributions really matter?



Moving and my math classes

Haven’t blogged in quite a while now. School year ended and most of my energy has been consumed by writing (we’re doing our own Geom text here at my school!) and by preparing to move. The apartment moving adventure brought me (over and over again) to our local Lowe’s hardware store. The trips to Lowe’s got me thinking about my classroom – especially about the Geometry classes I’ll have in the fall. Let me try and make sense of why.

When he was a young man my dad worked as a carpenter for Macy’s in NY. I inherited NONE of these skills. I am lucky that my wife seems to find me handsome since I am certainly not handy at all. I have had the occasion to go looking for shims, looking for shelf pins, buying paint, etc. What I have noticed is that I have NO ability to understand the basic architecture of this store. I know that there are clues about where to find certain items but I cannot decode them. This got me to thinking about my students as they try to navigate their studies. I made myself go and seek help. I was confident enough to be able to describe what I needed and in some cases brought evidence with me of what I needed to buy. In some cases, as with the purchase of paint, I was asked questions that I was not prepared to answer. Since I knew that I needed to paint to make my children happy about their new rooms, I was determined to make sure that I completed my purchase. When I was asked questions about what kind of finish I needed I was able to simply admit my near total ignorance and simply take the advice of the salesperson. This did not make me feel good about myself, but I knew it was important enough to make myself suffer through this relatively minor indignity. I started thinking about my students and how difficult this situation is for a person’s ego. The difference between my level of knowledge and the level of expertise that the workers at Lowe’s possess created an uncomfortable tension for me. Not because of their behavior, but simply because I felt bad about my lack of ability to process the information expected of me. How different is this from the situation that many of our students find themselves in every day? I would say that there are a few big differences.  I knew I needed to work through this to make my move possible. I am not convinced that my students are always able to impress upon themselves the importance of the struggles they are experiencing. Part of that becomes my responsibility to convince them of this. The more important difference as I see it is the fact that I have no continuing relationship with the workers at Lowe’s. I don’t have to face up over and over to my lack of ability to process what seems natural and obvious to them. My students have to see me every day for 9 months or so. It is easier for them not to admit when they don’t understand. It’s easier for them not to ask for help in navigating a mysterious terrain. 

When i start up again in the fall I need to remind myself of the feelings I had at Lowe’s – especially with my younger students in Geometry. It was important for me to be reminded of how uncomfortable it is to be confused. It was important for me to revisit that feeling. I will be processing that this summer and try to make sure that I can create both a sense of the importance of our academic mission for my students and a sense that it is okay to admit when the fog of confusion settles in. I need to be aware of helping my students understand the structural architecture of their studies so that they can help themselves more than I was able to help myself in Lowe’s.

Bragging About My Students

Two things I want to share tonight. One of them has multiple parts.


One of my international students shared a lovely gift with me yesterday. It’s a food treat that her mom sent here for her to share. I have a few food allergies so I was concerned but did not want to tell her because it felt rude. Luckily, there are a number of boys in my from who can translate the ingredient list for me. Pretty cool. Oh yeah, I’m allowed to eat it – no nuts.


I blogged a couple of days ago about a problem on my calc BC final. Here is the problem

For your final problem on your final Calculus test, we will play with number bases. Consider the following passage from Lewis Carroll’s Alice in Wonderland:


“Let me see: four times five is twelve, and four times six is

thirteen, and four times seven is — oh dear! I shall never get

to twenty at that rate!”


 Explain, in terms of your knowledge of number bases what is happening in this pattern. Explain how four times five is twelve, and how four times six is thirteen.    Guess what she will say four times seven is and make it clear to me why she won’t be able to get to twenty.

Twenty-four students took this final and a number of them really did a wonderful job in explaining their reasoning. I’m going to present a handful of the best responses here.

  1.  4 x 5 = 12  is number base 18. 4 x 6 = 13 is number base 21. 4 x 7 =   she will say 14 in number base 24.   4 x 12 = 19 number base 39, 4 x 13 = 20? number base 42. If following the pattern, as one of the numbers remains constant 4, another increasing by 1 each time, we get the product increasing by 1 each also. This is possible for number base 18 , increasing 3 each time. This 4 x 13 = 20 in number base 42 accordingly. However, 4 x 13 in number base 42 is one full round with another 10, which in base 42 as there exists another symbol for that suppose A, thus 4 x 13 = 1A and will never equal to 20 this way.
  2. She is using number base to calculate it. Every time 4 times initial number plus 1 and number base that is increased by 3. It will never get to twenty because the number base is always growing as number grows. Number base is growing faster than the number we multiply by. (Note – this one was accompanied by meaningful, but scribbly, calculations)
  3. (This answer starts with all the calculations hinted at in the first answer i presented)  Because the base in continually increasing by 3 and the answer is only increasing by one, the answer will never be able to get out of the ones digit.
  4. The base is increasing and in order to get to 20 the result of the calculation must be EXACTLY twice as big as the base., which is not possible.


All of these were accompanied by calculations on the side. We spent about two and a half days talking about number bases and I must admit I was really impressed by the patience that my students had with this problem. Nice way to end the year!


Thinking About How to End a Year

So, this morning both of my AP classes took their final exams. I have some questions for the world about this process, but first I want to share something fun from the Calculus BC final. You should know that my students understand that the word fun, when I use it in class, means that a problem is challenging, thought-provoking, unusual, or some other words that they might use but I won’t type. I believe that I mentioned that we ended the year after the AP test with a quick tour of some interesting topics that many high school students don’t get to see. I included a small unit on different number bases. I always start this by writing a series of addition and multiplication facts on the board. However, they don’t know that these facts are base 8 number facts. I usually reveal the secret by showing a picture of Lisa Simpson and tell them that these facts are how Lisa would compute. It’s a fun conversation to have. So, for our final exam I told my students that there would be ten problems. Nine of them would be Calculus problems taken from old tests. they have all of their old tests (all ten of them) so they could be very well prepared for that. I also told them that one problem would come from our last two weeks. After discussing this with my friend Richard – a former math teacher – he sent me the following passage from Alice in Wonderland

Let me see: four times five is twelve, and four times six is

thirteen, and four times seven is — oh dear! I shall never get

to twenty at that rate!    

I played around with this for a while and fell in love with this as a final problem for the year. This is how I presented it to my students:

For your final problem on your final Calculus test, we will play with number bases. Consider the following passage from Lewis Carroll’s Alice in Wonderland:

“Let me see: four times five is twelve, and four times six is

thirteen, and four times seven is — oh dear! I shall never get

to twenty at that rate!”

            Explain, in terms of your knowledge of number bases what is happening in this pattern. Explain how four times five is twelve, and how four times six is thirteen.    Guess what she will say four times seven is and make it clear to me why she won’t be able to get to twenty. 


I will sink my teeth into grading finals tomorrow since I am on dorm duty tonight. I did browse through four or five of the test as they were turned in and two students really nailed the problem and provided beautiful, detailed answers explaining the pattern. I won’t spoil it here, I’ll let you work through it if you wish to do so.


I don’t know how your school works, I do know a bit about the four schools where I have worked. All of them have been independent schools that emphasize the idea that we are college preparatory schools. Each school I have worked at has had a statement in their handbook about the importance and significance of final exams as a college preparatory experience. However, I know that tonight many of the seniors in the dorm will tell me that they do not have any finals left. Today was the first of three and a half days of final exams and many seniors won’t have any more after today. There is a pretty common feeling that final exams will not be pretty and are of questionable usefulness with our seniors who are days away from graduation. This is not just a feeling at my current school. But I really wrestle with this. We say we believe that taking a final exam, preparing and organizing a large body of information for a one-day thorough examination, is a useful skill AND one that is important for college. However, it is those students who are closest to college who are the most likely to have been excused from a final exam. In some classes the final experience is a paper or a presentation that happened last week. But our school, and others where I have worked, carve out quite a bit of time for final exam administration. I wonder whether we could use our time in a more meaningful way. I wonder whether the idea of a final exam makes sense only in certain disciplines or for certain age levels. Is it reasonable for us to ask our freshmen to take exams under the same circumstances that we ask of our juniors and (sometimes) seniors? I don’t see AP scores for my students until July. I like the idea of some capstone where I check in with them in one last, broad examination of ideas. I feel pretty old-school in that regard.

I want to make my assessments meaningful for me and for my students. I am really beginning to doubt whether exam week is such a positive way to do this. I would love to hear from others about how they deal with this question. Are many of you bound to a policy that your school or your department has mandated? I want to be smarter about this and I’d love your help.

Professional Growth in a Connected Age

I’ve been teaching for a long time now. It makes me feel old when I realize I am in my 27th year in the classroom now. I joke with my students that I have been teaching longer than any of them have been alive. When I started teaching the predominant models of professional development were the inservice days at school where the school administrators decided how we needed to grow, the weekend workshops or summer workshops that I would scramble to find funding for, or the one or two day workshops that would cause me to miss school. It’s a different world now. I know I’m preaching to the choir if you are even reading this but this world of twitter, of blogs (both writing them AND reading them), of online simulcast workshops, or improv EdCamps, the list goes on. In this day in teaching I am fully convinced that if you want feedback and you want connections to help you think about your craft and to expand your toolbox – if you really want it – there is an ocean of resources at your fingertips. Literally (since I’m typing this right now!) at your fingertips. Not all of it fits everyone. I know that I am still wrestling with the timing and pace of my twitter feed, but I think I’m getting better at it and I KNOW I’m growing as a result of it. I spent a long time reading blogs, then commenting on blogs before I felt confident enough to launch my own. I have two kids at home so I know how tight time can be, but I also know that the past two Saturdays (that I blogged about separately here and here ) where I spent a combined 16 hours out of the house were worth the time and effort. Luckily Mrs. Dardy is kind and flexible and supportive of this pursuit.

I’ve been thinking quite a bit about this, about how different my life as a teacher is in the past few years. I’ve been in regular communication with one of my former colleagues, Gayle Allen. Gayle (@GAllenTC) hired me seven years ago when my family left Florida and we landed in New jersey for a while. Gayle is a remarkable, energetic thinker and was a great boss. She and I have been engaged in a long conversation about professional growth and one of the results of this conversation is an article that got posted today over at a website called Getting Smart. I know that I am not unique in this journey, but I also know that there are still many of our colleagues who have not taken this plunge. Some because they are not interested in doing so, some because they don’t know where to start. I’m pleased to be able to give shout outs of thanks to Dan Meyer (@ddmeyer) and to Sam Shah (@samjshah) through that article and I’m pleased to be connected again (even if we are nearly 3000 miles from each other) with Gayle.

This summer will see a trip to OK to take part in TwitterMathCamp. This would not have happened if Tina Cardone (@crstn85) had not reached out to me and asked me to join in on the fun. This summer will see me finish an in-house Geometry text for our students. This project would never have happened without the encouragement and advice of Jennifer Silverman (@jensilvermath). This summer will see me work on plans to help a brand new teacher in our high school take the leap from teaching Algebra I in the middle school to teaching Honors Precalculus for the first time. All of these experiences will help me grow as a professional. 27 years at it now and I feel like I still have an awful lot to learn. I hope to be smarter this time next week about this craft than I am right now.