That Visible Learning Business Again

I gave my AP Calculus BC kiddos some rough problems to deal with on HW last night. On one of the problems they were asked to consider the greatest integer function – to be called the floor function from here on – and they were to graph the relation (floor(x))^2 + (floor(y))^2 = 1. A tough problem for sure but a couple of students nailed it. Most did not, so I fired up Desmos to show them the graph thinking we’d spend some time discussing why the graph was what it was. Desmos gave a nice graph but one of my students insisted it was not quite right.


So, I told them (kind of bragging, honestly) that I had met Eli this past summer and I would tweet out our issue. My last blog post was about sharing my learning with my students and here was another great opportunity. Eli tweeted back with another go at the graph – found here


This happened while we were still in class! I got to show my students this graph and it still was not quite right. I wrote back to Eli again and his response was awesome. He tweeted back that this problem would be a lunch time conversation at Desmos World Headquarters. How fantastic is this? I get to share with my students tomorrow that a problem we shared out to the world was going to result in some brilliant guys reprogramming their fantastic tool so that it would tackle this thorny problem. Don’t know that it gets much better than this. Oh yeah, as an added bonus I was sent another take on the problem by Christopher Danielson. Here is his take


Days as a teacher don’t get much better than this. Oh yeah, I should note that I got so wrapped up in these tweets and talking to my Calc students after school about this problem unfolding that I was late to get my children from their bus. A friend had to call me to get me out of my classroom building to go grab them. For one afternoon math seemed more exciting than seeing my own children.

Making My Learning Visible

I am in my 28th year of teaching high school math (with some overlapping years of middle school math thrown in as well) and some would think that I should have it all down pretty well but his point. Luckily for me, this is only partially true. Also, luckily for me I have a great community of support online (I’m looking at YOU #mtbos.)

In the spring of 2010 when I was interviewing with my current school I was told that one of my tasks was to teach AP Statistics. I had never taught stats before at any level. There was a time in my life – in 2001 as a matter of fact – when I stopped talking with a school about a position because they needed a stats teacher. This time I was more confident and more interested in the school so I took on this challenge. I enrolled in a week-long summer institute taught at Fordham in Manhattan by Chris Olsen. I’ve really enjoyed teaching this class but I still feel far less confident in Stats than I do in any of my other classes. I gave our first quiz of the year last Friday over Section 4.1 of the Starnes, Yates, and Moore 4th Edition of The Practice of Statistics. A number of my students were engaged in a pretty heated debate outside the classroom. I was pretty sure that I knew what the answer was but students in favor of two different answers both made compelling arguments. In the past, I might have dug in my heels and stood by my initial guess. Or I might have thrown he question out. Or I might have given everybody credit regardless of their answer. In any of these situations I would have felt pretty unsatisfied and I would not have been any smarter. I was tempted to go to the AP Community page where I have found some pretty helpful folks. However, the feedback cycle there is not particularly rapid and I have to remind myself to go back and check in. Twitter to the rescue! I sent out a plea to Hedge (@approx_normal) and to Bob Lochel (@bobloch) sharing a link to my quiz and begging for help. Hedge replied in a series of about 8 tweets and Bob replied as well. Hedge suggested that I also reach out to Shelli (@druinok) Temple for help as well. In the end, I felt smarter, I realized that my students who had a misconception (a) had a very reasonable misconception and (b) had that because of something I had said earlier. I now know to be more careful with my use of vocab, I know that there are folks who have my back. I was able to show my students the twitter transcripts of these conversations so that they can (a) see that learning keeps on going on even when you are a supposed expert like they see their teachers to be and (b) they (hopefully) see that I am trying my best to be clear and fair in how I evaluate their work. The fact that Bob suggested that each of the two hotly debated answers should be accepted certainly helped.


Back in the Saddle … or Back in the Classroom

Yesterday was my 5th opening day here at my current school and my 28th overall as a teacher. I was speaking with a senior who was a little melancholy that this was her last first day of high school. I let her know it was my 32nd first day of high school so she might have more ahead if she chooses.

There are quite a few changes this year for me. Last year our parents association agreed to help fund an AppleTV cart for my room. I now have an iPad that I am trying to learn to navigate and I have a lovely color TV to project GeoGebra, TED Talks, my new Geometry book, class notes and activities, etc. I rearranged my room for maximum viewing space, got rid of two clunky old file cabinets, and I fully intend to get rid of at least one bookshelf soon to create more space. I rearranged my AP Stats curriculum based on helpful suggestions by Josh Tabor and @majorfstats. I have never been much of a room design guy but the fantastic @mathymeg07 designed some fantastic posters for me so my room is adorned with messages for my students. The biggest change and challenge for me by far will be my Geometry course. I blogged about this recently. I wrote a text for our school’s students to use. This summer was taken up with trying to blend years of thoughts about Geometry with recent advances in my own understanding of GeoGebra. Thanks to the help of some twitter pals (especially @jensilvermath, @mathhombre, @mathbutler, and @a_mcsquared) I’ve been making some progress in my mastery of GeoGebra. I hope that my flashy new AppleTV, my PDF text with link outs to applets on GeoGebraTube and to blogs with great activities, and my enthusiastic younger students will lead to a great year in this course. I already have my first piece of editing advice from a student about one of my diagrams. I’m going to work hard at not being thin skinned and looking at all suggestions as ways to make this a terrific text.

So, we’re only two days in but I already have had a number of really terrific conversations with my students. A couple of Calculus BC kids have especially wowed me already. I hand out a problem set on the first day and ask them to work in small groups. I’m not a fan of going over a syllabus on day one. It bores me to tears and I suspect it saps a good deal of enthusiasm from my students as well. This problem set is designed simply as a way to shake off some rust and give me an opportunity to eavesdrop and begin to understand how my Calculus students think. I wrote last year about how I had one VERY quiet class and one interactive class. This year I only have one group of BC students and I think that they’ll be willing to share. I did two of the ten problems on the board and students did the other eight. I was thrilled to see some students use old precalculus knowledge on the ellipse problem and I saw a couple of different approaches to the logarithm problem. My favorite work that I saw from them today involved a minimization problem and a square root curve. I had a student solve this minimum distance question with no calculus at all. He wrote the appropriate distance formula and made a substitution so it was a square root with a quadratic function inside. He then completed the square to factor the quadratic and said ‘I know that this square expression is never smaller than zero, so the distance is smallest when the square quantity equals zero.’ Lovely, lovely work and I appreciate that approach rather than the automatic reaction of differentiating. Don’t get me wrong, I want my students to remember their calculus from last year, but this kind of analysis really makes me happy.

I start my day with my younger Geometry students and they too seem more than willing to engage in conversations so far. It is fun to teach a class that doesn’t feel quite as serious and important as the AP classes can sometimes feel. I’m especially excited to think about the fact that I’ll et to spend time with them watching them develop into better thinkers and then I may get to see them on the other side as they prepare to graduate. I have not had that opportunity as often recently as I want to.

So, nothing major here (yet) but I am certainly happy to be done with meetings and get back to the classroom.


How I Spent My Summer Vacation…

One of the committees I serve on at our school is a group consisting of department chairs and other academic leaders at the school. Late in the winter of this past school year our administration approached this group with an offer/request. A little background first will set the stage. A number of years ago our current science department chair wrote his own set of class notes for his Chemistry Honors class. Since I have been at the school I have seen students with three 3-ring binders (one for each trimester) that contain their Chem Honors notes. These are legendary at our school by this point. Our administration approached this curriculum group and asked if anyone was interested in creating something similar for one of the courses in their department. My department was already in the process of searching for new texts for Geometry and for Precalculus Honors. There were some precalc texts which made us happy enough, but there was little agreement for the Geometry search. I have also been trying to move myself in our curriculum so that I would teach some younger students. I have mostly been teaching AP Statistics and AP Calculus in my four years at our school. This offer seemed like a way for me to work with younger kids AND begin to create a Geometry book that we would be happy with. One that would grow and adjust as we worked with it. So, I took on this challenge and spent most of my summer writing a Geometry book. I leaned heavily on the wisdom of our MTBoS army and I scoured the web for activity ideas (many of which are living in my Virtual Filing Cabinet here) to try and enrich the book. I have always been bothered that students seem to treat their math texts as simply a source of HW exercises and not as a resource to help learn the material. So, I chose to include very little in the way of practice exercises in the text. I will be working with two of my department colleagues to put together HW assignments and activities to help make this course come alive for our students. Writing this was a great deal of work and I am reasonably proud of it at this point. It’s not a work of art yet, but it is my hope that this text will morph and become more meaningful and beautiful in the next few years. This is where you, my dear readers, come in. I want to share my public dropbox link to a PDF of this text. I want to encourage all of you to look it over when you can, to borrow any ideas that seem helpful, to point out where I goofed, and to share your experiences and activities that will make this a better experience for my students. I will keep updating this text based on our experiences teaching it and based on the suggestions/comments.compliments/complaints I receive from students, from colleagues (both here at school and out in the world), and from the parents in our community. 

I am excited to launch this project. I am thrilled to share it with some of the people who have shared so many ideas already with me. I am anxious about the warts we’ll discover as the year unfolds. I am too exhausted by the whole project to have a clear eye for it at this point.

Thanks in advance for any wisdom and advice you are able to share. August 25 is when we launch – this will surely be a regular topic of conversation on this blog for the upcoming academic year.


I would be severely remiss if I did not make a special thanks to Jennifer Silverman (@jensilvermath) for her inspiration throughout this project. She gave me the first significant nudge down this path.

TMC14 – A Newbie’s Reflection Part 3

This will be my final installment of trying to process out lot the experience I had last week in Jenks, OK


There are two aspects of the camp I want to process that I have not mentioned. The afternoon sessions and the after-hours social life of the camp are aspects I’d like to touch on this morning. As I mentioned before, I was fortunate enough to help facilitate a morning session and I worked alongside Tina Cardone and some fantastic math teachers who are trying to make the Precalculus experience a meaningful one for their students. After lunch each day there were many afternoon sessions. Each of us got to choose two sessions to attend. There are multiple choices for each session and the most obvious indication of what a great camp this was is the fact that these were HARD choices to make. I’ve been to conferences where I choose the least objectionable session to attend. Here, it was a matter of passing up interesting opportunities to find the BEST possible session to sit in. On the first day I sat in on Michael Pershan’s session on complex numbers and I had the great joy of sitting with Eli Luberoff and Jed Butler while we worked through a stunning activity on Desmos (you can find it here) that tied together transformations on the plane with the introduction of complex numbers. I was so thrilled by the activity, by working elbow to elbow with two brilliant people, by trying to anticipate where the activity was leading us. I’m still knocked out by it and I will work with my Precalculus teachers this year to see if they are comfortable leading their students through this activity. I then attended Megan Schmidt’s session on using NRich activities in your classroom. Again, great math, great conversations, and a solid takeaway for my classes this coming year. Megan said she was nervous – it sure did not show. On Friday I went to Jed Butler’s session on transformations using GeoGebra and Andy Pethan’s session on activity-based stats. Jed is someone who I have admired from afar as he has been part of the team that has really encouraged me to step up my GeoGebra game and his session did not disappoint. Deep math here and great conversations about pedagogy happened in the room. Justin Lanier asked a probing question about student struggle and how much we should try to relieve it through the way we set up approached to problems. I’m still bouncing that question around in my head. Andy is a very your, very enthusiastic teacher who has some serious programming chops. He led us through a great activity where, in teams, we drafted ultimate frisbee players and he ran our teams through a simulated season. Our team was the regular season champ but we lost in the one game finals. On Saturday, I went to the GeoGebra team session hosted by Audrey McLaren, John Golden and Jed Butler. I pushed myself to tackle a problem from Five Triangles and I am pretty pleased with where I am on the project. I’ll post about that with a link to my solution once I am satisfied. I finished my day there with a brainstorming session led by Lisa Bejarano where a group of us got together to talk about how to try and translate the ideas form this camp – and the experience of building this virtual network of support – back to our home schools. No solid solutions, but great conversations. The thread through all six of these experiences – and the ten morning sessions that were offered – is that this was a group of excited, excitable, and thoughtful educators. We were talking math and thinking hard. We were challenging each other and ourselves. It was pretty exhilarating. The fact that no one was getting reimbursed for their work in facilitating these sessions is pretty inspiring.


This was my first time going to this camp. I was aware of it in the summer of 2013 when it happened about two hours from my home. At that time I was not blogging yet, I was not yet on twitter and I only knew about it from reading the blogs of other involved individuals. I did not sign up to go because I felt I would have been a bit of a fraud attending at that time. Even this year, when I have been blogging for about a year, I’ve been on twitter since October (I think) and I have been actively engaged with a number of individuals I still had doubts about whether I belonged or was worth one of the 150 spaces. This is all on me – it’s how I was processing my thoughts. When Tina invited me to work with her I was incredibly flattered and said yes as soon as I spoke with my wife about the trip. So, as the trip got closer I was a bit intimidated. I’m not exactly shy, but I am more comfortable in social situations when I have a bit of an anchor. There was only one person of the 150 that I had met in person so I was a little nervous. I knew that there were certain people I really wanted to meet but overall I was unsure of how this was all going to work out. While I was there I was struck by two competing feelings. I loved the interactions I had with accidental meal partners and stumbling into conversations in the back area of the hotel where people gathered each night. I made some real connections – people I know I will be directly communicating with during the school year (I’m thinking of you Meg Craig and Lori Likens and Megan Schmidt and Jed Butler and Audrey McLaren and Jasmin Walker and John Golden and Brian Miller and Lea Ann Smith and and and …) The camaraderie at the Waterfront Grille on Saturday night and Sunday brunch was inspiring. The trip to the chili and hamburger joint was delightful. The gathering of the tribe in terminal B of the airport gave me the good cheer to make it through the rest of what turned out to be an annoying day later on. However, a part of me checked my tweet deck at night and was sad about other social opportunities I was missing. I think it ends up being a similar feeling to that I had about the working sessions. A mix of disappointment about what I was missing and joy at what I was experiencing. That seems to be a pretty strong endorsement of the whole experience, isn’t it? If I could be so happy about what happened AND so disappointed about what I missed it must mean that there was a whole lot of goodness (or at least potential goodness) going on.


Thank you SO much Tina for making me jump in. Thank you to the whole team that organized this event – most visibly Lisa Henry. Thanks to the Jenks school and greater community – most visibly Shelli who seemed to be everywhere at once.

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.