Why Do We Teach our Students (_________________)?

A quick post here as I get ready for our first full day of staff meetings. Yesterday, at a lunch with department chairs for our lower school and our upper school, one of my colleagues raised a nice question. We were discussing our goals for this afternoon’s joint department meetings and we were bouncing around some topics related to summer reading, content alignment, etc. We are a PK – 12 school but we are on two campuses separated by three miles so we do not see each other as often as we would like. Meetings like the one we will have this afternoon are few and far between. So, the question raised by one of the chairs was this – ‘Why do we teach, fill in the blank?’ In other words, can our English teachers say something similar to each other about why we teach English? Can my math team say something coherent and cohesive about why we teach math? I titled this blog post the way I did in honor of Glenn Waddell (@gwaddellnvhs) who reminded us this summer at TMC15 that it is important to remember that math is the subject we teach (at least all of us there!) but we teach people.

So, what would you say in response to the question ‘Why do you teach math to young people?’ Why do you teach chemistry to young people?’

I think I have an elevator speech in mind but I realize, when pressed, that it is not as succinct as I would like it to be and I am not convinced that there is enough overlap between my reason and those of my departmental colleagues. I think that, for the benefit of our students and for coherence in our program, that we should probably share this question with each other. I would love to hear in the comments or over at twitter (I am found there @mrdardy) what you think of this question and what your answer is.

Off to full faculty meeting starting in 33 minutes!

Another Lovely Learning Day, Part Two

The other day I wrote about my morning session for the summer conference of PCTM (The Pennsylvania Council of Teachers of Mathematics) and today I will write about a session I attended. Unfortunately, child care considerations meant I had to leave at lunch time on Friday but I did have the pleasure of attending a session led by Dr. Daniel Ilaria of West Chester University in Pennsylvania. Professor Ilaria had one of his students, Kaitlin Nora Silard, along with him. She is about to begin her student teaching in the fall. The presentation was titled Learning to Persevere in Problem Solving and its description sort of sang out to me as something that would be of interest to me personally and that might be a great talking point for my department this year. They did not disappoint.

One of the highlights of the discussion – as it should be at a valuable PD event – was the opportunity to share ideas with other educators. There were about a dozen of us in the room and Prof Ilaria was terrific at allowing space for us to discuss and debate. His student Ms Silard was excellent as well. At one point after presenting a problem on the projector Ms Silard walked by and saw I had a solution and I had put my pen down. She looked at my work and simply said ‘Why don’t you try to find another way to solve this?’ Awesome for her and an important reminder for me. I should not let myself get away with a ‘Do what I say, not what I do’ type of attitude. Below is the problem we were working on at the time:

Boaler peg

My original approach to this problem is what I would call an additive approach. In figure 2 I saw a row of 3 on top of a rectangle that was 2 by 3 and then 2 left over on the bottom row. In figure 3 I saw a row of 4 on top of a rectangle that was 3 by 4 with 2 left over on the bottom row. In figure 4 I saw a row of 5 on top of a rectangle that was 4 by 5 and then I saw 2 left over on the bottom row. I generalized this as (n+1) + (n(n+1)) + 2 which simplifies as n^2 + 2n + 3

I shared this solution and a number of people shared slightly different – but all additive – solutions, What I mean by this is that all of the techniques of decomposition were rooted in adding up pieces of what we saw. A lovely explanation was presented by Ms Silard that involved moving pieces to simplify a rectangular image. It was great fun working this out and hearing different responses. It was important to both feel good about my solution AND to hear others explain their point of view. It reminded me of Christopher Danielson’s keynote at Twitter Math Camp 2015 (you can view it here and you should, it is fantastic!) When Prof Danielson was speaking about the prompts from his upcoming book and from the great website http://wodb.ca He talked about the importance of having students face a problem that has multiple ‘correct’ answers and developing the ability for them to explain their answers. He also talked about the experience of listening to a different solution and how that can inform your own problem-solving skills. I definitely benefited from listening to the other session participants. But I have to admit that I benefited the most from Ms Silard pushing me to think of another solution. This one was a subtractive solution. I saw a figure that was not really there and then pulled pieces away to create what WAS there. For figure 2 I saw a 4 by 4 square that was missing 5 pieces. For figure 3 I saw a 5 by 5 square that was missing 7 pieces. For figure 4 I saw a 6 by 6 square that was missing 9 pieces. I generalized this as (n + 2)^2 – (2n + 1) and, magically, this turns out to be the same answer. My table mate appreciated this approach more and it was interesting to me that this was the only approach that looked at taking pieces away (pieces that were not really there to begin with) and that this relied on a more stable shape (the n+2 sized square.) The experience of hearing other people’s voices, the experience of thinking of another way to do it myself, and the experience of explaining myself to my table mate and then the whole groups were all powerful experiences. I thank Prof Ilaria and Ms Silard for their careful management of this presentation. I need to revisit these feelings throughout the year and commit myself to trying to create a careful space for my students to experience these powerful feelings as well.

PS – I knew when I read the program for the PCTM conference that Prof Ilaria’s name seemed vaguely familiar. I realized this morning why that is. Back in 2004 I started on a doctoral program at the College of Education at Florida Atlantic University. In 2007, after finishing my course work, I moved away with the intent of finishing my dissertation (titled A Case Study of the Intentions and Actions of Secondary Mathematics Teachers with Regard to Questioning in the Classroom) and an article that Prof Ilaria wrote back in 2002 is in my list of references for my work. Kind of cool to have met him in person now and he was kind enough to share his slide presentation with me via email.

Another Lovely Learning Day

This morning I had the pleasure of presenting a session at the summer conference for the Pennsylvania Council of Teachers of Mathematics. I know a few people in the organization and in a twitter conversation with Bob Lochel (@bobloch) I was encouraged to submit a speaker proposal. I wrote up a proposal for a session called Escaping the Tyranny of the Textbook. As my loyal readers know I spent the summer of 2014 writing a Geometry text for use at our school (you can find the 2nd Edition here) and I wanted to talk about the relative ease (compared to 10 or even 5 years ago) of  putting together your own curriculum. I was not there to advise that people write their own books, just to point out the wealth of resources available on the web and through twitter. I was fortunate enough to have my proposal accepted and I was originally slotted to lead a small roundtable conversation for about 30 minutes. Well, about a week ago Bob reached out and asked if I was willing/able to move to a 50 minute time slot. I agreed but figured that this required a bit more advance planning on my part. I sent out a call for help and received numerous helpful tweets and I put together this Power Point slide show. I had a group of about twenty people in the room and we had a lovely conversation. People asked some great questions, they seemed to appreciate the resources I organized with the help of the MTBoS tribe, and I have already had folks reach out to me via email. The common theme that emerged was a desire on the part of the participants to have support in navigating the world of resources available. A few people told tales of schools giving them technology tools but no PD support to help them use those tools. It is clear from my experiences, and from the voices in this room, that this is a long process. Finding trustworthy connections both in your building and out in the wide world, takes time, energy, and support. I positioned myself as a cheerleader for MTBoS and for collaboration – both virtual and face to face. I am optimistic that I will strike up a correspondence with some of the folks in the room this morning and that some of them will take the jump this year to reaching out of their classroom and tapping in to the world of resources available to them.

TMC 15 Reflections, Part Two

So, in my last post I gathered some of my thoughts about the days at TMC up through the post-lunch keynotes. I did not touch on the My Favorites sessions, the afternoon sessions, or the evenings. I’ll tackle those in this post.

Each morning and afternoon we were treated to quick presentations by volunteers (well, we were all volunteers there, really) who wanted to share something brief about their classroom practice, about some project they were working on, or to share some other insights. These were all fun nuggets and I don’t want to ignore any of the treats that were shared, but I do want to single one of them out. Glenn Waddell talked about his commitment to high fives with his students. When I was a young teacher (a LONG time ago) I was encouraged to stand outside my door between classes. I try to remember this and use this as an opportunity to greet my students and other students passing by in the hallways. However, I do not have the kind of commitment to this and energy that Glenn shared. He greeted each student with a high five and discussed how he broke down the resistance of a particular student who finally joined in the fun. He talked about how on days when he had to miss class, the students would ask for two high fives the next day. I had the pleasure of picking Glenn’s brain about all of this one evening and he told me that would tell his students that he was congratulating them because they were about to do something awesome. He also said that he greeted each class with a rousing ‘Good Morning!’ no matter what time of day he met them. His enthusiasm was infectious and I could not resist giving him a high five every time I saw him the rest of the week.

I went to two afternoon sessions each day and I want to highlight two particularly fantastic ones. Dylan Kane (@math8_teacher) conducted a session called Arithmetic to Algebra: Key Building Blocks in Abstract Thinking. Dylan shared a series of fantastic questions that he had developed to extend student thinking and give him insights into their thought process. Dylan is bubbling with energy and enthusiasm and he asks great questions. He has been on fire on his blog lately – if you have not read his stuff you should change that and visit him regularly. A great conversation ensued in the room and I am looking forward to using some of his questions as class warm ups this year. Meg Craig (@mathymeg07) conducted a session called Function Transformations Without Tears. I have raved about Meg and her blog recently where she has been on a tear recently sharing tons of great files. She conducted a thought provoking session on how to approach function transformations together. What was as impressive as the thought that went into her presentation was the energy of the conversations in the room. A number of us were sharing ideas about what language to use and how to incorporate this into our classroom. Meg gave us room to debate/discuss even though it may have taken time away from what she was thinking of doing in her session. Both Dylan and Meg stepped back and let us, their ‘students’, take the conversation where we needed/wanted it to go. Nice modeling of positive teaching behavior on their part!

This just leaves dinner time and evening fun to recount. On Thursday I had the pleasure of meeting my brother for dinner. It has been well more than a year since we had seen each other. He has lived in LA for nearly twenty years and just recently bought a house outside the city. We met in a nearby town that was roughly equidistant from his home and my hotel. We were able to spend nearly four hours catching up before I started winding down and headed back ‘home’. When I arrived at the hotel a little after 10 PM I walked into the hotel courtyard to find about 40 or so conference members hanging out and catching up with each other. I was greeted warmly by three of four folks as soon as I walked through the doors and my energy was lifted. I spent some time chatting – especially with Brian Miller (@TheMillerMath) who I met last summer. Brian became my car buddy for a number of events over the time there and I tried to help him with an unfortunate battery problem his car developed. I was struck once again by how meaningful friendships can feel even though we had seen each other in person for only three days in Oklahoma. Meaningful conversations with Alex Overwijk (@AlexOverwijk), Jasmine Walker (@JazMath), and others over the time there made the hotel courtyard a warm and wonderful place in the evenings. One evening featured a barbecue at a nearby park that was funded by the wonderful folks over at Mathalicious and prepared by local hosts. It was a beautiful night to be outside, great food, good conversations, and wonderful views of the mountains nearby. Having spent most of my adult life in Florida, I am taken by mountains.

On my final night there I ended up having dinner and a cupcake run with a group of 8 terrific women (lucky me) before retiring to the hotel courtyard again. I had a very early flight back across the country but I kept myself up until almost midnight talking with fascinating folks before finally retiring for the night.

I realize I am just barely scratching the surface of the energy and spirit of the camaraderie of the folks there. I will try to capture it this way. I left for California on a Wednesday and the rest of my immediate family left the Friday before for an adventure. I was away from my wife and kids for nine days and missed them terribly. While I was at TMC15, instead of primarily dwelling on missing my family (which I still did, for sure) I felt so immersed in a different kind of family. Sure, we are not bonded by blood, but we are bonded by a love of idea, by a love of our work, and by a generosity and, dare I say it, love for each other. Although I have spent six days or less in the physical presence of this community, I feel a deep bond of trust and friendship. I have seen over and over again how a blog post about challenges and questions in my job results in thoughtful replies and sharing of resources. I have seen how a tweet sent out about a question results in quick, thoughtful replies. I have seen how the simple accident of sitting in the same place in an auditorium for three days in Oklahoma leads to a sense of camaraderie that spans the width or height of our continent. I have also seen how my wife is already making plans for me to attend TMC16 in Minnesota and how to turn it into an extended family adventure. She sees what I clearly feel. This professional community helps me grow, helps me feel supported, and tickles my brain.

Next challenge – a new school year starting in seventeen days!

Overdue TMC15 Reflections, Part One

So, after spending four days in California in the midst of the inspiring MTBoS crowd at TMC15 I flew to Florida to catch up with my family and spend a week on the Gulf there. Got home on Sunday and I am finally clearing my head out enough to throw in my two cents on the whole experience.

Many of you reading this are familiar with the format of TMC either because you were there or have already read some of the other reflections out there.

I was fortunate enough to be the co-leader of a morning session this year with Lisa Bejarano (@lisabel_manitou) exploring Geometry. We had a great group of eight colleagues who were energetic and eager to share ideas and questions about teaching this course. I had a blast working with them – and with Lisa, who is fantastic! – and I certainly hope that our participants felt the same way. We worked together Thursday, Friday, and Saturday mornings from 9:30 – 11:30. I walked away with a commitment to working in the rolling cups activity into my curriculum next year and to playing with proof blocks this year, I have two new colleagues on my Geometry team at school this year and our Geometry team (five of us at my school, including the middle school Geometry teacher) have already been having some lively conversations via email. My brain is bubbling with questions and energy that I normally do not have at the beginning of August. Nothing like spending six hours with creative, enthusiastic colleagues to stimulate my mind!

After our morning sessions we have lunch and the week held some great treats. On Thursday I sat to call my wife after my morning session and Megan Schmidt (@veganmathbeagle) summoned me to accompany her and two other pals to a lovely vegan restaurant nearby. One day I went to In n Out and got coached by John Stevens (@Jstevens009) on the proper Cali way to order. One the third day I sat on the sidewalk eating my food truck food while engaging in fantastic conversations with Jed Butler (@mathbutler) and Michael Fenton (@mjfenton) On all of these days I was so taken aback by the warmth and friendliness of people I had only met for a day or two last summer or had never met physically before at all. I was also taken aback by the lovely weather. After living in FLA for 35 years and enduring summers of unbelievable heat AND humidity this felt like heaven. The sky was blue, the air was warm but not heavy, and the evenings were cool and clear.

After lunch we were treated to keynote speeches each of the three days. Two of those speeches knocked me out. Ilana Horn (@tchmathculture) spoke about professional communities and the differences between local communities and virtual ones. At least these are the points that resonated with me. She spoke of the different ecology of our home school/district than that of the conversations among the MTBoS, through our blogs and on twitter. This really has me thinking. As a department chair I have seen sharing resources as one of my responsibilities. Ilana talked about being sensitive to differences in our local culture and after her speech Max Ray-Riek (@maxmathforum) made an analogy that is still resonating in my head. He made a comparison between trying to implement an activity from a source such as NCTM’s Illuminations and implementing an activity from a virtual colleague in the MTBoS. He talked about how activities from the MTBoS come with a sense of personality and context that is missing from something from a source such as NCTM. However, it occurs to me that if I share a resource from a MTBoS colleague, my local colleagues have no sense of context the way I do. These activities may be no more meaningful to them than an activity from a publisher or Illuminations. I need to be careful here and note that Max was not in anyway bashing NCTM, nor am I. He was just pointing out that activities are more meaningful when there is some context in which to place these activities – some way to envision what carrying out this activity looks like. If I borrow (steal?!) an activity from Kate Nowak or Sam Shah or Dan Meyer I feel like I have some basic understanding of what these people are about and why/how these activities are written. If I share these with my colleagues without providing some background context then I am not supporting my colleagues as well as I should. I also came away from her speech with the concern that I am spending time and energy in developing my online relationships and that by doing so I am taking energy away from my home team. I want to pick Prof Horn’s mind about these questions and I need to work hard to find the appropriate balance this year.

Friday’s keynote was delivered by Christopher Danielson and it was delightful. He spoke about finding what it is we love about teaching and making sure that we spend more time and energy on that. He spoke of his love for ambiguity and talked lovingly about finding open spaces and places where there are multiple ways to see a problem, where there may not be A correct way to do things. His phrase opening the presentation was – Find What You Love, Do More of That. A simple and powerful message. One I hope to be able to bring back home – tying in to my reaction to Prof Horn’s speech the day before.

Afternoons were filled with choice sessions and I will write about those tomorrow I hope.

Been Too Long…

It has been quite a while since I have posted anything here. The end of the school year is one main reason, the summer days spent with my kiddos is another reason.

I have also been working on editing the Geometry text I wrote last summer. I have to admit that it has not been as rewarding as the initial writing was. I mostly spent time undoing silly typos, trying to clean up some explanations, and formatting so that it is easier to read along with. I received feedback from our students who made it through the maiden voyage of this text and my teammates were great at catching some silly typos and giving me constructive feedback regarding layout. So, I am happy to share this link to my dropbox where you can find a PDF of the 2nd edition of my text as well as a folder containing all of the HW assignments we wrote last year for the text. We did not make it through all of the sections, so there are some HW gaps. I am hoping we can pace ourselves better this year. If we can, I’ll be uploading any new HW docs as we move along. I will also try to keep track of any other documents we create along the way as classroom practice, as explorations, as review notes, etc.

I have to give credit to a series of amazing blog posts recently by Meg Craig (@mathymeg07) where she has been sharing a goldmine of classroom resources. You should hope over to her website – http://www.megcraig.org to see what she has been sharing. Your life will be better after you do!

In five days I will be heading out west to twittermathcamp15. I will report back after that.