Category: Uncategorized

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!

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:

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.

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.

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.

Whenever I finish a new post I will tweet out a message about it and I often encourage people to drop on by and share some wisdom. I definitely need some tonight.

Had a great conversation with a colleague today about his Algebra II Honors class. They are examining exponential functions and are ready to talk about logs. He came by with what seemed like a straightforward question – but it no longer feels like it is. He sketched the graph off y = 2 ^ x and marked pi on the x-axis. He talked about working his kiddos through the argument that there must be some power of 2 that yields pi as the answer. He talked about a method of exhaustion making better and better guesses to get closer and closer. We talked about how this conversation could be approached as carefully as possible. I talked about the intermediate value theorem but advised that it not be named yet. I talked about temperature during a day and speed on a car’s speedometer. But as he pushed me I realized that all of these arguments rely on comfortably knowing that this function is continuous and that if 2 ^ 1 = 2 and 2 ^ 2 = 4 that there MUST be some value of x between 1 and 2 so that 2 ^ x = pi. We ended the conversation – because I had a committee to run to – with this questions: How do we convince Algebra II students that this function actually does have to have an input x that yields every output y in a region? How do we recognize a function as continuous? What are the markers? This feels like a question that I should have had a better answer for and maybe in October when my brain is smarter I might have. So, I ask you out there – how can we convince Algebra II students that there is some real x so that 2 ^ x = pi?

Last week it felt as if my Geometry class as making great progress in examining radians, looking at areas of regular polygons, dealing with a new vocabulary word (apothem), and generally doing a nice job of making connections – especially with the right triangle trig that we are now leaning heavily on. Well, Friday morning we had a conversation that I am still unpacking. On their HW from Wednesday night one of their tasks was to complete the table below:

When my students asked me to review this problem I started by drawing an equilateral triangle and its apothem. I know that it is more efficient to use our knowledge of the triangle area formula, but I wanted to reinforce our new formula that the area of any regular polygon is half of the product of the apothem and the perimeter. Unfortunately, when I drew the picture they asked me why I drew what I drew. They wanted instead to draw an equilateral triangle and its altitude. I followed this suggestion by drawing the following figure:                                       

They were quick to identify that this is not, in fact, an example of an apothem. Pretty much everyone agreed to this quickly. Instead they instructed me to draw the following figure:

I have been trying to figure out why so many made the same mistake and, after talking to one of my Geometry teaching colleagues, I have a theory. I keep drawing pictures like this one: 

I fear that seeing this drawing repeatedly has somehow convinced some of my students that the altitude of a triangle is simply the apothem. I know that I pointed out the similarities between the terms when I was trying to help them remember this new word. I talked about how altitudes and apothems were each perpendicular to a side (and that they both begin with the letter a). However, I do not know why some students would simply draw an altitude and figure that, for some reason, we are now calling it an apothem. So this was a disappointing way to start the day. I think I feel a little better now that I have a feeling where the triangle mistake came from. Have any of you experienced something similar?

The other disappointment came when they asked me to address the next HW question. They were again asked to fill out a table, Here is the second HW problem:

I suspect that most of you see some similarities between the first two HW questions. My students did not. I understand that when you are learning something new it is hard to step back and see big picture things going on. However, I also know that this type of HW problem is tedious and I hope that my students are thinking of ways to streamline the process. Instead of recognizing that the triangles in the two problems are in a 1 to 2 ratio and using our knowledge of similarity and scale factors, many students simply reset and did the problem from scratch. I try really hard to build in these sort of  connections in my assignments and my tests. I do not expect students to recognize that in October, but I would hope that they do by May. I’d love some advice about how I can better help my students look for these types of connections. How can I help them step back a bit and see these connections?

Back at it 8 AM tomorrow. I’m optimistic we’ll have a good week. Test on Wednesday – I want them to really kill it on this one.