It’s Sunday afternoon, and Friday seems like a long time ago. Typical of intense self-contained experiences, time has taken on a paradoxical quality. To paraphrase a colleague of mine here at PCMI, I feel as if I just got here, and if I’ve been here forever. And I know that just as quickly, I will be going home. So I’m glad that every day seems like a week, and that the intense experiences of the weekend make Friday recede into the past.
Speaking of Friday, during morning math, we were joined at our tables by the students attending the High School Math Camp, as we experimented with pouring salt and the shapes thus formed. They worked hard and enthusiastically, using what they knew to make the connections that solving the problems we are given each day require. Their presence added some excitement to the room, I thought; as exciting as it is to access our inner explorers as we problem solve each day, it is even more exciting to see a student to light up with discovery.
We finished our week of Reflections on Practice by grading sample student papers, and crafting feedback that would hopefully move learning forward for the student who had made either a conceptual or mechanical error. Seeing the actual student handwriting on the papers, and witnessing errors I have seen frequently in the work I grade ‘for reals’, added a touch of reality to the work we are doing. In pairs we discussed not only how would we grade the work in front of us, and what feedback we would give, but whether that feedback would actually have an impact – we all agreed that students, receiving their papers, would immediately compare grades, calculate percentages (when grades were not out of 100 points), and, in the case of failing papers, crumple and toss. Regardless of our shared occasional pessimism, I know that going through these exercises re-energizes all of us [participants] to go back to school (when it is time – not yet! not yet!) with the energy and positivity to bring our best game once again to our classrooms, to engage in those best practices we discuss every day, with the goal of __________. (Fill in: raising student achievement, creating lifelong learners, increasing conceptual understanding and appreciation of math, whatever your personal pedagogical goal may be.)
After some ‘pre-lunch plank’, followed by the meal, our working group continued to progress towards a consistent product with which to complete our task for this conference – content for an online Geometry course for 8th grade and high school teachers shifting to the Common Core standards. It’s been a challenging process, not without its frustrations, but I personally feel like we have completed a lot of work in a short amount of time, and that perhaps the task is too large for the allotted time in which we have had to work, and I would like to have the opportunity to continue this process to its fruition, which will surely be after July 18. We presented our work to some visitors, who asked pointed questions about our goals and intentions, our resources, and even our vision of the final product.
The cross program activity on Friday afternoon was a presentation on the Math Behind Game Shows by none other than the Amazing
Kreskin Kerins. Bowen’s presentation modeled perfectly what makes an engaging lesson – a topic about which you are intrigued and want to know more, enthusiastic participation from those engaged in learning, and PRIZES.
The final ‘official’ activity I participated in last week was a meeting with two teachers from Monument Valley High School in Kayenta, Arizona (Kayenta is a census-designated place which is part of the Navajo Nation and is in Navajo County, Arizona); they had reached out to PCMI through Herb Klemens of the University of Utah for help with rich tasks to engage their at-risk students. A group of us had compiled a list of tasks and resources to share with them; we were aware of some of the challenges these teachers and their students face, and we were also aware that their issues and needs were, for most of us, beyond our purview. But we met, we shared, we discussed, we offered our suggestions and insights, and they were accepted graciously. At the close of the meeting, Herb suggested that lines of follow-up communication and sharing be created and maintained; I certainly hope that is the case. Although I am aware that the circumstances in which these teachers work and students live present huge challenges, I would like to find some way in which my experiences and knowledge could be of benefit to them. It was a humbling experience.
And then the weekend began!
Friday evening brought a lovely dinner with some new friends at The Farm at the Canyons, a restaurant that took its ‘farm to table’ philosophy very seriously. After we finished a deliciousdinner, we wandered around a bit and into the lobby a hotel which had decor that actually competes with our home-away-from home, the Zermatt. The staff was so cordial that they didn’t seem to mind that I was snapping photos of their light fixtures. Hmmm. When we returned to Midway, I caught the tail end of a repeat karaoke session, and had the pleasure of being part of a highly dramatic performance of Bohemian Rhapsody.
Saturday brought even more fun – thanks to MaryAnn Moore, I had the opportunity to volunteer at an outdoor concert at Deer Valley and see Smokey Robinson with the Utah Symphony for free (plus a t-shirt)! It was a beautiful night, and my job was to distribute programs as patrons entered the park, a plum job – everyone was excited to be there, and happy to receive a free program. My official duties were by the intermission (which was before Smokey came on; the first few numbers were entirely orchestral), so I saw the concert from a lovely perch on the hill. And he is amazing – spry and cheery and wearing a metallic green tux as he danced around the stage! The crowd was, of course, loving it, and many people got up and danced. It was another special evening.
And to top off a wonderful weekend, I hiked Park City Mountain (I couldn’t quite figure out whether that mountain has another name) with an intrepid crew. The hike went much further than we initially thought, but we persevered with good spirit and patient navigation, and made it up to the only running chair lift on the mountain, which also happened to be the one at the highest altitude. I had to fight with myself to keep going – I’m not normally outdoorsy and have a couple of decades on the people I was hiking with – but I persevered and accomplished yet one more thing on this trip that I had not before.* If you’re still reading, I apologize for saying this again, but I continue to be grateful for this experience.
And so the final week begins.
We were reminded this morning that tomorrow is the halfway point of PCMI! How can this be? Time has begun to take on that quality of seeming to speed up when you want to slow it down. This post may be a little shorter because (a) I don’t want to spend time writing when I should be experiencing and (b) I only want to write what might be meaningful to say. I also want to point out that anything I write here reflects only my experience at PCMI, and how I am processing all the activities in which I am participating and everything that I am learning. Just a caveat to my readers.
We were put into new groups in morning problem-solving, which I have just reminded myself is called Developing Mathematics. At my new table, I met someone who teaches in the same building in which I used to teach in Prospect Heights, Brooklyn (the building houses 4 schools), and the stories this teacher has heard recently about the principal I and most of my colleagues ran away from in 2011 continue to substantiate everything I remember as driving me out of the school. Luckily, he teaches in the most functional school in the building. Although hearing about the continued outrageous behavior of this woman is somewhat validating, there are 400 students who are being educated under her misguidance, which is sad and unconscionable on the part of those who supervise her. The work pace at my new table is quite different – I am sitting with a group of people who appear to work much faster than I do, and with more on-going discussion. My discomfort is a challenge to me – how can I produce good work and participate in this group in the most meaningful way possible? And how can this discomfort illuminate my understanding of student behavior and performance when I want them to work together in groups I have selected? One of the many wonderful features of this conference is the many layers of learning that are presented to us. The actual problem set is terrific – we are exploring complex numbers as rotations, an idea I have never completely understood until now, and the skill with which the progression of problems has been designed takes our understanding deeper as we work through them.
Meghan’s Eye View of the Crew
Our Reflections on Practice session was a little shorter today because of the group photos being taken. – Nonetheless, it was very fruitful with takeaways large and small. We shared new ways to collect evidence of student understanding – Plickers, Desmos Teacher Tools, Jeopardy – and how some of those tools provide data from more open-ended questions than others. Plickers does not require technology except for the teacher’s phone, but it is collecting answers to multiple choice questions. One teacher suggested a tweak on the speed-dating activity using whiteboards (another suggested call the activity Meet and Greet to avoid discomfort with the ‘speed-dating’ title) for sharing solutions to a range of problems, and yet another teacher pointed out that good old fashioned eavesdropping on student conversations about their work could provide meaningful formative assessment as well. We looked at the activity “Two Truths and a Lie” as applied to mathematics: students are directed to write two mathematical truths and a lie in any order; the lie must be well written, and in a way “which would challenge your last year’s teacher”. As we reviewed some examples in our groups, we saw how rich discussion could result from this activity. We finished the session by learning about a “Hinge point” question: it is a carefully crafted check for understanding midway through a lesson to see if students grasp the central concept, need to have it briefly clarified, or need the teacher to start all over again. This idea, written about by Dylan Wiliam in his wonderful book, Embedded Formative Assessment, is simple yet powerful. As teachers, we check for understanding throughout our lesson, but how intentional are we each day when we do this? By examining this practice, and crafting our hinge questions (and potential responses to the results) while planning our lessons, we can do way more than pay lip service to the idea of regular formative assessment.
The features of good hinge questions as described by Wiliam are:
- They are concise; students should be able to answer them in under 2 minutes, and teachers need to decide on a response to the results in 30 seconds.
- The questions can’t be answered correctly for wrong reason; common errors and misconceptions should be visible.
- Teachers should be able to see responses from every student using some method of data collection.
As Cal Armstrong put it, the goal of the Hinge Point question is a quick read of the classroom, “taking the temperature, not doing an MRI.”
[so much for a briefer post]
I finally feel on track in our professional development group, and managed to craft a solid framework of a draft for our portion of the course. My partner and I are still working on adding activities to our lesson, but the structure exists – essential questions, learning outcomes, and content. I am relieved; it felts as if we were floundering for a few days, but I suppose that was part of the process of allowing our ideas to coalesce.
The rest of the day was filled with enrichment: a visit to the gym, a Cross Program Pizza and Problem-Solving session in which the High School Math Camp participants gave us a big run for our money, and a Line Dancing Workshop run by David Nacin, a professor at William Patterson University and participant in the Undergraduate Faculty program here at PCMI. Who knows – maybe there will be a talent show at the last night in which we can perform our group tango!
And shameless promotion of my super-talented child (or a minute of video entertainment for my readers):
Day 6 of problem-solving aka Heavy Rotation was filled (as usual) with rich problems moving in a different direction: transformations. There were seriously intriguing problems involving unusual combinations of transformations that resulted in figures that looked different but still had mathematical relationships, and a dilation of a parabola that had simultaneous intuitive and counterintuitive results. Once again, there is a promise of patterns that will emerge throughout the week; today’s set provided a great hook. The group I am currently working with (or was working with until the end of today’s session) had a good work symbiosis. We seemed to complement each other well, working solo and then sharing, always respectful of each other’s work pace and need to discover solutions on our own. I hope my next group works as smoothly.
In Reflections on Practice, we switched rooms, tables, and mixed with other people, but the routine of working problems, sharing strategies in small groups, and then with the whole room seems familiar and flows easily. Today we examined a range of student solutions to a fraction problem, in which they needed to identify 1/4 of a compound shape, and then modeled an activity designed surface both evidence of understanding and misconceptions, in which students were given solutions and asked to design problems to accompany them. The activity – entitled Jeopardy was a well-designed bit of work; it included problems on elementary, Algebra 1, and Algebra 2 levels, and was a much richer task than the usual ‘Jeopardy’ that is frequently used for exam review in classrooms.
The afternoon working group session finally felt (to me) like concrete progress was being made. We split into two groups, and each had the opportunity to present the overall plan for the online course to a different professional development group; the act of having to explain our structure and strategy was enlightening in itself. The feedback we received was helpful and clarifying, and pointed us toward a wealth of new resources, including Henry Picciotto’s website, The Math Ed Page. When the Online Geometry Course group reunited and shared our feedback, the path we need to take to solidify the proposed first three days of the course finally became clear to me. It was helpful to discuss the strategy for the first of the four days, the portion of the course for which my partner and I are responsible; we had been going back and forth regarding the level of familiarity with the transformations content we should assume on the part of the participants (my partner is a sixth grade teacher and I teach high school), and some outside perspective lent clarity to that as well.
The rest of the day included optional activities, starting with attendance at a Cross-Program lecture on problem posing by Judah Schwartz; he talked about the power of visual representations, and using them in unexpected ways to explore problems. Schwartz says the essence of teaching is posing provocative and engaging next questions at the proper moment – such a succinct and articulate phrase, and such a huge challenge. His demonstrations opened a little bit of a door in that direction. Following the lecture, people from the Teacher Leadership Program met in small ‘special interest’ groups. I gravitated toward the group for ‘At Risk Students,’ hoping to share stories and resources for dealing with my most challenging kids. What I found instead was the opportunity to participate in a unique collaboration – two teachers from Monument Valley High School – a school on the Navajo reservation in Kayenta, Arizona, will be visiting PCMI on Friday in the hopes of gaining some resources and support for teaching math to their highest need students. These teachers had shared a list of standards for which they wanted resources, but as we began to discuss how we would assist them, larger questions emerged – for example, what resources had they already tried? What cultural issues did we need to be aware of? Finally – how could we help them without hearing them speak about their students and their need first? We agreed to collect ideas in a Google Doc, but not to share anything until we had the opportunity to speak to them directly and hear their story. I am looking forward to meeting these math teachers, who are willing to travel far to meet with us in order to find solutions for their students. I do not feel like any kind of an expert, but I know that having someone listen and really hear you, and then make suggestions, can be enormously beneficial and supportive.
The evening was just plain old hands-on fun (at least for math teachers and students). Carol Hattan, one of the directors of the program, sponsored a Math Building Party – basically Origami Fun Time. We began to create modular structures – first cubes and then icosahedra constructed of similar units. As she instructed usin the folding of the basic unit, Carol pointed out how much mathematical language could be used with students in completing this soothing and satisfying activity. And as we chose our colors and folded our papers, we shared stories of school and home, deepening the bonds that have been forming since the program began. Despite my quilting, folding paper is not something that comes easily to me, so I am very proud of my progress, and am looking forward to completing my icosahedron next week. Of course, I have to figure out how to get it home….
[NOTE: After spending the better part of Sunday morning writing this post, WordPress decided to delete it when I began to upload photos. If you have any dissatisfactions with this post, I assure you, the previous version addressed them all. I am rewriting this post as I comfort myself with a completely decadent cinnamon roll from the Zermatt Bakery.]
It seems impossible that we have been here at PCMI for a week. Impossible because we just got here a minute ago, and equally impossible because so much has happened – so much thinking, connecting, bonding and laughing – it feels as if it has been much longer. And in case you are wondering whether any of that gratitude for having made it here has faded, it has not.
In our final problem-solving session of the week, the connections between the problems given to us by Darryl Yong and Bowen Kerins had been working on all week deepened, sometimes in intriguing ways. For example, the same side lengths of a prism that give an equal surface area and volume (disregarding squared versus cubed units) are the same regular polygons (by number of sides) that when abutted around a single vertex add up to 360˚. WHAT? Why? This is a delightful mathematical mystery (at least, to me it is) that I am still pondering. Some teachers wondered whether all the apparent connections were meaningful – as my new sock shopping buddy, Jennifer Osgood articulately put it, “Just because my shoes fit you doesn’t mean my shirt will fit you as well.” In the coming week, we will be joined by some high school ‘Math Camp’ students; I wonder whether this line of inquiry will continue or resurface – what do Darryl and Bowen have in store for us all?
We became a group of 6th graders during Reflections on Practice, up on our feet, working in pairs, creating graphs with chart paper and post-its (for data points) given a set of scoring constraints about a soccer tournament. Once our graphs were complete, we were charged with determining which of the resulting 9 graphs depicted the ‘fairest’ tournament, in which teams were most evenly matched. It was a great activity, hitting multiple learning modalities – visual, kinesthetic, logical – and Cal Armstrong eavesdropped on our conversation to determine the direction and depth of our understanding, just as any teacher would (or perhaps should). We debated what fairness meant in terms of a soccer tournament, and how to ‘calculate’ that fairness given the graphs in front of us, and being 6th graders, our language wasn’t always mathematical. Finally, Cal asked us if there was a mathematical method for definitively describing the level of fairness in each scenario. When the term “Mean Absolute Deviation” was offered, there was the briefest of silences as many of us thought – 6th grade? Mean Absolute Deviation? But the fact is that this concept is included in the 6th grade math curriculum – at least it is in New York. It seems as if this is a huge idea for 11-year-olds to find through inquiry, but as one teacher pointed out, it may be more important to get students to want a tool than for them to actually discover it, echoing Dan Meyer’s recent series, “If Math Is The Aspirin, Then How Do You Create The Headache?”. This activity beautifully created the headache for finding reliable statistical measures, and modeled how students might function in a similarly well-designed activity, and how powerful the results might be – a huge takeaway to finish the week.
Our working group struggled with the sequencing of the online geometry course we are developing, as we debated the topics on which teachers might need direction given the curriculum shift that will accompany the full implementation Common Core standards in Geometry, and the logical order for those topics to be addressed. Clearly crafting an effective learning experience with these objectives in 4 sessions is a huge challenge. On Monday we will be meeting with teachers from other professional development groups for feedback, and I anticipate that the ‘outside’ perspective will be instructive. My partner and I are working this weekend to clarify our own session, which is the first one of the course. How much to include when introducing a topic, and what groundwork needs to be laid to prepare the participants for the second session are large questions we are trying to address.
The final formal session of the week was the Cross Program Activity: How to Make Sculptures of 4-Dimensional Things, featuring Henry Segerman, an assistant professor in the Department of Mathematics at Oklahoma State University. I must confess that
most of the presentation was beyond my mental capacity at the time, but the images he presented were quite beautiful, and I plan to attend one of his workshops next week in which I will hopefully learn how to use the incredible 3D printer that is available to us during the Institute.
And thus the weekend began.
After a leisurely dinner in Midway during which I was forced to rein in my New York tendency to insist on immediate and swift service (mostly because we were ignored for 30 minutes! – to my credit, I think I behaved well), I participated in the largest game of Cards Against Humanity I have ever witnessed – close to 20 participants! Each turn took close to 5 minutes, and one of our players (who may or may not have been one of the authors of the Common Core State Standards for Mathematics) had an uncanny knack for offering the most appropriate answer for each of the game’s inappropriate cards. The game left us all with an odd sort of bond – we ALL think nasty thoughts – and we hear them with good humor and tolerance. A perfect lead in to an evening of karaoke, skillfully hosted by Bowen, during which we discovered that not only do many of us have hidden (and not so hidden) performers within, but that also among our numbers are truly wonderful singers, rappers, interpretive dancers, and go-go girls.
On Saturday, the work on our parade ‘float’ came to fruition as we participated in the Park City Annual 4th of July Parade. I haven’t been in a parade since I was in a marching band uniform, which is more years ago than a lot of my colleagues have been around, but I don’t remember EVER having as much fun as I did yesterday. The Park City parade is a bit eccentric and very enthusiastic. Among the floats and participants included a snowplow, an antlered hedgehog, and grown-ups in tutus. Where else would you find a band of 60 to 70 math students and teachers, walking along waving geometric structures, chanting math slogans and being cheered on by the spectators with shouts of “Math Rules!” and “I love math!”? I was ‘officially’ a Marshall for our group, keeping us a pace, but you know what they say about herding cats? Luckily, the group immediately in front of us in the parade line-up was a 6 piece New Orleans jazz band, which not only kept us stepping in time, but provided musical accompaniment to which some our members danced and leapt for joy (truly!). After a lunch to rest our feet (and arms – waving Zome tool stars takes a lot more energy than you might think), I spent the rest of the day roaming around town, helping the local economy, which I actually don’t think needs my help. It was one of the loveliest 4th of July’s I can remember.
After a dip in the pool, and a little quiet time (what’s that?), the day finished, appropriately, with viewing a fireworks display from atop the crater across the road. There was a local show, and off in the distance we could see four or five other firework displays like small showers of sparkles across the dark horizon. It was a perfect end to a great day, and a great week. At the risk of maudlin redundancy, I am thrilled to be here, to meet these people, and do this work. I can’t wait to see what the coming week brings.
Today, after 4 whole days, things are beginning to feel comfortable and familiar. Even though we switched groups during our morning problem-solving session, our routine of working in pairs and trios, and then solo resumed quickly. Geometry was always my first love in math (hence my quilting), so the growing connections among the problem sets this week are deeply satisfying. When Bowen intoned “Take a break!” at the midpoint [ouch!], I wasn’t ready to stop working, nor again when it was time to transition to our next activity. I need to think about the takeaway from this feeling – is it possible to recreate this drive to explore in my classroom, and for more students than just my uber math geeks?
During lunch today, I had a conversation with several teachers who were either National Board Certified, or working towards that through their Math for America fellowships. Obtaining this certification is a large undertaking with many facets, not the least of which is a whole class video, something of which many teachers are leery. Today’s Reflecting on Practice session was a perfect example of why this is so. Although neither of the other classroom videos we watched depicted a perfect lesson, today we witnessed a host of missed opportunities in a TIMSS video of an eighth grade math class from 1999. It’s so easy to spot problems from the safe position of being an observer. But even as I ticked off the moments in the video when I saw areas in need of improvement, I knew that I make just as many grievous errors in my own classroom, sometimes in the name of expediency or pacing or classroom management. We worked on plans to remedy the shortcomings in the lesson we saw, and Cal Armstrong thoughtfully reminded us that in spite of any criticisms we might have, the teacher in the video opened her classroom for the entire world to see and learn from. A word to the wise: the video is 16 years old; what goes on the internet stays on the internet.
Our afternoon working group has begun to assume a productive structure after exploring some big ideas about transformations; my partner and I are outlining the first day of the proposed online course, in which the definition of rigid motions is clarified and teachers are provided with some exploratory activities for themselves and their classrooms. Every step in our planning requires conversation, debate, pushing back and forth to refine and clarify the experience we envision our lesson to take. Two hours of work flies by pretty quickly, and each pair in our team seems to be going through the same process. I know we won’t have a completely finished product at the end of PCMI, but I feel like we will have a solid framework of high quality.
In the later afternoon, I attended the Cross Program activity (open to the undergrads, graduate students, Research program participants and the teachers), a talk by Evelyn Lamb, a professor at the University of Utah, on the value of the online math community. Evelyn is an unassuming and entertaining speaker, and she spoke with insight and humor. Her presentation was organized into the purposes of Online Math Communication – talking to ourselves [reflective blogging, for example] and talking to others [chatting on twitter, reading and commenting on blogs]. She broke this second category down into further subdivisions: talking to other math researchers and educators, and then talking with people who either wondered about the life of a ‘math person’, or wanted to relate their own personal tale of math horror. She told some stories about unlikely connections she made through her blogging and tweeting, and shared some personal truisms and myths of the online math experience, two of which I am paraphrasing here (as well as I remember them):
- Myth all bloggers need to remember: If you write a blog post, people will read the whole thing. Remember – readers are not your students, and have no obligation to do so!
- Myth of writing posts: “Any sacrifice of accuracy, rigor or generality is unacceptable.” This is a good thing to remember [Wendy…] for those of us who agonize over every sentence as if it were our graduate Statement of Purpose.
Here’s my personal favorite of Evelyn’s Big Ideas about Blogging: Why should the math internet be different from any other internet? Use cats liberally to please and entice readers.One final note related to Evelyn Lamb’s talk – the Ballroom had some prime examples of Zermatt decor, such as a trompe l’oeil tapestry, and this Merry Band of stained glass windows, pointed out to me by the observant Ashli Black.
The day finished with the final touches of 4th of July parade preparation, complete with tangrams, pun posters and all manner of Zome creations. Who is this masked man in the patriotic Zome tutu? You’ll have to attend the parade on Saturday to find out!
I’m writing this for myself as much as for you, dear readers, because each day is filled to the brim with mathematical, pedagogical and interpersonal stimuli. I am certain that when I return to Brooklyn, I will not remember half of what seems so clear and immediate today. As the days go by (all three of them), fewer people show up early to breakfast. I love coming in and sitting with someone different at each meal. Today the conversation ranged from state assessments to finding thrift stores in Park City (I found some kindred spirits ready to go on the hunt with me).
During today’s morning math session, I felt that patterns that have been spiraling the problem sets each day begin to take shape in my mind, and had a burst of insight. When we discussed some examples as a whole group, I saw that there were more mathematical dances going on in the work than I had currently grasped. I think I need to take some other time out of the day to continue working on these sets, but choosing what NOT to do in order to find that space in the schedule? Hard choices, but lucky ones to have to make. Tomorrow we switch groups, and the work habit adjustment will begin again. I am discovering that I like to work through things by myself, I think. It is challenging to work on a math problem, develop insight into it, and formulate a solution at the same pace as anyone else, and the admonishment ‘not to teach our colleagues’ in the norms we were given proves difficult to follow. I wonder if there will be a conversation about this at some point during these morning sessions.
Instead of watching a video of students solving problems in our Reflection on Practice session, today WE were the students, and worked first solo and then in pairs on several problems that appear regularly in middle and high school classes. I unwittingly provided a sample of a typical student misconception in the first problem (it involved working backwards using fractions and a bag of marbles, a word problem subject of frequent choice), and got a completely wrong solution, much to my chagrin. As a group we explored different types of solutions – mostly correct ones – and discussed the conflict between how the Common Core standards view fractions (along a number line) and the area or ribbon models which can be effectively used to create visual representations that are not pizzas. Our task for pair work was to develop both explicit and recursive sequences for the Handshake Problem, and then to choose from among the various solutions three to present to a class (and to justify those choices) to further a particular lesson goal. Tomorrow we are moving on to formative assessment (I should be doing my homework reading right now…). I’m looking forward to really digging in to that conversation.
There was no formal programming this afternoon, and buses ran back and forth to Park City. As lovely as the Zermatt Resort is, it was equally lovely to move off the grounds of this Tyrolean Wonderland (with its varied amenities, including goats and footmen) into the semi-real world of Park City. My colleagues and I roamed the center of town and made our contributions to the local economy, and enjoyed speaking to the shopkeepers, many of whom were transplanted from elsewhere but did not live in town due to the high real estate values. Main Street in Park City is a super clean version of a Western town – a quaint and upscale commercial strip with unusual boutiques and art galleries. My friend Irene almost scalded herself trying to embrace this fellow in the 85˚ heat, but found someone a bit more accessible in one of the shops.
The Kimball Art Center was a highlight of my day; it is a cultural center which runs, among other things, art classes for children and adults, exhibits of photography, mixed media and other art, an annual art festival and other events. The current exhibitions included a series of reproduction WPA-commissioned posters for the National Park Service and a quilt and felted wool exhibit by Faith Hagenhofer, an artist who explores issues of culture and contested land, and uses wool from her own sheep in her art work.
On the bus back to Midway, our bus driver advised us to keep our eyes open as we drove back; he claimed he had seen a lot of wildlife on the move that evening. And sure enough, we were treated to an elk. We shared stories of our day in Park City, and in doing so, shared even more of ourselves, moving from tales of the classroom to tales of our lives. What a rich, rich day it was.
What an auspicious beginning to my day with this cup of coffee!
Another beautiful day in the mountains began with sharing over breakfast – swapping school stories (good, bad and humorous) with teachers from different cities, and different types of schools. The underlying commonality we share includes being committed to improving our practice regardless of our length of service, and giving our students, thus, a better experience in the classroom. It is clear from talking to my colleagues that our desire to help our students goes beyond their understanding of mathematics, although this is the key we use.
Our morning problem-solving session felt more comfortable today; the sequence of the daily questions not only builds on ideas in each set, but also hearkens back to emerging patterns from that of yesterday. I was again reminded that the personal adjustment period in our experience mirrors the same phenomenon in our classrooms. I discovered that another teacher at my table felt a discomfort similar to mine with respect to the different speeds and work habits of everyone at the table, and knowing that, felt more comfortable. How often have I told my students not to be afraid to ask questions because if they have a question, at least one other student, and probably more, has the same one.
We ‘observed’ a Calculus class in today’s Reflection on Practice, examining student response, participation, and body language as the teacher led them through a problem solution. We sympathized with the silent girl in the video who we all felt was lost in the exercise but clearly didn’t feel safe or comfortable enough to voice any of her confusion or ask any questions. We collectively cringed when the teacher asked, “does everyone understand?”, knowing full well that a student who didn’t understand was unlikely to admit that in a room full of peers who apparently did. As we worked at our tables to craft questions that would probe and push student understanding, I thought how much easier it was to do this exercise with another colleague at a workshop but perhaps not quite so simple on my feet in the classroom.
Although the specific objectives are a little unclear in my afternoon group – creating an online Geometry course for teachers, my partner Irene and I had a blast exploring as many possibilities as we could in reflecting a line segment, so much so that we found it difficult to stop when we were told to take a break. Irene is a 6th grade teacher, and I love how her style, which is well-suited to the explorations her students need to make in learning this content for the first time, illuminates my own. It forces me to be more methodical in my approach, which I can see is necessary for my own high school students, many of whom may not recall the content well from middle school or who may not have mastered it at that time.
The special treat of the afternoon was a session with Bill McCallum, one of the authors of the Common Core State Standards in mathematics. Bill walked us through the re-design of the Illustrative Mathematics website, and tried to answer our questions about the implementation of the Common Core standards and curricular resources. He is an eloquent and engaging speaker, although the theme that the standards are not curriculum seemed to emerge after every question or two. From a teacher’s perspective (if I may be so bold to speak for some of my colleagues as well as myself), the implementation of the new standards, especially at the high school level, is hampered by a lack of choice in curricular resources that are fully aligned and easily usable. [Note: I use the word ‘easily’ because while engageny.org clearly has fully developed rigorous curriculum for Algebra 1 through PreCalculus, many teachers find the organization of the materials challenging, and the scope of each lesson ambitious for most classrooms.] The frustrations that were surfaced through this session, however, serve as action points for teachers seeking solutions for their classrooms and students. We were exhorted by the inspiring Gail Burrill to advocate for change, particularly around creating support for students who begin to fall behind in elementary school. I left this optional session feeling ready to tilt at a few windmills.
The day finished with something New Yorkers love to do – shop in a supermarket outside of New York City! The Riddley’s in Midway was not only spacious and filled with all manner of comestibles not frequently seen in my local grocery store, but also had some 4th of July treats. Interestingly, everyone we met knew we were with “the math group.” Huh!
A day that begins with this view on the way to breakfast is full of promise. And my first day at PCMI was full of a lot of math and a lot of collegiality. I’m still trying to sort through it all before the next one begins. It’s late, and this is unedited. I’m trying to document this whole experience for myself, so I apologize for any typos or rambling sentences. I always try to make my posts ‘sing’ a bit, but I’m don’t think I’ve hit that mark.
The morning session – “Some Applications of Geometric Thinking” – had participants working on some interesting problems, looking for patterns in simple problems which were clues for more challenging questions. I love the way the problems were scaffolded. It was interesting to note the different work habits and paces of my colleagues at the table, and I encountered my own frustration at having some ideas and discoveries shared before I was able to work towards them. An important part of this process is not only to do math, I think, but also to simulate a problem-solving classroom environment, so this frustration provides an opportunity to gain some insight into the student group work experience, as well as my own ability to counter it in myself. I’m looking forward to digging into it again tomorrow.
The second half of the morning was spent on “Reflections on Practice”, the theme of which is Eliciting Evidence of Student Thinking. We analyzed a video-taped rich middle school classroom activity in which students explored how changes in the shape of a quadrilateral affected its area, similar to a problem we had explored in the morning. Student thinking at several levels was apparent throughout the video, and we discussed and shared this evidence in both small groups and as a whole class. We examined some written student work, noting the difference in our own styles of analysis. I’m looking forward to seeing where this conversation goes.
In the afternoon, we were broken into working groups – I have been assigned to a small group that has been tasked with framing an online Geometry course for teachers to address the some of the content shift in the Geometry Common Core standards – to alleviate some teacher anxiety, address some possible misconceptions or murky areas, and provide ideas for weaving these ideas into the classroom. Today we
played explored transformations with patty paper – trying to stay away from coordinates – and examined how transformations could be used to prove theorems. The challenge of reshaping proof as it is taught in high school to include transformations as well as forms other than two columns is another big issue for many teachers, and a topic we are going to continue to address.
The rest of the day was devoted to self-replenishment and fun. After a rousing session of creating stellated figures out of Zomes for the PCMI float in the Park City 4th of July parade, I managed to get to the gym at this lovely resort and take a dunk in the pool (although at 95+ degrees all day, the water is more like a warm bath). The opening dinner was followed by even more bonding over Zome building.
There are so many nuances that I am processing – the wide range of attendants at this huge math event which from undergraduate math students barely older than my own, graduate students and math professors, the teachers from all over the country who readily share their stories and ideas, and the warm and welcoming organizers of this event who make even the newcomers like me feel as if they have been attending for years. But I have 3 weeks to make sense of this wonderful experience, so I’ll stop here for now.
So I got on the plane, took my happy pills, drank some wine, and just by putting one foot in front of the other, arrived in Salt Lake City. Being late was actually a hidden bonus – the rooms are not ready at the resort, and the folks who came on the 7 am flight have been sitting in the hotel lobby since noon.
The Zermatt Resort in Midway, which is all things Tyrolean, is approximately an hour away from Salt Lake City. I rode over here in a van filled with undergrad and graduate math students, who were interested in hearing how a high school teacher keeps her students engaged in an age of cell phone ubiquity. I shared my strategies for using cell phones for instructional purposes (which, truth be told, only work some of the time), and had the pleasure of turning them on to Desmos and Kahoots.
Once we arrived at the resort, I immediately began chatting with other teachers – a surprising number from New York – who were waiting for their rooms. (The front desk keeps giving away banana bread in lieu of room keys.) Conversation was easy and interesting – we all teach math, we’re all engaged in our teaching of math, and motivated to share and improve the quality of the experience we give our students. The math students, on the other hand, are busy doing math already – reading papers, sketching out problems and proofs. It’s a major nerdfest here in the lobby of the resort, and I’m loving it.
Barely decompressed from the end of a harrowing school year, I am off to Utah for PCMI, the mega Math Teacher conference. I’ve been fortunate to be accepted with a scholarship after my second go round at applying, and thus am waiting at the airport for my rebooked flight to Salt Lake City. Why rebooked? A series of unfortunate events, as they say – a wrong turn on the way to airport, almost putting me on the Verazzano Bridge rather than the Belt Parkway, which made me miss baggage check, and a double scrutiny by security – those graphing calculators, you know – caused me to miss my flight, the first I have taken in many years. Yes, dear readers, I am a flying phobe. I had picked the earliest direct flight possible to avoid having an expanse of time at the airport during which my anxiety could begin to brew.
A kind agent rebooked me on a flight not too far in the future, and waived the rebooking fee as I somewhat hysterically told her my tale of woe at security. My husband returned to the airport with my large bag (he was going to somehow figure out how to ship it to me), which as it turned out was overweight. We tried removing things, but worn out by the mornings events, I opted to shove everything in to the bag and pay the fee.
So I have re-checked in to my flight, bag all tagged and on the vast conveyor which will hopefully put it on the right plan, using my complimentary 30 minutes of wi-fi to write my first post in 2 months. I purchased a book of Sudoku which are too difficult at the moment, and I am debating the timing of using my happy pills so I can be preferably asleep for the bulk of the flight.
It’s been a year in which I tried to do too much – way too much – and achieved a bunch of it at the expense of my free time and personal life. I completed an online Multivariable Calculus class, earning me enough higher math credits to apply to the Math for America Master Teacher program. I met several financial goals required to maintain my home [long non-bloggy story] through a lot of private tutoring and per session work for the Department of Education. I improved the Regents performance of my Algebra 2 students – a goal which was important to my school, and to my students’ transcripts. And my Geometry Fundamentals class – all three sections of it – completed the Mondo Choice Board final, further refined from last year’s launch. Oh – and last but certainly not least, I have made my attendance at PCMI happen.
As I mentioned, this all came at the cost of my personal life, the reparation of which will take time, but the summer is long (although never quite long enough…). I’ve learned a lot this year about what works and what doesn’t, what I can do and what I can’t, and I’m ready, for the moment, to set all the what ifs and ‘what I should be doing’ thoughts, and immerse myself in what I am sure will be an unforgettable personal and professional experience.
I just have to get on the damn plane.