Wheels starting to turn

1837452I found out on Tuesday that I would not be teaching Geometry this fall – because of ‘department needs’, I will instead be teaching an elective course for juniors and seniors who need one more math credit and who have not been successful in other classes (along with my Algebra 2/Trig classes).  The structure of the class is completely up to me, which is kind of wonderful, or would be, if school wasn’t starting next week.

I’ve decided to teach this as a problem-solving class, using the structure of the book Crossing the River with Dogs; the book features a different problem-solving strategy in each chapter, strategies which are tiered and accessible.  I met a teacher at PCMI this summer – Evelyn Baracaldo – who developed and taught this class at the NYC iSchool, and generously shared her materials with me.  I probably shouldn’t be putting together YET ANOTHER NEW COURSE, but somehow I can’t resist this one.

The challenge – in the few days remaining before the start of school – is to envision the [realistic] classroom culture I want for this course.  So the ideas are spinning around in my head, giving me more than a bit of anxiety, although I know this is part of my process.  This is what I’m thinking about so far, and I would LOVE FEEDBACK.

  1. The weekly warm-up sheets will be perfect here – estimation180, Which One Doesn’t Belong, visual patterns, Would You Rather?.  I’m thinking that the students will save these in a folder all term, and then do an end of term reflection/assessment of their progress.
  2. This class will also lend itself naturally to Number Talks, which I have been reading about all summer.
  3. Our notebooks may not be full-blown INBs, but they can still have a structure – sections for each strategy (tabs!) with worked problems.
  4. Group work, large whiteboards, vertical non-permanent surfaces – these are modes of working which are excellently appropriate for this content!

Sounds great, right?  But here is what is still unclear to me:

  • How will students be accountable?  With all that group work, how do I ensure that everyone is working?
  • How will students be assessed? (I am responsible for grading them, and for developing a grading policy; this is engrained in how my school functions.)
  • How will I introduce new strategies?  Will we just work on problem sets?  One 3-act per strategy? Math Forum PoWs?

I’d like to NOT reinvent the wheel.  There are so many wonderful resources out there (and in my house and on my computer and in my files at school….); I want to pull together all the applicable nuggets of brilliance that I have already squirreled away and teach those kiddies some math!

I welcome your thoughts –

Asking the right questions

I was thrilled when I received this message from Le Sam Shah last week:

Screen Shot 2015-08-30 at 12.31.43 PM

At PCMI, I set a personal goal of improved formative assessment this year through ‘hinge point questioning’ (see this post), and had been wondering how I would maintain this new practice all year; many September initiatives, despite my best intentions and aspirations, fall by the way side as planning, grading, and college recommendations pile up, and as report cards, observations and parent conferences encroach on the regular day-to-day work.

But what Sam created a collaborative and open space, betterqs, in which teachers can share and reflect on their questioning practices, and by inviting me to participate, provided me with the checks and balances to stick to my goal.  And even better, the blog betterqs is open to anyone who wants to share.  As we all know, the more you participate in the #MTBoS, the greater rewards you reap.  So here are the details:


Hope to see you out there!

It all hinges on the question

hinges-1I had the privilege of attending PCMI earlier this summer, and the further privilege of attending a presentation by Dylan Wiliam on formative assessment.  One of the big takeaways from all of that awesome professional development, and something I have committed to as a personal teaching goal for this coming school year, is the implementation of regular hinge point questioning in my lessons.   In the ‘middle’ of the lesson (or the hinge point), the teacher poses a question which is designed to elicit evidence of understanding of the key ideas [presumably] taught thus far.  The students should be able to answer the question in 2-3 minutes, and the teacher should be able to evaluate the responses and make a decision on how to proceed in class in 30 seconds – whether to continue forward, clarify or re-teach, or remediate in some way (review a requisite skill, for example).   For further clarification or description, watch this video by the Great Dude of Formative Assessment, Dylan Wiliam.

I love this simple and brilliant idea, even though I’m uncertain of its implementation.  My classes are only 45 minutes long.  What happens if the evidence shows the students haven’t gotten the idea at all?  What if half of the students have gleaned the lesson goal?  My favorite idea from the video above is that we should design our lessons with hinges in them, rather than designing them as airport runways, building in feedback loops.  I teach in New York, and always have a Regents exam breathing down my neck.  But I know that this technique will help build better outcomes and deeper understanding for more students if consistently and intelligently used.

I committed this goal to the video archive of PCMI teacher goals (note no link shared here!), and am committing it to the InterBlag; thanks to Sam and Rachel for creating this collaborative space.

The Anxiety of Influence: PCMI, TMC15, and [Trying to keep] Moving Forward

IMG_5793I returned from Utah 2 weeks ago, and am about to embark on my ‘real’ vacation – 2 weeks on Lake Dunmore, Vermont (aka heaven).  My final recap and reflection has been on the to-do list since July 18, but my return to Brooklyn was accompanied by a delayed crash from the end of the school year, some home disaster and drama, and my envious observation of TMC15 as it appeared in my Twitter feed (which I couldn’t always bear to look at) and blog reader.11782342_10205542168927228_3163533859352772846_o

11412004_10205487182992614_2989034904136514571_oPCMI was a gift for which I am deeply grateful on many levels.  The scholarship enabled me to attend, got me to confront a long-lived phobia (flying), placed me in the company of like-minded teachers who became dear friends that I know I will see again, and restored my teaching soul more fully than I thought possible.  The challenge I face in the coming year is maintaining that restoration and energy in an environment which, while personally supportive, pretty much works in opposition to many of the mathematical and philosophical teaching practices to which I aspire.11709954_10205462851144333_210506142907498134_o

As wonderful as PCMI was, the emotional backdraft from Twitter Math Camp is overwhelming. One by one, I have been reading everyone’s blog posts, getting a keen feeling of what I missed in California.  (The top hit for this, by the way, was Fawn Nguyen’s keynote, the slides and notes of which I read in their entirety, IMG_5880hearing Fawn’s voice loud and clear.)   But there has been a bonus for me – everyone’s heartfelt and articulate writing has left me with some TMC takeaways, and I wasn’t even able to attend.  And these takeaways reflect and magnify my experience in Utah, creating some kind of ultimate fractal/exponential virtual math professional development effect (there’s an acronym or hashtag in there somewhere – I’m sure of it).  Here’s what I’ve gleaned and what I continue to ponder:

  • Mary Bourassa wrote so eloquently about the growth of her participation in the MTBoS and how she finally felt she made a ‘real’ contribution with the Which One Doesn’t Belong website.  I’ve participated in workshops Mary has run (at Exeter), and thought she was the cat’s meow way before Which One Doesn’t Belong.  I share resources and participate as constructively as I can generously on Twitter; I would love to be able to create something as broadly useful.
  • Darryl Yong was awesome as the co-designer of the morning math problem sets at PCMI, but his post on Radical Inclusivity spoke directly to my heart.  I have a personal goal to speak out against the pervasive tacit racism at my school, and to try and start a diversity committee. Darryl’s post eggs me on.
  • Julie Reulbach and Anne Schwartz, two women who I really would like to see again after meeting them in Philadelphia at TMC13, both wrote about hesitancy in attending TMC after personally and/or professionally difficult years, and the restorative power of this community.   I know that at the end of the school year, I felt quite spent in both of those arenas, and approached PCMI with trepidation.  I was thrilled to find myself recharged within days (even amongst ‘strangers’), and I was glad to hear the same was true for Julie and Anne.
  • Megan Schmidt, my favorite person who I have never met, looked another scary issue squarely in the face in her post Safe Spaces at TMC15.  Megan’s posts speak directly to me, every time, and this post was no different.  The personal issue I dealt with at PCMI was the baggage I carry and the isolation it occasionally forced on me; it was difficult (I found it impossible, actually) to talk to anyone about the ravage alcohol abuse has wreaked on my home life or the ever-present watch for cancer recurrence that has become routine for me.  I’m not sure how to create those safe spaces wherever I go, but I love that Megan articulated the need for them, because there are times when group hugs just don’t do it.

In addition to the aforementioned keynote by Fawn Nguyen, there were two other quotes (below) that appeared numerous times in the blog posts that sum up, for me, not only both conferences – PCMI and TMC – but how I view my involvement online and why I continually seek these opportunities, and relish them so.

from Lisa Henry: “It’s the community, stupid!” (quoted from Dylan KaneIMG_5780

from Christopher Danielson:  “Find what you love.  Do more of that.

Thanks to everyone who shared, and you can bet you will see next summer!



PCMI Day 12 Here Come the Kids (and then there was the weekend…)

just another gorgeous shot in utahIt’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.

salt pyramidSpeaking 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.salt ellipse

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 aspens[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.)

planksAfter 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.

game show screenThe 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.grant deal no deal

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!

nutmeg lampFriday 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 deliciousperfect eggdinner, 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.

Now, THAT'S a chandelier!

Now, THAT’S a chandelier!

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 smokeyconcert 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.

aspensAnd 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.  crew on a mountain 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.

Mathness on a Mountain

PCMI Day 10 and 11: Tripling your fun

Hint of a rainbow

Hint of a rainbow

This post will be brief but I want put down some notes, if only so to remember the details of this great day myself.  Working backward from a jam-packed evening:

I have just returned from Line Dancing taught by one of the Undergraduate Faculty participants, David Nacin.  He has been generously giving his time in the evenings, giving us the opportunity to bounce around the dining hall doing jazz boxes, Monterey turns, and coaster steps.  We’ve been having a great time and trying to figure out the appropriate time and venue in which to show what we’ve learned this week.

hotsnlots photoPrior to that, we had a Sharing session, which was an opportunity for participants in the Teacher Leadership Program to share some of their best practices with the rest of the group.  I led off the evening (which I was pleased to do, being pretty nervous) with my end of term differentiated assessment for my Geometry class.  It was wonderful to be able to share this project with an audience of appreciative peers.  Getting my presentation out of the chris plickway allowed me to focus on the interesting ideas my colleagues had to share, including a demonstration of Plickers, a program to enlist students as teachers, engaging ‘synthesis’ tasks which require multiple problem-solving strategies as well as broad content knowledge, and an engaging marketing/statistics project which resulted in Bacon, Mac and Cheese pizza being conceptualized by 8th grade students and test produced by Pizza Hut. I am in awesome company here.
getting there foldAnd prior to that, Ashli Black, math teacher and folder extraordinaire taught us how to fold fabric in order to geometric designs.   So therapeutic to work with your hands, and so engaging and challenging that it is fabric and math simultaneously!  I think I may have more of a chance finishing this in a timely fashion than a quilt.

What a finished product might look like.

What a finished product might look like.

All of the above happened after 4:30 this afternoon.  That’s how full the days are.

Before 4:30, we had our regular programming – morning math problems, Reflections on Practice, lunch, professional development groups.  And just before lunch, a group of teachers decided to do three minutes of plank for core strengthening.  (This took place in a corner of the dining hall.)  At first, I volunteered to be the timer, but watching their effort, I had to join in – so I only had about 1 and 3/4 minutes of plank.  Hopefully tomorrow I’ll make it for all three.

I should probably be writing about all of the math we’ve been doing (discovering the relationship between complex numbers and Pythagorean triples, and constructing ellipses and hyperbolas), all the reflection and sharing that has been going on (examining feedback and types of listening the teacher can do in the classroom, developing the all important hinge question), and the progress we are making – inch by inch – in our professional development group.  But the hour is late, and I’ve got a bunch of math to do in a few hours!

The Zermatt Resort has footmen.

The Zermatt Resort has footmen.

Day 9: Almost Hump Day!

rainWe 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 inversemisguidance, 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

Meghan's Eye View of the CrewOur 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 2 truthsteacher”.  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.

RNDC2-624x841The 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):