PCMI Day 8: Hooks and Provocative Problem Posing

quilty mountSo week 2 begins.  Today was jam-packed with activities – the regular classes, some enrichment, some special interest grouping, and some hands-on geometric fun – math teachers at play!


Dan loves gummi candy while he does math.

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 ourthsthe 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 ourJen 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 MPj03988190000[1]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.

phiolaThe 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 uslotsa foldingin 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….





      • Maya Quinn

        Ah; he is a proponent of experimenting with the x-y plane and graphs of the form “y = mx + b” by visualizing the lines in a corresponding m-b plane as points.

        For example: The graph “y = 2x + 5” would be graphed on an m-b plane using the point with coordinates (2, 5).

        (A good challenge: What does a line in the m-b plane correspond to in the x-y plane?)

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