Park City Mathematics Institute
Secondary School Teacher Program

Reflection on Practice Class: Day 5
Akihiko Takahashi

The group spent the first half an hour looking at the eleven posters which were displayed in the hall. We were asked to choose the poster which best reflected the criteria, which are listed in Day 3. We were also asked to "explain why we chose this group's poster as the best among the class."

After we voted (via the internet) we reconvened into three groups for a discussion.

A summary of the discussion follows:

In our discussion, we focused on two topics. We first talked about the group process of choosing an open-ended problem to put on the poster. One group, at least, had hoped to be able to present the poster to the group and to explain it. They didn't realize at the beginning of the process that the poster had to be self-explanatory. One group divided up the task and came up with six possibilities for the poster. They then agreed on a particular one. One participant wondered "Is there a clear method for obtaining open ended problems?" and felt that the group used a non-methodical process. Which task was the easiest? Which was the hardest? One participant said: "It was easy to decide student response, hard to choose math level." Another stated that it was "harder to find something rich in mathematical content." Another said that it was "hard to address certain criteria, like prescribed outcomes." Another participant said that "the open ended approach is good for introducing topics." Other comments were "Can a problem be too open?"; "Wording is very important"; "Think about the goal and the task"; "Think about your objectives"; "Take into account students understanding of the question"; "All had a beginning or place to start " and "It was a challenge to determine was who is "the class" and to restrict the idea based on that."

Our second topic considered the posters from the ten other groups. Participants remarked that there was a "diversity of problems." One looked for posters where the "question is clear and interesting. Does it grab you?" Another was looking for posters that were "accessible - to both low and high levels." Another participant looked for those which showed a "good job of anticipating student responses." One looked for "questions that require student analysis." Another looked for ones which were "challenging for both individuals and for groups." One participant focused on those which were "well organized and visually well done."

Other comments were: "There were no repeated topic ideas - lots of variation." "Judging with 'high school' eyes or perspective made it difficult to decide the 'best'." The "grade level content was appropriate and was rich with higher content." "Simple questions led to rich math which could be a good springboard to other areas such as proofs or arguments." The "quality of the display did not equal the quality of the question." One participant expressed surprise that "geoboards could be used in high school." "Scaffolding could be provided by asking further questions based on the original question or by showing student work as the class progressed." "The poster was not meant to be a lesson plan, it was to provide the main ideas for the rich task."

It was hard for some participants to quickly access the mathematics underlying the themes of the posters. Others felt it was "hard to judge multiple entry points." One participant emphasized the need to 'look at the math' and not the 'art' or presentation." One noted that some problems needed to have clearer statements. Another noted that "some problems lent themselves to both geometric and algebraic solutions."

Posters from each table:

Table 1: Watex
Table 2: Problem
Table 3: Who's Afraid of the Big Bad Wolf!
Table 4: Going the Distance
Table 5: Triangle Tangle
Table 6: Mission
Table 7: Polygons
Table 8: Maximum
Table 9: 4-Mula 4 Area
Table 10: Path Problem
Table 12: P3: Perusing Possible Parallelograms

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This material is based upon work supported by the National Science Foundation under DMS-0940733 and DMS-1441467. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.