## Visualizing Functions Summary

### Monday - Friday, June 26 - June 30, 2006

Participants:
Matt Bracher
Matthew Carpenter
Douglas Lutz
Gregory Monson
Craig Morgan
Rebecca Neighborgall
Monte Saxby
Mario Shaunette

Monday, June 26:

We had Rita Kabasakalian, Harvey Keynes, Gonzalo Riera from DIPD program join us today. After introductions, we dove into an activity involving graph complexity. We answered open-ended questions related to three applications of graphs: networks, routes and scheduling. This gave us enough mathematical background to define a notion of graph complexity. Specifically, participants were asked to define a function that assigns nonnegative numbers to graphs; higher numbers are interpreted as more complicated graphs. At the end of the two hours, four sample graphs were presented. Each small group was asked to come up with functions that could be used to explain why one of the example graphs is the most complicated of the four.

Group A: Monte, Doug
Group B: Harvey, Rita, Becca, Beverly
Group C: Craig, Matt C., Greg
Group D: Mario, Matt B.

Tuesday, June 27:

During the 90 minutes, we continued our investigation of graph complexity from yesterday and each small group presented its ideas to the rest. The complexity measures that came up were the diameter, number of distinct vertex degrees, and measures related to the number of edges in the graph. In the last 30 minutes, we discussed the reason why we pursued the graph complexity activity, briefly looked at the main ideas from the "Habits of Mind" article by Cuoco, Goldenberg and Mark, and described the purpose of the working group - to come up with some sort of product related to functions that could be used in a classroom.

Each participant was asked to cull the PCMI web site for ideas from previous years and to come to the next meeting ready to put forth a few ideas, no matter how well formed.

Thursday, June 29:

We spent the first hour brainstorming about possible ideas for projects. After the first hour, small groups of individuals formed to further explore different ideas before committing to any project plans. Beverly, Doug, Mario, Monte worked with probes. Matt C. surfed the web to look for ideas. Craig, Greg, Matt B., Becca, and Darryl discussed functions on things other than numbers.

Friday, June 30:

Darryl had organized the brainstorming from Thursday into three big ideas:

• know what functions are
• modeling with functions
• relate symbolic expression for function to graph

The group then expressed which idea they felt most attached to. Some found it difficult to commit to exclusively to one. Based on this, three groups, Mario and Greg, Matt C and Doug, Matt B, Becca, Craig and Monte, were formed to come up with possible project they could pursue.

Towards the end, we met as a whole group to debrief.

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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.