



Participatory simulations stand to introduce fundamentally new and significant forms of reasoning and insight to schoolbased curricula. They also have the potential to be transformative of students' experience and understanding of the topics at the core of the traditional curriculum. We present here an elementary example of an activity well connected to the current curriculum that illustrates this potential &emdash; the Function Activity. The Function Activity has already been used in a number of different settings including both an urban middle school and an urban high school. The Function Activity sequence begins with a seemingly random collection of points visible on the upfront projection system (see Figure  panel; a). The teacher begins the conversation by asking students if there is any pattern in this collection of points. Although students will occasionally qualify their comments with the observation that there "might" be a pattern, the general consensus is that there is not an obvious pattern. The teacher then "hands out" one of these points to each of the students using the network. A single point is then visible on each student's screen (panel b). The student finds the location of the point by moving the calculator cursor, the "+" shown in panel b, to the point and reading off its x and ycoordinate. The teacher now gives the class a rule by which to move their individual points.
The Function Activity sequence for the rule, "Move until your yvalue is two times your xvalue."
The rule illustrated in the panels is, "move until your yvalue is twotimes your xvalue." When the students have found this location for their particular point, they press ENTER on the calculator and the network software collects the points. Either before or after the students move their individual points, the question can be asked, "What do you think will happen if we now display all the points together?" Students will give a range of answers, such as "a 'V' will form", or "some kind of line." Teachers then ask the students to explain why they made these predictions. Students will say something like "I just guessed" or "it is a hunch". The teacher can then display the results. Figure c illustrates what appears if every student follows this rule.
Every time this activity has been run in a classroom, we have observed that students are quite invested in locating their individual points (panel d). Initially, there have been a significant number of points "off the line". If a particular student realizes that his or her point is "off" the emergent pattern, s/he will sometimes offer an explanation for what happened. What is encouraging to hear and observe in the videotapes of the classroom interactions is that consistently students are willing to try and make sense of other students' reasoning. Even for points off the line, students will say something like "oh, that person must have just switched his y and his x," or "she didn't multiply by the negative." Rather than simply ridicule "wrong" answers, students will assume their colleagues had a reason for what they did and that they can identify with these other forms of reasoning. One hears comments like, "I almost did that but.…" The classroom discussion for both students and teachers centers more on the reasoning the learners brought to the task, and less on "right" or "wrong". Both the teacher and the students have a "snapshot" of how the entire class's reasoning is working. The teacher can help shape the spontaneously generated conversations that occur. Other function rules have been explored using this sequence (e.g. move until your y value is five more than your x value or move until your y value is the absolute value of your x value). The link from the rule they used to move their own embodied point to the analytic function can be explored. By using this participatory approach, cognitive scaffolding is provided for moving from "my point" to big ideas related to the concept of a function and to various analytic forms of expressing the functional dependency.


Uri
Wilensky 
Walter
M. Stroup 