(Page 4 of 7) - A Taxonomy of Generative Activity Design Supported by Next-Generation Classroom Networks authored by Stroup, Walter., Ares, Nancy. and Hurford, Andrew.

multiple
paths

Agreed
endpoint -

e.g. Same
as 4x

2x+2x
100x/25
etc.

Multiple Path, Agreed Endpoint

Figure 3. Visual representation of multiple path, agreed upon endpoint task.

Nominally generative tasks can often be made significantly more generative by simply reconsidering the form

of the question. For example, rather than asking two students to simplify 2/4, ask them to create ten fractions that
are the same as 1/2. Rather than asking students to simplify 1x+3x, ask them to find five functions that are the same
as 4x. Better use is made of the uniqueness and creativity of each student in the class (everyone is involved in
creating new functions) and attention to mathematical structural issues is highlighted. With all of the student
responses, or more often a selection of each student’s favorite, projected in front of the class, structure-related
questions like the following can be asked: “How can all these functions that look so different be the same?” or “If I
added another example, how would you know if it was the same as or different from 4x?” Inviting students to
extend the patterns they’ve observed to work in one context to a novel context also highlights mathematical
structure. After completing the 4x activity, we’ve asked late elementary aged students to create functions that are
the same as 4sin(x). (And here is an example of one of their responses.) Attention is now drawn to the form or
structure of mathematics that “works” across contexts.

Asking students to pursue multiple paths-agreed upon endpoint tasks allows a larger mathematical space to

be explored than would be the case with nominally generative tasks and supports participation in a way that has the
potential to significantly advance the groups’ engagement with mathematical structure. Additionally, the teacher
gets a quick snapshot of where student thinking is. For example, if none of the equivalent expressions involves
decimals or negative numbers, the teacher is able to both see the space of kinds equivalence they are comfortable
with and areas they may need to explore more. When this happened in some of the classrooms we worked with in
Roxbury, Massachusetts the teacher could immediately adjust the flow of the activity in the classroom. One teacher
simply asked his students, who were supposed to be familiar with these ideas, “How come?” in a playful/joking
way. He then went on to encourage them to include, in the next “round” of expressions, examples of equivalent
expressions involving decimals and negative terms.

Unlike the nominally generative tasks described earlier, a final observation we can make based on our work

in classrooms is how mathematical creativity, flair, and insight are more likely to be acknowledged and celebrated
with this type of task. 8x-4x is certainly acceptable (and was praised by the teacher) as an example of an expression
that is equivalent to 4x, but 1,000,004x – 1,000,000x and 100x/25 are seen as more interesting by the students
themselves. We know this, in part, because once these examples were projected for the whole class other students
quickly worked on creating similar examples to share. Mathematics serves to structure the social activity of the
group – students create, discuss, and share expressions that embody the idea of equivalence – and at the same time,
the social sense of knowing and legitimate participation serve to structure the mathematical activity (Stroup, Ares &
Hurford, in press). Sixth graders, within ten minutes of working with graphing calculators for the first time, created
these and many other examples and our experience is that the next time an activity like this is done, students want to
be the ones creating more interesting examples to “show off” to their peers.

In collaborating with pre-service and in-service elementary and secondary teachers, we began working on this

kind of design well before the latest generation of highly interactive classroom networks was developed (cf., Stroup,
1997). Teachers find it useful to think about turning answers (“4x” as in “1x+3x = 4x”) into questions (Can you find
expressions that are the same as 4x?). Highly interactive networks support this kind of generativity by allowing
expressive artifacts to be readily shared, displayed, recorded and aggregated. Students can submit responses
anonymously and then decide later if they want to take ownership of a particular solution (Davis, 2002).

4.3 Modeling - Multiple Paths and Endpoints Where Fit with Data is Central

Modeling has the potential to be a multiple pathway and multiple endpoint activity. Learners can create

different models that yield distinct outcomes. A central feature of the subsequent conversation is