Toward a Classroom as Learning Organization
Generative - having the power or function of generating (bringing into being), originating, producing, or reproducing
Activity - a pursuit in which a person is active, an organizational unit for performing a specific function
(Merriam Webster - http://www.m-w.com/dictionary.htm/)
A generative activity is an activity intended for use in a classroom.
Students are asked to create objects or outcomes that are the same or
alike in some mathematically significant sense and that these responses
can be arrived at or built from this sameness in a wide range of ways.
The same-ness gives coherence to the task and allows it to be an "organizational
unit for performing a specific function." The generativity requires
that the activity produce or give origin to a diversity of responses. This
diversity is then used to explore patterns in the responses and in the mathematical
ways in which the responses might seen as related to one another. A simple
way of thinking of this is to ask questions with more than one right answer
and then think hard about the ways in which some mathematically significant
connections or insights might be seen as embodied in the 'data' learners
produce. Ideally the space of responses will be large enough so that the
kinds of answers students create can also give the teacher significant insight
into the ways the students are thinking about the task or understanding
some important mathematical idea. The activity should be 'thought revealing'
in this sense, and it should also be capable of giving rise to additional
rounds of generative exploration and/or detailed investigation. The best
examples of such activities can often be stated very simply and yet produce
complex patterns of interrelations in the student responses. A generative
activity can be made up of a series of generative tasks.