John Dewey . . . asked a class, “What would you find if you dug a hole in the earth?” Getting no response, he repeated the question; again he obtained nothing but silence. The teacher chided Dr. Dewey, “You’re asking the wrong question.” Turning to the class, she asked, “What is the state of the center of the earth?” The class replied in unison, “Igneous fusion.” (Bloom, Engelhart, Furst, Hill, & Krathwohl, 1956, p. 29)
Did the class know about the center of the earth? The point of this little parable, of course, is not really. Being able to correctly recite the words ‘igneous fusion’ on cue does not make evident that these words had meaning to the students.
Meaning. An ultimate goal of learning, to enrich our lives with meaning.
From this perspective, Dewey’s question was the right one. It connected school learning to meaning in the world.
How do we do that in math class, the place with the most deep-seated rituals of recitation and mindless calculation? How do we move from what English mathematician and philosopher Alfred North Whitehead called ‘inert ideas’ — those which have been received and not utilized or tested or ‘thrown into fresh combinations’?
I am about to go teach a class of pre-service teachers to engage this very set of issues. We are looking in math class at the nature of the tasks being used through the lens of Margaret Smith and Mary Kay Stein’s work on this topic.
I already anticipate some of the arguments that will come up about the richness of particular tasks. Over the years of using this framework with both new and experienced teachers, I can boil the issues down to these questions:
1. What kind of thinking will students do as they engage in this activity?
2. Is there sufficient ambiguity that they can approach it in different ways or re-think familiar ideas in new ways?
I would love to hear from others about your ways of judging good mathematical tasks, as well as your favorite resources for finding them.