Disequilibrium is Piaget’s term to describe when what a learner already knows comes into conflict with new information. Learners must work through the confusion to reconstruct new knowledge. How does the process feel to a learner? How as a teacher can we respond during a number string when students demonstrate disequilibrium?
To explore these ideas, I present a few moments in a number string led by one of my pre-service students, Nadine, at Chapman University’s College of Educational Studies. Nadine wrote the problem 6 x 20. Betsy, also a pre-service teacher, began by admitting that the arrays were still confusing for her, but that she wanted to try to solve the problem with an array. She gave her answer as 72. Noemi began to sketch the array that Betsy was describing: a 6 by 2 array connected to a 6 by 10 array.
Another pre-service student, Leia, disagreed. Leia said that Betsy’s array would not work, that the array had to be two 6 by 10 arrays. Nadine drew this:
Betsy continued to try to puzzle out her array. After about a minute of thinking, she burst out, “ it does work, you just need ten of the first array.” As Nadine began to sketch that, which looked like this,
Leia was doubtful. “It won’t work,” she insisted, “What, do you really think that 2 times 6 ten times will be the answer? Like 12 + 12 + 12 + 12 . . . ten whole times? That won’t work.” Betsy insisted and a minute later, Leia broke out into laughter once she realized that the Betsy’s second array also worked.
This moment helps us see how critical it is that learners grapple with moments of disequilibrium. Betsy forced herself to think with the array, and in so doing she thought additively instead of multiplicatively. You can see additive rather than multiplicative thinking in the array that is really 6 x 12 rather than 6 x 20. This is an example of what happens, even with adult learners, when you do not spend sufficient time developing the model through context. And in Leia’s delighted laugh when she realized she was wrong, we can recognize that disequilibrium can be joyful.
Betsy and Leia were allowed these moments because their teacher, Nadine, allowed them to take the active role in resolving their own disequilibrium. Nadine represented thinking using powerful models, and then stepped back and let the learners do the difficult, joyful work themselves.
Number Strings 