The example space
These examples were inspired by the idea of Doing-Undoing described by Mark Driscoll in his book Fostering Algebraic Thinking.
Students often find it difficult to 'undo' processes (and perform inverse operations), perhaps because we do not devote enough time to this as teachers. I find asking students to reverse a process has a huge impact on improving understanding, and often include questions that require undoing as plenaries.
The interesting thing about 'undoing questions' is that they often have more than one answer, leading to what John Mason calls a conjecturing atmosphere. I regularly use Magdalene Lampert's technique of writing the full range of student responses on the board without evaluation, and then discussing which ones make sense. In this way, all students have a chance to participate, and the absence of evaluation results in students being more happy to offer solutions.
Of particular interest were the responses given below:
My role here is to orchestrate a discussion which will reveal to students why these solutions may or may not make sense in relation to the graph. I start by asking students to discuss each response in pairs, and then we discuss them as a class, trying to reach an agreement on which equations make most sense and why.
The students agreed that A makes sense in terms of the roots, although a couple of students pointed out that the graph would need to be symmetrical given this equation.
The students who conjectured equation B admitted that they hadn't thought about roots when forming their equation, and had only considered the y-intercept. We then checked whether the equation had a positive and negative root. Student were generally happy with a positive and negative root, although some were not happy that the roots (-2 and +1) do not match the graph exactly. We then discussed the level of accuracy required in a sketch.
It was agreed that equation C was partially correct, in that there was a positive and negative root, but that the intercept would be -8 instead of -2.
All students agreed that equation D was perhaps the best conjecture in the sense that if fitted the graph most accurately.
I could see that the students who offered equation E wanted to revise their solution at this point, and I was very happy to give them the opportunity to revise their solution in public.
The creation of undoing questions is an important and simple instructional design that helps develop student understanding.
Additionally, these types of questions often lead to a number of possible solutions , promoting a conjecturing atmosphere, and providing content for productive mathematical discussions.