TITLE: Ambiguous Questions and Wrong Answers AUTHOR: Eugene Wallingford DATE: July 15, 2014 3:19 PM DESC: ----- BODY: Pianist James Boyk recently shared a story about mathematician Andrew Gleason with me. Boyk had studied a bit with Gleason, who was known for his work in cryptography and in helping to solve Hilbert's fifth problem. Gleason also had a strong interest in early math education. Once, after observing first-grade math teaching for some weeks, Gleason said:
I never saw a kid give a wrong answer. I heard a lot of ambiguous or erroneous questions, but never a wrong answer.
As Boyk told me, there's a lesson in this attitude for everyone. So often in my own courses, I start to mark a student's answer as incorrect, step back, and realize that the question itself was ambiguous. The student had given a correct answer -- only to a question different than the one I had in my mind. Not all my questions are ambiguous or erroneous, of course. Sometimes the student does give an incorrect answer. For a while now, I've been trying to retrain my mind to think in terms of incorrect answers rather than wrong answers. Yes, these words are synonyms, but their connotations differ. The meaning of "incorrect" is usually limited to the objective sense of being not in accordance with fact or standards. "wrong" has a wider meaning that also can include being immoral or dishonest. Those words seem a bit too judgmental to be applied to my students' sorting algorithms and Scheme programs. In the case of erroneous answers, I find I'm usually more effective if I focus on the factual incorrectness of an answer. What misconception or missing piece of knowledge led the student to this conclusion? How can I help the student recognize this flaw in the reasoning and reason more accurately in the future? That seems to be the big lesson of Gleason's comment: to keep the teacher's focus on how a student's answer is the correct answer, given what he or she knows at a given point in time. The teacher's job is to ask better questions and lead students to a better state of knowing and doing. -----