It’s been my privilege, recently, to sit in on several algebra and pre-algebra classes being taught by master instructors.
Wrong Answers — Teacher perspective
Think about an experienced algebra teacher going over a set of practice problems with her class. It’s yesterday’s homework assignment that she wants to correct at the start of class today, so she and the students can get quick feedback on how they are doing before she moves on to new material.
She asks for the students’ attention. She reads each answer out loud, repeating it if requested, while each student marks up his or her own paper. At the end, she asks, “Do you want me to work through any of the problems on the board?”
Because she’s experienced, she really doesn’t need the students to tell her which problems to go over, but she would rather they do so. It’s one way she knows she’s developing rapport with the class, when students feel comfortable enough to raise a hand and ask out loud, “Can you please do #11?”. But, if no one does ask, she’ll almost certainly volunteer, “Well, I think I’ll just work through #11 and #13, before we go on. How about that?” And she is likely to see grateful head nods around the room.
Out of any ten problems, she’s quite certain which two most students are likely to miss or, at least, get wrong unless/until they checked their answers. These problems are the dreaded “hard ones” that students will ask each other about if they have time and permission. “What did you get on #11?” “How did you do #13? I couldn’t even figure out how to get started.”
As with all skillful teaching, what is going on here is a process that is both straightforward and quite complex, simple at first glance yet with a lot going on under the surface. If it was well constructed, the whole problem set was exercising a combination of old skills and the new ones just taught. The dreaded “hard ones” are there to push the edges of the student’s skills envelope in some way — they require him to remember an easily forgotten exception-to-the-rule or to combine old and new skills in a way that can be tricky to get right.
Having students get those problems wrong and then ask the teacher to go over them creates a quintessential “teaching moment.” The appropriate rule or the correct way to combine skills can now be repeated, but to an audience that is, she hopes, more prepared, more open to absorbing the information than it was when the material was first introduced.
Wrong Answers — Student Perspectives I
I’m guessing most readers of this post were good students and fairly amenable to those teaching moments the instructor was trying to create: Oh, darn. Got that one wrong. How did she do it? Hmmm. Ok, I guess I see now.
Not me. I was a good enough student but clueless about the purpose wrong answers played in the classroom process. Wrong answers just struck me as a personal affront, and not one I handled very well. If I had gotten a “hard one” right, I was impatient with all that going-back-over the teacher felt he needed to do. And if I got it wrong, I’ll admit to being the student who would sit there, waving my arm in the air, waiting to ask/challenge the teacher about why his answer for #11 was right and why mine was wrong. I suspect I derailed more than one good teaching moment through the years, not taking “because this is how it is done” for an answer.
Those years of arguing math rules with every teacher I ever had were probably part of what fueled my enthusiasm for the abstract algebra and number theory classes of my undergraduate years. I was always delighted to hear Ernst Snapper admit, as he often did, that, while everyone else is sure of it, mathematicians really can’t explain why 1 + 1 must equal 2. Nor can they actually assure you that it will always be so. I doubt he had haystack math* in mind when he said that, but I can imagine his big toothy grin if he had ever heard it described.
* Haystack math: If you add one haystack to another haystack, what do you get? One big haystack, of course! So, in haystack math, 1 + 1 = 1.
But the perspectives on wrong answers, for both instructors and students, are much different in the developmental math classes conducted at community colleges.
See my next post for more about that.