This is the second of two posts on the role that wrong answers can play in the algebra classroom. The student in a developmental math class generally doesn’t react in either of the two ways described in my previous post.
Wrong Answers — Student Perspectives II
It is easy to picture the developmental classroom as being filled with undirected young slackers who just didn’t try hard enough when they were taught algebra the first time around in public school. Clearly this is the theory of the many state legislators around the country who are restricting funds for developmental education in their post-secondary education systems.
Ask a developmental education instructor, though, and what you will consistently hear is, “Our students have complicated lives.” Whatever the back story of an adult learner sitting in a remedial pre-algebra class, you can be pretty certain that some aspects of their current situation are not that pretty. Most of them are making an extraordinary effort to pull themselves up by their bootstraps.
So, while the instructor is up there at the board, working through that hard one, a thousand things unrelated to the correct math process are probably running through the students’ minds.
I knew I had that wrong. I always get those wrong. I HATE math. I’ve always hated math. What am I even doing here, anyway? I’m sitting here getting math problems wrong instead of being home putting the kids to bed and baking Sammy’s birthday cake. I’ll be too tired when I drag home at 9. And I already have to get up early tomorrow morning to make up that extra hour of work I missed for the last two-hour exam. Wait, what is she saying? I missed that — damn, now I still don’t know how to do that dumb problem and I’m not about to raise my hand to ask her to repeat herself. Who am I kidding? I’ll never be a nurse; I may as well just give up now, drop this stupid class, and be home spending time with my kids. They’re only going to be little once.
Christ! 20 minutes late again. God I hate being late. The prof is okay about it. But now my seat in the back row is taken. I HATE having to sit in the middle — I really need to keep the whole room in front of me or I just can’t concentrate at all. Oh, Jody just moved over so I can have the seat on the end of her row. Jody is okay; I think she understands. I can deal with being on the end of the row; I just have to turn in my chair a bit and I can see everyone. F*** that knee hurts when I sit down. Can’t wait for surgery during spring break. Doc says it will take a bunch of PT to make the leg work better afterwards. Christ, where am I supposed to find the time to get to physical therapy four times a week? And where’s that damn homework paper; I know it’s in my pack somewhere. Ok, got it. Now which friggin problem is she working up on the board? Duh, Jody’s tapping on her paper with her finger. #11. That was one of the hard ones. So what’s it look like when the prof does it? Hmm. Not like mine, that’s for sure. Damn; I thought I got that one right. I sure worked on it long enough. It’s hard enough to get all the body parts working together in the morning anymore but it’s like I just can’t get the mind parts working together at all. Can’t drop this stupid class though; I need the credits to qualify for my GI benefits. And I gotta get paid or else I am out on the street on my ass. What’s she saying? Ready? Yeah, prof, sure. I’m ready. Sling some more mud on me and I’ll wade right through it — got no choice.
Wrong Answers — Developmental Math Perspective
Using wrong answers in the normal way in the classroom presents something of a dilemma for college developmental math instructors. Their students need those teaching moments every bit as much as younger students learning algebra the first time around. In fact, they surely need them more.
Developmental students don’t just ‘not know’ math. Sure, there is plenty they haven’t learned yet. But the real obstacles for them and for their instructors are not what they don’t know yet but what they know wrong. No students “enter the classroom as blank slates.” [Booth, et al, 2011] For older remedial students, those slates are not just much-written-upon and oft-erased but also gouged and chipped with misconceptions and misgivings.
For many remedial students, every wrong answer on a problem set or a test is simply an affirmation of the negative image of math and of themselves that they bring to the course. Developmental instructors consistently talk about “creating a safe environment,” “making students comfortable with mistakes,” and “encouraging them to at least try a problem even if they don’t know how to do it.” For these teachers, just getting to the point with a class where the students will actually ask questions is an achievement. Hearing that quiet girl in the second row pipe up, after she gets a quiz back, and say, “Hey, I think my future may be looking up!” can be a real triumph.
Remedial instructors already use many techniques to build students’ confidence and comfort level with their own mistakes, trying to get students to see those mistakes as truly part of the process and not as roadblocks. The instructors work a lot of problems out on the board. They have students work problems in groups. They have students work problems up on the board. They encourage students to check each other’s work and help each other with wrong answers. They really, really encourage students to spot mistakes in the teacher’s own work up on the board. In every way possible, these instructors try to convince the students to both “give it a try, even if you are not sure” and “check your work!”
So, is there another technique these educators could integrate into their classroom practice? One that could offer many of the same benefits of the traditional teaching moment based on the students’ wrong answer but without the emotional baggage that is involved with the students’ own mistakes?
According to education researchers, there is such a technique. It is the use of “correct and incorrect worked examples” and it will be the subject of a future post. Meantime, my next post examines the difference between a mistake and a misconception.