by Sarah Lonberg-Lew
NOTE: This blog is a companion piece to an article Sarah wrote earlier this year called “What’s Right with Wrong Answers?”
As much as students seem to hate word problems, there is no way around the fact that they are a necessary part of the curriculum. Setting up and solving word problems is a vehicle for learning to analyze information and reason logically. It is an essential skill for taking high school equivalency test and for success in college and real world situations requiring math —this in spite of the fact that the contexts for these problems are often silly and occasionally absurd. The funny thing is that students who groan when I bring up word problems also suddenly appear to care very deeply about how many more plums Joy has than Jonathan (even though they have never met Joy or Jonathan and plums are out of season). When I pose such a problem in class, I hear a chorus of answers, some of them right and some of them wrong. What seems to be most important to the students is knowing as soon as possible who came up with the right answer and what it is. This concerns me because what really matters is the process of problem solving and not the answer itself. However, I’ve seen students stop thinking once they know the answer. They write it down on their papers and move on to the next problem even if they have no idea how it was obtained.
This makes sense considering their histories with math education. In a traditional math classroom, students can get the idea that their purpose is to find the right answer, present it to the teacher, and thereby accumulate stickers or check pluses or “A”s. They are just as happy to get the answer from the teacher or a classmate as to rely on their own reasoning and intuition. The former is usually easier.
Because of this, focusing on the correct answer in math class seems counterproductive to me. Learning doesn’t come from knowing the answer; it comes from finding it. If I shut down the discussion of a problem as soon as I hear the right answer, I also shut down learning for all those students who didn’t find it themselves — and possibly even for those students who did. For example, a student who comes up with a right answer because she remembers from high school that similar-looking problems are to be tackled using a certain procedure hasn’t really learned anything from the exercise, either.
Sometimes, in order to take the focus off the right answer I leave an exercise unfinished. Maybe we’ve worked hard on a problem and arrived at the conclusion that we have to multiply 207 by 48 to get the answer. At this point I feel that we have done the important work of problem solving. We’ve figured out what computation will get us the answer, and the answer itself is not that important. Even though there would be some satisfaction in finding the actual number, the tension of leaving it unfound can help to make the point that the value is in the process instead of the solution. Even when we do math in authentic contexts, the process is more important than the answer. Knowing how much a fictional driver will pay to lease a car is not useful to a student. Knowing how to figure out the cost of leasing a car, on the other hand, is very valuable.
One sad consequence of a classroom culture that values right answers above all else is that students can start to become like “Clever Hans”. Clever Hans, the student of a mathematics teacher in early twentieth century Berlin, could answer all kinds of questions correctly. Clever Hans was a horse. He tapped out the answers to questions with his hooves and amazed people with his mathematical skills and knowledge. It wasn’t a hoax. Clever Hans could answer questions correctly even when people besides his trainer asked them. However, his conceptual understanding was eventually found to be severely lacking. He wasn’t able to get the answer if the questioners themselves didn’t know it. That’s because Clever Hans wasn’t reasoning at all. He was reading the faces of the people asking the questions and they were communicating to him (whether they wanted to or not) when he had arrived at the correct answer. Clever Hans had learned something very interesting in developing the ability to read those cues, but he hadn’t actually learned what he appeared to have learned. As teachers, we also can be fooled into thinking our students have learned what we were trying to teach when they come up with right answers and we don’t look beyond them.
The researchers who worked with Clever Hans found it impossible to control the micro-expressions that the horse was reading. The only way they could keep from communicating the right answer to him was to put up a screen so that Hans could not see their faces. That is an impractical solution for the classroom and our students are cleverer than Hans. We may not be able to control communicating when we have heard the right answer, but we can work to create a classroom culture where we focus more on the process than the outcome. One thing we can do is make sure we always ask for the reasoning, whether the answer is right or wrong. We can train ourselves to say “why?” instead of “yes” or “no”. My students want to hear from me, the authority, whether their answer is right or wrong, and it frustrates them when I refuse to answer. But when I push them to explain their reasoning, they usually figure out for themselves whether they had it right or wrong. I hope that by being consistent in asking for reasoning, I will help my students learn that their math education is not about coming up with the number that will satisfy the teacher, but about thinking deeply and feeling proud of their hard work.