Burt Furuta and I met long ago on one math education list or another. I always enjoy his perspective, and wanted to share it. Here’s Burt:
Sue often writes about math being challenging, engaging, fun. This post is about math in our schools, which is rarely engaging or fun for most kids. I want to explore why this is so. I think Sue was right when she said in this video that math in school is all about getting the right answer. Not that we don’t want our students to get the right answer. Of course we do. And we want them to understand the concepts we are trying to teach. Some people will say that we have no time in the school day for children to play with math ideas. If students understand the concepts and can solve the problems, then we've done our job. I don‘t think that way of thinking is the real problem - I’ll explain why later. I think the root problem is that we’re all graduates of the same system.
The vast majority of us, including those with the power to shape reform, believe that if we can compute the answer then we understand the concept; and if we can solve routine problems, then we have developed problem-solving skills. Being products of the system, we generally don’t have an appreciation of what conceptual understanding and problem solving really involve. We think we know what understanding is, but unless we've thought deeply about it, it is likely that we don’t. That could be because the word understanding has many levels of meaning - see this article by Skemp for more.
In this post, I’ll just focus on the fallacy that computing answers to problems means we understand the mathematical concepts related to the problem. At a conscious level, most of us get that computation and conceptual understanding are two related but different things. But at an unconscious level, we often treat them as the same thing. When students can compute the answer, we think they understand.
When we talk about conceptual understanding, what we often consciously or unconsciously mean is that students know how to use procedures to compute answers. This is evident in the majority of Liping Ma’s interviews with American teachers described in Knowing and Teaching Elementary Mathematics. The TIMSS video study reported in The Teaching Gap by Stigler and Hiebert describes how our math teaching is dominated by "memorizing definitions and practicing procedures." As Eric Mazur explained, this happens even with our best students at our most prestigious institutions—students compute answers to problems, we think they understand the concepts related to those problems, but in fact they don’t understand.
I am saying that just talking about conceptual understanding doesn't overcome the long history that we have in thinking that getting the right answer indicates understanding. We need to consciously focus on, and think deeply about, the difference between understanding procedural concepts that allow us to calculate answers to problems versus understanding the underlying math concepts. That understanding tells us: how elements of the problem are related to one another, why certain computations are appropriate, when they may not be appropriate, what the significance of that computed answer is, and to what purposes we may apply that information.
Conceptual understanding involves more than associating cues in a problem situation with a known computational procedure and correctly plugging in numbers to crank out the answer, which is how Mazur described his students' behavior in the above linked video. Conceptual understanding always involves a network of connected concepts that are not simply related to computation. When we understand concepts, we can recognize relationships in problem situations, think logically about how the elements of the problem are related, and see how one element may change as another changes.
Last summer I mentored three 7th graders in math for several weeks. I say mentored instead of tutored because they were all straight A students from three of the best private schools in the state. I gave them a small set of scores and asked them to find the median and arithmetic mean, and to define the concepts median and mean. They had no trouble. They agreed on definitions that are typical: for a set of scores the median is the middle value (with as many scores above as below it) and the mean is the sum of scores divided by the number of scores.
Then I asked if the mean was also a kind of middle value. They didn't think so. I asked them to do some things that made them change their minds - the mean is also a middle value. We discussed how the median is a middle value with ordinal data, and the mean is a middle value with interval data. (If all you care about is which is bigger, you’re talking ordinal data. If you care how much bigger, that’s interval data.) Concepts like the mean are often only taught as a formula or procedure. Is the purpose of the lesson limited to computing a value? If we teach the median as a middle score doesn't it make sense to also teach the mean as a kind of middle score? I wonder if there were a simple formula to determine the median, would we just give the formula and not mention that the median is a middle score?
In the first paragraph, I said the problem is not in thinking that the traditional teaching approach is more efficient in teaching math and that we don't have time to play with math ideas. Let me explain what I mean. Everyone wants our children to understand concepts. No one is saying that computational fluency is sufficient without conceptual understanding. People "know" that traditional methods are more efficient than playing with ideas - students get the right answer, therefore they “understand”. Arguing about teaching methods only brings heated words and hardened positions. The critical factor in reform is not teaching methods, but rejection of the belief that computing the right answer means understanding the concepts.
It is when people realize that students often don’t understand the concepts, despite being able to compute answers, that they will seek change. Eric Mazur's experience is a good example. His Harvard students did great with the computations, but they still didn't get Newton's laws, which form the conceptual foundation for all those computations. For example, his students did not really believe that the forces a light car and a heavy truck exert on one another in a crash are equal. He knew that a true understanding of Newtonian mechanics would make this a simple conclusion, and sought a better way to teach the concepts. Until then, he had thought he was doing a good job teaching. The realization that his students actually did not understand the concepts is what brought significant change to his teaching methods.
If the difference between computation and concept is not made clear, then to improve the system people will focus even more on getting students to compute the right answers. We are seeing that now, with the emphasis on improving standardized test scores. This is squeezing the life out of learning. We need to help people understand what conceptual understanding really is, and then they will see that real understanding can only come from engagement in activity, making guesses and mistakes, thinking hard, questioning and arguing with oneself or others, testing ideas—in short, some kind of purposeful "play" with math. And this is what is missing in school today. When we only memorize procedures to compute answers, math is boring. On the other hand, true understanding is inherently interesting. There’s decades of research on competence motivation, e.g. see this 10-minute video. Real, meaningful learning is fun.
Talking abstractly about concepts is not enough. So let’s think about this example, and what understanding concepts means:
Sarah was paid $10/hr for her summer job; while her sister was paid $12/hr at her job. To earn the same amount of money as her sister, Sarah work 60 hours more than her sister that summer. How much money did each girl earn? Before reading on, think about how you would solve it.
The typical response is to look for topical associations or simple relationships that cue the use of known procedures. Not finding any, an equation is set up and solved. This would be an easy algebra problem, solved almost mechanically. Those who don’t know algebra might try guess-and-check. Get the answer, then move on.
Let’s do more than get an answer. What are the relationships in the problem? Even if you use algebra to get an answer, look back on the problem. Using the relationships in the problem, could you solve it in other ways? What are some relevant math concepts here? How difficult is it to see the concepts and use the relationships to logically solve this problem without algebra? How much time do we normally spend on analyzing relationships in problem situations, and relating what we find to prior discussions of those concepts?
One last question: For what grade level do you think this problem is appropriate? You might think it's too hard for most students, whatever their grade level. But third graders can solve problems as challenging as this when given a curriculum that helps them develop the necessary thinking skills. Jean Schmittau ran a research project in an elementary school in New York which used a curriculum originally developed by the Russian educator Vasily Davidov. She found that these third graders "were able to analyze and solve problems that are typically difficult for US high school students." Davydov's curriculum is impressive, and it shows what young children are capable of learning, but we don't need to clone it. What we do need to do is to get past the major obstacle in our own system, which is believing that computing answers to problems is all that is needed to understand concepts. When we teach for real understanding and real problem solving, we will put the joy back in learning and the meaning back in understanding.
*Schmittau, J. (2004). Vygotskian theory and mathematics education: Resolving the conceptual-procedural dichotomy, European Journal of Psychology of Education. Vol. XIX. I: 19-43