Wednesday, June 12, 2013

Guest Post: Understanding is Misunderstood

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


  1. I noticed that I find putting it into algebraic terms much easier. When I first started to reason it out without algebra, I got a bit muddled for a bit.

  2. I definitely do those with a "relative speed" kind of approach, just like a lot of those classic motion problems: 60 extra hours for Sarah means $600 extra, and her sister gains $2/hr relative to her, so her sister worked 300 hours for $3600 (and Sarah worked 300 + 60 extra for $3600 also, as a check).

    There's also the nice Singapore-style number-line/rod/length diagrams to do the same thing as I did a bit more numerically here.

  3. I teach algebra, but often reason in arithmetic:

    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.

    60 hours at 10/hr? She's making up a 600 difference. Each hour created a 2
    Difference, so they must have worked 300 hours before she worked the extra.


  4. @Sue. Thanks for saying that reasoning out a solution can get muddled. My reason for not giving any solutions in the post is for readers to see how easily they could recognize relationships and see the solution through conceptual analysis. I didn't readily see any solution, but after I found two solutions and saw that they were not difficult, I realized how biased my thinking was against a conceptual analysis of problems.

    @ Jonathan. Thanks for posting a solution. I'd like to elaborate about the concepts.
    You're making an additive comparison between Sarah's numbers and her sister's, which gives what is the same between the numbers and what is the difference.

    Sarah's hours has two additive parts, what is the same as her sister's hours and the 60 extra hours. Her sister's hourly pay also has two additive parts, what is the same as Sarah's and the extra $2. The amount Sarah earned working 60 hours more makes up for and must equal the amount her sister earned working for $2 per hour more than Sarah. They each earned $3600, 300 x 12 = 3600, and 360 x 10 = 3600.

    Will someone please post a solution using a multiplicative comparison between Sarah's and her sister's numbers?

  5. @Joshua. Sorry I didn't reply sooner. Yes, this problem is just like the motion problems where one car is slower than another car but has a head start, and you need to determine how far they are from the starting point when the faster car catches up to the slower car.
    Have you ever tried solving these using multiplicative comparisons?

  6. sarah makes five sixths what sue makes, so needs to work six fifths as long to make the same money. So the 60 hours is the extra fifth...

    60 is one fifth of 300...

    How's that?


  7. Sister Sue, huh? (I was wondering how I got in there!)

    That is lovely. I wasn't sure at first that it made sense to me, but it does. I'm not sure I'd trust it without algebra...

    Sarah: R*T=P (rate times time equal total pay)

    Sue: 5/6R * 6/5T still = P

  8. Yes. Since the total pay is the same, we have an inverse relationship between hours worked and hourly pay--the greater the hourly pay, the less hours needed. If the numbers were different and Sarah earned half as much per hour as her sister (or one-third as much), then it would be easy to see that she would need to work twice as much (or three times as much)--the inverse ratio. With the ratio being 5/6, we don't readily see that she needs to work 6/5 as much, but the math is the same.

    When reading the Sarah problem, we can easily set up an algebraic equation to solve it. That is not something we could easily do when first learning algebra, but we learned how to do it. Could we as readily see the additive, multiplicative and inverse relationships, or did we simply not learn how to do it, that is, we didn’t learn to think about relationships?

    When Polya recommended Looking Back after solving a problem, it was to increase understanding. Do we really give enough importance to that purpose? How often do we look back and be able to see the solution “at a glance” as Polya says, because we see the mathematical structure, all the relationships and how they fit together? If we had been taught to do that, as we were taught to use algebra, then I think both conceptual solutions would have been as easy to see as the algebraic solution.


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