If “mathematics is the gate and key to the sciences,” as English scientist Roger Bacon put it in 1267, then many of today’s public school students are being given an ill-fitting key to unlock the gate of the sciences. In the Third International Mathematics and Science Study, U.S. twelfth-graders ranked 19th out of 21 nations on a test of general mathematics knowledge. In advanced mathematics, U.S. students were 14th out of 15 other nations; in physics, they were 15th out of 15.
Mathematics is the same the world over. But it is taught differently in different countries, resulting in children with markedly different mathematical capabilities.
Psychology professor Harold W. Stevenson and his research group at the University of Michigan have been investigating this matter since 1979, conducting a series of cross-national studies of children’s academic achievement–primarily mathematics achievement–in Japan, Taiwan, Hungary, Canada, and the United States.
As well as having served as president of a number of professional psychology organizations, including the International Society for the Study of Behavioral Development, Stevenson has received numerous awards for his work. These include the American Psychological Association’s G. Stanley Hall award for research in developmental psychology and the American Psychological Society’s James McKeen Cattell Fellow Award in Applied Psychology.
Together with James Stigler, Stevenson is author of the 1992 book, The Learning Gap. He recently spoke with School Reform News Managing Editor George Clowes.
Clowes: What does research say about the best way to teach mathematics?
Stevenson: One of the most impressive comparative studies shows that the East Asians have learned all kinds of really fascinating approaches to teaching math, much more so than the West. If you look at the TIMSS data, all but one of the countries in the top five were East Asian countries.
Their emphases are quite different from ours. Whereas we’ve emphasized advanced math and tried to put our best teachers in our high schools, they’ve tried to put some of their best teachers in the early years. They think it’s very important to get a proper basis for understanding mathematics.
We’ve just finished a study on China and the U.S., and found that kindergartners in China already were better than our elementary students in their functioning in mathematics. We also found that four-year-old Chinese preschoolers were doing about the same as American kindergartners.
It’s these kinds of comparisons that lead one to believe that the East Asians have developed certain techniques that are very, very effective in teaching mathematics. They don’t have extensive instruction in mathematics in preschool or kindergarten, and so what we’re trying to do is find out how the parents teach their children and how the teachers teach their students.
One of the things that comes out is that their approach is very different from the didactic approach used in the West, where we teach children the operations of mathematics as a lesson. The East Asians are more likely to teach the operations very casually, very incidentally–which seems to be a very effective approach with young children–and then gradually they get into the more abstract aspects of mathematics.
Clowes: Is that similar to one of the reforms that’s popular over here–incorporating math in with the other lessons, so that students learn math along with their social studies and other subjects?
Stevenson: Not really. That would not characterize the East Asian teaching process. They might have stories which embed mathematical concepts, but it’s not that they’re trying to teach mathematics in the context of reading or spelling or other subjects.
We have everybody from Clinton to Wilson in California talking about reforms in school, and they leave out the critically important aspect: Getting teachers who know the subject matter and know how to teach it. For example, adding teachers in California didn’t improve the math scores. That’s no surprise at all. You can write the most brilliant play, but if you don’t have good actors presenting it, it’s not going to go very far.
If you talk to the teachers–even those in school districts that perform well–the teachers say, “I just don’t know the mathematics. I just took two beginning courses in mathematics in college, and I can’t teach it effectively.” So over and over again, you get the sense that not only is it the style of teaching, but also the content of what our mathematics teachers teach. They really don’t understand all of the kinds of fundamental issues in mathematics that would be helpful to them and to children.
Clowes: So one priority for improving U.S. mathematics education would be to hire teachers who know more about mathematics. What about the way the subject is taught?
Stevenson: In the early years, the teachers use a lot of concrete objects and manipulation of objects. One big thing is the use of tiles and the base ten, and they create many, many interesting, elaborate lessons on the basis of the tiles manipulated by the children.
But there’s a big difference in the way that material is presented. American teachers often begin by verbalizing the lesson, which is the most complex aspect of learning. For example, graphing would be introduced by the teacher saying, “Now we’re going to learn to make a graph. This is called the ordinate and this is called the abscissa.” The East Asian teachers, on the other hand, begin with a concrete representation of the problem, and then say, “We have to work on this; we have to solve this.”
We gave an example of this in The Learning Gap, where a teacher comes into the classroom with a big bag and starts taking containers from the bag without saying anything. Right away, she has the children’s attention. They’re saying to themselves, “What is she going to do with those containers?”
Then the teachers tells the class, “I’m trying to figure out which one of these containers is going to hold the most water. What do you think?” After she’s gone around the room, getting the children’s suggestions, she says, “You’re all giving me different answers. We have to figure out a way to get the same answer. How are we going to do that?” And she keeps asking questions to get the children to come up with the answer.
“You have to measure the water.”
“How are we going to do that?”
“With a cup.”
“Any old cup?”
“The same cup, filled to the same level.”
The teacher then hands out the containers to different groups for them to measure the volume. It turns out she’s teaching them a lesson in graphing and sequences of numbers. She gets the children to tell her which container holds the most on the basis of the number of cups of water. She then graphs this and gets a picture of the different numbers of cups of water that the children have measured to fill the different containers. That is a very indelible lesson for young children.
Clowes: Once the students have an understanding of basic arithmetic, how soon can they go on to start doing algebra and calculus?
Stevenson: I think the goal in the Chinese system is to be able to do increasingly complex problems in algebra by the seventh or eighth grade, and to get into calculus in high school. But the critical thing that differentiates their curriculum from ours is that theirs is a linear curriculum–that is, one that successively introduces more and more abstract concepts.
They begin with very concrete representations; then you have written representations, where you have just an algebraic formula, a story problem, or an algebraic question; then, you end up with the verbalization of the rule. In the U.S., they often begin with the students trying to verbalize the elements of the rule, and more often than not they end up confused.
Clowes: Is the East Asian approach similar to the idea of ‘discovery learning’ that’s sometimes used here?
Stevenson: The typical East Asian lesson would be this: “Here we have these things. We want to figure out the volume, or area, or length of the diagonal, or something. I want you to come up with as many different ways as you can of solving this problem. Don’t worry about getting the correct answer.” Then the students take a while to work on it, often in small groups, sometimes by themselves. They come up with many different solutions.
The teacher then asks each of the students to explain their solution to the other children. She asks the class, “Is what this person said right?” Finally, she asks, “Which do you think is the clearest solution?” It’s a continuous, interactive process, always with the goal of understanding–not just being able repeat a rule and then ending the whole lesson.
One value of this approach is that it introduces the idea that it is better to produce ideas and not worry about making errors than it is to be perfect and worry about making errors. There’s a lot of emphasis on the importance of being able to generate ideas, whether or not they end up being acceptable. That makes it interesting.
Clowes: If everyone will have a calculator and be able to look things up on a computer, why do children need to learn basic math facts?
Stevenson: When the basics are presented effectively, that teaches students to understand the fundamentals of mathematics. Calculators and computers are ineffective if they detract from that understanding. In a mathematics class, the goal is to teach the conceptualization of mathematics–to understand mathematics, not to get the correct response. If you want to get the correct response, we will also teach you how to use the computer and the calculator. But that is something that is outside the math class.
For example, mathematics teaches concepts such as 12 minus 9 is really 2 plus 1. That’s because 12 is 10 plus 2, and 9 is 1 fewer than 10. So you have a difference of 2 plus 1, which is 3. The East Asian students understand the use of base 10 and how it is used in solving the problem. The American student punches in 12, the minus sign, and 9, and gets 3. The issue is: Can the person who’s done that then explain how you can solve it the first way?
Clowes: What would you recommend to policy makers to improve mathematics education in the U.S.?
Stevenson: The first thing is to provide opportunities for teachers to refresh or develop their knowledge of mathematics. If it is approached sensitively, teachers are enthusiastic about it. If teachers are just challenged as not knowing something, then, of course, it produces defensiveness.
The second thing is to improve the way that mathematics is taught. It’s not only having teachers who know the content, but also in the way that they present the information. Unfortunately, I don’t see any big movement in America to improve the instruction in mathematics.
Another concern is that we don’t even allow our children to have a textbook that they can mark in. Other countries think this is very important so that the child can make notes, and underline, and so on.
Clowes: What can parents do to make sure that their child is learning math appropriately in elementary school?
Stevenson: All the states are setting up mathematics standards. The simplest way would be to check the standards and see what is expected at different developmental levels. When do you introduce fractions, when do you introduce fractions with different denominators, when do you add fractions, when do you multiply fractions–it’s a whole sequence of events that should be delineated clearly in the state standards.
Parents also could check the Japanese standards, which are being used in a lot of places to help create the state standards. They’re written very clearly and don’t start talking about things before they’ve been defined.
Another way that parents can help their children is with practice books, which are very difficult to find in the U.S. That’s another advantage that children in other countries have. Last year, I went to Czechoslovakia, which was one of the top-ranked countries in mathematics in the TIMSS study. I went to look at mathematics textbooks in a moderately sized bookstore, and I was surprised to see so many books on sequential teaching of mathematics. There were whole racks of these books, and in fact the whole bookstore was simply books on mathematics–concepts, illustrations, practice problems, and textbooks. I’ve never seen anything like it in the U.S.
Clowes: What about those who say that international comparisons are unimportant since the U.S. obviously is developing people who can come up with good ideas?
Stevenson: That’s OK if you want to have an elitist culture in which a small percentage of the population get very excellent educations and the rest don’t. That is not a sound basis for creating the kind of scientific applications and developments that are going to win out in the global competition ahead. We’re talking about the future. We’re not talking about the present.