By Thomas C. O'Brien
We cannot go back to basics as the critics demand, Mr. O'Brien points out, because we've been there all along. And the fact is that the back-to-basics approach, not the activity-based approach the critics abhor, has failed us badly.
A SMALL but vociferous group of very well-organized critics is espousing a return to "parrot math." These critics believe that mathematics education in elementary schools should be confined largely to arithmetic and that mathematics should be taught by the force-feeding of inert facts and procedures shorn of any real-life context. They have no tolerance for children's invented strategies or original thinking, and they leave no room for children's use of estimation or calculators.
The critics claim that their approach is the only correct approach. Although some of their most vocal leaders have no apparent expertise in mathematics and no experience teaching mathematics at any level, they say that anyone who criticizes them is not a mathematician or doesn't understand how students learn mathematics. Their understanding of how children learn mathematics gives short shrift to the notion that knowledge is a personally constructed network of ideas, information, images, and relationships that tends toward coherence, stability, economy, and generalizability.
They criticize new approaches to the teaching of math -- approaches that can be summarized by saying that math should make sense to children and that children should be thinkers rather than storage bins for thinking done by others. They also argue that constructivism is a fad -- this despite 80 years of empirical research, replicated worldwide, on the construction and growth of children's thinking about essential mathematical and scientific ideas, such as number, space, logic, causality, classification, and contradiction. The main findings of this body of research -- that the development of knowledge comes from an interaction between knower and known, that children's thinking is very different from adults' thinking, and that social interaction is a major cause of intellectual growth -- are foreign to them.1
In the field of children's learning of arithmetic, there is significant research to show that the force-feeding of computational procedures is harmful.2 But the critics continue to insist that arithmetic -- and knowledge in general -- is inert stuff to be transmitted and stored.
They employ emotionally loaded labels. (Children's classifying, inferring, generalizing, hypothesizing, and other basic acts of thinking are dubbed "fuzzy math"; thus I see my use of "parrot math" as only fair.) And their major argumentative weapon is the anecdote. For example, 11-year-old John Jones is in an activity-based math class, and he cannot compute 10% of 100 without a calculator. Parrot math, we are told, will save him (and the nation).
The news media, bombarded by a relentless blitzkrieg of neatly packaged press releases from the critics, report their claims and give network television time to John Jones' unhappy mother. Among other questions that are overlooked, no one asks whether John Jones' class is really doing activity-based math, what math approaches John had experienced prior to his present class, how long John has been in an activity-based class, what education and support John's teacher has had in what is almost certainly a new approach, or how John is doing in his other classes.
The critics have been interviewed on national television and on National Public Radio, and they have gone so far as to testify before an education subcommittee of the U.S. House of Representatives. One wonders what it is about children's mathematical learning that gets the critics so hypercharged. Here are a number of hypotheses that I have heard.
While the critics may be good at seeking publicity, they are not so good at analyzing evidence about the effects of parrot math. And when they criticize activity-based approaches for today's low test scores, they are woefully uninformed regarding the pace with which change takes place in America's public schools. Despite what the critics would have us think, the activity-based math they fear is unknown in the majority of American classrooms.
A similar series of events took place not long ago with respect to "whole language." Critics blamed falling reading scores in California on whole language, despite informed estimates that at most 20% of California classrooms used the method. By the time the National Research Council presented findings to rehabilitate the image of whole language, the approach had been devastated by critics and a subservient press.3
What the Research Says
The back-to-basics approach to learning has been dominant in U.S. math classrooms throughout this century. As early as the 1930s the math education researcher William Brownell saw parrot math as dominant and criticized it heartily, pleading for children to be allowed to find meaning in math.
But the view of math as isolated bits of information to be transmitted to passive receptors continues to be dominant in America's schools. In January 1998 I contacted some 20 colleagues around the country -- education and publishing experts in daily contact with entire states and even regions. I asked them to tell me the proportion of activity-based, constructivist-minded elementary school classrooms in their areas. Many respondents replied, "There are none," and the highest figure cited was 20%. Reporting on the analyses of the video component of the Third International Mathematics and Science Study (TIMSS), James Stigler and James Hiebert state that U.S. eighth-graders spend almost all their time practicing routine procedures transmitted by the teacher, while in Japan students are asked to think. And to judge from what classroom teachers remember, not much has changed in decades. (See "What Teachers Recall," which accompanies this article.)
We cannot go back to basics as the critics demand. We've been there all along. And the fact is that the back-to-basics approach, not the activity-based approach, has failed us. Let's take a look at some of the evidence.
Many people have raised objections, some of them serious, to the sampling procedures and attempts at curriculum matching used in these international comparisons, especially the early ones. So let me offer some test items and data from a purely American study, the National Assessment of Educational Progress (NAEP), "the nation's report card," which has assessed mathematics achievement periodically since the late 1960s. The problems and the data I report here are taken from studies conducted in the early 1980s, a time I chose deliberately so that results could not be attributed to the "new math," which had disappeared by that time, or to the "fuzzy math," which had yet to appear. The problems I cite are examples of a vast body of NAEP results that show the disastrous effects of parrot math.
Estimate the answer to 12/13 + 7/8. You will not have time to solve the problem using paper and pencil.
*Figures add to 97% as in original NAEP document.
The fact that both of the fractions were close to 1 -- and that the answer must thus be close to 2 -- escaped 76% of the 13-year-olds and 63% of the 17-year-olds. So much for blindly applying rote procedures. The respondents seem to have thought, "Addition of fractions has something to do with adding numerators (thus the 19) or denominators (the 21)."
Here's a combinatorics problem:
1 2 3 4 5
In a game you are given these five cards. A rule says you must select two cards and form a two-digit number such as:
How many different two-digit numbers can you form, including the one above?
At grade 7, only 85% of the pupils responded. At grade 11, 93% responded. And here are the percentages of responses at both grades:
In general, the NAEP results show that American pupils have difficulty solving problems that they have not previously met. Here's another example:
An Army bus holds 36 soldiers. If 1,128 soldiers are being bused to their training site, how many buses are needed?
The Army bus problem was answered correctly by 23.9% of the 13-year-olds who tried to answer the question (25% skipped it). Of those who did answer, 46.4% simply lopped off the remainder and reported 31 buses or reported 31 buses plus some fraction of a bus. When allowed to use a calculator, only 7.1% of the 13-year-olds who responded answered correctly (31% did not answer at all). Calculators do parrot math very efficiently. But when kids use a parrot math calculator to assist their already barren parrot math knowledge, their performance gets worse. Similarly, a thousand dollar pen doesn't make a writer more thoughtful or more eloquent. Here's another example:
A store is offering a discount of 15% on fishing rods. What is the amount a customer will save on a rod regularly priced at $25?
At age 13, 14% of the children answered correctly. At age 17, just 44% of responses were correct. Here's another example, in which merely figuring out what to multiply by what presented eighth-graders with plenty of difficulty:
What is 10% of 50?
What is 60% of 50?
What is 75% of 12?
Thus we can see that even straightforward arithmetic -- the hallowed goal of the back-to-basics zealots -- doesn't fare well.
In the 1980s my colleague Shirley Casey and I published the findings of a simple research study dealing with grades 4, 5, and 6 in which math was being taught as present-day critics would wish. Pupils in these grades were asked, "What is 6 x 3?" Virtually 100% of the answers were correct.
But when we asked the children, "Give me a real-life situation or a story problem for 6 x 3 = 18," 74% of fourth-graders and 84% of fifth-graders could not do so! Furthermore, more than half of the erroneous stories involved straight addition: "On Monday I bought 6 doughnuts. On Tuesday I bought 3 doughnuts. So 6 x 3 is 18." So much for the alleged value of the parrot math approach to instruction.
But perhaps the most telling outcome of the memorization and parrot-like drill that have held sway in most of our elementary classrooms for generations is the large number of American adults who, when they find out that one is a math teacher, volunteer unashamedly, "I was never any good at math. I hate math. I am still hopeless at math." What damage parrot math has done!
At a time when test results have the attention of school administrators and the public alike (and when have they not?) it should be widely understood that the tests show that parrot math has failed utterly. Perhaps now, activity-based approaches -- supported by a constructivist philosophy and involving the real basics of classifying, inferring, generalizing, hypothesizing and so forth -- can get a fair trial.
2. For background on the harmful effects of the direct teaching of arithmetic algorithms and on the success of pupil-originated procedures, see Constance Kamii and Ann Dominick, "The Harmful Effects of Algorithms in Grades 1-4," in Lorna J. Morrow and Margaret J. Kenney, eds., The Teaching and Learning of Algorithms in School Mathematics, 1998 Yearbook (Reston, Va.: National Council of Teachers of Mathematics, 1998), pp. 130-40; and Constance Kamii, Young Children Continue to Reinvent Arithmetic, 3rd Grade (New York: Teachers College Press, 1994), pp. 33-48.
3. For a description of "the political campaign to change America's mind about how children learn to read," see Denny Taylor, Beginning to Read and the Spin Doctors of Science (Urbana, Ill.: National Council of Teachers of English, 1998); and Jacques Steinberg, "Reading Experts Suggest Teachers Mix Two Methods," New York Times, 19 March 1998, p. A-1.
What Teachers Recall
In the summer of 1997 I taught a small class of 10 veteran elementary school teachers. At the outset I asked them to describe what they had experienced in math as children. The experiences they shared were in line with those of the majority of teachers whom I have met over the years. After 35 years of working with teachers and with children in classrooms all over the country, I can say with some authority that the comments reported below describe the mathematics experience in a majority of American elementary schools.
"Math was all taught in isolation from everything."
"It was straight arithmetic."
"No higher-order thinking or skills. It was a bunch of tricks."
"This is the one right way to do it. My thinking was not seen as useful or usable."
"There was no different way to do things. There certainly was no room for 'my' way, even though it worked."
"No one was patient with me."
"Here it is. Why can't you understand it?"
"If you don't get it, tough. We've got to move on."
"I was sick for my 7 and 8 times tables, but the teacher wouldn't back up. I got yelled at for trying to figure them out. I still don't know them."
"I got told off for using my fingers, and I remember that to this day."
"It was all right or wrong, yes or no. There was no room for being on one's way toward understanding."
"There was no connection at all with real life."
"There was no enjoyment whatsoever."
"The first thing outside of memorization that I remember was in grade 7, when we tried to figure out the probability of certain events involving a deck of cards."- TCO'B
This Is Math?
A subhead reads, "Suddenly math becomes fun and games. But are these kids really learning anything?"1 Time reporter Margot Hornblower had visited a fifth-grade classroom in California and found the children working on the following problem:
Hornblower saw the class as pretty chaotic:
I talked with Kathy Pullman, the fifth-grade teacher whose class is described in the Time story. Here is her take on the events in the story:
The handshake lesson reported on by Time, Pullman said, took about 20 minutes. "No one with any knowledge of education would think that a problem like that gets wrapped up in 20 minutes. On the next day, many of the children were still at work inventing ways to solve it, and many of them succeeded." Pullman said that she could have pulled out any of a hundred lessons that would have "gone as smooth as silk." But she wanted to give the visitor an honest impression of what the curriculum she was teaching was like, and the handshake problem was the next item. So the Time reporter was judging the direction of American math education from a 20-minute warm-up problem-solving session. And four million readers of Time had no way of knowing that what was reported was way off the mark.
"I've been teaching for 30 years," Pullman said, "and I've seen a lot of changes. When a new project has been going for three years, just when it is ready to begin to show progress, someone impatiently tosses it out because it hasn't yet shown wonderful results."- TCO'B
1. Romesh Ratnesar, "This Is Math? Suddenly, Math Becomes Fun and Games," Time, 25 August 1997, pp. 66-67.
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