Is English hazardous to mathematics?
Miura and friends observe that the 'superior performance of Asian students in abstract counting, in understanding Base 10 concepts, and in mathematics achievement is already apparent by first grade, before teaching effectiveness and other school-related factors can account for such large variations." They go on to point out that various Asian languages derived from ancient Chinese, including Korean and Japanese, differ from European languages in the way they represent numbers.
The Asian languages use words that point directly to place value. The European languages do not. In English, French, and Swedish, for example, there is no relationship between the word for 10 and the words for 11, 12, 20, and any number of other numbers: ten, eleven, twelve, twenty; dix, onze, douze, vingt; tio, elva, tolv, tjugo. While Swedish and English words for some later numbers bear a relationship to earlier numbers and to base 10 (e.g., four and forty or, in Swedish, fyra, fyrttio), French children must cope with words like soixante-dix (sixty-ten) for seventy and quatre-vingt (four-twenty) for eighty.
In Japanese, however, the word for I is ichi and for 10, juu. Eleven is juu-ichi -- ten-one. Twenty-one is ni-juu-ichi -- two-ten-one. And so forth. The relationship of all numerals to those from I to 10 and their position in the base-10 system is always seen in the words used for them. Similar relationships appear in the Chinese and Korean counting systems. Do children who speak such place-value-oriented languages understand essential mathematics concepts better than children whose language does not reveal place value? Miura and her colleagues attempted to answer this question.
In the first half of the academic year, children in America, France, Sweden, Korea, and Japan were given tests involving their cognitive representation and their place-value understanding. American students were tested at the end of the year on one task because earlier work had indicated that they could not perform the task at that time. The samples were from middle- and upper-middle-class families, "as based on standards of their respective countries." It is thus possible that the groups are not comparable, but the researchers note that the American school they studied, a private school in a San Francisco suburb, "was selected because entrance is competitive, the curriculum is considered academically rigorous, and there were no bilingual children in first grade."
In the first task, children were shown a card with a numeral on it and asked first to read the numeral aloud and then to construct it using the commercial base-10. blocks. For the second task, children were shown numerals and asked to point to the ones and tens positions. They were later shown a numeral constructed with the base-10 blocks and asked what number the blocks made.
On the various aspects of the cognitive representation, the Japanese and Korean students scored higher than the American, Swedish, or French students. When the students were then asked to use the blocks to make the same number in a different way, American students were much less able than were the French and Swedish students, who, in turn, were less able than the Japanese and Korean students.
Japanese and Korean students also bested the American and European contingents by a large margin in their understanding of place value. Five problems had been presented in ascending order of difficulty. The first was answered correctly by 33% of American first-graders, 39% of the French children, 9% of the Swedish children, 67% of the Japanese children, and 100% of the Korean children. On the fourth problem, the percentages were 25%, 9%, 17%, 46%, and 92% respectively. Only on the final problem did the Korean performance decline substantially -- to 58% -- still far higher than that of any other country except Japan, which came in at 46%. The Koreans consistently outscored the Japanese, which implies that something other than the representation of numbers in language is at work here.
Qualitative differences in the way the students constructed numbers appear to corroborate the researchers' hypothesis. Japanese and Korean children typically represented numbers with blocks representing tens and ones, while American, Swedish, and French students preferred to use a collection of units. This suggests "that they [American and European children] comprehend the quantity represented by a whole numeral as a number of individual units, do not as readily see tens and ones in written numbers, and may lack an understanding of the meaning of the individual digits as tens and ones."
Miura reports that the researchers attempted to obtain data on the cognitive representation of numbers from American and Korean kindergartners. The American children could not perform the task; the Korean children could. Clearly, in order to be first in the world in mathematics by the year 2000 we need to completely restructure the first five years of life in this country, creating a radical, break-the-mold childhood.
Tips for Readers Of Research
HERE'S A TIP that will stand you in good stead: when looking at graphs, bar charts, and the like, try to "see through" the pictorial representation to determine what the data really mean. Graphs can be misleading -- and so can their labels. An example of the latter occurs in Workforce 2000. One graph therein is headed: "Productivity Has Declined Substantially Since 1965." But when one looks carefully at the graph, one finds that it is not productivity that has declined, but the rate of productivity improvement: in recent years we have continued to become more productive, but more slowly than before.
The problem of the graphs themselves being misleading is more common, however. Here are two examples.
In several articles and in The Learning Gap, Harold Stevenson and his colleagues present graphs showing, they claim, that American mothers believe school success comes from ability, while Asian mothers believe it comes from effort. As pointed out in "The Third Bracey Report on the Condition of Education" (October 1993 Kappan), the graph does not show this: American mothers and their children rate effort as more important than ability and rate it almost as important as the Asian mothers (and their children) do.
There is, however, an important aspect of the graphical presentation of the data that is misleading. The results are shown on a scale of ratings that goes from 0 to 5, implying that a 5-point scale is involved. The largest difference among the mothers is about one point. One point on a 5-point scale might be quite significant. However, the text reveals that the scale is actually a 10-point scale, a fact that significantly reduces the relative importance of the difference. Had the full scale been used on the graph, the difference would have looked much smaller as well.
Similarly, many representations of trends in Scholastic Aptitude Test (SAT) scores have been shown on a scale that runs from 400 to 500. In some respects this is reasonable. The SAT averages were around 500 when the long decline began, and neither the verbal nor the mathematics subscore has fallen below 400. Thus a scale that runs from 400 to 500 can show all of the data. However, on such a 100-point scale, the decline in SAT scores looks dramatic, and scores are often described as "plunging" or "plummeting." Such wording should be scrutinized carefully.
If the decline in SAT scores were shown on a graph that displayed the full 600-point range (from 200 to 800), the decline would look more like a blip than a dive. Readers can get some feel for what such a graph would look like by examining page 26 of the May/June issue of the Journal of Educational Research. This page contains the decline as reported in the Sandia Report. The Sandia engineers decided to show on the same chart trends in averages for the individual subtests and for the total scores. Thus they needed a scale that runs from 400 to 1,000. On this 600-point scale, the declines do not seem so large.
My point is not that the representation chosen by the Sandia engineers is more accurate. It is not. Rather, I wish to stress that the visual impact of charts and graphs often suggests an interpretation that may or may not be in line with the data. Readers should ask, "What does this information mean?' The answer may be something other than what the graph implies.
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|Author:||Bracey, Gerald W.|
|Publication:||Phi Delta Kappan|
|Date:||Dec 1, 1993|
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