July 21, 2005

Of four parts, one

A few days ago, I posted about the latest fuss from Tokyo's buffoon of a mayor, Shitaro Ishihara, who is being sued for responding to some insubordinate language teachers by asserting that French "is disqualified as an international language because French is a language which cannot count numbers", and that "guys desperately clinging to such kind of language are lodging opposition for the sake of opposition". I haven't heard that Mayor Ishihara has gotten any letters of support from Sergei Brin or George W. Bush -- though they might well be sympathetic to his conclusion in this case -- so I thought I would pitch in and produce one myself.

Well, not really. I might have defended William Safire against Steven Pinker and even against David Beaver, but Mayor Ishihara is going to have to deal with the French teachers without my help. However, his remarks about number names and counting, while foolish, are related to some real psycholinguistic results, which he may perhaps have heard about.

Apparently Ishihara's concern is with the irregular patterns of certain French number names, such as quatre vingt ("four twenty") for 80, soixante quinze ("sixty fifteen") for 75, and so on.

The theory that transparent number names are cognitively helpful was proposed in 1987, in a paper by K.F. Miller & J.W. Stigler ("Computing in Chinese: Cultural variation in a basic cognitive skill", Cognitive Development, 2, 279-305). This study found that four- to six-year-old Chinese children could count higher, with fewer errors, than U.S. children. Because the Chinese number system is more regular (e.g. eleven is "ten-one", twelve is "ten-two", and decade names are similarly compositional, the conclusion was that "systematically organized number names facilitate Chinese children's understand of counting."

In 1999, I.T. Miura and others (I.T. Miura, Y. Okamoto, V. Vlahovic-Stetic & C.C. Kim, "Language supports for children's understanding of numerical fractions: Cross-national comparisons", Journal of Experimental Child Psychology, 74, 356-365) found that similar linguistic transparency appears to help with the understanding of fractions. Korean, Croatian and U.S. children were tested, all first- and second-graders who had not gotten any prior classroom instruction in fractions. Each trial involved showing a picture like this one:

A fraction name (like 2/3) was then read aloud in the child's native language, and the child was asked to circle the picture corresponding to the name. The main result was that the Korean children out-performed their Croatian and American counterparts. The suggested explanation is that in Korean (as in Chinese and Japanese), the names of fractions explicitly mention the idea of fractional parts, so that the Korean name for "one fourth" is (in word-for-word glosses) "of four parts, one".

Jae Paik and Kelly Mix ("U.S. and Korean Children's Comprehension of Fraction Names: A Reexamination of Cross-National Differences", Child Development 74(1) 144-154, 2003) started out by replicating the findings of Miura et al. In the new study, 40% of Korean first graders got >5 out of 8 fraction problems correct, while among American first graders, only 31% did. This was a smaller difference than Miura et al. found -- Paik and Mix speculate that the relatively poorer performance of their Korean students, compared to those in the Miura et al. study, might be due to the socio-economic differences, and also might be due to the fact that their experiments were not performed by the children's classroom teachers.

Now comes the fascinating part. Paik and Mix also did an experiment with American students in which artificial English fraction names were used. Specifically, they tested five fraction-wording conditions, shown in the table below. Below each wording is the percentage of first-grade students getting at least 5 of 8 answers correct.

  Denominator then numerator Numerator then denominator
Part-whole relations explicit
"of four parts, one"
67% >5 of 8
"one of four parts"
57% >5 of 8
Part-whole relations not explicit
15% >5 of 8
25% > 5 of 8
Standard English fraction name
"one fourth"
31% >5 of 8

But the Korean first-graders' score, on the very same test, was only 40% >5 of 8 correct! When the idea of fractional parts was made explicit, the American students did substantially better than the Korean students -- even though the best-performing fraction name "of four parts, one" has an Elizabethan flavor that must seem quite strange to American six-year-olds!

Paik & Mix offer the following explanation:

In English, the word parts is in children's everyday vocabulary. By the time children enter grade school, they usually understand the word parts, even if they have not mastered the idea of equal parts. In contrast, the Korean word for parts that is used in fraction names (i.e. boon) is not in children's everyday vocabulary before formal schooling. This word is actually borrowed from Chinese and is not introduced to children until they are taught fractions in school. Until then, children use informal words to refer to parts in their daily lives (e.g. jo-gak). ...

An analogy might be if we called 1/4 "of four morceaux one" in English.

(There's that numerical inadequacy of French again!) Anyhow, these results suggest that the problems with French as a mathematical language, asserted by Mayor Ishihara, are real but entirely superficial. The lack of transparency in number names will surely make things harder for learners, and conceivably even for adults -- but it's easy to change things by using positional naming (e.g. "sept cinq" rather than "soixante quinze") or other expedients (see this earlier post for some evidence of entirely non-linguistic methods).

And of course the bad educational consequences of the absurd Japanese orthographic system trump any possible disadvantages of French number names...

Posted by Mark Liberman at July 21, 2005 09:15 PM