“The Best Language for Math”—Is there such a thing?

Oct 10, 2014 by

[Thanks to Stephane Goyette and Ekaterina Lyutikova for their help in working on this post!]

American school students are not great at math, at least that’s the common wisdom. According to 2012 PISA results, U.S. students rank #36, with the Maths mean score of 481, well below the OECD of 494. Several recent studies, reported in an article by Sue Shellenbarger in the Wall Street Journal, claim to have found the culprit—the English language. The studies compared elementary math skills of English-speaking American children with those of Chinese, Japanese, Korean, and Turkish kids. Their conclusion: the U.S.-Asian math achievement gap is due to linguistic properties of English vs. Asian languages such as Chinese, Japanese, Korean, and Turkish. Shellenbarger explains: “Chinese, Japanese, Korean and Turkish use simpler number words and express math concepts more clearly than English, making it easier for small children to learn counting and arithmetic”. By “simpler” she means “more transparent” in “identify[ing] the place value of the numbers”. Some examples will help us understand what is wrong with English number words; according to Shellenbarger:

“The trouble starts at “11.” English has a unique word for the number, while Chinese (as well as Japanese and Korean, among other languages) have words that can be translated as “ten-one”—spoken with the “ten” first. That makes it easier to understand the place value—the value of the position of each digit in a number—as well as making it clear that the number system is based on units of 10.

English number names over 10 don’t as clearly label place value, and number words for the teens, such as 17, reverse the order of the ones and “teens,” making it easy for children to confuse, say, 17 with 71, the research shows. When doing multi-digit addition and subtraction, children working with English number names have a harder time understanding that two-digit numbers are made up of tens and ones, making it more difficult to avoid errors.”

PISA_math

Before we analyze whether the difference in numeral vocabulary of different languages, let’s consider the PISA results once again. Math scores of South Korea and Japan are indeed high: 554 and 536, respectively. But when it comes to China, the scores are given to three most prosperous areas: Shanghai (613), Hong-Kong (561), and Macau (538). Besides wondering how other parts of China would score, one is left with a nagging question: if language is the problem, what accounts for the fact that the discrepancy between Shanghai’s and Macau’s scores (75 points) is larger than that between Macau’s and USA’s scores (57 points)? Turkish, another language claimed to make it easier for children to work out counting and arithmetic, does not seem to help after all, as Turkey’s PISA math score is well below that of the USA—448. Yet another obvious problem with blaming the English language for children’s math woes is that students in different English-speaking countries do very differently in PISA tests: Canada scores at 518, Australia at 504, Ireland (where much of the education is in English) at 501, New Zealand at 500, UK at (the OECD average) 494, and USA at 481. Thus, the gap between Canadian and American scores is almost as large as that between Japanese and American scores. And if you are wondering whether Canada’s higher score is due to the French-speaking children that might be helped by their native language, French, as we shall see below, is no better than English in the “simplicity” of its numerals, quite the opposite in fact.

Let’s now consider the numerals in English and the Asian languages considered in Shellenbarger’s article. Although she is not very clear on what it is about English that holds American kids back as far as math is concerned—occupational hazard when non-linguists dabble in language-related research—from her examples we might deduce that three characteristics of English numerals are to blame. First, some numerals, such as eleven and twelve, are “unique”, as she calls them, or more technically, are opaque as to their morphological make-up. Historically, eleven derives from Old English endleofan, literally ‘one left’ (over ten), and twelve has a similar derivation. But over time, these words—like much-used coins—got rubbed around the edges resulting in today’s opaque forms. This process is due to high frequency of use, as well as the fusional nature of the English language, whereby meanings are expressed by morphemes that are not merely stacked one after another, as in isolating language like Chinese or agglutinative languages such as Japanese and Turkish, but tend to meld with one another, changing themselves and transforming their neighbors in the process.

The second property of (some) English numerals is the non-transparent order of the morphemes for ‘ones’ and ‘tens’, as in seventeen ‘7+10’, compared to the Chinese, Japanese, and Turkish numerals, which are structured as ‘10+7’. Let’s note that this property is independent of the opaqueness issue described above. For example, in Russian, numerals odinnadtsat’ ‘11’ and dvenadtsat’ ‘12’ are as transparent as semnadtsat’ ‘17’ or devjatnadtsat’ ‘19’: all of them consist of a numeral for ‘ones’ (odin, dve, sem’, devjat’) followed by nad ‘over’ and dtsat’ from desjat’ ‘ten’. In other words, all the numerals from 11 to 19 are ‘x-over-ten’. Note, however, that the order is the same as in the English numerals: the ‘ones’ precede ‘10’ rather than follow it as in Chinese, Japanese, and Turkish numerals.

The third property that allegedly makes English so hard for learning math concerns numerals like ‘27’: in Chinese and Japanese, it is expressed as ‘two-ten-seven’, whereas in English the ‘20’ part is expressed by another “unique” (i.e. non-transparent) word. Note that Turkish patterns with English in this respect, as it too has ‘twenty-seven’ rather than ‘two-ten-seven’, while Russian patterns with Chinese and Japanese in having dva-dtsat’-sem’ ‘two-ten-seven’.

To recap, three properties of numerals in a given language appear to be relevant to easing the acquisition of math literacy, and they do not correlate, as can be seen even from our small sample of languages.

transparent ‘11’, ‘12’

transparent ‘17’

transparent ‘27’

English

No

No

No

Chinese

Yes

Yes

Yes

Japanese

Yes

Yes

Yes

Turkish

Yes

Yes

No

Russian

Yes

No

Yes

 

One could argue, however, that English numerals for ‘20’, ‘30’, ‘40’, etc. up to ‘90’ are somewhat transparent, with –ty being the “worn off” version of ‘ten’, as in forty (ignore the spelling!), fifty, sixty etc. Some languages, such as French, are even less “transparent” when it comes to these numerals. For example, ‘80’ in French is not some (“worn off”) form of ‘eight-ten’ but is literally ‘four-twenty’ (with the ‘20’ part being another “unique”, non-transparent word); ‘97’ in French is ‘four-twenty-ten-seven’—a mess from the point of view of language-math parallels. A better study might look at English- vs. French-speaking children’s abilities at math. It would not be hard to find children speaking these languages schooled in the same system: look no further than Canada!

vigesimal

While French uses the base-20 (or “vigesimal”) system just for ‘80’ and ‘90’ (and arguably for ‘70’), some other languages use such a system throughout their numeral vocabularies. For instance, in Basque hogei ‘20’ is used as a base for numbers up to 100. Thus, ‘75’ is hirurogeita hamabost, literally ‘three-twenty-and ten-five’. While this system may seem bizarrely at odds with the decimal math that we are used to, it is not as rare as one might think. It is found in a close relative of English, Danish, where numbers from 50 to 99 are based on 20: for example, tresindstyve literally means ‘3 times 20’ (i.e. 60). Even in English, a vigesimal system was used historically, as in the famous opening of the Gettysburg Address “Four score and seven years ago…”, meaning eighty-seven years ago. According to the Wikipedia article, “in the Authorised Version of the Bible the term score is used over 130 times although only when prefixed by a number greater than one while a single “score” is always expressed as twenty.”

Curiously, historical remnants of an earlier vigesimal system pop up even in otherwise very “transparent” languages, such as those in the Chinese family. Thus, Cantonese and Wu Chinese often use the single morpheme meaning ‘20’ (Mandarin niàn, Cantonese yàh, Shanghainese nyae or ne) in place of a fully decimal two-morpheme combination meaning ‘two ten’ (Mandarin èr shí, Cantonese yìh sàhp, Shanghainese el sah).

Any research that claims language to be the culprit for math vows of its speakers would need to show which specific properties of that language’s numerals is to blame and to what extent. Given the non-correlations between language and PISA scores, discussed above, my hunch is that the blame is to be placed on social and cultural rather linguistic factors. The Guardian report on the PISA scores, cited above, seems to agree. After all, purely linguistic factors cannot explain why “boys scored higher than girls in maths” or the “correlation between a higher GDP per capita and a successful performance”, or between “spending on education” and performance in PISA tests.

Crucially, the studies reported in the WSJ fail to control for any such social or cultural factors. A “cleaner” study should consider speakers of various (and very different in relevant respects) languages who are schooled in the same social and cultural environment. For example, one might study Russian- and Tatar-speaking children: Tatar has the same numeral system as Turkish and thus differs from Russian in two important respects (see ‘transparent 17’ and ‘transparent 27’ columns in the Table above). Assuming one can control for socio-economic and cultural factors, such study has three possible outcomes: (1) Russian-speaking children perform better than Tatar-speaking ones, (2) Tatar-speaking children perform better than Russian-speaking ones, and (3) the two groups perform equally well. The first outcome would should that the transparency of ‘27’ is the relevant linguistic property, while the second outcome would mean that the transparency of ‘17’ is the key. The third outcome would mean that the morphological properties of numerals in a given language are irrelevant to math abilities of its speakers.

Another way to tease apart the effects of language vs. culture would be to compare math abilities of children growing up and schooled in the same system in the U.S. but speaking different languages. One good comparison would be between English- and Hindi-speaking children. The latter are known (alongside Chinese-speaking kids) to be better students in elementary schools, in large part due to their math results. If the research reported by the WSJ is on the right track we would think that Hindi is more transparent than English in its number words, just as Chinese, Japanese, and Turkish are claimed to be. This couldn’t be further from the truth, though, as Hindi numerals for ‘20’, ‘30’, ‘40’ etc. are all completely opaque: for example, ‘20’ is bīs, while ‘2’ is do and ‘10’ is das; ‘30’ is tīs (cf. tīn ‘3’), ‘40’ is cālīs (cf. chār ‘4’), ‘50’ is pacās (cf. pāṅc ‘5’), ‘60’ is sāṭh (cf. chaḥ ‘6’), ‘70’ is sattar (cf. sāt ‘7’), ‘80’ is assī (cf. āṭh ‘8’), and ‘90’ is nabbē (cf. nau ‘9’).

All in all, the differences in math ability seem to be due more to cultural, social, and economic factors than the languages themselves. Back to the drawing board, then!


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  • John Cowan

    The recent switch in official Welsh from one of the most opaque systems in existence to simple decimals probably has helped Welsh-speaking children do elementary mathematics better. The old system, after unique words from 1-10, went with ‘one/two/three/four on ten’ (11-14), ‘five-ten’ (15), ‘one/two on five-ten’ (16-17), ‘two-nine’ (18), ‘four on five-ten’ (19), and then a unique word for 20. The system then repeats with ‘one on twenty’ and so on until ‘two twenty’ (40), then ‘two twenty and one/…/nine’ (41-49). Then there’s another shift: ‘half a hundred’ (50), then ‘half a hundred and one/…/nine’ (51-59) then ‘three twenties’ (60) and a repeat of the 20-39 system, followed by ‘four twenties’ and another repeat of the 20-39 system, taking us up to 99. Then unique words for 100, 1000, 1,000,000, and 1,000,000,000. And all of this with standard Welsh initial mutations!

    • 康邁克

      Okay, I’d like to learn Welsh. But maybe not let my kids go near it.
      On the other hand, it would be a fun maths/linguistics game for children to play with and try to figure out the numbers of Welsh.

      • Ivan Derzhanski

        And indeed such problems (aka puzzles) have been composed and assigned many times at linguistic contests.

    • Thanks for sharing this, John. I suppose one could study Welsh vs. English with respect to the speakers’ math skills — if one wanted to proliferate such pointless research, that is.

      • Ivan Derzhanski

        It is quite conceivable that converting numbers from their spoken names to decimal notation and back takes a little longer in Welsh than in English, and a little longer in English than in Chinese, which may affect the speed of doing sums in one’s head. But (1) the difference needn’t be great and (2) doing sums in one’s head is a vanishingly small part of the science that’s mathematics.

        • Excellent points, Ivan! I am also wondering whether doing arithmetic “in one’s head” necessarily involves number words — several people I talked to about this, commented that it does not.

          • Ivan Derzhanski

            I’d say the same. You use them when talking to people, though, so if someone asks you what’s fifty-two plus seventeen, you’ll have to convert the words (Chinese or Hindi, say, _wŭshí’èr_ or _bāvan_ to 52, _shíqī_ or _satrah_ to 17), calculate 52+17 and then convert 69 back to its name, sixty-nine (_liùshíjiǔ_ or _unhattar_ — note that the latter actually means `1 to 70′), and there’s no question but that this is easier to do in Chinese than in Hindi. But so what?

          • Precisely!

          • Lahesha Mann

            Doing math quickly is not a “so what”….it is actually of importance in daily life. And no, it probably does not affect PISA scores given that the student is given so much time to do the test, but it certain helps to be able to do calculations fast in real life at the supermarket, at work, etc. Furthermore, the easier logical system probably means it takes less effort to both teach and learn basic arithmetic. Unfortunately, children have a tendency to shun difficult things…. the easier to learn, the less resistance there will be to learn.

    • John Cowan

      Apparently the newer system using ‘ten and four’ (14), ‘two-ten and one’ (21), ‘three-ten and five’ (35), etc. first appeared in Patagonian Welsh, where it was probably calqued from Spanish. It was then adopted in Standard (British) Welsh.

  • Guest

    This
    author overextended when talking about Cantonese. The word “yàh” (廿)
    is a direct phonetic and visual contraction of the characters for twenty
    “yìh sàhp” (二十)which can also be thought of as two tens (十十) hence the
    character. It is a stretch to use as evidence of a vigesimal substrate,
    when it seems more likely that the word derives from the decimal. I
    can’t talk about the other Chinese languages mentioned, but I would not
    be surprised that it is the same.

  • P Yu

    This author overextended when talking about Cantonese. The word “yàh” (廿) is a direct phonetic and visual contraction of the characters for twenty “yìh sàhp” (二十)which can also be thought of as two tens (十十) hence the character. It is a stretch to use as evidence of a vigesimal substrate, when it seems more likely that the word derives from the decimal. I can’t talk about the other Chinese languages mentioned, but I would not be surprised that it is the same.

    • Thank you for the correction. It’s not a vigesimal system but the contracted form is exactly the sort of non-transparency that the authors of the WSJ article ascribe to “twenty”, so in this respect Chinese isn’t much different from English.

    • Ivan Derzhanski

      Cantonese _yah_ 廿 seems to be at the intersection of two systems. One
      is an old set of names of tens, with Chinese characters and cognates in
      other Chinese languages and Middle and Old Chinese reconstructions;
      these involve 20 廿 (Cantonese _yah_ < Old Chinese /*njɯb/), 30 丗
      _sai_ or 卅 _sā_, 40 卌 _se_, and it stops there. The other is a system
      of contracting names of tens in Cantonese: 30 _sà'ah_ < _sāam-sahp_,
      40 _sei'ah_ < _sei-sahp_, and so on.

    • 康邁克

      Actually in response to other Chinese languages, Taiwanese (Southern Min) does it too, but the two syllable structure appears in the hundreds when followed by a unit of ten. So 350 san-pah-goo-tsap becomes saa-goo with a long /a/ but with higher tone on the second half. This is also an abbreviated form.

  • Ivan Derzhanski

    Worse, in the Hindi numerals that contain ones as well as tens, the two terms undergo such morphophonological changes that there’s no substitute to learning all numbers up to 100 by heart. _do_ 2 + _pacās_ 50 = _bāvan_ 52. How much more opaque does it get? And then the Indians (okay, they didn’t speak Hindi then, but Sanskrit sandhi aren’t for the faint at heart either) went and invented the positional notation, which was to conquer the world in time. This is just how bad they were at maths—Sue Shellenbarger, here’s a wish you were as bad at anything as they were at maths.

    • Oh that, that’s perfect! And yet Hindi-speaking kids are among the better students in math in the US, alongside Chinese and Japanese kids!

      • Ivan Derzhanski

        Yes, there you have it—within the same educational system, native speakers of better-for-maths Chinese and worse-for-maths Hindi both outdo the native speakers of English, the language of instruction. Sounds like a hint that the reasons are of socio-economic, and then perhaps cultural and historical, rather than linguistic nature. But then again anyone with a clue would have thought of that in the first place. Or does the desire to have a sensational paper printed in WSJ beat all other considerations?

        • You’ve pretty much summarized my post. I am glad to see that you agree!

          • Ivan Derzhanski

            Btw, as the same colourful diagram in _The Guardian_ shows, Shanghai and Singapore and Hong Kong are world leaders in Reading and Science, as well as Maths. Anyone care to write a paper on how Chinese isn’t just the best language for maths, but also the best language for science and, _mirabile dictu_, for reading?

          • Oh this is a BRILLIANT point, Ivan!

  • Ivan Derzhanski

    Funny how Greece, Serbia, Turkey, Romania, Cyprus and Bulgaria form a compact group.

    • Yes. And math scores is probably not the only way in which they do…