The influence of language in mathematics

Secondly, during the child's development, language serves as a temporary scaffolding for the acquisition of concepts. It facilitates the acquisition of the concept of exact number by enabling children to learn to count a precise number of objects, as shown by a whole series of experiments with the Mundurucu and Pirahã Indians of Amazonia, or with deaf children in Nicaragua. Elizabeth Spelke speculates that this learning depends on mastery of the rules of generative grammar, and that it is the syntax of the natural language that makes it possible to understand large numbers. According to this hypothesis, speakers of different languages - for example, with different numbers ("vingt" as in French, versus "deux dix" as in Chinese), a different base, or a different word order ("drei und zwanzig") - should possess distinct mental representations - Elizabeth Spelke is therefore defending a "Whorfian" theory. However, the behavioral and brain-imaging data available to us do not support this, at least in adults. Rather, they suggest that mathematical concepts are encoded in a non-verbal, universal "language of thought". My own findings suggest that, during development, a rich and stimulating linguistic environment helps children acquire the concepts of arithmetic, not because these concepts are encoded verbally, but because linguistic constructs draw children's attention to the most appropriate or effective non-verbal mental constructs. Language thus acts as a catalyst for non-verbal concepts, bringing them together and suggesting how they might be combined.

Thirdly, in adults, linguistic recoding of information can serve as a mental tool in certain tasks. Memory for numbers, like that for multiplication tables, benefits from additional storage in verbal form. Thus, aphasic patients, while retaining most of their arithmetic faculties, may have difficulty with multiplication tables - and bilinguals, with transferring their knowledge of exact calculation from one language to another.

Finally, and most importantly, contrary to Whorf's ideas, interlanguage variations have only a modest impact on perceptual and cognitive abilities. As predicted by Leila Gleitman, it's mainly when the task directly or indirectly involves language that language has an impact - and even then, it remains small. For example, children learn to count more quickly in Chinese than in English; this effect seems to be linked to the greater simplicity of the Chinese numbering system. The brevity of Chinese number names also has a positive impact on memory. Finally, the inversion of tens and units in Germanic languages has a small effect on calculation, but only when the stimulus or the answer is presented verbally: no effect is observed when the problem is presented in Arabic numerals and the answer is typed on the keyboard, showing that this is not really a linguistic influence at the conceptual level.

In conclusion, language is not the medium of thought, but serves as a catalyst. Mathematics relies mainly on non-linguistic representations, on a purely internal, non-verbal "language of thought". However, learning mathematics can be facilitated by the structure of language. Linguistic interactions focus attention on the most relevant mental objects. Base-ten numeration, for example, is made transparent in Chinese oral language - and perhaps it would be interesting to experiment with the use of pseudo-Chinese numeration with French kindergarten pupils, having them temporarily name the numbers "dix cinq" instead of "quinze" or "deux dix" instead of "vingt"? Even if these effects of linguistic catalysis are not strictly "Whorfian", they could be important on a practical level. From kindergarten onwards, teaching should combine (1) the manipulation of concrete, proto-mathematical non-verbal objects (collections, constructions, board games, shapes, tangrams...), but also (2) explicit verbalization that makes thought processes obvious and gives the pupil clear, precise learning objectives.