“Eins, Zwei, Drei, Vier,…”.

This is how I count.

And “Zwanzig Prozent von Neunzehn-Dreiundsiebzig?” is how I think about tipping the bar tender.

It’s mildly weird, because although I grew up bilingually and went to school in Germany, I have lived the larger portion of my adult life in the US.

For the past 7 years most of my conversations have been in English. I think in English. I dream in English. When I count or perform simple algebra, however, I invariably switch into German.

On the surface, there are a couple of reasons for how small language differences might result in greater comfort for performing simple arithmetic in one language over another, since even little peculiarities might have real influences on how we learn or even think about basic mathematical concepts. For example, counting in Chinese is easily learned by memorizing the numbers 1 through 10, together with a simple combination rule. After learning these, all the remaining numbers can be generated according to the principle that 11 equals “ten one”, 12 equals “ten two”, 21 equals “two one”, etc.

In comparison, counting in English requires also learning the unique words for 11 and 12, as well as the names of the tens numbers.

Similarly, the number 42 is thought of as “two and forty” in German, but can be obviously read left to right as “forty-two” in English.

While these may look like very trivial differences, a series of experiments in the mid 90s suggests that American students at the time lagged behind Chinese students in their counting skills precisely because of this kind of language difference.

Even smaller differences in the language specific lexicon and syntax for math seem to similarly unsuspected impacts: For example, Chinese and English differ in the length of the simple number words, with Chinese words number words being particularly short. This has consequences that cognitive scientist Stanislas Dehaene in his 1997 book The Number Sense presents as follows:

“Read the following list aloud: 4 8 5 3 9 7 6. Now close your eyes and try to memorize the numbers for 20 seconds before reciting them again. If your native language is English, you have about a 50% chance of failure. If you are Chinese, however, success is almost guaranteed. As a matter of fact, memory span in China soars to about nine digits, while it averages only seven in English.”

The difference Dehaene cites for this particular type of memory task, is hardly due to a differences in average intelligence, or even differences in working memory between English and Chinese speakers, but instead, it has everything to do with the time it takes to read the number digits out aloud, where the shorter time it takes to speak the digits in Chinese aids in keeping the numbers in memory. In fact, Dehaene points to the same pattern in other languages, including Welsh, Hebrew, Arabic and Japanese, to argue that the above correlation between digit memory span and the time it takes to pronounce numbers is reproducible and not due to other differences specific to the Chinese and English language. What is more interesting, the effect that word length has on digit memory, seems to extend also to the speed in which children perform mental arithmetic, thus contributing to Chinese and Japanese children being able to significantly outperform their American peers, when it comes to simple calculations.

At the neural level, our mathematical thinking also appears tied to language. Mainly because of how early arithmetic memory networks are formed in the brain when we first learn mathematics. Here it is worth noting that most of the mathematical tasks we perform in everyday life rely on what may be called “arithmetic facts”, solutions to simple mathematical problems which do not need to be computed, but which can be routinely recalled from long-term memory. For example, you probably don’t have to actually calculate 4 times 3, but can rely on your knowledge of multiplication tables to identify the correct solution as 12.

Interestingly, many of these kinds of arithmetic facts (multiplication tables as a prime example), are learned by verbal rehearsal, and thus exhibit a strong link with the language in which they were originally encountered. In fact, studies with bilinguals have repeatedly demonstrated that arithmetic facts are more efficiently accessed in the language in which they were initially learned. And—as has been recently demonstrated by University of Texas researchers Elena Salillas and Nicole Wicha - language dominance plays only a negligible role in how arithmetic facts are recalled from memory. Instead, their recent findings, which are now published in the journal Psychological Science, highlights the pivotal role that early learning plays in shaping memory networks for arithmetic.

Memory networks for arithmetic are stylized representations (underpinned by actual neural networks) of how the brain organizes, and uses arithmetic facts. Similar to other conceptual networks, arithmetic memory networks form extensive connections between individual arithmetic concepts, and essentially provide a map of how concepts, facts and meanings are related to each other. The way these networks are shaped, and how separate facts are connected to each other within the network is important for problem solving. For example, research shows that people are quicker to identify whether a suggested solution to a simple multiplication problem is correct, than they are in identifying whether a suggested solution is incorrect, and argue that this is because the correct solution is more closely linked to the simple multiplication problem within the arithmetic memory network, thus making it easier for the brain to identify the correct solution as “belonging to” the problem, and judge it as correct.

More tellingly, if presented with two incorrect solutions, humans take longer to identify suggested solutions as wrong when they are more closely connected to the problem at hand, than when they are farther removed on the network. For example, the number 13 is more readily identified as the incorrect solution to the problem 2 times 3, than is the number 15 (which is at least connected to 3 by being one of its factors). The underlying principle at play for both of the above observations, is that once one element of a network is activated inside of the brain, this activation spreads to related elements. Arithmetic facts which are close to each other in the memory network are therefore more readily activated jointly, and thus more quickly identified as belonging together. In contrast, distant facts do not enjoy a lot of co-activation, and thus are quickly interpreted as unrelated. As a result of this, when subjects are asked to judge whether two arithmetic facts belong together (e.g. as in when subjects are asked whether 15 is the solution to 2 x 3) the brain is quicker at identifying unrelated suggested solutions as incorrect, than related ones.

Cognitive neuroscientists believe that electrophysiological measurements for subjects viewing simple math problems further highlight that the brain responds to how closely related arithmetic concepts are. In particular researchers focus on a particular neural signal (N400) that is emitted shortly after a problem is presented, and are able to show that this signal varies in strength depending on the closeness of arithmetic facts being evaluated: When a suggested solution to a problem is correct, the N400 signal is small, and when the solution is incorrect N400 is large. However, when a presented solution is incorrect and unrelated the N400 signal is larger still than if the solution is incorrect but related (Indeed, this finding is not limited to arithmetic, but has also been demonstrated in experiments using semantically related versus semantically unrelated word pairs).

Focusing on the N400 brain response in bilinguals during presentation of related and unrelated arithmetic facts, Salillas and Wicha have recently shown that the above described graded response to potential solutions occurred exclusively when problems were presented to bilinguals in the language in which they had originally learned basic arithmetic (or in digit format). More importantly, the researchers were able to show, that the relation between brain response and presentation of arithmetic facts was independent of language dominance, concluding that “arithmetic facts are organized into strong associative memory networks for problems presented in [the language in which they were originally learned] and digit format, with a weaker network for [other, even more dominant languages]. … Thus a connection between language and math is established at the time of learning and maintained into adulthood independently of natural language dominance.”

Hence, the proclivity of bilinguals to slip into a particular language for math, is not merely a matter of habit, but may be due to the qualitatively different way in which the brain appears to manipulate arithmetic depending on the language in which it is presented, with the brain treating the language in which arithmetic facts were initially learned as preferred, even if this language is no longer the speakers overall preferred language.

The researchers intend to conduct further experiments to examine the effects of explicit retraining, but believe that their results “may have important implications for teaching and testing young bilinguals and highlights the idea that bilinguals should not be treated as two monolinguals in one brain”.

So for now, “Zwei mal Drei macht Vier, und Drei macht Neune”.