Language helps you find out just how weird kids are

My daughter M, my wife, and I were visiting family on the east coast about a month ago. One night, M was whining a little bit before bedtime, and after some investigation, my wife figured out that M's pajamas – a new lighter-weight set that we brought because the weather was still hot – were bothering her. The following dialogue ensued:

Mom: "are your pajamas bothering you?"
M: "yah."
Mom: "are they hurting you, sweetie?"
M: "yaaah!"
Mom: "where do they hurt you?"
M: "'jamas hurt mine face!"

Now I don't know where M went wrong in this exchange – does she not understand "pajamas," "hurt," or "face," or does she just think that hurting your face is the ultimate insult? – but there's clearly something different in her understanding of the situation than we expected. One more example, from when I returned from a trip to the mountains last week ("dada go woods!"):

M: "my go woods see dove!"
me: "yeah? you want to see a dove?"
M: "see dove in my ear!"
me: "in your ear?"
M: "dove go in my ear go to sleep."
me: "really?"
M: "dove going to bed."

M is now officially a two-year-old (26 months), and it's been a while since I wrote about her – in part because I am realizing as I teach her the ABCs that it won't be that long before she can read what I write. But these exchanges made me think about two things. First, her understanding of the world, though amazing, is still very different then mine (there are many other examples besides the painful pajamas and the dove in her ear). And second, it's her rapidly-growing ability with language that allows her to reveal these differences.

Children spend a short, fascinating time in what's been called the "two-word stage." There was an interesting discussion of this stage on the CHILDES listserv recently; whatever your theoretical take, it's clear that children's early productions are fragmentary and omit more than they include. Because of these omissions, this kind of language requires the listener to fill in the gaps. If a child says "go store," she could be saying that she wants to go to the store, or commanding you to go to the store. If she says "my spill," you have to figure out what it is she just spilled (or wants permission to spill).

Since the listener plays such a big role in understanding early language productions, they are plausible by definition. There's almost no way for the child to express a truly weird sentiment, because the adult listener will tend to fill in the gaps in the utterance with plausible materials. (This can be quite frustrating for a child who really wants to say something weird.) M's language, in contrast, is now at the stage where she can express much more complex meanings, albeit with significant grammatical errors. So in some sense, this is the first chance I've had to find out just how weird her view of the world really is.

Can we improve math education with a 5000-year-old technology?

(This post is written jointly by my collaborator David Barner and me; we're posting it to both his new blog, MeaningSeeds, and to mine). 

The first calculating machines invented by humans – stone tablets with grooves that contained counting stones or "calculi" – are no match for contemporary computers in terms of computational power. But they and their descendants, in the form of the modern Soroban abacus, may have an edge on modern techniques when it comes to mathematics education. In a study about to appear in Child Development, co-authored with George Alvarez, Jessica Sullivan, and Mahesh Srinivasan, we investigated a recent trend in math education that emanates from these first counting boards: The use of "mental abacus."

The abacus, which originates from Babylonian counting boards dating back to at least 2700 BC, has been used in a dozen different cultures in different forms for tallying, accounting, and basic arithmetic procedures like addition, subtraction, multiplication and division. And recently, it has made a comeback in classrooms in around the world, as a supplement to K-12 elementary mathematics. The most popular form of abacus – the Japanese Soroban (pictured below) – features a collection of beads arranged into vertical columns, each of which represents a place value – ones, tens, hundreds, thousands, etc. At the bottom of each column are four "earthly" beads, each of which represents a multiple of 1. On top is one "heavenly" bead, which represents a multiple of 5. When beads are moved toward the dividing beam, they are "in play", such that each column can represent a value up to 9.

When children learn mental abacus, they first are taught to represent numbers on the physical device, and then to add and subtract quantities by moving beads in and out of play. After some months of practice, they are then asked to do sums by simply imagining an abacus, rather than using the actual physical device. This mental version of the abacus has clear – and sometimes profound – computational benefits for some expert users. Highly trained users – called "masters" by those in the abacus world – can instantly encode and recall long strings of numbers, can add two digit numbers as fast as they can be called out in sequence, and can compute square roots – and even cube roots – almost instantaneously, even for large numbers. Most startling of all, these techniques can be practices while simultaneously talking, and can be mastered by children as young as 10 years of age with record breaking results (see also here, here, and here).  If you haven’t ever seen this phenomenon, take a look at the YouTube video below. It is truly remarkable stuff. 

In our study we asked whether this technique can be mastered to good effect by ordinary school children, in big, busy, modern classrooms. We conducted the research in Vadodara, India, a medium sized industrial town on the west coast of India, where abacus has recently become a popular supplement to standard math training in both after-school and standard K-12 settings. At the charitable school we visited, abacus training was already underway and was being taught to hundreds of children starting in Grade 2, in classrooms of 70 children per group. To see whether it was having a positive effect, we enrolled a new, previously untrained, cohort of roughly 200 Grade 2 kids and randomly assigned them to receive either abacus training from expert teachers or extra hours of standard math training, in addition to their regular math curriculum.

Even in these relatively large classrooms of children from low-income families, mental abacus technique edged out standard math. Though effects were modest in this group, they were reliable across multiple measures of math ability. Also, children attained the best mastery of mental abacus best if they began the study with strong spatial working memory abilities (to get a sense of how we measured spatial working memory take a look at this video).

Why did abacus have this positive effect? One possibility is that learning a different way of representing numbers helped kids make generalizations about how numbers work. For example, the abacus – like other math manipulatives – provides a concrete representation of place value – i.e., the idea that the same digit can represent a different quantity depending on its position (e.g., the first and second 3 in “33” represent 30 and 3 respectively). This better representation might have helped kids understand the conceptual basis of arithmetic. Another possibility is that the edge was chiefly due to the highly procedural nature of mental abacus training. Operations are initially learned as sequences of hand movements, rather than as linguistic rules, and according to users can be performed almost automatically, without reflection. Finally, it's possible that it's this unique mix of conceptual concreteness and procedural efficacy that gives the abacus its edge. Children may not have to learn procedures and then separately learn how these operations relate to objects and sets in the world: Abacus may allow both to be learned at the same time, a welcome tonic to the ongoing math wars.  

Right now it's uncertain why mental abacus helps kids, and whether the effects we've found will last beyond early elementary school. Also, the technique has yet to be rigorously tested on US shores, where it's currently being adopted by public schools in at least two states. This is the focus of a new study, currently underway, which will test whether this ancient calculation technique should be left in museums, or instead be widely adopted to boost math achievement in the 21st century.