Children are beautifully attuned to patterns in the world around them. They notice repetitions and regular relationships. That’s how they discover basic laws in the physical and social world and learn to generalize, predict, and think logically.
The number system that is used around the world—the decimal, or base-ten system—is built around a wonderfully simple pattern. There are ten numerals and they keep repeating, always in the same sequence. You go 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, and then, to go past 9, you put a 1 in the “tens place” and start over--10, 11, 12, 13, 14, 15, 16, 17, 18, 19—and then you put a 2 in the tens place …. and so on, and so on, and so on. Once you’ve got the system down you can do it forever if you like; you could spend your whole life at it and never come to the end.
The most crucial thing to understand if you want to do any calculations with this number system is its base-ten repeating nature. The algorithms that we teach children for adding, subtracting, multiplying and dividing are all necessarily founded on an understanding of that pattern. If children don’t understand the base-ten pattern, they have no way of making sense of the procedures they are taught for calculating. They memorize the procedures without understanding, which means that they haven’t really learned them. They’re just going through motions, mechanically, without thought or reason. They take it on faith that the system works, because a teacher told them it does. That’s not how to become a mathematician!
Children in the United States are notoriously bad at mathematics. I suppose there are a number of reasons for this, but here is one that might be near the top of the list. We hide the base-ten system from our children. We make it hard for them to discover, appreciate, and incorporate into their way of thinking about numbers.
The first problem has to do with the names we give to our numbers. We go one, two, three, four, five, six, seven, eight, nine, ten---and then we go eleven, twelve. Why eleven and twelve? Those words provide no hint about the repeating nature of what is happening here. The numerals repeat, but the names do not. There's nothing in these names that tells us explicitly that eleven is one added to ten and twelve is two added to ten. And then we go to thirteen, fourteen, etc. Even many adults don’t know that the teen here means ten, so thirteen is literally three and ten.
Contrast this to the number words in Chinese, Japanese, and Korean. Those number names (translated literally into English), beginning with ten, go ten, ten one (for 11), ten two (for 12), etc., and then onto two-tens (for 20), two-tens one (for 21), and so on. The words repeat in exactly the way that the numerals repeat. For a child learning to count in any of these languages the base-ten system is completely transparent. It can’t be missed.
To understand how all this is relevant to learning arithmetic, consider a child learning to add two-digit numbers, say, 34 plus 12. English-speaking children pronounce the problem (aloud or to themselves) as “thirty-four plus twelve,” and the words provide no hint as to how to solve it. Chinese-speaking children, however, pronounce the problem (in effect) as “three-tens four plus ten two,”and the words themselves point to the solution. In the two numbers together there are four tens (three tens plus one ten) and six ones (four plus two), so the total is four-tens six (46).
One line of evidence that number words indeed do make a difference comes from a study showing that children in the United States learn to count to 10 at the same age, on average, as do Chinese children but fall behind Chinese children in learning to count beyond 10, which is where the languages begin to diverge (Miller et al., 1995). By age 4, most of the Chinese children studied could count to 40 or beyond, whereas most American children could not get past the low teens.
In an even more telling study, 6-year-olds in the United States and in France and Sweden (where number words contain irregularities comparable to those in English) were compared with 6-year-olds in China, Japan, and Korea on a task that directly assessed their use of the base-10 system (Miura et al, 1994). All the children had recently begun first grade and had received no formal training in place value. Each was presented with a set of white and purple blocks and was told that the white blocks represent units (ones) and the purple blocks represent tens. To make it clear, the experimenter explained, “Ten of these white blocks are the same as one purple block,” and set out ten whites next to a purple to emphasize the equivalence. Each child was then asked to lay out a set of blocks to represent specific numbers—11, 13, 28, 30, and 42.
The results were striking. The Asian children made their task easier by using the purple blocks correctly on over 80 percent of the trials, but the American and European children did so on only about 10 percent. For example, while the typical Asian child set out four purples and two whites to represent 42, the typical American or European child laboriously counted out 42 whites. When they were subsequently asked to think of a different method to represent the numbers, most of the American and European children attempted to use the purple blocks, but made mistakes on about half of the trials.
So, even before they begin any formal training in numerical calculations, the Asian children already have a head start—they implicitly know the base-ten system.
Another handicap we hand to children in the United States lies in our ridiculous units of measurement. Nearly all the rest of the world now uses the metric system, where units relate to one another in ways that mirror the base-ten number system. There are ten centimeters in a decimeter, ten decimeters in a meter, and a thousand meters in a kilometer; ten grams in a dekagram, ten dekagrams in a hectogram, ten hectograms in a kilogram; and so on. Every time a child measures anything and sees a conversion from one unit of measurement to another, the base-ten system is reinforced. Not only does this make measurement easier, but it also provides the child with a concrete understanding of the base-ten system, so it isn’t just an abstraction, it’s something you use every time you measure your height, or weight, or work out the amounts of ingredients to put into a recipe. In contrast, our children learn about inches, feet, yards, and miles; ounces and pounds; and cups, quarts, and gallons—none of which relate to one another in multiples of ten and few of which bear any logical relationship across measurement dimensions.
The base-ten numerical system is a beautiful thing; it’s too bad we mess it up with our language and our units of measurement and thereby hide its beauty from our children, and from ourselves too. That’s an effect of history and the power of tradition, which very frequently trump logic and aesthetics.
What do you think? Does it seem plausible that our number names and our refusal to shift to the metric system may be giving our children a disadvantage in learning arithmetic and the math to follow? What was your experience learning arithmetic? Did you understand the decimal system, so the procedures made sense to you, or did you just memorize the procedures, with no understanding? It would be especially interesting to hear from people who grew up speaking Chinese, Japanese, Korean, or another language that makes the base-ten system transparent.
This blog is a forum for discussion, and your stories, comments, questions, and disagreements are valued and treated with respect by me and other readers. As always, I prefer if you post your thoughts and questions here rather than send them to me by private email. By putting them here, you share with other readers, not just with me. I read all comments and try to respond to all serious questions, if I feel I have something useful to say. Of course, if you have something to say that truly applies only to you and me, then send me an email.
For much more about children’s natural ways of learning, and the conditions that best help them learn, see Free to Learn.
 Portions of this post were taken from P. Gray, Psychology (6th edition, Worth Publishers).
 Miller, K. E., Smith, C. M., Zhu, J.,& Zhang, H. (1995). Preschool origins of cross-national differences in mathematical competence: The role of number-naming systems. Psychological Science, 6, 56-60.
 Miura, I. T., Okamoto, Y., Kim, C. C., Chang, C., Steere, M., & Fayol, M. (1994). Comparison of children’s cognitive representation of number: China, France, Japan, Korea, Sweden, and the United States. International Journal of Behavioral Development, 17, 401-411.