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 begin with units at 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 place value and 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, only to come un-stuck when understanding is required and the teacher is not there.
Australian students are notoriously bad at mathematics. Just look at the recent Trends in International Mathematics and Science study outlined in SMH 2016. 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 say one, two, three, four, five, six, seven, eight, nine, ten—and then 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 maths languaging 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 America 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. These results are also seen in Australia.
In an even more telling study, 6-year-olds in the United States, France and Sweden (where maths languaging contains irregularities) 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 (units). So, even before beginning any formal training in numerical calculations, the Asian children already have a head start—they implicitly know the base-ten system.
The base-ten numerical system is a beautiful thing; it’s too bad we mess it up with our language 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.
Maths Australia’s metric-based multi-sensory maths program builds a foundation of understanding through explicit teaching and showing maths through our custom made Integer Blocks. Students see, touch, feel and understand the language of maths based on a concrete-representational-graphic process. Place value and the base-ten system we use are easily learned as the basis for understanding all maths, preschool to Grade 12. To see more, visit www.mathsaustralia.com.au
What do you think? Does it seem plausible that our maths languaging may be giving our children a disadvantage in learning arithmetic and understanding maths? What was your experience learning arithmetic? Did you understand the decimal system, so the formulas made sense to you, or did you just memorize the formulae, 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.
 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.