# What s Sophisticated about Elementary Mathematics?

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6 Understanding Place Value Many teachers, rightly in my opinion, believe place value is the foundation of elementary mathematics. It is often taught well, using manipulatives such as base-10 blocks to help children grasp that, for example, the 4 in 45 is actually 40 and the 3 in 345 is actually 300. But despite the importance of place value, the rationale behind it usually is not taught in colleges of education or in math professional development. That s probably because the deeper explanation is not appropriate for most students in the first and second grades, which is when place value is emphasized. But it is appropriate for upper-elementary students who are exploring number systems that are not base 10 (which often is done, without enough explanation, through games) and it is certainly something that math teachers should know. So here it is: the sophisticated side of the simple idea of place value. Let s begin with a look at the basis of our so-called Hindu-Arabic numeral system.* The most basic function of a numeral system is the ability to count to any number, no matter how large. One way to achieve this goal is simply to make up symbols to stand for larger and larger numbers as we go along. Unfortunately, such a system requires memorizing too many symbols, and makes devising a simple method of computation impossible. The overriding feature of the Hindu- Arabic numeral system, which will be our exclusive concern from now on, is the fact that it limits itself to using exactly ten symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 to do all the counting. Let us see, for example, how counting nine times is represented by 9. Starting with 0, we go nine steps and land at 9, as shown below But, if we want to count one more time beyond the ninth (i.e., ten times), we would need another symbol. Since we are restricted to the use of only these ten symbols, someone long ago got the idea of placing these same ten symbols next to each other to create more symbols. The most obvious way to continue the counting is, of course, to simply recycle the same ten symbols over and over again, placing them in successive rows, as follows *This term is historically correct in the sense that the Hindu-Arabic numeral system was transmitted to the West from the Islamic Empire around the 12th century, and the Arabs themselves got it from the Hindus around the 8th century. However, recent research suggests a strong possibility that the Hindus, in turn, got it from the Chinese, who have had a decimal place-value system since time immemorial. See Lay Yong Lam and Tian Se Ang, Fleeting Footsteps: Tracing the Conception of Arithmetic and Algebra in Ancient China (Hackensack, NJ: World Scientific, 1992). Historically, 0 was not among the symbols used. The emergence of 0 (around the 9th century and beyond) is too complicated to recount here. In this scheme, counting nine times lands us at the 9 of the first row, and counting one more time would land us at the 0 of the second row. If we want to continue counting, then the next step lands us at the 1 of the second row, and then the 2 of the second row, and so on. However, this way of counting obviously suffers from the defect of ambiguity: there is no way to differentiate the first row from the second row so that, for example, going both two steps and twelve steps from the first 0 will land us at the symbol 2. The central breakthrough of the Hindu-Arabic numeral system is to distinguish these rows from each other by placing the first symbol (0) to the left of all the symbols in the first row, the second symbol (1) to the left of all the symbols in the second row, the third symbol (2) to the left of all the symbols in the third row, etc Now, the tenth step of counting lands us at 10, the eleventh step at 11, etc. Likewise, the twentieth step lands us at 20, the twenty-sixth step at 26, the thirty-first step at 31, etc. By tradition, we omit the 0s to the left of each symbol in the first row. That done, we have re-created the usual ninety-nine counting numbers from 1 to 99. We now see why the 2 to the left of the symbols on the third row stands for 20 and not 2, because the 2 on the left signifies that these are numbers on the third row, and we get to them only after we have counted 20 steps from 0. Similarly, we know 31 is on the fourth row because the 3 on the left carries this information; after counting thirty steps from 0 we land at 30, and one more step lands us at 31. So the 3 of 31 signifies 30, and the 1 signifies one more step beyond 30. With a trifle more effort, we can carry on the same discussion to three-digit numbers (or more). The moral of the story is that place value is the natural consequence of the way counting is done in the decimal numeral system. For a fuller discussion, including numbers in arbitrary base, see pages 7 9 of The Mathematics K 12 Teachers Need to Know on my Web site at berkeley.edu/~wu/school mathematics1.pdf. H.W. AMERICAN EDUCATOR FALL

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