# What's Sophisticated about Elementary Mathematics? Plenty—That's Why Elementary Schools Need Math Teachers, by Hung Hsi Wu; American Educator, Fall 2009; American Federation of Teachers

## Transcript

6 Understanding Place Value In this scheme, counting nine times lands us at the 9 of the first Many teachers, rightly in my opinion, believe place value is the row, and counting one more time would land us at the 0 of the foundation of elementary mathematics. It is often taught well, second row. If we want to continue counting, then the next step using manipulatives such as base-10 blocks to help children grasp lands us at the 1 of the second row, and then the 2 of the second that, for example, the 4 in 45 is actually 40 and the 3 in 345 is row, and so on. actually 300. However, this way of counting obviously suffers from the But despite the importance of place value, the rationale defect of ambiguity: there is no way to differentiate the first row behind it usually is not taught in colleges of education or in math from the second row so that, for example, going both professional development. That’s probably because two steps and twelve steps from the first 0 will the deeper explanation is not appropriate for land us at the symbol 2. The central break- most students in the first and second grades, through of the Hindu-Arabic numeral system is which is when place value is emphasized. But it is to distinguish these rows from each other by appropriate for upper-elementary students who placing the first symbol (0) to the left of all the are exploring number systems that are not base 10 symbols in the first row, the second symbol (1) (which often is done, without enough explana- to the left of all the symbols in the second row, tion, through games)—and it is certainly some- the third symbol (2) to the left of all the symbols in thing that math teachers should know. So here it the third row, etc. is: the sophisticated side of the simple idea of place value. 00 01 02 03 04 05 06 07 08 09 Let’s begin with a look at the basis of our 10 11 12 13 14 15 16 17 18 19 The most so-called Hindu-Arabic numeral system. * 20 21 22 23 24 25 26 27 28 29 basic function of a numeral system is the ability to 30 31 32 33 34 35 36 37 38 39 . . . . . 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 90 91 92 93 94 95 96 97 98 99 we go along. Unfortunately, such a system Now, the tenth step of counting lands us at 10, the requires memorizing too many symbols, and eleventh step at 11, etc. Likewise, the twentieth step makes devising a simple method of computation lands us at 20, the twenty-sixth step at 26, the impossible. The overriding feature of the Hindu- thirty-first step at 31, etc. By tradition, we omit the 0s Arabic numeral system, which will be our exclusive to the left of each symbol in the first row. That done, concern from now on, is the fact that it limits itself we have re-created the usual ninety-nine counting to using exactly ten symbols—0, 1, 2, 3, 4, 5, 6, 7, numbers from 1 to 99. † 8, 9—to do all the counting. Let us see, for We now see why the 2 to the left of the symbols example, how “counting nine times” is repre- on the third row stands for 20 and not 2, because the sented by 9. Starting with 0, we go nine steps and 2 on the left signifies that these are numbers on the land at 9, as shown below. third row, and we get to them only after we 9 → → 3 → 4 → 5 → 6 → 7 → 8 2 1 → 0 → have counted 20 steps from 0. Similarly, we know 31 is on the fourth row because the But, if we want to count one more time beyond the 3 on the left carries this ninth (i.e., ten times), we would need another symbol. information; after Since we are restricted to the use of only these ten symbols, counting thirty steps someone long ago got the idea of placing these same ten from 0 we land at 30, symbols next to each other to create more symbols. and one more step The most obvious way to continue the counting is, of course, lands us at 31. So the 3 of to simply recycle the same ten symbols over and over again, 31 signifies 30, and the 1 signifies one more step beyond 30. placing them in successive rows, as follows. With a trifle more effort, we can 0 1 2 3 4 5 6 7 8 9 carry on the same discussion to 0 1 2 3 4 5 6 7 8 9 three-digit numbers (or more). The 0 1 2 3 4 5 6 7 8 9 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 * This term is historically correct in the sense that the Hindu-Arabic numeral system numbers in arbitrary base, see was transmitted to the West from the Islamic Empire around the 12th century, and pages 7–9 of The Mathematics 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, on K–12 Teachers Need to Know who have had a decimal place-value system since time immemorial. See Lay Yong http://math. my Web site at Lam and Tian Se Ang, Fleeting Footsteps: Tracing the Conception of Arithmetic and berkeley.edu/~wu/School Algebra in Ancient China (Hackensack, NJ: World Scientific, 1992). . mathematics1.pdf † Historically, 0 was not among the symbols used. The emergence of 0 (around the 9th –H.W. century and beyond) is too complicated to recount here. 9 AMERICAN EDUCATOR | FALL 2009