(Beyond "Because the teacher said so, that's why)
In my last post, I argued that understanding place value is an essential part of a comprehensive mathematical foundation. The key word here is understanding. What does it mean to truly understand a concept like place value? I would guess that anyone reading this blog "understands" that the 4 in 45 is in the tens place and represents 40. But why? What is the purpose of place value?
A digression: Our number system happens to be based on only 10 symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. To count more than 9 objects, we have to start repeating. For example, if we wanted to count 29 shells on the table, we might count as follows:
0 1 2 3 4 5 6 7 8 9
0 1 2 3 4 5 6 7 8 9
0 1 2 3 4 5 6 7 8 9
The issue of course is how do we keep track of which row we are on? In other words, how do we distinguish between the 6 in the second row and the 6 in the third row? That’s why place value was developed. By using two places rather than one, we can now distinguish between columns AND rows.
00 01 02 03 04 05 06 07 08 09
10 11 12 13 14 15 16 17 18 19
20 21 22 23 24 25 26 27 28 29
Now it makes sense why the 2 in 23 means 20. It is because the row beginning with 2 follows 2 rows of tens - in other words, two rows of ten have already been counted. And with an understanding of place value, we can now count to ANY number, no matter how large, using only these 10 symbols.
I would argue that even a child can understand this, and even more, should explicitly be taught about place value as he learns to count and recognize numbers beyond nine. This naturally happens when learning numbers in other languages. For example, in Chinese, 45 is 四十五or “four ten five” and 382 is 三百八十二 or “three hundred eight ten two.” Place value is embedded in the language. For English language learners, we need to teach them that the “teen” in eighteen does not just mean that there is a one in front of the eight, but that it represents ten. Similarly, “forty” means four tens, so 45 is really four tens and five (ones). And more importantly, we need to teach them why 45 is four tens and five ones and not just because the teachers says it is.
The beauty of truly understanding place value is that a child is then no longer limited to only understanding a base-10 counting system. She can now easily grasp a base-2 counting system that only uses 2 symbols: 0 and 1. It obviously takes a lot more time to teach a child true understanding rather than just saying “this is the way our number system is,” but establishing a strong mathematical foundation early will pay-off in the long run.