When you think of operations as having a front office component and a back office component, where the front office component provides the public face and the back office component provides the implementation, as we first introduced in the post about *classes*, a new view of quantities emerges. In traditional school math, for example, there is little talk about cleanly separating issues of representation and implementation from the result that is to be produced. There is no perceived need. As a result, we tend to think of “14” as a number, rather than the representation of a number. We tend to look at “06/13/2009” as a date, rather than the representation of a date, and we tend to take for granted that “06/13/2009” has nothing to do with division, let alone two divisions. Similarly, “123-45-6789” looks like a Social Security Number, and not a set of subtractions. “34×32” looks like a blue jeans size, rather than a multiplication. In a sense, I think it is rather amazing that people, adults and kids alike, aren’t more confused than they are about notational issues. Typically, it isn’t until you run into an instance of real confusion that it becomes clear that there is something to be confused about.

The way we denote money amounts is one of those where there appears to be widespread confusion. An amount like $2.69 is easily read and interpreted as 2 dollars and 69 cents. The “$” symbol announces the dollars, and the “.” symbol announces the cents. This is a reading that is consistent with many other constructs: the symbols that announce can come before, in the middle, or after the number involved. In the construct 2’3″, which you are supposed to read as 2 feet and 3 inches, the symbol ‘ comes after. In the date 06/13/2009, the slash separates the month from the day, telling us the 06 is the month and the 13 is the day (following USA conventions), and the second slash separates the day from the year. In fact, the leading zero is completely unnecessary, and I can write or 7/4/2009 or 7/04/2009 or 07/04/2009 interchangeably without confusion.

So when a child writes “2 dollars and 9 cents” as follows: $2.9 *we* know that this is wrong because the “.” really denotes a decimal point. But how is the child to know this? Unlike the case with dates, suddenly the leading zero is critical. But it is only critical for the cents, not for the dollar part.

In the American system for denoting money values, there is a subtle clue the child gets when hearing “2 dollars and 69 cents”. The clue is that we are talking about value, and not about currency. When hearing “2 dollars and 69 pennies” the child knows we are talking about currency: 2 dollar bills, and sixty-nine penny coins. In contrast, “2 dollars and 69 cents” may come as 2 dollar bills, 2 quarters, 2 nickels, and 4 pennies – or in many other ways. There is no standard notation for 2 dollars bills, 2 quarters, 2 nickels and 4 pennies. What this suggests is that the value is considered important, but not the make-up in terms of particular coins and bills. This matches the convention that any number of coins adding up to 2 dollars and 69 cents will be accepted by the store. Yet it doesn’t match the convention that Coca Cola machines take quarters but not pennies, and that machines that take bills rarely accept bills over $5, and rarely accept 2-dollar bills at all.

When the store rings you up and announces the total amount you are to pay, they present you with the *value interface*. $2.69 can be paid in many different ways, and you expect the store to accept it all. You give them Susan B Anthony dollar coins, Kennedy 50-cent pieces, and expect them to accept it, though perhaps not with a smile. You expect to be able to pay them with a $20 bill and get change back. Where the model breaks down is when you present them with a $1,000 bill (I have to admit I have never even seen one, but I’m told they are in circulation). Conversely, if you present them with a $20 bill, they are allowed to give you the change as a huge stack of pennies, even if they will rarely do so.

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