Gas station without pumps

2013 October 7

Thinking about decimals

Filed under: Uncategorized — gasstationwithoutpumps @ 10:14
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In Money and decimals [TDI 2] | Overthinking my teaching, Christopher Danielson brings up some questions about whether money is really a good example for teaching about decimals.  Unfortunately, he has closed comments on the blog post, preferring to have tiny snippet discussions on Twitter or having people join some course website on instructure.com, neither of which appeals to me.  So I’ll discuss his points on my own blog instead.

The most interesting question he raises is the following:

 Is it possible, as Max Ray suggests below, that the conception people tend to carry in their minds is of dollars and cents as separate units, as they do feet and inches?

@Trianglemancsd though I am curious what students use to make sense of cents prior to fractions. A dollars unit and a pennies unit, right?

— Max Ray (@maxmathforum) September 27, 2013

I report my height as 6 feet 1 inch. I do not report it as 6\frac{1}{12} feet, although I know that I could. Likewise I don’t think of 1 hour and 5 minutes as 1\frac{5}{60} hours, although I know this to be correct.

Is it possible that many people think of $1.25 as 1 dollar and 25 cents, rather than as 1\frac{25}{100} dollars?

Maybe students are thinking of dollars and cents as different units that have a nice conversion rate, rather than of dollars as the natural unit and cents as a partitioning of that unit.

Follow-up questions: (a) Might Max’s insight help to explain the errors in the gallery of misplaced decimals? (b) What are the implications of this for using money to teach decimals?

I think it is certainly the case that most people think of money in terms of dollars and cents, not in terms of dollars and fractions of a dollar, and many of the notational mistakes one sees in ads supports this interpretation.

The Mad Hatter's hat was priced at "ten and six"—10 shillings and 6 pence, which was half a guinea.  Incidentally, I've never really understood why there was a unit for 21 shillings, when 20 shillings was a pound.

The Mad Hatter’s hat was priced at “ten and six”—10 shillings and 6 pence, which was half a guinea. Incidentally, I’ve never really understood why there was a unit for 21 shillings, when 20 shillings was a pound. (According to the Wikipedia article about guineas, it started out as a one-pound coin, but fluctuated in value according to the relative prices of gold and silver, before getting fixed at 21 shillings in 1717.)

It used to be that many monetary systems were not based on a decimal system—I’m old enough to remember the former British system of pounds, shillings, and pence (20 shillings to the pound, 12 pence to the shilling) with prices given with a slash separating shillings and pence.

The colloquial use of “and” to separate shillings and pence or dollars and cents is also indicative of the general thought pattern that these are separate units, neither composed nor divided in the way that Christopher likes to think of units. People don’t think in terms of dollars and fractional dollars, nor in terms of all prices in cents, but in terms of mixed unit of dollars and cents. (This may also help explain why 99¢ seems much cheaper to people than $1, while 99 and 100 are close—though the tendency for people to truncate rather than rounding is probably a bigger part of the story.)

Because people think of dollars and cents as separate units, using money as a metaphor for explaining decimal notation is not as useful as it might be. With separate units, the conceptual process is one of unit conversion and scaling, not of place value. Unfortunately, more useful models (like using cm, mm, and meters; grams and kg; mL and liters) are not open to US educators, because of the archaic units (inches, feet, and miles; tons, pounds, and ounces; pints, cups, tablespoons, and teaspoons) still used here.

A lot of teachers claim that decimals are hard—harder than fractions—but I’ve never understood that claim. Place value always seemed easy and natural to me, so having more places to the right seemed like a trivial extension.  None of the arithmetic algorithms changed (other than having to keep track of where the decimal point ended up), so decimals always seemed like a simpler extension to whole numbers than fractions were.

Of course, fractions are a more powerful and more fundamental concept than decimals—not all rational numbers can be expressed as simple decimals, and the notational conventions for repeating decimals are not obvious nor easily explained. So the rather complicated relationship between fractions and decimal notation is understandably difficult to teach.  But I would have thought that the problem comes more from the fractions than from the decimals.

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