On Monday I was walking with one my colleagues (Paul Scutt) as we tromped around Cranbury, NJ, on an excursion to the local historical museum and the store fronts of the town’s main street. We’d just had a bit to eat at Teddy’s, a local diner. Paul was wondering about a possible discrepancy in the bill and was calculating what the price of one drink must be if the total we’d been charged for three drinks was $5.25.
Paul was working out the quotient aloud in the style of long division. “3 into 5 is one with 2 left over…” Going through the steps was tricky—there was some retracing steps and checking involved—but things carried through and the answer came out as $1.75. This squared with the prices on the menu. All was well.
Through some of this I was only half listening. My thoughts were distracted by something or other; perhaps I had already begun my alternate calculation. Soon enough I was sharing my thoughts with Paul: “$5.25 is 5 and 1/4 or 21/4. 21/4 divided by 3 is 7/4, and that’s $1.75.”
It was an occasion to recall that over the course of my years of teaching math—and middle school math, in particular—I’ve developed a preference for fractions over decimals. Decimals, I’ve been inclined to argue, are just a special case of fractions—fractions with powers of ten in the denominator that we can suppress with a bit of ingenious and pragmatic notation. And decimals are a very slick trick—calculating with them is practically verbatim from whole-number arithmetic.
But then again, seen in a certain light, so is calculating with fractions—whole numbers that we calculate with while carrying them around in separate buckets. And since calculations with fractions have their own shape to them—their own kind of mental juggling—their meaning isn’t sedimented and murked up with their relation to ordinary place-value arithmetic. What is gained in ease is often purchased with opacity. This is true throughout math, and I’ve certainly found it to occur with decimals in particular. From these experiences, I’ll claim that it’s easier to help kids to think about fractions than about decimals, and that a teacher with a particular bent of mind can safely leave decimals as an afterthought. (Note: not “should” be.)
The ability to think flexibly about both decimals and fractions and to move between them comfortably is certainly the ideal. But how do we help students get there? What are the paths that we carve?
This post was prompted by an exchange on Twitter:
What thoughts do you have to add? How do you think about the relationship between fractions and decimals? How do you talk about this with your students, or how might it affect your classroom silently? What is your own relationship with computations involving fractions and decimals—any preferences, for any reason? And what are good examples of their relative strengths and weaknesses?
My Algebra 2 students have a [maddening] preference for decimals, and I link this back to their dependence on calculators. For example, they much prefer expressing 1/2 as .5 in a linear equation (as slope, for example), and will consistently refer to it as ‘point 5’ rather than ‘five tenths’. I don’t think fractions being cumbersome has anything to do with it, and I have a sense that there may be a lack of conceptual understanding at play here. (Wish I could be more articulate about it, but there’s my 2 cents – or two hundredths – worth!)
I like fractions for computations and decimals for size. If I need to compare two numbers, decimals are certainly easier for me. If I need to report something especially a probability, decimals (and thus percents) are easier there, too, I think.
As you mention above, 71164/100000 is fine for mathematical understanding, but if you were telling me that number, I would lose track with all the words there. To the \point\ I can more easily understand you if you say “point seven one one six four” and know the relative size of that thing.
In general, I tell my students to leave fractions as answers since they are more accurate. With calculators all around us these days, if decimals are requested/needed they can be calculated and rounded to necessary precision from there.
Reading the comment on percents/percentage I think makes the point of the post – students really do not get that a percentage is parts per 100. They can calculate it in a single direction all day long, but applying it in it’s various forms leaves them befuddled. As for the example, why not discuss it as 0.7 parts-per-million?
As I read the problem of $5.25 divided by 3, my mind went with: That’s $1 plus something per drink. The $2.25 remaining is 9 quarters, so each drink costs another 3 quarters, or 75 cents, thus $1.75.
Forget the math, Justin, I just need to remember to visit Teddy’s when I’m in the area. :)
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