Via Mark Chu-Carroll I just read this article, from the USA Today, about a mathematician at the University of Pennsylvania who believes that fractions have no place in the elementary and middle school mathematics curriculum:
A few years ago, Dennis DeTurck, an award-winning professor of mathematics at the University of Pennsylvania, stood at an outdoor podium on campus and proclaimed, “Down with fractions!”
“Fractions have had their day, being useful for by-hand calculation,” DeTurck said as part of a 60-second lecture series. “But in this digital age, they’re as obsolete as Roman numerals are.”
The speech started a firestorm, particularly after the university posted it online.
“There were blogs and rants, and there were some critical e-mails,” said DeTurck, who is now dean of the college of arts and sciences at Penn. “They’d always boil down to: ‘What would we do in cooking and carpentry?’ ”
DeTurck is stirring the pot again, this time in a book scheduled to be published this year. Not only does he favor the teaching of decimals over fractions to elementary school students, he’s also taking on long division, the calculation of square roots and by-hand multiplication of long numbers.
Since I am suspicious of the USA Today‘s ability to report accurately on this sort of topic, I will withhold final judgment until I have read DeTurck’s book. However, based on what is reported in this article I must tentatively conclude that, sadly, he is out of his mind. Mark has already said most of what needs saying, but I have a few disagreements with him as well.
For the moment, let us pass right over the specific and proceed directly to the general. Every so often someone comes along and argues that since calculators can carry out many of the computations children often struggle to learn to do by hand, traditional arithmetical lessons are now obsolete and should be abolished. The idea behind this seems to be that the only reason for carrying out a computation by hand (or in your head) is to obtain the correct numerical answer to something, and if calculators allow you to do that more reliably and with less stress then so much the better.
Overlooked in this argument is the simple truth that strong arithmetic skills are the foundation of any future work in mathematics. You have absolutely no hope of handling algebra or geometry (much less calculus and beyond) unless you can perform basic arithmetic proficiently, accurately and, yes, quickly. I have seen this time and again in my own teaching experiences. All branches of mathematics, from the basics taught in grade school to the highest echelons of research mathemativs, present you with a nearly endless stream of arithmetical puzzles. If you are constantly hesitating or reaching for your calculator when this happens then you will never grapple successfully with the underlying logic of the situation. Facility with arithmetic also develops number sense, which is crucial in higher branches of mathematics.
Returning now to the specific, the idea of teaching only decimals (presumably because they are more readily entered into a calculator) and not fractions is especially ridiculous. Fraction notation and decimal notation are simply two different formalisms for the same underlying concept. Both are enormously useful, which, indeed, is why most mathematicians recommend learning all about each of them. Fraction notation, however, strikes me as far more natural for young children to learn. I don’t even know how you would teach a child about decimals until they already have some facility for fractions. How do you explain what .5 is without using the phrase “one half”?
Decimal notation is based on the idea of place value, and that is a concept children often find abstract and confusing. Fractions, by contrast, are immediately linked up to things in their everyday life. Children can get the hang of the idea that if you have three eighths of a pizza on your plate, then you should picture a pizza sliced into eight equal pieces, three of which are now on your plate. I can’t imagine how to explain to a child who doessn’t know fractions that he has .375 of a pizza on his plate. Furthermore, you learn a great deal about numbers from understanding, say, why adding fractions involves finding a common denominator while mulitplying them involves going across the tops and bottoms as two separate little multiplication problems. How Professor DeTurck expects students to, for example, evaluate deriviatives without being able to mainpulate fractions is unclear. (Perhaps he would recommend just memorizing the rules for finding derivatives, without giving any consideration to where the rules came from.)
That said, there are a number of standard tropes of how fractions are taught that really ought to go. For example, the words “numerator” and “denominator.” Refer to the top and bottom of a fraction and everyone knows what you are talking about. Numerator and denominator, by contrast, mean precisely nothing. (Actually they descend from Latin words meaning, respectively, “He who numbers” and “That which is named.” Terribly important to keep those words around.) Or how about the nonsense that when you are adding fractions it is terribly important that you don’t simply find a common denominator (a simple process carried out by multiplying together the bottoms of the two fractions), but instead you must find the least common denominator (LCD)? Where did that come from? Use something other than the LCD and your final answer will not be in lowest terms. The horror! If that bothers you then reduce the fraction after you carry out the addition.
(Incidentally, the absurdity of obsessive fraction reduction was brought home to me not long ago when I was pawing through a set of drill bits looking for the appropriate size. Who ever stamped the bit widths onto their sides could not bring himself to write 4/32 or 6/32. No. They were 1/8 and 3/16. Some of my bits were measured in eights, some in sixteenths, and some in thirty-seconds. Very annoying, but at least I was able to do the conversions quickly in my head. Woe to the would-be driller who learned arithemtic by the DeTurck method…)
Happily, the article also includes some contrary thoughts:
Questioning the wisdom of teaching fractions to young students doesn’t compute with people such as George Andrews, a professor of mathematics at Pennsylvania State University and president-elect of the American Mathematical Society. “All of this is absurd,” Andrews said. “No wonder mathematical achievements in the country are so abysmal.
“Arithmetic is the basic skill. If children do not know arithmetic, they can’t go on to algebra, which leads to calculus. From there you go on to other things,” Andrews said. “It’s fine to talk about it, but this is not a good pedagogy.”
Others see value in both fractions and decimals. To Janine Remillard, associate professor of education at Penn, the decimal system is “incredibly powerful.” And fractions can be a powerful steppingstone to understanding decimals, she says.
“Fractions, if taught well — and that’s a huge caveat — can actually help kids understand the value of the size of the pieces,” Remillard says.
Now for the part where I disagree with Mark. The article closes as follows:
Penn State mathematician Andrews says he believes DeTurck’s ideas will “unfortunately”” gain traction because of the misguided belief that math education can somehow be made easy:
“Math is hard. The idea that somehow we’re going to make math just fun is just a dream.””
On the other side, many of the responses I’ve seen have been like this one from the USA Today article: “Math is hard. The idea that somehow we’re going to make math just fun is just a dream.”
Math is fun! It’s idiots like DeTurck and friends who ruin the fun of it, by turning it into nothing but repetitive rote exercises that don’t mean anything. Anyone who says that math can’t be fun should be eternally banned from teaching math.
Last year, I went to my daughter’s first grade class, and did a project with them, where each kid made a four-column abacus. Then I showed them how to add big numbers on the abacus. They were so excited! The idea of being able to do it was thrilling, and the idea that they made this thing that let them do it, they were so happy, having so much fun. That’s how math should be. Of course there’s rote – just like there is for reading. You’ve got to memorize some things, you’ve got to learn the skills, and practice them. And practice isn’t always fun. But teaching math should make time for the joy of being able to do something new – and make sure that it’s taught as something fun.
I look at my daughter’s second grade class now. She’s got a wonderful teacher. And the teacher really does make math fun for the kids. Sure, it’s hard sometimes. But it’s also fun, and she’s great at making the kids see that.
Both DeTurck and his quoted opponent don’t believe that you should do that. DeTurck thinks it should be nothing but rote. And his opponent thinks it should be hard, not fun.
They’re both idiots.
This, I think, is very unfair to Professor Andrews. Mark has simply overlooked an important word in Andrews’ statement. Andrews said that the idea that we’re going to make math just fun is a dream. He is not saying that we should make no efforrt to make math fun for kids, and he certainly isn’t saying anything that would justify Mark’s comment that he would object to Mark’s lesson about the abacus.
His point was simply, and reasonably, that the drudge work of learning your multiplication tables and other bits of arithmetical formalism is an important part of becoming educated. The only way I know of really learning these things is by rote memorization and drill. This is boring, tedious and time-consuming. The difficulty of learning arithmetic in its various forms is sometimes damaging to a child’s self-esteem. There is no teacher on Earth creative enough to make it interesting. It is critically important nonetheless. Yes, of course, do creative activities that engage the students. Just realize that that’s not the whole story.
In short, it looks to me like Andrews was saying exactly what Mark himself said in the passage above.
I would probably go a bit further. My impression is that elementary school math classes have tipped the balance too far in the direction of “wonder” at the expense of making sure that students can actually do stuff. I see students every day who can’t factor quadratic equations with any facility because the problem of, say, finding two numbers whose product is 56 and whose sum is 15 is not something they can do quickly. I suspect that is because there was an insufficient quantity of drill and rote memorization in their past. Frankly, I’m less concerned with making math fun for kids than I am with making sure they can do the basics of the subject. Send me students with some facility for arithmetic and I’ll take care of reawakening their imaginations.