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.
I had a high school student tell me today that 1/4 is bigger than 1/2 because .25 is bigger than .5 because 25 is bigger than 5. (I had docked points for a misplaced line on graph paper. I'd assumed it was a minor error, might've docked more points if I'd realized the thinking behind it.) I drew a bar, chopped it into four pieces, shaded in 1/4 and 1/2. How could I have demonstrated .25 and .5 visually? I rewrote .5 as .50, but this seems way more abstract.
I have students who don't know how to tell when a number is divisible by 2 or 5 or 10 because they've never had to pay attention. I have students who want to turn every number into a decimal: fractions, roots, pi. They insist that the decimal value is more precise, and it is a struggle to convince them otherwise. More significantly, they are uncomfortable manipulating symbols, want to be rid of them as soon as possible.
"The only way I know of really learning these things is by rote memorization and drill."
Drill yes, but not necessarily memorization. I found that to be the dullest thing *ever*, and was constantly told that I was no good at math because I couldn't memorize tables.
I homeschooled one of my children for several years, and I really think that 1 - you should just give the kids a multiplication table and let them use it. They will memorize it through use much better than in preparation for a test (after which they promptly forget it). 2 - Math books that they use in schools really stink. (I used Singapore math books with my son, and he learned just fine with not as many problems and a lot more fun activities. And they also teach metric...)
I agree that fractions are necessary for decimals (or ratios!) to make sense. Fractions are also critical for trig and geometry. Very few things in those classes would make sense without fractions.
I don't get the reliance on calculators at all. I managed to get a degree in physics and the only classes in which I used my calculator were the basic calc series.
I have no problem with the idea of letting children use a multiplication table when they are starting out. That seems like a perfectly reasonable way of doing things. I intended simply that at some point the multiplication tables have to be not just committed to memory, but actually become automatic.
Are fractions obsolete? Half of me says they are, 50% of me says they are not.
We cannot give up fractions in education! While my kids are learning the basics, I've come to realize that math is a matter of pattern recognition. They struggle so much more in school than I ever did, and I so want them to see the patterns the way I always did. So while the schools teach them the curriculum, I try to show them how everything they do involves some kind of math. My older child hated math until one day, all of the sudden, it clicked. She saw the patterns, and how they are everywhere. Now she loves math, but she never would have without all the work PLUS the extra weird stuff I added.
And "hard" and "fun" are not opposites. The hardest things we do can be the most fun... and the most rewarding.
How does anyone teach decimal notation without teaching fractions? After all, ".123" is simply 1/10 + 2/100 + 3/1000 = 123/1000. Decimal fractions are themselves fractions. Who doesn't get this?
Here's another need for fractions where decimals won't do. Suppose you're the navigator on a sports-car rally, and you need to turn miles per hour into minutes per mile or seconds per tenth-mile. Knowledge of fractions is crucial to figuring out who to divide or multiply by whom. Or maybe you're buying gas for your car and want to turn odometer reading and gallons purchased into miles per gallon. With fractions, it's easy to figure it out by just dividing whatever by whatever makes the units come out looking sensible.
It's sad people have trouble with fractions.
I find that any time I'm solving a typed equation (physics being the archetypical example, but there's others), if I stay with fractions as long as I can, I get much fewer errors and it's simply easier to keep track of what I'm doing.
I'm quite surprised nobody has raised the point that fractions and decimals do *not* represent the same things.
If you consider only finite decimals, most fractions can't be represented at all; the best you can do is an approximation. It's important for kids to understand the "real" mathematics before they understand the compromises we make to help our calculators. 33% is not one third, and if a math teacher isn't equipping her students to understand that a cake really can be cut into three equal pieces is failing them...
If you allow infinite decimals, then we wander into the scary differences between the rationals and the reals. I find it truly weird that we're contemplating a world in which this would be the first point at which a student learns about fractions.
You could limit yourself to repeating decimals to get similar expressiveness, but if you can explain to a fourth-grader that 0.9999,,, add 1.0 are the same number, then you shouldn't have any trouble explaining fractions as well.
"(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...)"
If the US converted to the metric system of units, this rubbish wouldn't be necessary.
Roy Clouser covers some of the history of math ed in his book The Myth of Religious Neutrality. For those in education the book is a fascinating read on this topic.
This question goes to the very nature of mathematics. We see "1+1=2" as a straight-forward formula. But when we say "5-8=(3)" then we see a relationship determined by the debt of three on the right side. The elimination of fractions, likewise, eliminates the relational character of a mathematical formula. It matters not whether we're talking metric or english, the problem remains the same.
There is also the matter that fractions are used in symbolic expressions as well. (x + y)/z is a fraction, but obviously cannot be represented as a decimal. If students have already learned numerical fractions, the use of fractions in symbolic expressions can be straightforwardly built on that previous learning. If they have only learned decimals, not so much.
Wow, no wonder math achievement in this country is abysmal.
I always loved fractions, especially for advanced calculations in Trig because they are much more elegant and accurate than decimals, and I do believe that my high math achievement in high school resulted from that. But the students I now tutor are completely lost on them, and make that impulsive grab for the calculator at the first sign of a fraction. I have had to deny them use of it to force them to do the problem as a fraction. But that little bit can't undo the cult of the calculator they get every day for twelve years.
All the basic math skills and algebraic rules are being thrown out the window, and students are using their calculators earlier and earlier in their schooling. But they are being crippled mentally because they are not learning the basic foundational concepts of math. It is sad that I can beat them on their calculator in many calculations, especially those involving fractions, because I had a very strong foundation in math and use of mental math techniques.
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 would aurgue this as I am a math major in colege studing such topics as ordinary diffrential equations and linear algebra despite the fact that i didnt get good at additon untill high-school, and still have to work out multiplication with scrach. (like 7 * 9 = 7 * 7 + (7 * 2) = 49 + 14 = 63?)(rather, I would say that a understanding of the propertys of additon and multiplication, is what is nessary.)
On the other hand I agree that fractions are totaly nessary, and i was teaching my-self algebra in elementry school so genralizing from me is a bad idea. and I dont own a calculator. (and have yet to need one). (if i was a enngeniering studing then I would need a calculator, but for straight math, "Bah!" I say. sqrt(pi/2)-2*sqrt(e) is a perfectly good answer)
(spelling is also not my strong suit)
As a child in the 80s and 90s, I had to make the realization myself one day that fractions and decimals and percentages and division were all the same thing -- none of my teachers ever told us this, it was just a big list of things with names that we were being taught separately. I think it was the fact that the division symbol and the percent symbol both looked like fractions that tipped me off -- but again, nobody ever even mentioned what the symbols might mean either.
I turned out fine, finishing up a PhD now, but it surprises me sometimes that I came through okay.
Wow. It really sounds like he expects that everyone will grow up to be waiters and janitors. People don't necessarily need to understand fractions to do their jobs and be successful, but tech, medical, manufacturing and engineering jobs (i.e. higher paying jobs) often require a lot more math than primary school arithmetic.
It seems like the default position would be to give them the skills they need to be as successful as possible, not to teach them the bare minimum they need to know to scrape by.
When I learned fraction we added them like you suggest Jason. Just cross multiply the tops, add the new tops, and then reduce.
I am sorry to say, converting to metric would not remove the problem. There a a lot of stuff out there using the old measurements, and it won't all be replaced for hundreds of years, if that would even be desirable.
I live in Denmark, where we have been metric for - well I don't know how long, lets say forever, and we have wrenches in both systems - a the wonder of it all.
My music theory teacher frequently taunts my class by alleging that his generation made sure my generation would suck at arithmetic in order to make sure that we never figured out that we were being screwed, and to make sure that we never overtook government scientists.
I'm beginning to think that he's right. I'm off to do a simple calculation ("what's the frequency of the nth partial of a given musical pitch?") that will take a pathetically long time to do in my head.
(Wow, that was some interesting grammar...)
The old-fashioned carpenter's bits (the ones with the squared tops and the pointy lead screws) are in fact numbered consistently. Each is traditionally labeled with a number that is the "top" of the fraction whose "bottom" is 16. Thus, a 3/8" bit (or 0.375", if you insist) is labeled "6".
And a friend once told me that his father or uncle or some older relative would read the fractions on a carpenters ruler as the number of inches (they're all written on the ruler) plus the number of "bigs," mediums," and "littles," referring to the length of the dividing lines. I can't reproduce that system for myself, but have on occasion found myself saying, e.g., "one-half plus one sixteenth" without mentally calling it "nine sixteenths." If you're just going to reproduce a measured length, then the mental LCD conversions aren't important, IMHO. (This is true all the more when you're into 32nds, 64ths or 128ths.)
i had a student bring that article in to me. a 7th grade student who is studying fractions in my class, right now. and a host of examples of why you should teach fractions came to me almost immediately.
i'd like to see someone teach the concept of dividing a whole into 7 parts to a 10-year-old without fractions. with fractions, on the other hand, that concept is not fundamentally different from dividing a whole into 5 parts.
one important interpretation of a fraction is as a ratio, which is a hugely important concept. slopes, for instance, are ratios. without a background in fractions, the idea of reducing a ratio is unconnected to any other concept.
nothing hammers home the importance of the concept of a prime factorization better than showing how you can add (4/33) + (2/77) without even knowing what the least common denominator is, at least until the end.
the simplest proofs of the countability of the rationals is based on the fact that they can be written in the form of a fraction.
basic trigonometry requires being able to count in units of pi/6 and pi/4. if you never learned about fractions, that is not an easy concept.
does this man seriously think people are going to stop writing division with a horizontal bar between a dividend on top and a divisor on the bottom? as long as we have that notation, it's ridiculous to consider 2/9 as meaning nothing other than the number 2 divided by the number 9, which has a place on the number line only because the result of the division is .22222....
need i go on?
Rather than pile on this obviously kooky idea, what are we missing? The guy clearly isn't an idiot. Is there some kernel of truth, or something useful, some principle he is overextending that I am missing?
It really sounds like he expects that everyone will grow up to be waiters and janitors. People don't necessarily need to understand fractions to do their jobs and be successful
"Waiter; is it possible to have half a portion?"
"Janitor; the lose board is three quarters of the way up the stairs."
My hypothesis is that DeTurck is ignorant of the way elementary schools actually do teach mathematics; instead of trying to solve problems which do exist, he's trying to solve a problem which he imagines to exist. Notice that he opposes teaching how to extract square roots by hand. I wasn't taught this, nor were my fellow grad students and the post-docs I work with; judging by the comments I've received, it went out of style circa 1980.
[Andrews'] 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.
It's more fun if you're a synaesthete. ;-)
The best part about learning how to divide fractions, back when I was a wee pup, was after the test having a pineapple upside-down cake party to celebrate.
I'm being a nitpicky prick, but numerator and denominator have perfect meaning when considering the lower part of a fraction to denote the fractional denomination and the upper part, which acts a multiplicand, to denote the number of denominations present. Referring back to your Latin definitions, the words express relationship between top and bottom as the "one who numbers" the "named" fractional value.
All pedanticism aside, I do think top and bottom are much simpler, but the meaning behind the former terms isn't entirely without value in understanding for those seeking deeper insight into fractions.
Yes, fractions are obsolete -- at least, mixed integer and ratiometric representation (such as 2 7/8; please, either call it 23/8 or 2.875, or some rounded version).
Keep the ratio form for exact work with abstract figures (and don't worry too much about putting things into lowest terms, which can always be done later); but everything in the real world already works in limited-precision decimal notation. Filling up your car: 26.7 litres of unleaded at Â£1.096 costs Â£29.26 (after rounding). Putting up some shelves: We want 4 shelves evenly spaced, the height of the ceiling is 2.25m. and we want the bottom shelf 1m. from the floor, so they will have to be (2.25 - 1) / 4 = 0.3125m. apart, which we can round off to 313mm.
Notice from that segue in the last example, by the way, that prefixes (as applied to measuring units) are really just another way of denoting fractions (or multiples). There is a good reason why we don't label a standard 19-piece set of drill bits as "1/1000", "3/2000", "1/500", "1/400", "3/1000", "7/2000", "1/250", "9/2000", "1/200", "11/2000", "3/500", "13/2000", "7/1000", "3/400", "1/125", "17/2000", "9/1000", "19/2000" and "1/100" -- even though it would be entirely mathematically correct to do so! "m" can be thought of as just an abbreviation for "/1000", "Âµ" is just an abbreviation for "/1000000", and so on.
There something I don't understand here. When I was in high school, Casio came out with this calculator, the FX82super FRACTION. Notice that last word? This cheap calculator could do fractions perfectly 15 years ago. And now kids should not be taught fractions because they're obsolete in 'the digital age'?
This cheap calculator could do fractions perfectly 15 years ago. And now kids should not be taught fractions because they're obsolete in 'the digital age'?
The point is this:
By the time you get to the stage of a calculation when you are ready to use a calculator (i.e. when all the abstract algebra is already done, and the only thing that remains is to substitute-in figures for variables in your formula) you presumably are working with some real-world quantity. Which is naturally going to be best expressed in decimal notation, simply because measurement units always go up in powers of ten anyway.
In fact, that's what the ENG(ineer's Notation) button on a calculator is for: it re-scales the mantissa so as to force the exponent to a multiple of 3 (i.e., the multiples for which there are named prefixes). For instance, if you calculate something that happens to be a capacitance and it comes out as 1.424E-7 farads, a press of ENG will change the displayed value to 0.1424E-6; or a press of INV followed by ENG will change it to 142.4E-9.
Someone explain to me why I have to rationalize the denominator?
My daughter (four years old) is learning addition, subtraction, multiplication, and division at her preschool (a genuine Montessori school). There's rote, but they don't emphasize memorization. They use "number rods" to get a strong visual feel for the quantities, which can be grouped into sets of ten and stacked to form squares and cubes, usefully demonstrating powers and simultaneously demonstrating why volume scales up so much faster than, say, width. The emphasis at this stage is on demonstrating the patterns that emerge from similar arithmetic problems. My daughter is kinda bored with it (reading is more her "thing"), but she's grasping it much better than I did at age seven, when I was memorizing my times tables.
So I'm not sure it's so important to make it *fun* but it is important to find a way of teaching it which actually helps the kids to understand. And there is definitely such a thing as too much rote. I remember doing over a hundred nearly identical problems in a typical assignment, due the next day. That was excessive, and taught me that math is stupid gruntwork. When I took calculus in college and loved it, I felt cheated. I'd actually rather do one problem that takes two hours than 120 problems that take thirty seconds, even though the latter is supposedly easier and would take the same amount of time. Hell, I'd rather do the single two-hour problem than sixty thirty-second problems, even though the latter would be done more quickly.
Fraction are obselete?
Fractions are utterly precise while decimal notation is sometiems simply an abstaction.
23/29ths as a deciaml is? As a fractions, it's...23/29 ths. Not very difficult is it?
1/9 as a decimal is 0.1111... when using base 10 and while is expressed pretty much as well as one can in base 10 is innacurate. Change the base and we can express it accurately as non fraction (0.14) .
I think it very easy to answer what 1/7th of 1/317 th is as a fraction, good luck to the average middle school student trying to get that right in decimal notation. My calcualtor gets it wrong everytime.
Interesting article. As a 42 year old returning to university to do calculus as part of my aeronautical engineering degree, I sometimes wish all this happened by magic and I didn't have to calculate anything.
I have realised one thing though. I will put my fractions up against the decimal system any day. If we both had to write Pi I figure Mr. decimal would be there forever whilst I'll be home in a flash for tea and medals.