A bunch of people have been sending me links to a USA Today article about a math professor who wants to change math education. Specifically, he wants to stop teaching fractions, and de-emphasize manual computation like multiplication and long division.
Frankly, reading about it, I'm pissed off by both sides of the argument.
On one side, you've got Professor DeTurck, who wants to abolish fractions, in favor of teaching children only decimals. This is a perfect example of an out-of-touch academic making idiotic proposals.
To be abundantly clear, I don't think that academics are, in general, out of touch, clueless, ignorant fools. I'm not a subscriber to the "ivory tower" view of academia. But like any other group, academics have idiot members who do their best to reinforce the negative stereotypes about the group.
But Professor DeTurck has clearly not actually looked at how kids learn about numbers. In fact, I don't think he's looked at how normal people actually understand numbers. Many (most?) adults actually don't really understand decimals. Try asking a random adult what 20% means - you'll get some astonishing answers.
Back in grad school, my wife and I moved between two different apartments in the same complex. The rent in one was $600. The rent in the other was $650. We moved on the 12th day of january. When we moved, the rental agency said we needed to pay the pro-rated difference between the two rents. By their calculation, that meant that we owed them $60. I tried arguing with the agent: how can pro-rating 19/31 of $50 be larger that $50? And the idiot agent kept answering: "I plug the number into my computer, and that's what it says." No matter how many times I kept pointing out: "the difference between the rents is $50. If we moved in on the first, that meant we needed to pay $50 for the month. If we moved in after the first, we'd by paying part of that $50." And the agent agreed with me. So we moved in on the 12th. That means we should be paying part of $50. Again, the agent agreed. So how much do we owe? $60. Is $60 part of $50? According to the agent "My computer says it is.". Eventually, I gave up in disgust.
Dr. DeTurck argues that the scenario above is fine and dandy. He doesn't think that understanding what parts means, or how numbers work is worth teaching to kids. Just let them use calculators and decimals.
The understanding of numbers starts with whole numbers. From there, you move from whole to parts. Fractions are parts. Kids understand fractions in terms of a very tactile representation of parts. 1/2 means what you get cut something into two equal pieces: each piece is one half of the original. 1/3 is what you get when you cut it into three equal pieces. And so on. It's very concrete, very tactile. It matches our natural intuitions about things: "I want a cookie for a snack, but my mommy says I can't have one because it's too big. So can I have a piece of one?" That's how my kids learned the basic idea of fractions.
According to Professor DeTurck, they should have learned "Can I have 0.5 cookies?"
Abandoning fractions to teach only decimals does exactly the opposite of what we should do in teaching math. The biggest problem, in my experience, with how math is taught is that we focus on mechanics to the exclusion of understanding. Switching to pure decimals without fractions is carrying that to a ridiculous extreme. What does 0.3 mean? It means 3/10 - three parts of something split into 10. If you do away with the fraction, then the decimal representations are meaningless. You can't explain what they really mean without using fractions - because they're just an alternative way of writing fractions. So what Professor DeTurck's proposal comes down to is: teach math without providing any intuition for what things mean. Just throw the decimals at kids, and let them solve problems with calculators. None of it means anything. It's just fiddling around with meaningless symbols.
How is that improving math education?
DeTurck also wants to de-emphasize manual computation. He doesn't think kids need to know how to do long division, or multiplication, or square roots. After all, why should they do those things by hand? A calculator can do it!
The reason to know how to do it by hand is because doing it by hand teaches you to understand what it means, and how it works. If DeTurck's proposals are accepted by schools, what will happen is that kids will end up with even less of a sense of what numbers all mean!
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."
Argh!
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.
Education & Careers
Comments
This reminds me of the moronic column by Washington Post columnist Richard Cohen where he advised someone that algebra wasn't important. The mainstream media is full of schmucks like this.
Posted by: SLC | February 1, 2008 2:31 PM
I observed one of our graduate students averageing 30 measurements. The measurements ranged from 6 to 8. The student used a calculater, and very confidently wrote down 9.3 for the average. I pointed out that this was unlikely, and explained how you do that kind of calculation in your head. The student did so and got a reasonable answer.
Posted by: Jim Thomerson | February 1, 2008 2:37 PM
That USA Today article reads like an uninspired exercise in he-said, she-said reporting. Small surprise. It also trims the statements of DeTurck's critics down to such tiny nubbins that I'm not confident about extrapolating their views on the subject (which is why my own typically snarky statement didn't discuss them).
You might be right, but I'd read his statement with the emphasis on "just fun", which makes his sentiment resemble that of this post. (The problem is conflating fun with easy; like that rascal Thomas Paine said, "What we obtain too cheap, we esteem too lightly.")
I'd argue that punching numbers into a calculator with no notion of meaning isn't fun, either.
Posted by: Blake Stacey | February 1, 2008 2:37 PM
It's well to remember that your experiences, and views derived from them, are unrepresentative and not applicable to most people. You're a software engineer for Google, and I'm sure your wife lies within a small neighborhood of you for intelligence -- as do the parents who live in your real-life neighborhood.
Because intelligence is heritable, your kids (and the kids of those around you) will be pretty smart too. Abstract and intuitive approaches may be just fine for them -- but try teaching these things to kids who aren't smart, meaning average or below-average. Average for the entire population, not average for a kid with smartie parents.
Even those who are smart enough differ in personality traits, which really express themselves in adolescence and early adulthood. Little kids may be able to get excited by math, or anything else -- but try it with teenagers! Only nerds will find it fun, and there's no changing that. Non-nerds may study it because they want to go into finance, but that's who most of your audience will ever be: smarties who see math as solely a means to an end (usually a monetary end).
These are the constraints that the universe has foisted upon our optimization. Ignoring them can only lead to folly.
Toward that end, I am glad there are calculators -- anyone who isn't smart enough to do math at the college level will at least know how to do tedious calculations easily. They still need to know what to do -- add or multiply, subtract or divide, and so on -- but they should off-load as much math as they can to the computer / calculator rather than struggle to keep it crisp in their brain.
Ultimately, whether or not to teach fractions (and with what weight) depends on reality. We know that students differ immensely in intelligence and personality traits, OK. But are some ways more natural than others? That can only be answered by empirical investigation.
Posted by: agnostic | February 1, 2008 2:44 PM
Intuition aside, I don't see how you can get very far in math without fractions. Fractions are the "real deal" - decimals are kind of the fake version. (My friend had a physics teacher in high school who started the semester with a question - "What is 1 divided by 3?" The answer is 1/3, not 0.3333...)
In algebra and everything that uses algebra, you have to use and understand fractions continually, including fractions that contain variables. You can't write x/10 as 0.x or in any kind of decimal notation that I'm aware of.
I remember what a big deal it was when I learned that fractions were the same thing as division. That's a fantastic piece of intuition that requires understanding fractions.
That is just totally wrong-headed.
Posted by: Tam | February 1, 2008 2:45 PM
We have a prof like that in the Netherlands as well: Kees Hoogland.
He thinks that students should rise up against learning antique math techniques. They should protest against having to learn long divisions and that they should instead be taught to be experts on the calculator/computer.
Luckily, not many people agree with him.
@Agnostic:
In the Netherlands all children are first taught basic maths, which is then followed up by a few short classes on how to do the same thing on a calculator. Out of the eight years they spend in basic education, maybe one week is spent on the calculator. Why? Because it's so darn easy. But also because a calculator is absolutely useless without knowing what it all means.
If we'd only teach our kids how to use the calculator, it'd be like going to a carpenters apprentice and saying: "This is a hammer, this is a saw, oh and that's a power-drill, be careful it's sharp." Nothing more.
I'm with Tam: the whole idea of fractions is indispensable in all maths above the level of seventh grade.
Posted by: Cailin Coilleach | February 1, 2008 3:06 PM
I agree that if Dr. Andrews always thinks math is hard he's a moron. Maybe if asked he would put a qualifier in his first sentence,such as "(Sometimes) Math is hard". Hopefully we can all agree with the qualified statement. And sometimes it's fun. We can hopefully all agree with that. What Andrews seems to have a problem with is the idea that it can always be fun. Don't you think that some topics are just hard to make fun, like some mathematics or the Holocaust, or any number of other topics?
Posted by: whitney | February 1, 2008 3:06 PM
Yeah, let's get rid of fractions. What a 0.5-baked idea!
Posted by: Neal | February 1, 2008 3:25 PM
@Neal - I don't often laugh out loud when reading, but you got me with that one.
Posted by: pough | February 1, 2008 3:39 PM
Speaking as a math grad student who has TAed many a course I wanted to say that your remark reminds me of the attitude every math grad has *before* they teach. Ultimately the problem comes down to the fact that not everyone will like math no matter what you do (short of drugging them).
The fundamental issue is that whether we like something or not largely comes down to whether we are good at it or not. Did you like sports as a kid? I sure didn't and sports literally are a game. If people don't like things that were created from the ground up to be fun and are relativly low stress (no one holds you back in school for sports) what chance does math have? Unfortunately once a child decides they hate math and gives up they really are lost to true understanding because they will never reach out and try to figure it out.
Now you can disagree with my analysis but not the fact that 99% of pre-college math is taught in a rote manner. Hell most math teachers don't know enough to do otherwise. Given that they won't understand anyway why not reduce the pain so maybe a few more kids won't already despise math by the time they make it to college.
Yes, I really think most math teaching is worse than nothing for real understanding.
Posted by: Truepath | February 1, 2008 4:16 PM
Oh agnostic, you break my heart! Yes, certainly, adolescents are more challenging to teach than younger kids, because it's so 'uncool' to be excited about learning. But this is just as true in an English or history class as it is in a math class. Perpetuating the myth that math is somehow more difficult to teach and learn than any other subject helps no one and merely excuses the all-too-frequently low quality of math education in the States. There is no math gene, and it is patently untrue that only nerds can learn and excel at math. Certainly, the environment that inevitably surrounds a kid from an educated family/community fosters academic success, but that just means that kids from less educated families/communities need a little more effort and attention from their teachers to get excited about learning all subjects, including math.
Posted by: Megan | February 1, 2008 4:17 PM
Agnostic:
I think "heritability" of IQ may not mean what I think you think it means. I don't want to hijack this thread into an IQ discussion, so I will provide a link in case you haven't seen it, and leave it there:
http://www.cscs.umich.edu/~crshalizi/weblog/520.html
Posted by: yagwara | February 1, 2008 4:19 PM
And what Megan said (our comments crossed in the wires).
Posted by: yagwara | February 1, 2008 4:26 PM
Truepath:
I've taught both math and CS classes, and they didn't change my opinion one bit.
First - by the time you're teaching at the college level, a lot of attitudes have become pretty solid. If you've been taught to hate math, to think that it's miserable and hard and boring, it's going to be very hard to convince you otherwise when you're 18 or 19. Not impossible, but hard.
Second - I've seen people do the turnaround. Watching someone change from "I hate this useless stuff" to "Hey, you know what, that's *cool*" is amazing. And I've seen it happen.
Third - I've seen some very direct evidence of the idea that a lot of math haters don't really hate math - they've just been taught to hate it. When I was in college, at one point, I went to visit my brother. My brother was studying music. He hated math, and firmly believed that he was horrible at it. But one evening while I was visiting, I was watching him to his music theory homework. He was analyzing a serial composition. And what he was doing was, basically, forming a matrix from the intervals in a passage, and taking its determinant. This is a guy who never got near things like matrices in real math classes, because they told him he wasn't good enough at it. But he was doing linear algebra! Not just determinants, but a bunch of different kinds of algebraic manipulations of vectors and matrices.
He didn't love music theory, but he did think it was interesting, and even (sometimes) fun. It was math - he just didn't know it. But if you tried to teach him linear algebra, he wouldn't be able to do it. He'd just been so thoroughly convinced that math was miserable, hard, and dull, and that he couldn't do it and didn't want to.
Posted by: Mark C. Chu-Carroll | February 1, 2008 4:26 PM
Even if we're only talking about kids who will never meaningfully learn algebra or anything beyond, not teaching fractions is still stupid. Fractions are way more useful in everyday life than decimals are. The only useful decimals are the ones in money, and everyone understands those.
Posted by: Tam | February 1, 2008 4:33 PM
Oddly, I learned decimals years before I learned to handle fractions properly. I had so much trouble with fractions that at one point I routinely converted fractions to decimals, worked the problem in decimal and then figured out the equivalent fraction so as to write down the answer they wanted for the question.
Don't ask me why I could do that but couldn't simply work the fractions directly. Multiplying and dividing them just didn't work for me until my older sister finally explained them to me. The teachers' explanations never 'clicked' for me.
What was interesting was that my base understanding of math was strong enough that once I learned to handle fractions, I destroyed their grading scale. They had a system based on 'pre-tests' to determine if you needed to learn something and 'post-tests' administered after training to show you had mastered the material.
The grade was based on the number of post-tests you took and passed. Once I mastered fractions I never took a post-test because I passed all of their pre-tests the first time. They had the very strange situation of my passing every test I took (dozens of the pre-tests) while technically failing the course according to their grading criteria because I wasn't passng any post-tests (because I never took any after passing the dividing and multiplying fractions post-test).
Posted by: Benjamin Franz | February 1, 2008 4:37 PM
It seems we have terribly low expectations about kid's ability to learn math: I whole- heartedly recommend the following:
The Harvard Math circle has a very interesting approach to math teaching...
http://www.msri.org/communications/vmath/VMathVideos/VideoInfo/1913/show_video
also fantastic is their book:
http://www.amazon.com/Out-Labyrinth-Setting-Mathematics-Free/dp/0195147448
Posted by: tbell | February 1, 2008 4:43 PM
Take a look at what the axiom of choice can do.
From a creationist mathematician. His published papers appear to be ok but goes off the deep end with his limited selections. His math goes well for virtual reality constructed universes.
http://www.serve.com/herrmann/main.html
Gary Ehlenberger
Retired Staff Scientist, Member of the Technical Staff, Motorola
Posted by: Gary Ehlenberger | February 1, 2008 4:47 PM
Well, hell, without fractions, how on earth can you do, say, algebra? Or calculus? Or a boatload of other advanced math? I am SO GLAD one of my later grammar school teachers realized the entire class had somehow missed how to work with fractions and tossed out her original curriculum to devote a few months specifically to fractions for us...otherwise, calculus would never have made any sense.
Mark--The music theory example is great. Music theory is *full* of mathematics.
Posted by: OmegaMom | February 1, 2008 4:57 PM
Fractions are the "real deal" - decimals are kind of the fake version
So what if we abandon decimals and teach only fractions?
Posted by: Coin | February 1, 2008 4:59 PM
Coin:
My point is, the meaning of decimals is fractions. Decimals are just a useful compact notation for writing fractions. If you don't teach fractions, how can you teach what a decimal *means*?
I wouldn't endorse *not* teaching decimals. But I still argue that fractions are the fundamental thing.
Posted by: Mark C. Chu-Carroll | February 1, 2008 5:09 PM
Sorry for coming late to the comments -- but what do you do with 1/3, 1/7, 1/9, etc.? Simple as fractions, but as decimal fractions??
Posted by: andy | February 1, 2008 5:21 PM
But you know, learning is not 100% fun, all of the time. Sometimes, one must master drudgery to get at the "fun" stuff.
(e. g., algebraic topology does provide some cool things, but going through the nitty gritty to make it work isn't always a thrill a minute).
But thanks for this blog post.
Posted by: ollie | February 1, 2008 5:30 PM
But if they just use calculators, how will they know when they have made a mistake? I had a student last week struggling with a hash function. She did not get it. I gave her an example: 72%23 (% as the modulo operator) and asked her where this hashed.
She grabbed her calculator, punched in 72 / 23, got 3.1304348, subtracted 3, and then multiplied by 23, getting 3.0000004. She then complained: I thought modulo was supposed to give us an integer.
I want to ban calculators, and let's throw out decimals while we are at it. An earlier exercise was to make a data type Rational. So many ended up with methods like
public double multiply (Rational r1, Rational r2) {...}
I must have explained the use of rationals 15 times.
I found this paper to be the eye-opener that cured me of double-love as a student:
"What Every Computer Scientist Should Know About Floating-Point Arithmetic", David Goldberg, Computing Surveys, March 1991. (copy at http://docs.sun.com/source/806-3568/ncg_goldberg.html).
Posted by: WiseWoman | February 1, 2008 6:07 PM
When I hear about revising math pedagogy because nowadays calculators do all the hard work, I can never decide what I think. I'm of the generation that didn't get to punch keys until late high school, even university, so there's a whole lot of basic math skills that I learned by hand. Of course, once I got a calculator (hell, once I got a slide rule) I rarely calculated anything on paper or in my head. But I have this feeling that having learned it all manually left me with a gut feel for the numbers and the concepts that I would not have otherwise (not that I'm a brilliant mathematician or anything, but I have managed to wrap my head around a fair bit of stuff over the years).
An anecdote that may be relevant: I did a Masters degree (Systems Engineering) in my 30's -- ie. a good 10+ years older than the class average. When taking up one mid-term (which I had aced), the prof scolded the students who had lost marks for what were obvious arithmetical errors -- especially things like answers obviously off by factors of a hundred or more. "You just believe whatever comes out of your calculators!" At this point he paused and asked how many of us knew how to use a slide rule (a few of us old fogeys raised our hands), before going on about the virtues of mentally sanity-checking your answer's order of magnitude -- something you *had* to do when using a slide rule, because it only gives you the mantissa.
So, I want to hear from you younger folk -- the calculator and PC generation: do you agree with me that one needs to master a math skill manually, before out-sourcing the donkey-work to a machine? Or am I OTL?
Posted by: Eamon Knight | February 1, 2008 6:08 PM
andy: Of course, we use some 5-10 decimal approximations. After all, nobody ever needs more than 3 digits precision for anything, do they?
</sarcasm>
Posted by: Mikael Vejdemo Johansson | February 1, 2008 6:16 PM
Tam said:
Fractions are the "real deal" - decimals are kind of the fake version. (My friend had a physics teacher in high school who started the semester with a question - "What is 1 divided by 3?" The answer is 1/3, not 0.3333...)
Well... Let's say you are on a team of 12 people who are working for a $500 payment, and it will be split equally among you. So the foreman says, "People, you will all get $500/12. Just think what you will be able to do with $500/12--why, that's almost as much as $600/14!". You'd feel a little less than satisfied with your contract, and then your buddy would tell you, "Pssst, it rounds to $41.67. "
So, in that sense, decimals are pretty real too. Common fractions like 1/2, 1/3, 1/4, 2/3, or 3/4 are pretty easy to think in terms of, but once you get into your 8/11, 2/7, 1/13, etc., it is much easier to use decimals and just see them as approximately 0.73, .29, .08 and so forth.
For me, fractions are just a placeholder for a division operation. Writing 2/5 means the shortest way to notate "Divide 2 by 5". The great thing though is once they are down in that way on the paper there are all these great operations which allow you to do math fast. To do 2/5 * 3/4 you just multiply across and quickly get 6/20, which goes to 3/10, but imagine having to do it by "Multiply the result of dividing 2 by 5 with the result of dividing 3 by 4".
Both fractions and decimals should obviously be taught.
Posted by: cm | February 1, 2008 6:37 PM
I teach mathematics at the junior college level at a couple of different schools here in Cincinnati. I have degrees in both engineering and business. I conduct math trench warfare daily, and I could not disagree with Professor DeTurck more strongly.
One of the skills of numeracy is the ability to quickly estimate approximate answers to math problems quickly. To do this, you need to be able to do some basic arithmetic in your head, simply from what you remember. From what I've seen, we don't even teach children math facts, let alone methods! And whoever is teaching children math these days seems to succeed primarily at instilling fear, loathing, and intimidation, rather than wonder and curiousity.
Fractions lead to proportions, and proportions can be used not only to solve basic problems in their own right, but also percentages and the conversion of units of measure. If we restrict our learning to decimals, how do we perform these tasks? Memorize decimal constants instead of times tables? And how do we learn how to do algebra? Can we express a polynomial as x.abc?
My experience leads me to agree with a recent column in "Seed" that stated that "the US has the worst math and science education of any country with indoor plumbing." We need to inspire more and demand more, or our children will have even less chance of dealing with the world successfully than they do now.
Posted by: wmichaelb | February 1, 2008 7:32 PM
Did anyone actually read what Professor DeTurck says (as opposed to paraphrases of his views by the reporter or headline writer? He never says to not teach fractions or long division.
"DeTurck does not want to abolish the teaching of fractions and long division altogether. He believes fractions are important for high-level mathematics and scientific research. But it could be that the study of fractions should be delayed until it can be understood, perhaps after a student learns calculus, he said. Long division has its uses, too, but maybe it doesn't need to be taught as intensely."
One may debate whether delaying the teaching of these things is a guarantee of disaster, but if you're going to attack the man, at least attack him for what he says, not for a convenient distortion that misrepresents his views. Most of those commenting here have fallen into the standard Math Wars trap: someone proposes a change in emphasis or sequence in the "holy standard curriculum" and rather than address what s/he ACTUALLY proposes, critics immediately invent a far more extreme view and make that the point of debate. Those who weigh in are, almost without exception, arguing about something the original person never said, and the personal insults fly fast and think.
Frankly, as a regular reader of this blog, I'm very disappointed in you, Mark, for doing precisely what I've described above. You're far too bright for me to excuse your take on his proposal as simple carelessness: you've taken a provocative idea (whether it's right is open to debate) and turned it into something no one, including the person who is alleged to have said it, would likely care to defend. Debating straw men is a cheap rhetorical technique that has become so prevalent in the post-Reagan years that when I see someone use it, I'm sorely tempted to dismiss the person as a fool and an ideologue. I would prefer not to think that of you. I strongly suggest you re-read what the professor actually says and debate only that. Further, if you're wise and skeptical, as well anyone should be when it comes to issues about mathematics education these days, you should try to seek out his actual writing, rather than rely on what the reporter has said. DeTurck is no idiot, not by a long shot, and as he says, it's often by questioning the basic axioms that important, ground-breaking work comes to fruition. He's questioned an assumption, and he's not the first person to suggest we make serious mistakes in what we try to teach kids at certain points in their development. Fractions ARE a very slippery construct, not the least reason being that for the first time, kids are given two separate numbers, a/b, and asked to think about them as representing a SINGLE point on the number line, a single value, a single number. You think that's trivial for most kids at the age of 8 or so? If you do, you've not taught K-5. Many adults go their whole lives struggling to successfully understand both the meaning and the arithmetic of fractions. Decimals are not a walk in the park, but the are clearly an extension of our base 10 number system and hence are easier in some ways. Fractions are easier when dealing with ratios, and they are much easier if you have to take a reciprocal by hand or mentally, so I doubt very much that we would ever "do away with them." But then, no one ever said we should. What was discussed was WHEN to teach them, not whether to. I disagree with his proposal that they come after encountering calculus. That strikes me as far too late. But making them such a huge hurdle in elementary school as they are for a lot of students seems worth questioning. That many readers here are outraged by what no one has actually said is quite interesting, give the high degree of intelligence I would think being a serious reader of this blog calls for. If you left a comment that was a knee-jerk response to what was never actually said, you might want to ask why you are so quick to assume that the headline is true. It's not supported by what the man has actually said, is it? But it makes such a charming sound bite. Welcome to Truthiness, USA.
Posted by: Anonymous | February 1, 2008 7:39 PM
The previous comment was mine. I hit send too quickly. Sorry. I'm not a fan of anonymous posts.
Posted by: Michael Paul Goldenberg | February 1, 2008 7:40 PM
I think that primes should be taught before fractions. But don't move fractions back; move primes up!
My 5th grade daughter was struggling with reducing fractions. So I showed her the Sieve of Eratosthene and had her make some prime number cards. I showed her how she can build any of the counting numbers with the cards. (And, yes, you build a "1" by placing no cards on the table.)
Now to reduce fractions, she builds the numerator and denominator, finds what they have in common ("So THAT's the Greatest Common Factor!"), and writes down what's left over. She can even do it in her head after one evening's practice.
Posted by: Michael L Perry | February 1, 2008 8:08 PM
the funny thing is calculators have a fraction button these days, at least any worth having. It has become standard.
And yes math can be great fun, I always found it to be great entertainment.
Tam, comment #5:
"Intuition aside, I don't see how you can get very far in math without fractions."
Hmmm much math has no use of numbers after all Euclid did fine with no explicit mention of anything other than N.
Posted by: Starhawk Laughingsun | February 1, 2008 8:16 PM
The story about the $60 being part of $50 makes me feel like the battle tank in Restaurant at the End of the Universe. It makes me so mad I want to blow a hole in the ground and get myself killed. That kind of idiocy... I'll stop now while my computer is still intact.
Posted by: Jason Adams | February 1, 2008 8:22 PM
I looked at what DeTurck says, and I think I've been fair. He wants fractions removed from the basic teaching of math. He thinks fractions should only be taught after calculus, to math students in college. For teaching kids, and for people who never get to calculus, he thinks decimals only are sufficient.
The stupidity of DeTurck's proposal is exactly what I said it is: he's basically advocating totally abandoning any hope of understanding for kids learning math.
Posted by: Mark C. Chu-Carroll | February 1, 2008 9:41 PM
My views so agree with yours - http://webcatcher.blogspot.com/2008/01/fractions-should-be-scrapped.html
Posted by: Myth SilverStar | February 1, 2008 9:43 PM
The most scary thing in the world--stupid people have a "Dr." in front of their name
Posted by: Mgccl | February 1, 2008 10:07 PM
I am currently teaching basic maths (as far as precalculus) to adults, many of whom arrive extremely weak. A major problem seems to be, as someone said earlier, that weak students have no feeling for numbers. In a recent test I asked how many 2-ounce cups of juice could be poured from a 750mL carton of juice, to see if they could handle multiple conversions. Several came back with numbers in excess of 100, two managed 3,000 and 12,000 respectively.
In another course, students have used trigonometry to find the distance from an aircraft to a landmark correct to about 1/10 the diameter of the nucleus of an atom. It seems that for many mathematics has absolutely no relationship to reality.
Many students try to remember formulae without understanding, which results in the area of a rectangle being [(length x width)/2] x pi (seriously). I stress that understanding may require more effort in the first place but the result is that you need to remember less. This attitude is not helped by many of the textbooks which tend to present formulae without explanation. One that particularly irks me is that the 'original' form of the formula for variance is given then, in every basic statistics book I have looked at in recent decades, they write something like 'This can also be written in a form that is easier to use and here it is - splat!' This despite the fact that the procedure to get from one to another could be followed by any student in the course.
It is true that many are scared of maths. I try to alleviate it by pointing out that some of the concepts, for example of negative numbers, were not appreciated by mathematicians for 1000 years or more so they aren't to feel too bad if they don't get it in 20 minutes. In longer classes, which can be very stressful for some students, I try to bring in some mathematical puzzle that has nothing to do with the course, some anecdote or a curiosity like the number of rows of scales on a pine cone. I have several times had weaker students asking for the day's puzzle.
A problem with decimals is the way in which many TV commentators, especially economists who should know better, refer to them by describing 0.23 as point twenty-three. If a student does that, I ask if that is more or less than point three? Then what is 0.023? Finally, I agree with you, I don't see how it is possible to teach decimals without teaching fractions.
Posted by: Richard Simons | February 1, 2008 10:19 PM
Oh fuck no. You, Mr. Goldenberg, are missing the point entirely. Understanding decimal arithmetic is impossible without understanding fractions. I will repeat this in bold this because it is important: decimals cannot be understood without fractions. What is 0.73? Can you answer that question in full without using fractions? No? Then please shut up.
I'm not normally that short with anyone who posts here, but rarely do I see someone here advocating for ignorance. Can you imagine how dumbing down the level of math education in this country further would hurt future generations? How much easier would it be for con men to swindle people? How many more science and engineering jobs would go overseas?
America is already past its economic and scientific apogee, as far as I can tell. We don't need to hasten its descent, unless we're damn sure doing so would knock some sense into people.
Posted by: Josh in California | February 1, 2008 10:46 PM
Back around 1980 I was a know-it-all kid trying to get placed in school three grade levels above what would be normally appropriate for my age (by age, I should have been in 4th grade; my father was trying to get me into 7th). I took the position that decimals were "obviously" superior to fractions and would convert fractions to decimals to perform arithmetic on them. A problem like 13/28 * 7/13 would very likely had me do two onerous long divisions followed by a multiplication, all by hand, to finally report an answer probably close to 0.25.
The folks at the school who were evaluating me didn't buy it -- decimals were not superior to fractions, and I had better learn them and how to manipulate them. I wasn't bad at math (four years later, before I entered high school, I completed in one year an entire 3-year program of honors high school math with my lowest "course grade" of 96), but I was arrogant in assuming that since I knew how to manipulate decimals, I didn't need to know fractions.
Today I find fractions invaluable for everything I do using algebra, calculus, etc. Except when working with measured values I don't know the last time decimals were actually important to me, mathematically.
I could suggest some changes in how fractions are taught in elementary schools, such as not teaching the useless distinction between proper and improper fractions (a fraction a/b is proper if a
Posted by: Blaise Pascal | February 1, 2008 10:49 PM
Mrs. Krabappel: "Okay. Whose calculator can tell me what 7 times 5 is?"
Milhouse: "Ooh! Ooh! 'Low battery'?"
Mrs. Krabappel: "*sigh* Whatever."
Milhouse: *waggles eyebrows smugly*
Posted by: Skemono | February 1, 2008 11:06 PM
One thought that strikes me: DeTurck is implicitly advocating that the US go metric in more ways than it already is. For example, all carpentry would have to be done in metric (or else in decimal inches) since no one would understand things like "14 and 3/8 inches", and how to add or subtract such measurements (I think it's safe to say that future carpenters are unlikely ever to take calculus).
Posted by: Eamon Knight | February 1, 2008 11:14 PM
Mr. Chu-Carroll, you have given me the perfect excuse: apropos to basic misunderstandings of basic arithmetic, I send you this particular headache-inducing example of bad purely for my own sadistic amusement.
I wish I could say that I think it's a piss-take, but quite honestly, I can't.
Posted by: Luna_the_cat | February 1, 2008 11:48 PM
Benjamin, I still routinely change some fractions to decimals, often rather than finding common denominators, and I was thought odd for it last year in my class on how to teach math. Still, .375 would be pretty meaningless as an actual quantity if I did not know that it meant 3/8. Much to the shock of several people in class last year, there are still some fractions I would not attempt to work with as decimals, like anything non-terminating. I imagine that would be really fun for kids to attempt to compute.
Posted by: Jen | February 2, 2008 12:16 AM
I teach math to teenagers (algebra and geometry) and the biggest problem that I (and my colleagues) encounter is the lack of a concept of numbers. Understanding fractions is essential to understanding numbers. Those of us in the trenches can provide a great deal of evidence to show that students who lack a concept of numbers are lacking a fundamental understanding of parts of a whole (fractions), and this creates a difficult obstacle to overcome towards understanding algebra, and beyond, because they can't get past the LACK OF MEANING in the problems they must tackle. Also, when the numbers - the very tool of mathematics - hold no meaning for them, they can't even begin to determine what a reasonable approach (or answer) might look like.
Part of the problem is that these understandings are taught in the elementary schools by teachers who (more often than not) do not have a degree (or certification) in the area of mathematics specifically. They hold an elementary education degree (or certification), which (IMHO) does NOT qualify them to teach mathematics. Many people do not believe that this specific certification/degree is essential for teaching what they consider to be such elementary math (no pun intended), but I believe that the results of utilizing teachers without math certifications to teach math in the elementary schools are now speaking for themselves and pointing to the contrary. (For the record, I taught elementary school before teaching high school math - I have a degree in each - and I can tell you that there are, of course, some wonderful elementary school teachers doing a wonderful job of teaching math, as well as other subject areas for that matter. My point, though, is that their elementary education degree is not the reason why. Again, in my opinion, that degree is NOT a good preparation for teaching math).
P.S. To those that have suggested that they prefer decimal equivalents in calculating, this is not relevant to the issue. But let me give you an elementary example of some of the problems one can run into with this strategy: 1/3 of 21 is clearly 7 when calculated by using 1/3 as a fraction, and it's also arguably intuitive IF you understand that you are seeking only 1 out of 3 equal parts of 21. However, will you still get 7 when multiplying 21 by the decimal equivalent of 1/3? (Depends on what decimal equivalent you were using, right?...) In other words, if I owed you 1/3 of 21 dollars, how satisfied would you be if I insisted that $6.30 was the accurate amount (the result of calculating with .3)? Would you mark a student's paper correct for this answer when you asked for the number which represented 1/3 of 21? Or how about $6.93 for an answer (the result of using .33 for 1/3)? This is a bit closer to the actual and accurate answer of 7, but it's NOT 7. This is just a small example of the wrong and/or inaccurate answers arrived at with SOME decimal equivalents.
In a nutshell, even if the answers were the same using a decimal equivalent (as happens often), DeTurck is an idiot to hold such an opinion regarding the uselessness of understanding fractions in early mathematics. He clearly has no clue of the basic knowledge and understanding required to UNDERSTAND even basic mathematics, or of the necessity of such understanding for success in ALL math courses.
Posted by: mathteacher | February 2, 2008 1:51 AM
@ Luna_the_cat:
Wait, that's not your own writing, right? Because the author of that piece at Xee-a Twelve is making some basic mistakes.
Of course negative numbers don't make sense when they're applied to apples, because there's no such thing as a negative apple. Negative numbers need to be explained using stuff where they make sense, like money, temperatures, or height compared to sealevel.
Posted by: Cailin Coilleach | February 2, 2008 3:59 AM
It is unfortunate to find such a misguided educator. Decimals are a representation of fractions, a fairly limited representation that is almost always an approximation. Unless you know about roundoff errors and such, using decimals will yield inaccurate results in a calculator. Something as simple as (1/3) * 3 will prove unequal to 1, on many calculators, because you're comparing 0.999999999 to 1.0. The difference is insignificant, but a comparison for equality will fail.
Math has an undeserved reputation for being dreary, exactly because so many teachers insist on restricting their lessons to mere computation, instead of explaining what is actually going on. You are taught long division, or how to use a calculator, without being taught the reasoning behind it, without knowing the fascinating story of how someone figured out how to solve a problem in the first place. What a shame.
Posted by: Chiron | February 2, 2008 5:37 AM
What about 'whole' number fractions, like 25, 50, 75, and/or 03, 06, and 30, 60 or 33, 66 and what fraction would 132, 165 represent. Why would 33 divided by 11 equal the 'whole of 3 while it's composite parts individually divided by 11 then added equal (.2727... + 2.7272...) = 2.9999... Then 495 divided by 11 equaling the "whole of 45 while it's composite parts, 045 and 450 individually divided by 11 add up to 44.9999... Try 225 and 270, 315, and 180. Try (132 +165 +198) = 495
Then (054+540) = 594 and 594 divided by 11 equals the "whole" of 54. Maybe it is because the divisor 11 is (01+10)
Posted by: ray burchard | February 2, 2008 7:47 AM
Fractions are key to understanding proportion and variation, and nothing in nature makes much sense without those. Deprive students of fractions and you'll impoverish their understanding of the world around them.
Posted by: lurian | February 2, 2008 8:06 AM
I propose that we call Prof. DeTurck's proposed mathematical course "Verizon Math".
Posted by: Pseudonym | February 2, 2008 8:19 AM
I work in an elementary school but don't teach math, and I'm also a mom... given that most of us do not go on to higher math as engineers or physicists, I think the essential knowledge to take away from studying arithmetic is number sense-- having some idea of whether a numerical proposition is plausible, a point the original post made. The rental agent clearly plugged the numbers into the computer backwards or something. This means he doesn't understand the idea of what he is doing and can't therefor recognize his mistake. In my school system, kids are constantly (too much) being tested, and one of the domains of math knowledge is number sense. Fractions are a concrete way of understanding-- in elementary school, they show it with pizzas, arrays (relating it to division) and lots of other manipulative and visual approaches. It makes no sense to plug in numbers by some memorized algorithm. Besides, comparing my own facility with having memorized math facts with my kids' weakness there, I feel liberated from the drudgery of calculating, and can easily think more conceptually, which is the fun part of math. I mysef never got beyond algebra, and tend to make errors with calculators.
Posted by: mary | February 2, 2008 9:03 AM
Cailin Coilleach @#45:
You have no idea how horrified I am that you think it even might be. Dear god, what have I ever done to you?
That is a random page from a true nutter's website which I posted merely to torture Mark.
I've got to go drink heavily now. Cripes.
Posted by: Luna_the_cat | February 2, 2008 9:49 AM
Your comment at the end, why both sides are wrong, reminds
me of my general rule about discussions about primary mathematics
education.
One side:
Math isn't fun. Kids should have fun. Therefore, we cannot teach
children math.
The other side:
Math isn't fun. Kids should learn math. Therefore, we must make sure
the kids don't have fun.
Posted by: Russell | February 2, 2008 11:25 AM
I agree with cm @ #27.
Dunno about the age - but IMHO all useful methods should be more or less valuable.
The problem with such outsourcing is both laziness and bad psychology. People don't check order of magnitude (examples are given here) and tend to interact with calculators and computers as oracles, sometimes anthropomorphic such. ("Computers don't lie". But they do, very often as a matter of fact.)
Of course, if you don't understand the math, it will come back and bite you at such times you need the understanding, say in further math education. And using algorithmic devices you should really learn both the ideal computation and the real life algorithm your machine uses. (Or is supposed to use, remembering the common errors in math routines.) A tall order.
Posted by: Torbjörn Larsson, OM | February 2, 2008 11:34 AM
A fraction is just a fraction, whether it be expressed in decimal notation (0.25) or ratiometric notation (1/4). The two refer to the same thing. It's just a quarter -- if it's just a number. But if it happens to be a quarter of a pound, then it would be more properly written as £0.25 or 25p. And if it's a quarter of a litre, it would be better written as 0.25l or 250ml.
The ratiometric notation is generally superior for theoretical problems, where you want to preserve absolute accuracy. Whereas, the decimal notation is generally superior for real world measurements; where measuring instruments only have a limited precision and rounding is necessary.
What I absolutely do not see any point in, is rewriting something like "35/16" as "2 3/16". Such notation (integer and separate ratiometric fraction) is about as relevant today as Roman numerals. Either leave something as a ratio if you want it exact, or convert it to a decimal and round it.
Posted by: AJS | February 2, 2008 11:39 AM
I don't know if I agree with you since I borrowed an apple the other day. It all depends on the context where you are applying your math.
XT does another funny when (s)he glides over phase differences and claims that two sounds of the same frequency can't cancel each other (and probably will claim they can' t add either).
Perhaps (s)he needed a better math teacher.
Posted by: Torbjörn Larsson, OM | February 2, 2008 11:48 AM
Ah yes, Robert Herrmann, again.
Well, as I understand it he continues to claim that many or all formal models are realized as universes, and that humans can observe, understand and formalize nature in such a universe is a prediction of an all encompassing design theory.
As the first claim isn't verifiable or much sensible, and the later aren't unique predictions, I think the 2 minutes it took to dig up his theory from his mass of texts and links was totally wasted.
Except for the laugh when I noted that he leaves his 'predictive' theory wide open to all religious interpretations, YEC and OEC alike, simultaneously.
Posted by: Torbjörn Larsson, OM | February 2, 2008 12:13 PM
@ Luna_the_cat:
No disrespect meant, honestly. I just couldn't make up from the way you wrote the sentence over here whether you'd found it, or written it. Is all :)
@ Torbjörn:
Well, the point (s)he is trying to make is that there is no such thing as -440 kHz and of course -that- is true. The thing is that she/he/it starts off wrong since, when you're cancelling out two sounds, you don't use the negative frequency (which doesn't exist), but the phase changed sound. So due to a lack of physics knowledge, she/he/it gets it all completely wrong :)
Posted by: Cailin Coilleach | February 2, 2008 1:39 PM
AJS said
If a carpenter is making a fence 30 feet long out of boards 5 7/16 inches wide do you tell them to get approximately 66.207 boards, that they need 5760/97 boards or that they should use 66 boards and a strip 1 1/8 inches wide? You use whatever is convenient and should be able to convert from one to another.
Posted by: Richard Simons | February 2, 2008 2:09 PM
How can I say this?
As often is the case Mark is correct.
Anyone who thinks fractions aren't reqiured is ignorant and delusional, rrespective of how many Phds they may have.
As a simple engineer; fractions exist.
Decimals are simply an, often, inconsequentional and inaccurate representation.
As a former schoolchild, I can understand one seventh of an apple pie, what is 0.1482587 of an apple pie?
Well Duhhh DeTurck, is simply an example of a well educated fool with no understanding of reality.
Heaven help any department he leads.
Heaven help country that listens to him.
Posted by: Chris' Wills | February 2, 2008 2:42 PM
I am still thinking about this story. Part of me thinks that maybe he is on to something but ultimatley fractions are part of the real world. In order to learn aboout reality then we must learn them.
Maybe the real argument should be how much math students need learn not what we should leave out of basic math.
Posted by: Mathman6293 | February 2, 2008 2:50 PM
Tam
´"What is 1 divided by 3?" The answer is 1/3, not 0.3333...)´
Your professor is wrong. The answers are exactly the same.
Need proof?
Anders Eg
Posted by: Anders Ehrnberg | February 2, 2008 2:52 PM
I think overuse of calculators in elementary and middle school is a huge problem, that leads to innumeracy. (I've ranted about this on my own blog.)
Eliminating fractions is, for all of the reasons you state, a horrible idea! Decimals are meaningless without the foundations laid by fractions. And as many here have pointed out, you need to understand and be comfortable working with basic numeric fractions, in order to work with algebraic fractions later on!
My youngest son saw me reading the article, and asked what it was about. I told him. He replied, "That's stupid. Those things can be very useful!" (He's 6yo, but works with an older group for math, and has been doing a lot of fraction work lately.)
btw, Mark, any chance you would post a "lesson plan" or just more details on what you did with the first graders and the abacus lesson. I'd love to bring that to our kids' school, where they'd be quite happy to either let me present it or present it themselves.
Posted by: mathmom | February 2, 2008 3:42 PM
Posted by: AJS | February 2, 2008 3:49 PM
Just to add a bit to my thoughts above -- Dr. DeTurck wants to save fractions for after calculus, which I just don't see as possible.
As many have stated, a basic notion of fractions is required to understand decimals, and being able to manipulate fractions is required in algebra (and trigonometry, and calculus, etc.) so saving fractions for after calculus strike me as absurd.
I could see an argument for saving learning about most manipulations of fractions for pre-algebra or even algebra. When we teach 5th graders to add fractions with unlike denominators, or to divide fractions by using the rule "invert and multiply", they don't really get a good understanding of what they are doing in 99.9% of the cases. And if they don't really know what they're doing, the goal of increasing basic numeracy is lost.
What might be workable is teaching the basic notion of fractions early on (as is currently done). Teach simple fraction manipulations (like adding fractions with like denominators, and equivalent fractions). But then teach what decimals mean, how to convert fractions to decimals, and do computations using decimal notation. THEN in pre-algebra or algebra, teach fraction manipulations when you have a chance of the kids really "getting" what it is you are doing, why it works and makes sense, etc. And then they have the grounding for working with algebraic fractions, etc. Could work...
Posted by: mathmom | February 2, 2008 4:07 PM
Chuckle. This reminds me I'm still hearing people claim the US will 'never go metric' -- and yet, if you look at almost any consumer package purchase, what units do you find it using? I don't mean the decimal fraction units, I mean the integer units?
Right. And when the kids who grow up think of a container of milk, what units are they going to be thinking in?
Right.
Posted by: Hank Roberts | February 2, 2008 4:57 PM
Commenters here can ignore Mr. Goldenberg at their own risk. I think he is wrong on the whole, but he is good at pointing out real problems.
Fractions are hard, far harder than mathematicians and computer folks usually acknowledge.
We teach fractions, generally, poorly. When I ask older students and adults (not math folk) if they were ever good at math, most answer yes, and about half indicate "until fractions" or a closely related topic.
We need to learn how to teach fractions better. I do not believe that the "drill harder" people or the "let children construct their own knowledge" or now this "postpone indefinitely" guy make sense, at least on their own.
But we certainly need and should welcome proposals and ideas. In that spirit, what mathmom (#64) proposes is interesting (teach fractions early, but manipulations only with pre-algebra). I think the manipulations of fractions, practiced over time, prepare students for algebraic manipulations, and thus I would disagree.
My suggestion (and of course it is not the end of the discussion) is that a big chunk of the answer is in this thread: when Marc went into kindergarten he was doing something unusual. We need math smart people with creativity teaching little kids. And I think we have quite few of those.
Jonathan
Posted by: Jonathan | February 2, 2008 5:06 PM
"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."
Roman numerals are "obsolete"? ;-)
But math is hard, Mark, at least for some of us. The problem is, we have a culture that is impatient with process, intellectual exercise, and long-term goals. Math is one of the subjects that has been the most rewarding to study in my life, but it doesn't come easily for me. I'm still trying to conquer calculus on my own. It isn't "fun," but it sure as hell is rewarding when I finally overcome one of my many math blocks.
I'm sick of the "everything must be fun!" approach to education. People need to learn the difference between "fun" and satisfying.
Posted by: Kristine | February 2, 2008 9:41 PM
I'm still trying to wrap my head around what a calculus course devoid of fractions would consist of.
Last time I checked, [f(x) - f(a)]/(x - a) was a fraction. Good luck teaching people the definition of derivative using decimals only.
Posted by: holomor