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.
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.
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.
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.
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.
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.
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.
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?
Yeah, let’s get rid of fractions. What a 0.5-baked idea!
@Neal – I don’t often laugh out loud when reading, but you got me with that one.
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.
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.
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
And what Megan said (our comments crossed in the wires).
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.
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.
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).
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
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
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.
Fractions are the “real deal” – decimals are kind of the fake version
So what if we abandon decimals and teach only fractions?
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.
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??
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.
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).
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?
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>
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.
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.
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.
The previous comment was mine. I hit send too quickly. Sorry. I’m not a fan of anonymous posts.
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.
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.
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.
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.
My views so agree with yours – http://webcatcher.blogspot.com/2008/01/fractions-should-be-scrapped.html
The most scary thing in the world–stupid people have a “Dr.” in front of their name
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.
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.
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
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*
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).
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.
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.
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.
@ 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.
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.
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)
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.
I propose that we call Prof. DeTurck’s proposed mathematical course “Verizon Math”.
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.
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.
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.
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.
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.
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.
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.
@ 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 🙂
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.
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.
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.
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
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.
You tell him to chuck away his great-granddad’s tape measure, get a new one and build a fence 9.15 metres long out of boards 140mm. wide. And he will need 65 of them and a piece 50mm. wide. Easy innit?
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…
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.
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
“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.
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.
Anders:
´”What is 1 divided by 3?” The answer is 1/3, not 0.3333…)´
Your professor is wrong. The answers are exactly the same.
If they’re exactly the same, then why is it that 1/3 * 3 = 1, but .3333… * 3 = .9999…?
as an in-the-trenches math teacher, teaching fractions RIGHT NOW to 7th graders, i certainly think that fractions need to be taught relatively early. i gave a whole bunch of reasons on EvolutionBlog…feel free to read them there. suffice it to say that i doubt the author of that article has ever tried to teach mid-level algebra to students who understood fractions poorly. if he had, he’d surely know how important understanding fractions is.
Mark, and others who can help,
I completely agree that math is generally taught poorly and have mega-theories about how to better do that (non of which involve abandoning fractions or long division) which have not been tested except on one 5th grade subject whom I used to tutor. Anyway, I am extremely interested in learning about fun math-related teaching tools, such as your abacus activity for 1st graders, and would love to hear more about anything people have done and found successful. Feel free to respond via a comment at my blog. Thanks 🙂
@rm1ssret: Because 1 and .9999999….. are provably two ways of expressing the same quantity. There are a number of Web pages that present the proof much better than I could. I’ll try to find one later… there may even be one on this blog somewhere.
http://en.wikipedia.org/wiki/Proof_that_0.999…_equals_1
@rm1ssret, as Speedwell said with details.
X=.3333…
10X=3.3333…
9X=10X-X ; 3.3333…-.3333…=3
9X=3
X=3/9 ; X=1/3
Anders Eg
In the History of Mathematics, and the History of Physics, you will notice that the classical development of both arithmetic and geometry (and even, with Archimedes, an astonishingly early version of Integral Calculus) was codified and embedded in a Axiom – Diagram – Lemma – Proof paradigm by EUCLID in the most important and reprinted math book of all time (in over 1,000 editions!).
The Euclidean paradigm was ultimately (almost always) coincident with the ancient Greek conception of PROPORTION. Everything was ultimately expressed in proportion (a step beyond the “Egyptian” Fractions of an earlier culture, and the sexigesimal fractions of the even earlier Babylonian [still used by Fibonacci] that came from the friendly merger of Akkadian and Sumerian civilizations).
The paradigm was shattered by Pythagorus (and his grad students) with the shocking proof that the square root of 2 was NOT a rational number (i.e. not a proper fraction). They had a “hecatomb.” Today, we’d call it a gigantic barbecue party with lots of booze and dancing and screwing for the glory of God.
Nonetheless, when Sir Isaac Newton wrote the most important Physics book of all time (the PRINCIPIA) it used the Euclidean geometry of EUCLID and PROPORTION, making it hard to read today, as it is NOT in the modern language and paradigm and algebraic notation, despite it having forced Newton to invent FLUXIONS (simultaneously with Leibnitz doing the same and calling it CALCULUS).
If we don’t get teachers and students to REALLY understand FRACTIONS then we have thrown Euclid, Pythagorus, Archimedes, and Newton into the ashcan of History. Which makes one, in the most profound sense, an utter BARBARIAN and the enemy of civilization itself.
Pass me another spare-rib and another glass of wine. It’s Superbowl Sunday, and the RATIO by which the Patriots beat the Giants for their [prime number] perfect season of 19 victories is important in Las Vegas, but does not stay there.
QED!
Cailin Coilleach @#57:
Feh; my writing isn’t always the best, but I do not send out my own “headache-inducing examples of bad” to other people for sadistic amusement. Especially when I say “I don’t think this is a parody”. I stick my own bastard pieces of bad in a desk drawer never to see the light of day, or burn them. I do not flaunt them. I have *that* much good sense. o_0
On a separate note, sure you can have negative apples. Best explanation ever of negative numbers involves apples — read the fantasy novel The Blood Jaguar by Michael H. Payne (also the best treatment ever of the cognitive disconnect between science and religion, IMO).
Blast, post #76 was me.
It should be obvious that the limit of 0.999… as you keep adding more and more nines onto the end (effectively dividing by ten and adding nine-tenths each time) is 1. Because the difference between it and 1 (0.0001, 0.00001, 0.000001 … dividing by ten each time) is getting smaller and smaller all the time, tending eventually towards zero.
What is interesting (but not necessarily important, in the sense of “stuff they will use in the real world when they grow up”, for kids to learn) is that if you take a recurring decimal such as 0.142857142857… , the recurring part divided by (as many nines as there are digits in it) — so in this case, 142857 / 999999 — is the exact fraction. 142857/1000000 is an approximation for 142857/999999. And 142857142857/1000000000000 is a better approximation for 142857142857/999999999999. The second correct fraction is the same as the first, because both numerator and denominator have been multiplied by 1000001. The difference between the denominators keeps getting smaller (and therefore the approximation better) the more digits you include. At infinity, they will be equal.
I’m late to the game, but let me also chime in: I _also_ happen to be a math teacher, and just about every fall I have to teach a course for incoming freshmen of what would have been called in less debased days ‘remedial algebra’, but now has the title ‘Algebra for the College Bound’ or some such feel-goodism. Anyway, yes, yes, a thousand times, yes! My students lack a very basic number sense, and are terrible with fractions. If we happen to have a numerical calculation, say, 23×41, they have no idea that any derived figure over 1,000 is suspect. They don’t have the idea that any number over 1,500 is not possible. Similarly for reasoning that shows that numbers multiplied by five in the denominator have to be greater than the same numbers multiplied by seven in the denominator, or ten vs twenty-three.
I blame the parents, of course. We enrolled our daughter in a private program (Kumon) after we discovered that in the sixth grade, she was still counting on her fingers, didn’t reliably know that 8×7=56 . . . and was making straight A’s in her math courses.
Yes, I know that repetitious rote practice is soul-numbing work. But the proper comparison is not with another academic class like english or history; it’s like practicing scales in music, or repetitive set-ups in volley ball. Sure, you might know, intellectually, that the C Major scale has no sharps or flats, you might be able to recite the progression. But that is no substitute for actually playing C-D-E-F-G-A-B-C over and over and over on an actual instrument.
For every complex subject in life (reading,basic math (we often forget how hard these things are : we were young and malleable when we learned this, and life didn’t really want so much from us), a sport, playing an instrument, and then advanced math, computer programming etc.) you have some part of it that’s basically rote. Just memorize through repetition until it’s ingrained in you and you can’t pull the fact that 9×8 = 72 with a hammer from you. It’s annoying – yes, but it’s the foundation on which you’ll build the rest. I’d think of it as a initiation rite : if you don’t put in the time and effort to learn scales and rhythms, you’ll never be able to make it cry like Hendrix
Several thoughts:
First, if you want to convert a fraction to a decimal such as 1/7, instead of dividing 7 into 1.0000000… divide 7 into .999… (no carry).
Second, when I am trying to find the correct head size for my wrench, I do have to convert 7/16 into 14/32 before I know whether or not to move towards or a way from 16/32. The wrench set calls that 1/2.
In calculus (2nd term in particular) students mess up BECAUSE THEY CAN’T ADD FRACTIONS! It is worse in precalculus!
To teach fractions to really young ones, have them start counting by eggs: 1 egg, 2 eggs, 3 eggs, 4 eggs,… then by dozens: 1/12, 1/6, 1/4, 1/3, 5/12, 1/2, 7/12, 2/3, 3/4, 5/6, 11/12 one.
In the English units 1 foot is more natural than the comparably sized 30 cm. 1 foot since it is 12 inches is easily divisible by most smaller numbers (5, 7, 10 and 11) are exceptional (see above).
I can only think that the article got released because someone has to play the fool. The real problem in grade school is that many teachers are afraid of math, hate it, and think that 1st graders don’t need to know algebra. But a teacher who REALLY understands algebra can teach it using arithmetical facts: exemplify the theorem in well known examples, test the conjecture on other examples, and abstract the concept.
Examples:
2×2=4 => 1 x3 =3 =4-1
3×3=9 => 2×4=8 = 9-1
4×4 =16 => 3 x 5 = 15=16-1
5×5=25 => 4×6=25
6 x6 =36 => 5 x 7 =35 = 36-1
7 x 7 = 49 => 6 x 8= 48 = 49-1.
Is it true that 19 x21 =399?
Yes!
What about
21 x 29=625-16 = 609?
Yes!
Can you articulate this?
(A+1)(A-1)=A^2 – 1.
More generally,
(A+B)(A-B)=A^2-B^2.
Where is the school teacher that knows math, understands math and teaches math?
AJS: “if you take a recurring decimal such as 0.142857142857… , the recurring part divided by (as many nines as there are digits in it) — so in this case, 142857 / 999999 — is the exact fraction. 142857/1000000 is an approximation for 142857/999999.”
I NEVER KNEW THAT!
I knew about rationals becoming repeating decimals, of course. (Hello, 1/7) I even heard about some algorithm to turn the repeating decimal back into the original fraction. (Can’t remember what it is at the moment.) But I never heard that you can use the repeat as an exact fraction equal to the original fraction!
Is this consistent across all bases besides base-10? Can I assume that it’s 9s because that’s the last digit in base-10? How did you derive this?
Who are you, AJS, so I can give you proper credit?
You can’t write x/10 as 0.x or in any kind of decimal notation that I’m aware of.
Uh, .1x? Can I have my .5 cookie now? Actually, if a kid actually said, “Can I have .5 of a cookie,” he’d be pretty likely to get it from me!
Yep. Fractions are important, they should be taught. I struggled with them, and, for the most part, I don’t have trouble with them. I DO get very irritated when I measure anything in inches, because I’m sitting there counting the markings and trying to figure out if it’s a 16th or a 32nd or an 8th. Just pisses me off to no end. And then, if I’m working in Microsoft Word and setting margins or table cells, I have to convert that number to decimal… argh.
One of the the very coolest thing about fractions ever was when I was taking physics and understood what “inversely proportional” meant. And now I’m doing pre-calculus, and understanding that when you have y = x-1 / x2, you are going to have a horizontal asymptote because the denominator is getting bigger way faster than the numerator, meaning there is going to a limit to what y can equal… Good times.
So just use fricking millimetres already, you moron! If hitting your thumb with a hammer is so painful, maybe the solution is to STOP HITTING YOUR THUMB WITH A HAMMER?!
(Yes, except that IRL only scientists ever use more than 9 mm at once. So, 14 cm and 5 cm. After all, a cm is a pretty natural measure — the width of a thumbnail. Besides, “cm” and “mm” are symbols, not abbreviations, so they don’t get a period after them. They do get a space in front, though.)
Because the difference is infinitely small. In other words, zero. Infinitely little is the same as nothing.
Obvious typo (ARGH!)
5×5=25 => 4×6=24= 25-1=(5-1)(5+1)
In my physics program, my peers are wholly reliant on their calculators, which I find to be a crutch. They’ll be punching keys into their devices, while I’d be quickly jotting down a fraction. I feel that it speeds my calculations while allowing me to have a concrete notion of them, while my peers only have abstractions.
Funny I was always thought decimals were harder to visualize than fractions: 3/8 is easier than .375. Its easier for me visualize to sticking a little more on the end of 1/4 than visualize .375.
How do you teach decimals without long division?
mathteacher, I think you’ve hit the nail completely on the head in terms of my own difficulties with math. I’ve been saying for quite a while now that I have no metacognitive awareness of numbers. To explain by analogy, a metacognitive awareness of language is why a native speaker would know something was ungrammatical or “wrong” without necessarily knowing why.
To that end, Mark’s statement “None of it means anything. It’s just fiddling around with meaningless symbols,” caused me to think sadly, “That’s about how I feel about algebra.” Weirdly, I have very little problem with formal logic, but I tend to consider it as an alternate notation system for rhetoric.
So my question would be, what is the solution to a poor metacognitive awareness of numbers? Suppose one has, like I do, no sense of numbers whatever — how do you inculcate a sense of numbers into someone who doesn’t have one? Fix that, and you’ve probably got a ground-floor solution to fixing math education, and sluggards like me might be able to catch up.
(Anyone on this thread who says that Imperial units are easier than metric obviously didn’t grow up using metric. I still can’t figure out how anyone deals with Imperial linear measurements.)
Keep in mind that what .3 is defined to mean is: all the rational solutions to the inequality “10 times x is less than or equal to 3” (look up Dedekind cuts sometime). Whether you want to take the decimal .3 or the fraction 3/10 as your primary interpretation is a matter of convenience.
Posted by: AJS | February 2, 2008 11:39 AM 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….
Whereas, the decimal notation is generally superior for real world measurements; where measuring instruments only have a limited precision and rounding is necessary.
Don’t be confused by the notation, those aren’t decimals. When’s the last time you spent £0.251? You’re not using parts of a pound, you’re spending whole numbers of pennies (pences?). I imagine it’s the same for real world applications of millilitre.
@Hank, I probably shouldn’t feed the trolls, but my milk usually comes as one US gallon or half a gallon, just saying…
I am a bit late to this thread, but I wanted to throw in my $ 2/100. I have to agree with Mark on his major point. You need to learn the basics of math and this may include drills and learning by rote, but math can also be fun. Learning how to model the world around me with math and then derive new knowledge about the world based on math is very cool.
I also used to think of math as music, but a little different from what Mark described. I viewed learning to play notes and chords as analogous to learning the basics of mathematics (addition, multiplication, fractions, etc.) You needed to practice, practice, practice these basics. However, once you learned the basics you could start making your own music. That is why there has to be a balance between rote learning (neccessary but boring) and learning how to make music (fun but requires knowledge of basics).
BTW, when I was taking algebra as a kid I learned how to get rid of the fractions by multipling both sides of the equation by the reciprical of the fraction. So whenever I saw a vinculum (i.e. division bar) the first thing I did was get rid of it. This caused all sorts of problems when I should have been factoring and reducing first. It is still a bitter memory.
agnostic,
I did fine in math until I hit an absolute wall in 11th grade pre-calc. My first and only D, and I was trying hard. My husband quit math after trigonometry. Yet our son is a math whiz – at 9 he’s doing 8th and 9th grade work. He’s wired differently than either of us.
My husband is a teacher, and has found ways to make math fun for kids of all abilities in his class. It’s not so much a matter of the kids and their genes as whether the parents are supportive, how engaged the teacher is, and how willing the teacher is to find ways to teach a concept across a range of abilities.
I hope I’m not being pedantic but I thought I’d try to clarify something for the math-phobic: Decimal notation is just a way of writing a convenient series of fractions. Thas is, 0.1438 means exactly (1/10 + 4/100 + 3/1000 +8/10000). In the decimal, every column right is just another subdivision of the previous column into ten parts. It’s the same principle as dividing a foot into 12 inches and an inch into 16ths and so on as you desire more precision. Decimals are a compact notation with the convenience that everything is a power of 10, rather than arbitrarily varying ratios like 3ft:1yrd, 12in:1 foot, etc.
Decimals are convenient for adding and subtracting and thus for comparing the sizes of two numbers. Fractions (single fractions rather than sums of them a la decimals) are more convenient for multiplication and division. I don’t see how you could conveniently do without either once you get to algebra. You could convert (most) decimals to fractions but why bother? At some point it should sink in that an expression like (0.235 * 4)/1.32 is just a number and you can convert it as needed to a decimal or fraction when it is useful to do so. I don’t know the best way to teach this but I can’t conceive of doing algebra or calculus without understanding fractions.
Incidentally, I’ve TA’d a few undergrad intro physics courses as a grad student. The common complaint among TA’s and professors is that students rely on blindly applying memorized or written formulas so they can plug numbers into their calculators. Often they have no concept of what the formula means or how it relates to others. This is related to the fact that they are bad at algebra, trigonometry and calculus so they can’t easily derive and manipulate equations, nor can they mentally check their answers for reasonableness. There are also good students of course, it’s just an attitude that crops up and it seems like de-emphasizing fractions would, unintentionally encourage similar problems.
I repeat — it’s over. Metric won.
The heft, the thing, the chunk, the container, what kids think of picking up — it’s metric.
Check _any_ package these days. What are the integer units they’re using?
Nestle Spring Water 16.9oz Bottles –Convenient 5 Liter bottles
Vector 5 Liter Elite Console Travel Cooler
Brownie, Cake & Cocoa Muffin Mix 19.5 oz (550 g)
I grew up in a household where the universe sang to my father and older sister in number, and I was essentially tone-deaf. I never could get basic algebra, despite taking it over in summer school then trying again in college. It always felt rather like a moral failing to me that it *just didn’t make sense.*
Fast-forward another 40 years. I’ve had a successful career, follow a number of science blogs, read science books for fun, and my kids are doing in high school stuff that is beyond what I ever learned or could cope with. Now I’ve got a chance to go back to school, with no purpose but my own entertainment, and I’m retaking remedial algebra at my local community college. (the syllabus makes no bones about it being remedial, says that it teaches what students *should* have learned in high school).
for some odd reason, it’s mostly making sense now. My engineer husband is coaching me through, and I’m finding solving equations (except for pesky negative signs, which for some reason keep sliding around for me) rather soothing.
I don’t know why it’s making sense now when 40 years ago it didn’t. I haven’t had to use higher maths in my career (technical writing) so I haven’t learned it in any back-door way… maybe I just wasn’t old enough?
I’m not about to start a new career in hard sciences or engineering, or anything like that, but if the rest of the term continues on like it has so far, this won’t be the last math class I take, either.
Just wanted to toss out a few anecdotes that seemed relevant.
A) One of the most math-phobic classes I taught was “Math for elementary school teachers”. They were earnest, and well meaning (and worked very hard), but scared to death of math. It was a bit surreal.
B) My advisor used to say (quoting someone else I believe), that “Math is learned through the passage of time”. He wasn’t saying that one didn’t have to work… only that some complicate concepts and ideas make more sense when one goes over them again at a much later date. I’ve heard other grad students call this “soak time”.
C) I think math mastery is complicated by instructors not identifing their student’s misconceptions. A few mistaken ideas can render a simple idea almost incomprehensible. (Identifying misconceptions is almost impossible except in one-on-one situations so I’m not trying to blame the instructors… just outline what I feel is a serious problem) I think that empathetic computer programmers often make great math teachers because they are accustomed to breaking a process down into tiny pieces and are conversant in debugging– a process which adapts itself well to identifying where a struggling student is going astray.
Peter absolutely hit the nail on the head with the screwdriver!
“I think math mastery is complicated by instructors not identifying their student’s misconceptions”
Amen! That one sentence has more truth in it than the self-serving lunatic gibberish in the State of the Union and Intersection Speech about the “No Child Left Behind” neutron bomb co-constructed by Emperor Bush II and Teddy Kennedy (strange bedfellows), in that it merely destroys all students and teachers, but leaves standing the rotting underfunded textbook-deficient school buildings.
Hence it follows as the day follows the night, that the REAL purpose of an exam is NOT to be dumbed down and taught-to so as to lyingly claim that test scores have risen so please don’t shut down the school or withhold State and Federal funds.
No, the REAL purpose of an exam answer written by the student to a painstakingly crafted exam question (with not multiple choices pencilled on a Scantron) with actual equations and/or text and/or the right picture gets you half credit, or a narrative paragraph, is so the teacher can figure out WHAT IS GOING ON IN THAT SPECIFIC STUDENT’S HEAD AT THAT TIME, so the teacher can discover the student’s unique learning style, strengths, weaknesses, and what weirdly WRONG algorithm or axiom lurks, derailing the student, so that the teacher can regress the student back to the trauma of a BAD MATH TEACHER in the past, and debug the student’s little grey cells, and put the student back on the right track, heading towards the purpose of Math, which is not to graduate and get the teachers and staff and administrators underpaid, but INSIGHT!
As I mentioned in my blog entry on Terry Tao’s spectacularly good Math Blog, answering in my way “Does one have to be a genius to do maths?” and making a specific point that you can finally “GET IT” in Math at ANY age, even over 50:
Math is not usually life or death; but it can be in mission-critical engineering. Math is somehow simultaneously about nothing but itself [Formalism], about the universe [Realism], about a deeper and more fundamental universe [Platonic Idealism], and about our own personal quest for self-understanding and wisdom.
When interviewed several years ago, Paul R. Halmos (1916-2006) was asked: What is Mathematics to you? He responded:
“It is security. Certainty. Truth. Beauty. Insight. Structure. Architecture. I see Mathematics, the part of human knowledge that I call Mathematics, as one thing — one great, glorious thing.” A few years later, he was asked about the best part of being a mathematician. He said: “I’m not a religious man, but it’s almost like being in touch with God when you’re thinking about Mathematics.”
I am naturally good at math, although I’m pretty ignorant about most of it and out of practice with the rest (which is why I read this blog!).
The previous posts about “misconceptions” really hits home. I am going to need to take a lot of really grueling math soon toward an engineering degree I’m going to try to get. But after 25 years I’m still hopelessly bewildered by one stupid thing. I can’t seem to wrap my head around factoring polynomials.
My math teacher demonstrated factoring by telling us to “guess” the factors. I told him in absolute despair, “Guessing is not part of math!” (Imagine my dismay when I read about the axiom of choice, eh?) Would one of you more knowledgeable folks either tell me what in blazes the real algorithm for factoring is, or else ram it through my thick skull that guessing is, in fact, part of math? I’ll hold still.
I think we should leave math the way it is taught now. If you get rid of fractions and only teach decimals or change any other aspect of it, you are simply going backwards in terms of learning math. We should be progressing in math and want to learn more. That’s ridiculous that someone would want to change it like that!!!
i was thinking about this the other night, and I think DeTurck has a good idea. Consider the following way to teach numbers. Tell students that all numbers come in pairs x and y, one bigger than one and one less than one, so that xy = 1. Corresponding to the number 22 is a decimal y less than 1, with 22y = 1. The following example gives you y:
Solve 22y = 1.0000000…..
22 is bigger than 1, and bigger than 10, so we start by finding the biggest y_1 so that 22y_1 less than 100, and find y_1 = 4, and 22y_1 = 88.
Then 100-88 = 12, so we tack on another zero and find the biggest y_2 so that 22y_2 less than 120, and find y_2 = 5, and 22y_2 = 110.
Then 120 – 110 = 10, so we tack on another zero and find the biggest y_3 so that 22y_3 less than 100–we’ve already done this, and y_3 = 4.
Hence the inverse of 22 is .0454545….; we move the decimal point 2 places over because we started with finding 22y_1 less than 100 = 10^2…
I know some of you will argue that this algorithm is “really long division’; keep in mind that long division, like the algorithm I present above, is “really” just an implementation of the Euclidean algorithm.
Fractions are a particular notation for division; they are not division itself. There are many ways to implement division, and most of the arguments I read here are, essentially, saying that “fractions should be taught because that’s how I understand division.” If you had been taught something else, would you understand division differently?
@ Anders:
I know that 1/3 and 0.3333… are the same number. But at a certain stage of my math development, I would have always answered the latter and not the former. I think having a preference for decimals in answer to a question like “What is 1 divided by 3?” is undesirable. If you try to use 0.3-repeating in any calculations, you’ll probably round it off prematurely and get a wrong answer. If you can stick to 1/3 all the way through and then (if necessary) get a decimal representation at the end, it’s better. That’s all I meant.
@ inkadu:
In response to my comment that you can’t write x/10 and 0.x, you responded
That only works for numbers less than 10. 20/10 is not equal to 0.20. I was talking about the variable x, as in the equation
x/10 + 20 = 220
This is not equivalent to the (meaningless) statement
0.x + 20 = 220
Wow agnostic, I thought “The Bell Curve” thinking went out of style half a century ago. It’s a pretty sad state of affairs if we’re going to accept the grim notion that our students are so lacking in mental capacity that they cannot be taught basic math – and yes, fractions are basic math, just with an extra rule or two.
As I mentioned at EvolutionBlog, I tutor high school students in math and science. I don’t have a teaching certificate, but had very high math performance in high school, largely because my mother drilled me in non-calculator math and taught me tools to use in solving problems (such as fractions).
Even today I can calculate common trig ratios more quickly than my students can on calculators, and see a distinct correlation between understanding fractions and understanding higher math like trig and calculus. And the ability to comprehend and incorporate mathematical rules has positively impacted every other non-mathematical aspect of my life, including my field of study – Bioethics.
Because what is math but abiding by a set of rules and proceeding in an orderly fashion? If we think that human beings cannot learn that, we are in sorry shape.
Oh, and there is a difference between rote memorization and drilling to promote good habits. If you develop good habits on showing calculations and setting up equations in an orderly fashion, the brain will learn to remember common associations in short order.
Nobody should be allowed to touch a calculator until they have demonstrated proficiency with a slide rule.
Tam
My misunderstanding. I certainly agree that common fractions often are more useful than decimals.
Anders Eg
Speedwell,
While I’m certain that Mark could explain a bit more clearly, this site gives a quick tutorial on factoring polymonials.
http://www.purplemath.com/modules/simpfact.htm
you may want to read the section about factoring quadratics as well.
http://www.purplemath.com/modules/factquad.htm
My original comment didn’t really make my point clear. I really was trying to make two seperate claims.
1) You will never get 100% or even 50% of kids to really like math. Yes, I agree that teaching makes a difference and many people who might have liked math with a different approach don’t but the point is that we can’t expect to teach all kids as if they were math majors. Some kids will hate math because they suck at it and if they hate it you can’t make them think and reason on their own.
2) Given that many children hate math most of lower division mathematical education is going to be wholly practical. Only students who are interested and excited can approach it from a theoretical point of view. In other words choosing to tell grade school students the axioms of arithmetic instead of drilling them in rote tools would leave us with a few mathematicians/engineers/etc.. and a bunch of people who couldn’t make change.
3) Rote learning is exactly what make people hate mathematics. If we want more people to be open to doing mathematics in college we need to spend less time teaching them that math is awful rote memorization in HS and before.
Conclusion: Given that teaching mathematics as a reasoning based discipline in HS and before isn’t really an option we should minimize the amount of harm done to these students by throwing out any skills that aren’t critical to daily life for average people. Fractions are just such a skill.
Obviously if I really believed that the number of people who like math couldn’t be changed then I wouldn’t think it mattered what we did here.
TruePath:
1) You will never get 100% or even 50% of kids to really like school. So let’s cut school down to two hours a day. Or even better, make it options. And adults don’t like work, so let’s get rid of that too.
The point is that rather than succumbing to the idea that “kids don’t like math, so let’s reduce the math they have to do”, let’s look at how we teach math and how we present math in society. Speaking as a female, I know that it is considered unusual for a girl to excel at, much less enjoy math, but it was presented to me as a game and as a vital skill for life, so I studied it with vigor and excelled. When adults complain about math, we just reinforce the misconception that math is somehow a more arduous chore than any other subject in learning (which is not a chore, it is development and growth).
2) While I acknowledge that practical math skills will have more relevance, there has to be a solid understanding of basic concepts, such as “what you do to one side of an equation has to be done to the other”. Is that highbrow PhD-level theory? No. It’s a concept that any child with a sibling learns – if they get it, I get it too. Without a basic understanding of concepts, they will become robots as they mechanically work through problems according to instructions rather than algorithms.
3) See my above comment that repeating a task is not necessarily rote memorization. It’s called practice, and as Aristotle said, “Virtue is a habit”. They say that adults don’t learn until they have heard something at least six times; young people learn more quickly, but there still must be enough repetition to observe, attempt, retry, and reinforce. I was one of those kids who hated times table drills in school, but instead of slamming the teacher, my parents told me that it was part of developing good habits, and that the quickest way to get out of them was to show that I knew what I was doing.
When I see my students blindly groping for the solution to a problem with no idea how to use a method or a standard problem solving algorithm, it is clear that the harm is coming from lack of basic math training, not from excessive exposure to math, as you claim. And when one of my students was repeatedly moved to tears stating, “I’m bad at math; I’m a bad person”, it is apparent that parents, teachers, and society as a whole are failing our students by not stepping in to HELP them to better understand math rather than blaming everything under the sun (including them) for math being hard.
P.S. I find it ironic that you state that you had two claims, but listed three, plus a conclusion. Thank you for the demonstration of the importance of understanding basic mathematical theory.
P.P.S. I’m not even going to honor the ridiculous claim that fractions are not important in daily life with a response since my 3/8 of a leftover pizza is far more interesting.
SabrinaW–reading comprehension much?
Nowhere in TruePath’s post did he/she argue that the amount of math taught should be lessened; hence your dismissive “You will never get 100% or even 50% of kids to really like school. So let’s cut school down to two hours a day” is irrelevent. Did you actually read the post? TruePath argues for changing what we teach in math, not decreasing the amount of math taught.
Flex, that gives me a good deal to go on. I appreciate that very much. Good verification of the notion that the passage of time plus a different presentation of data can make the difference. 🙂
Bill, TruePath’s double syllogistic argument is as such:
First syllogism:
1) Kids hate math (erroneous conclusion)
2) You can’t teach math properly without drilling (erroneous premise)
3) Drilling is why people (including kids) hate math (possibly valid premise)
Second syllogism:
1) We should strip down math that is not essential to make it less hated. (premise)
2) Fractions are not essential (erroneous premise)
3) Therefore, we should cut out fractions. (conclusion)
My counterexample, while not perfect, does mirror the idea that if people don’t like something, we should reduce it to the bare minimum, or possibly eliminate it (which more accurately follows the first syllogism than does your interpretation of TruePath’s argument)
I see no constructive suggestions in changing how math is taught, except to advocate stripping out fractions because they are “non-essential” to everyday life.
And how about the inaccurate counting? The sum total was 150-200% off. 🙂
speedwell – you don’t mention it, but I wonder: was it ever presented to you why we teach factoring polynomials?
I discovered several years ago, when tutoring my younger sister, that no one had ever explained this to her. In fact, I don’t remember anyone ever explaining it to me either.
Anyway, in case an understanding of the ultimate purpose of factoring polynomials is useful, here it is:
The short answer is that we learn to factor polynomials, and practice it, because it is part of a frequently useful technique for turning one large problem into several smaller problems. (Ideally, into smaller problems which we can then solve)
So how does that work? It’s based on an interesting property of zero (*), that when you have two amounts, A and B, and know that:
A * B = 0
Then you know that either A=0 or that B=0. This lets us replace the big problem of “A*B=0” with the two smaller problems of “A=0” and “B=0”.
So that property of 0 can be considered like a natural law, and factoring is the engineering technique we use to leverage that natural law to do useful things.
The general technique for attacking a large equation with factoring is:
1) move everything over to one side so that you have:
BIG_BUNCH_OF_STUFF = 0
2) Factor big bunch of stuff somehow into two (or more) pieces:
(SOME_STUFF) * (SOME_OTHER_STUFF) = 0
3) Solve the two smaller problems SOME_STUFF=0 and SOME_OTHER_STUFF=0.
Step 2 requires factoring, which usually requires a bunch of fiddling with different numbers until you can make it work. It’s a little bit like that math game where you have a card with a bunch of numbers and have to combine them together in a way that makes 24 – it usually requires quite a bit of mental trial-and-error, and it’s the kind of thing that gets somewhat easier with practice.
(*) disclaimer for the anal: in fields, such as the real or complex numbers
Daniel, no, I don’t recall that ever having been presented to me before, though I think I did accidentally intuit something inadequately like it as I struggled through the rest of the class.
Fortunately I was never of the “math is hard” persuasion, even though I did frequently think “this process should really not be THIS hard.” Aside from rote memorization, which I always found to be a stumbling block, math always just made sense… in fact it was practically the definition of “making sense” until factoring was incompletely presented to me. Thank you for your great suggestions.
I think # 54 and # 94 expands nicely on the placeholder notion, while adding some territory of their own.
It is true that we must keep the different notions of idealized and realized methods apart as long as they impart different qualities on the solutions.
Of course, ideally (or realistically :-P) imprecision and uncertainty creeps back into idealized solutions when they are used in practice.
I could be satisfied with that. Provided a notion of “fun” doesn’t preclude it at the same time. 🙂
No, really, it is essential to get this. Thanks Kristine for bringing it up!
In that contect I could go on and on about rote memorization, but that is boring and not really satisfying. The less said and done, the better IMHO.
@ Cailin # 57:
Um? Of course I can define the negative frequency to exist all by it lonesome. But usually we use complex frequencies as it is more convenient, it lets us deal with all phases.
I remember hwo I was taught to manipulate fractions, how bad the method was, and how little I understood about what I was doing until, several years later, I began to learn algebra and learnt how to manipulate fractiosn algebraically. There, in the space of 15 minutes os class time, I finally understood what fractions were all about.
I wonder if DeTurck was taught fractions as badly as I was…
In answer to comment #25, I can tell you that you are absolutely right. Even in these times of hand-held calculators that can do algebra effectively, you do need to know what is going on. I won’t say that I mastered arithmetic skills before using a calculator (I did pass all the classes, though), but I did always have some idea of what was going on, and whenever I wasn’t sure of my intuitions, it was simple enough to plug an answer back into an equation and see if it was correct or not.
I actually learned a lot of math by fidling with my calculator. Not just how to use the calculator, but actual math. By seeing tens or even hundreds of examples of various things I developed a sense for what the answer to many problems was (from trigonometry to derivatives to complex numbers) sometimes long before I learned any of that in class.
This allowed me to sail through the introductions to those topics in class, and even through some of the more diffuclt problems, since I alreayd had an idea of what the answer might be, and could tell if I was on the right track.
So, I’ve had great experiences using calculators, but I did see many examples of people who blindly trusted their calculators and paid for it, and thus quite understood my teachers’ impassioned pleas to give up our calculators as much as possible.
That is the reason I agree so very much with what Mark is saying here; DeTurck simply advocates having children blindly trust their calculators while having no idea what the numbers they are typing or the answer that comes out mean.
I find it funny that professor DeTurck, who believes fractions are obsolete, has fractions written all over the board behind him. Check the picture in the USA link. Is this his board? I hope so. Irony can be beautiful.
@Daniel: you may be pleased to know I go through that explanation in class every freaking time we factor to solve a quadratic in my classroom (algebra 1 & geometry) [why factor? because when you are multiplying things and getting a zero, one of those things has to be a zero]
@speedwell: i’m not sure which flavor of factoring you have trouble with, but the process is basically – given an algebraic expression, find some things that you can multiply to give you that expression. here’s the way i teach factoring out a greatest common factor to 9th graders, and it works like a champ:
Factor: 12(a^3)(b^2) + 15(a^2)b
We are trying to find things that divide evenly into both.
Rewrite the terms as the products of their factors:
12(a^3)(b^2) = 2 * 2 * 3 * a * a * a * b * b
15(a^2)b = 3 * 5 * a * a * b
Circle any factor common to both. So in each row, I would circle a 3, 2 “a”‘s, and one “b”.
That’s the greatest common factor! 3a^2b
All the “leftovers” are part of the other factor. In the top row, that’s 2 “2”‘s, one “a”, and one “b”, and in the bottom row it’s a “5”
Rewrite the equivalent expression accordingly:
3a^2b (4ab + 5)
and check your work by distributing the 3a^2b to see if it matches what you started with.
Factoring trinomials (ax^2 + bx + c) still means finding stuff you can multiply to get the trinomial, but requires different techniques. I just wanted to share the GCF explanation because it has worked so well for my students.
BTW,
While I don’t have many examples of real world applications of factoring polynomials, there is one which occasionally crops up and is quite important.
Finding the zero roots of the polynomial equations used in control theory tells you whether the system you are designing is stable or could enter a amplifing feedback loop which will tear the system apart.
Of course, in this case we are talking about working in the complex plane, but the principles are the same.
Maybe knowing that factoring polynomials is important to prevent catastrophic failure would help encourage people to learn how to do it. (And frankly, I’ve seen a number of people do pole analysis very poorly.)
Speedwell: I’m not sure if you still need any help with factoring polynomials, but this is the algorithm I use. I don’t think I was taught it in exactly this way, but it’s how I picked it up, and it’s served me well.
So, you’re given a trinomial in the form ax2+bx+c, and you need to factor it into two binomials in the form (hx+i)(jx+k).
What you can immediately realize is that only *two* of the four variables in the answer contribute to a, and the other two contribute to c. Specifically, a = h*j, while c=i*k. It’s the b that’s a problem, because all four factors go into it, since it’s hk+ij.
What I do is find out what possible pairs of numbers multiply together to form a. This gives me the possible choices for h and j. Do the same thing for c, yielding the possibilities for i and k. You now have two lists of number pairs, any of which *might* be valid. All you have to do is find out which ones will successfully give you b when they’re smashed together. So, move through the lists and pair off one entry from each list, checking them against b, until you find the answer.
That’s where guessing comes in. You can do this mechanically, listing out all the factors of a (giving you all the possibilities for h and j) and all the factors for c (giving you all the possibilities for i and k) and then all the combinations of them until you find one that yields b. But once you’ve gotten used to this technique, you gain some skill at guessing which factors are likely to work, cutting down on the effort spent on this.
Simple example: factor 2×2+11x+12.
The possible factors for a are easy – it’s just 1 and 2. This gives you an immediate answer for what h and j are, though you don’t yet know which is which.
C is 12, giving you possible factors of (1, 12), (2, 6), and (3, 4).
Note that both of these actually allow negative numbers as well; (-1, -2) would be a valid choice for a, frex. However, the way the signs are distributed (a, b, and c are all positive) lets you know that all the factors have to be positive as well.
Now, just take up a pair from each of the lists, and find one that sums up to b. It turns out the the correct pair is (1,2) and (3,4), with the final expression being (2x+3)(x+4).
Remember that intuition is optional in a problem like this – I described a full algorithm that will always give the answer as long as all the factors are integers (if the algorithm fails, you know they’re not integers, and can use the quadratic formula instead). However, I can at least walk through *my* intuition in solving this problem.
The first thing I notice upon looking at it is that b is less than c. This is fairly rare, actually – c is a product of two numbers, while b is a sum of two products, so it tends to be larger. This tells me that the factors of a and c will probably be small (that is, close to the square root), as that will help minimize b’s value. So my first guess is for c’s factors to be (3,4). There’s only two possibilities here, which is pairing (1,3) and (2,4) or pairing (1,4) and (2,3). Going through both of them, I find that the former gives me 11, so that’s the correct answer. Intuition lets me cut out a lot of work immediately!
It’s tempting to immediately write that (x+3)(2x+4), but of course we must remember that this doesn’t multiply correctly. Instead we write down (x+4)(2x+3), verify it by multiplying it out again, and are done.
Once you get this basic concept (developing two lists of factor pairs, and combining them to find b), you can easily extend this to more difficult trinomials, such as ones with not all numbers positive or with different powers of x. All it does is increase the length of the factor lists, but the actual problem doesn’t change at all.
There’s a much more elegant (though, at root, equivalent) quadratic factoring algorithm: find numbers p and q that add to b and multiply to ac, then do “single-bracket” factorisation.
To use your example, we need to multiple to 24 and add to 11: 8,3.
2×2+11x+12 = 2×2 + 8x + 3x + 12 splitting middle term
= 2x (x+4) + 3 (x+4) factorising each pair of terms
= (2x+3) (x+4), extracting common factor (x+4)
This always works, and of course it’ll still work if you write 11x as 3x+8x (proof left as an easy exercise).
@Agnostic
I think you are going too far in emphasizing innate intelligence, and not taking into account effort and conditioning. Speaking strictly from the complexity of the task, speaking a language is far more difficult than math. Not everyone is a Gauss, granted, but anyone who can speak English is smart enough to learn Calculus. The problem is people tell themselves they cannot do it before they even try. They are told that math is hard and you need to be a genius or something to learn it, so they give a pathetic effort and give up when they do not learn it instantly. Math, like anything worth doing, is hard, but rewarding. This is a very deep problem, and there is no easy fix for it, but we cannot go for slipshod solutions like “let’s teach decimals” or “let’s give up on those not smart enough.”
Oh, excellent! I’m so gonna steal that! (Because it is easier to steal than, say, learning math.)
Anyone here that actually suggests that George Andrews is stupid or a moron or doesn’t understand math is a complete idiot.
Prof. Andrews is amongst the top echelon of number theorists alive. He has been a major contributor to the theory of partitions and also deciphering the cryptic works of Ramanujan. Prof. Andrews has more brains in his little finger than most of us have all together. I’ve actually had the pleasure of meeting him and attending a few of his lectures and he is one of the nicest old men you could meet. His lectures were also very clear and he was great at explaining anything that wasn’t obvious.
That being said, DeTruck is also pretty good in his own right for his work in Differential Geometry. He has even won awards for his teaching.
Neither are “idiots” , but DeTruck’s ideas are just not well thought out. You cannot do Calculus before fractions for many reasons, but one obvious one being the treatment of rational functions. Even taking the integral of a simple monomial becomes impossible without fractions.
There are obviously thousands of reasons why you need to understand fractions to do all sorts of real world and higher mathematics. DeTruck means well but he obviously has not thought this one out.
no you should ask a math teacher how to factor
its easy
You could never get away with just taking fractions out of schools, for the most part, they’re more practical than decimals anyway. The point of explaining decimals to a kid the, “.05 of a cookie” would be much more difficult than fractions. Kids already have learned how to share, and dividing amounts in equal quantities, making it easy to set up the idea of fractions.
Learning maths is really a unique experience. It uses a totally different language using symbols, numbers and expressions. It is more graphical than textual. Learners need to appreciate these maths elements to understand the “hidden” meanings within them.
Different students actually view maths differently. It involves many factor like past experience, interest level, etc. Therefore it is not surprising to find adults not being able to do simple fractional computations. Teaching maths needs patience to “dig” into each individual’s learning mathematical gaps and fill them up.
Ultimately, it is the attitude to maths that matters.
Roughly 45% of students are visual learners, who learn what they see. Talking about fractions to them is useless.
Roughly 30% of students are auditory learners, who remember what they hear. Showing pie-chart fractions to them is useless.
Roughly 20% of students are tactile/kinaesthetic learners, who learn what they touch and feel. Talking about or showing fractions to them is useless. They need to be folding, cutting, dividing piles of things into subpiles.
Few students are abstract thinkers, the way that our blogmaster is.
I may say more about teaching this stuff to gangsters, drug addicts, and the molested some other time. I’ve had some unhappy teaching experiences in the inner city, recently.
I am 8 year old boy. Just reading what my father has left on his computer screen before going to bathroom. Most of it i cannot understand. but my mother taught me fractions using chapati. it is fun. we then learn decimals. if you donot know chapati, google it in images.
saud
“It’s just fiddling around with meaningless symbols.”
But that is what “Formalism” tells us mathematics is!
CB
Weird, I found decimals easier to understand than fractions. 0.38523…blah blah is just going 0.38523… of the way to 1.