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Topic: Fractions... obsolete? (Read 3959 times) |
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SWF
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Re: Fractions... obsolete?
« Reply #1 on: Jan 30th, 2008, 8:42pm » |
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With his approach a recent problem that appeared in our Medium board would become:
Prove that
0.3333333...*pi = 1 + 0.2 - .142857... -.09090... + .0769230... + .05882352941176470...
I don't think there is a well known easy to type notation to write those numbers unambiguously, and decimals make it tricky to see the pattern.
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Sir Col
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Re: Fractions... obsolete?
« Reply #2 on: Jan 31st, 2008, 12:09am » |
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(Disclaimer: according the the article, which is the first time I've heard about it...) He isn't talking about abolishing fractions, rather delaying the introduction of them until a student shows sufficient mathematical maturity to grasp the significance of them.
I don't know how education works in other parts of the world, but being a mathematics teacher in a secondary (high) school in England I am entirely sympathetic to the basis of his idea. In England our students must study Science, Mathematics, English, ICT, and a Modern Foreign Language up to age 16; other subjects become optional at age 14. For a long time now I have been an advocate of stripping the compulsory components of the mathematical curriculum down to essential basics. The type of mathematics that is necessary for specialised careers should be taught as an optional subject; like Art, Music, Geography, and so on. Does everyone possess the skills necessary to study other optional subjects to the comparative level of mathematics? Of course not! So why do we grumble about the standards of mathematics when we force incapable students to study it for so long? Do we hear equivalent complaints about the standards in the optional subjects? If we did, we would have to question the wisdom of the school in allowing the child to choose that subject.
Please do not misunderstand me. I love mathematics, I see the value in it for every person beyond the utilitarian purpose that many would fight for. I fully understand the importance of mathematics in developing and demonstrating an aptitude for processing abstact ideas. But I honestly feel that a large proportion of what we teach to children is unnecessary and is more damaging than good. You won't believe the number of people I meet and when the conversation gets to, "I teach mathematics," the typical and predictable response is, "Oh, I was never good at mathematics at school." How disheartening is that!
But if I were given powers to reform the mathematics curriculum would I abolish fractions? No sure?! 
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towr
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Re: Fractions... obsolete?
« Reply #3 on: Jan 31st, 2008, 12:32am » |
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How will they ever properly divide a cake without fractions ?!
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mikedagr8
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Re: Fractions... obsolete?
« Reply #4 on: Jan 31st, 2008, 12:44am » |
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Use their visual-spacial skills? 
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JiNbOtAk
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Re: Fractions... obsolete?
« Reply #5 on: Jan 31st, 2008, 3:23am » |
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How about Imperial Units ? Decimal inches is far too strange...
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FiBsTeR
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Re: Fractions... obsolete?
« Reply #6 on: Jan 31st, 2008, 6:12am » |
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on Jan 31st, 2008, 12:32am, towr wrote:How will they ever properly divide a cake without fractions ?!
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I would be content in having whole cakes for everyone. 
on Jan 31st, 2008, 12:09am, Sir Col wrote:| (Disclaimer: according the the article, which is the first time I've heard about it...) He isn't talking about abolishing fractions, rather delaying the introduction of them until a student shows sufficient mathematical maturity to grasp the significance of them. |
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This is what I thought, too. On a second read, though, the professor mentioned in the article seems to contradict himself at times.
The article first quotes him as saying:
Quote:| "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." |
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Then later the article states:
Quote:| He believes fractions are important for high-level mathematics and scientific research. |
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I'm still in high school, so perhaps I can't yet comprehend the full scope of the arguments, but I don't see how algebra could be taught without fractions.
This was my favorite quote of the article; keep in mind that the article is written and is about a professor in the United States:
Quote:| Questioning the wisdom of teaching fractions to young students doesn't compute with people such as George Andrews, a professor of mathematics at Pennsylvania State University and president-elect of the American Mathematical Society. "All of this is absurd," Andrews said. "No wonder mathematical achievements in the country are so abysmal. |
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| « Last Edit: Jan 31st, 2008, 6:19am by FiBsTeR » |
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towr
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Re: Fractions... obsolete?
« Reply #7 on: Jan 31st, 2008, 7:01am » |
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Quote:| DeTurck said as part of a 60-second lecture series. |
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One has to wonder if one can really say much in a 60-second lecture that doesn't sound absurd, and still give people something to think on.
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Ghost Sniper
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Re: Fractions... obsolete?
« Reply #8 on: Jan 31st, 2008, 10:16am » |
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I believe that the main problem with using fractions is that it is harder to input a problem containing equations into a calculator, esp. mixed fractions.
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towr
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Re: Fractions... obsolete?
« Reply #9 on: Jan 31st, 2008, 10:20am » |
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on Jan 31st, 2008, 10:16am, Ghost Sniper wrote:| I believe that the main problem with using fractions is that it is harder to input a problem containing equations into a calculator, esp. mixed fractions. |
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Well, they shouldn't be allowed to use a calculator in the first place. You don't learn things by having a machine do it for you; well, not unless you have to teach the machine how to do it for you as well.
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FiBsTeR
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Re: Fractions... obsolete?
« Reply #10 on: Jan 31st, 2008, 11:13am » |
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on Jan 31st, 2008, 10:16am, Ghost Sniper wrote:| I believe that the main problem with using fractions is that it is harder to input a problem containing equations into a calculator, esp. mixed fractions. |
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This hardly seems like a reason to advocate for the elimination of fractions from a school mathematics curriculum. Since when have teachers sought to remove something from a course because it was too hard on students?
Not to mention, calculators nowadays are more than capable of accepting fractions, even mixed fractions, as inputs.
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Icarus
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Re: Fractions... obsolete?
« Reply #11 on: Feb 1st, 2008, 3:42pm » |
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Basic fractions are a fundamental concept, used though out the english language (and I imagine most other languages as well). Not to teach them to children so that they can understand what all these linguistic references mean would be a very bad idea. Also, fractions are a much easier concept for children to grasp than decimals. Just because actually calculating with them is harder, doesn't mean the idea itself is harder to understand.
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Sir Col
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Re: Fractions... obsolete?
« Reply #12 on: Feb 1st, 2008, 4:06pm » |
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Putting common sense aside - as this is a prerequisite in teaching today - you try to explain to a "mathematically challenged" student why 1/2 + 3/4 is not equal to (1+3)/(2+4) but 1/2 * 3/4 is equal to (1*3)/(2*4). After they've grasped that - yeah, right! - you can inform them that "predictably" (1/2)/(3/4) does equal (1/3)/(2/4), but it's much easier to do (1*4)/(2*3). That all makes perfect sense!
Compare this with the complexity of decimal fraction arithmetic... 1.2+2 = 3.2, 1.2*2 = 2.4, and 1.2/2 = 0.6?
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SWF
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Re: Fractions... obsolete?
« Reply #13 on: Feb 1st, 2008, 5:44pm » |
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How would one explain the meaning of a decimal number without the use of fractions? I guess by 0.5 = 5*10^(-1), but seem ridiculous to not explain how 0.5 relates to half.
Seems like that professor is trying a backhanded way of getting Americans to start using the metric system. Do people who grow up using SI units take longer than others to understand fractions?
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Icarus
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Re: Fractions... obsolete?
« Reply #14 on: Feb 1st, 2008, 8:07pm » |
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on Feb 1st, 2008, 4:06pm, Sir Col wrote:Putting common sense aside - as this is a prerequisite in teaching today - you try to explain to a "mathematically challenged" student why 1/2 + 3/4 is not equal to (1+3)/(2+4) but 1/2 * 3/4 is equal to (1*3)/(2*4). After they've grasped that - yeah, right! - you can inform them that "predictably" (1/2)/(3/4) does equal (1/3)/(2/4), but it's much easier to do (1*4)/(2*3). That all makes perfect sense!
Compare this with the complexity of decimal fraction arithmetic... 1.2+2 = 3.2, 1.2*2 = 2.4, and 1.2/2 = 0.6? |
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This is calculating with them. It is the concept of what a fraction - particularly an egyptian fraction - represents that is easier for young children to comprehend than the concept of what a decimal is.
You explain to a kindergardner what 0.3 apples is. I'll explain to them what 1/3 of an apple is. I'll bet I get the correct idea across long before you!
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Sir Col
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Re: Fractions... obsolete?
« Reply #15 on: Feb 2nd, 2008, 3:28am » |
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on Feb 1st, 2008, 8:07pm, Icarus wrote:| You explain to a kindergardner what 0.3 apples is. I'll explain to them what 1/3 of an apple is. I'll bet I get the correct idea across long before you! |
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I guess if I was forced to use fractions then I'd be more inclined to explain what 3/10 of an apple is.
But seriously, I am not advocating the abolition of fractions entirely. I am simply relaying the difficulty that most people face when trying to "do" arithmetic with them. Neither am I suggesting that decimal arithmetic is easy, but moving a learner from integer arithmetic to increasingly more involved decimal fraction cases is far more natural than the apparent "mystery" of fraction arithmetic.
In England it is common for schools to place students for mathematics in sets based on ability. For some reason, this year, I seem to have acquired almost exclusively "bottom" sets. It has been quite a revelation to see the students at almost every age struggling with the same concepts; fractions is a classic example. I find myself teaching these ideas as if they've never seen them before, with students saying things like, "Oh I get it now. I never really understood that before." This could be quite flattering until you realise that you actually taught the same concept a number of years ago to that particular student. I must also balance my experiences with the fact that I teach in a Grammar school, where students are selected based on passing entrance examinations. So I am dealing with at least the top 50% ability range.
Compare and contrast the way I would teach a "top" set and "bottom" set student division of fractions. With a top set student I would use the concept of equivalent fractions; to use the example in the previous post, (1/2) / (3/4) = (2/4) / (3/4) = 2/3. This relates immediately to their previous experience of adding/subtracting fractions and is readily transferable to an algebraic context. However, with bottom set we write down "points to remember" in red clouds: "Replace the divide sign with a multiply sign and flip the second fraction". I can now give both sets ten questions and most of them will get 10/10. However, one month later I'd have to reteach bottom set as if they'd never seen it before. But at least, "I get it now. I never really understood that before."
Please don't misunderstand my post. I am not criticising students for which mathematics does not come natural. Rather I am suggesting that much of what we teach is cruel and unnecessary for many students. Although we, as teachers, do our best to get them through exams, they spend the rest of their lives with impressions like, "I hated mathematics; I was never much good at it anyway."
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towr
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Re: Fractions... obsolete?
« Reply #16 on: Feb 2nd, 2008, 7:48am » |
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Well, my father had enough problems teaching some students that 0*1 isn't 1. I don't think the problem lies with fractions; some people just can't relate to numbers.
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Icarus
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Re: Fractions... obsolete?
« Reply #17 on: Feb 2nd, 2008, 10:25am » |
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I'm not saying that we should keep doing what we do now either. But abolishing fractions as this math professor suggests isn't going to solve the problem, and would leave students unprepared for the real world, where fractions - particularly the simple ones - exist in profusion.
And if you take his idea that fractions are obsolete because calculators make it so easy to manipulate decimals, well then, there are calculators now that are able to add, subtract, multiply, and divide fractions too.
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temporary
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Re: Fractions... obsolete?
« Reply #18 on: Feb 3rd, 2008, 9:51am » |
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Today they get rid of fractions, in year 3000 they get rid of decimal(the base, not the point) and use binary!
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JiNbOtAk
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Re: Fractions... obsolete?
« Reply #19 on: Feb 4th, 2008, 1:31am » |
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on Feb 1st, 2008, 5:44pm, SWF wrote:| Seems like that professor is trying a backhanded way of getting Americans to start using the metric system. Do people who grow up using SI units take longer than others to understand fractions? |
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I was only aware of imperial units in my teens, I used to think everyone used SI units !!
Personally, I prefer learning about fractions than decimals, although that is entirely personal. Some might find it differently though. I have to agree with Sir Col though, there's something wrong with the education system. I used to think it only happens here, but I guess it's true everywhere.
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