Dividing fraction - how do they teach? (2 Viewers)

oasfree · Jan 22, 2009

I'd like to get an idea how math teachers teach the kids to do division of fraction in modern times. I was shocked when I saw how professional teachers in a video clip offered in a HomeTutor product from Telstra Bigpond show kids how to do fraction division.

This is what they show

2/3 <meta name="Generator" content="Microsoft Word 11"><meta name="Originator" content="Microsoft Word 11"><link rel="File-List" href="file:///C:%5CDOCUME%7E1%5CDAO%7E1.OYS%5CLOCALS%7E1%5CTemp%5Cmsohtml1%5C01%5Cclip_filelist.xml"><!--[if gte mso 9]><xml> <w:WordDocument> <w:View>Normal</w:View> <w:Zoom>0</w:Zoom> <w

unctuationKerning/> <w:ValidateAgainstSchemas/> <w:SaveIfXMLInvalid>false</w:SaveIfXMLInvalid> <w:IgnoreMixedContent>false</w:IgnoreMixedContent> <w:AlwaysShowPlaceholderText>false</w:AlwaysShowPlaceholderText> <w:Compatibility> <w:BreakWrappedTables/> <w:SnapToGridInCell/> <w:WrapTextWithPunct/> <w:UseAsianBreakRules/> <w

ontGrowAutofit/> </w:Compatibility> <w:BrowserLevel>MicrosoftInternetExplorer4</w:BrowserLevel> </w:WordDocument> </xml><![endif]--><style>  </style> ÷ 3/4 = 2/3 x 4/3

Then they explain that 4/3 is the reciprocal of 3/4. Then they define the reciprocal of a number is whatever that times with that number to give you value 1. To find the reciprocal they tell kids to flip the fraction so that numerator becomes denominator. And this is where they stop the lesson.

Is this what they teach kids at HS these days? If kids are taught this way, how the hell will they really understand fraction? Those who don't challenge their teachers to fully explain will surely become morons who could only apply formula without understanding where the formula comes from.

I would also be interested in how they teach the same thing at coaching places. This will help me to understand if teachers at coaching places are any better than those at our schools!

kaz1 · Jan 22, 2009

What's wrong with flipping the fraction? It's easy to understand.

black_kat_meow · Jan 22, 2009

That's how I learnt... and everyone I know.

vds700 · Jan 22, 2009

I remember how my year 7 maths teacher taight us; "Invert and multiply"

Dont see why it need be any less simple than that

Trebla · Jan 22, 2009

Division by definition is the inverse of multiplication, which is where the reciprocal concept arises.

oasfree · Jan 23, 2009

Oh gosh, is that how you guys at HS think mathematics should be taught or learned? That way is simply the applying of a rule that teachers did not care to fully explore and explain how that rule was derived during the last 200 years. Wait until you guys get into University and if you guys ever enroll to learn pure mathematics where they start asking you to explore the foundation of mathematics, mathematical logic, number theory, set theory and the idea that mathematics is the closest to perfection among the fallible world of knowledge we acquire through the senses. Suddenly you start to doubt everything that you have ever learned before. I know perhaps I am asking too much of teachers. Many primary school teachers do not even learn mathematics at University (or even 3-4 unit maths when they were at HS).

When I read the NSW mathematics syllabus, it stated that kids at stage 2-3 in middle primary school should start to develop their own methods along with methods taught by teachers. I thought if kids are really like that, they would all become PhDs by the time they reach University. Perhaps that's the ideal world that never exists. I guided my own kid in this spirit, and the kid started to correct primary school teachers when they made silly mistakes (since middle of grade 3). Fortunately the teachers had been kind to my kid.

Honestly, any guy here between year 10-12 who have been taught to divide fraction that way (and have not been re-taught with a complete algebraic logical proof to derive the rule) can really trust that this method really produces correct results? Have you ever ask how they came up with such a rule? Have you guys tried to go back and derive the rule yourself to be completely sure that it's absolutely correct?

At year 7 in HS, they normally start to teach division of fraction. This happens at the same time when algebra is introduced. For gifted kids, they normally learn simple fraction at grade 2, fraction division at end of grade 3 and basic algebra at grade 4. The rule about division of fraction can be derived by using basic algebraic skills. I reckon it's a crime that teachers do not lead kids through this exercise by end of year 7 or early year 8. I think if kids do not completely master something, it's a learning gap. As they get older there is no time to bridge these gaps.

Mathematics is the only area of study that people can strive to be completely correct. It looks like this spirit isn't popular any more?

L · Jan 23, 2009

tl;dr

who cares
its yr7

black_kat_meow · Jan 23, 2009

oasfree said:
Oh gosh, is that how you guys at HS think mathematics should be taught or learned? That way is simply the applying of a rule that teachers did not care to fully explore and explain how that rule was derived during the last 200 years. Wait until you guys get into University and if you guys ever enroll to learn pure mathematics where they start asking you to explore the foundation of mathematics, mathematical logic, number theory, set theory and the idea that mathematics is the closest to perfection among the fallible world of knowledge we acquire through the senses. Suddenly you start to doubt everything that you have ever learned before. I know perhaps I am asking too much of teachers. Many primary school teachers do not even learn mathematics at University (or even 3-4 unit maths when they were at HS).

When I read the NSW mathematics syllabus, it stated that kids at stage 2-3 in middle primary school should start to develop their own methods along with methods taught by teachers. I thought if kids are really like that, they would all become PhDs by the time they reach University. Perhaps that's the ideal world that never exists. I guided my own kid in this spirit, and the kid started to correct primary school teachers when they made silly mistakes (since middle of grade 3). Fortunately the teachers had been kind to my kid.

Honestly, any guy here between year 10-12 who have been taught to divide fraction that way (and have not been re-taught with a complete algebraic logical proof to derive the rule) can really trust that this method really produces correct results? Have you ever ask how they came up with such a rule? Have you guys tried to go back and derive the rule yourself to be completely sure that it's absolutely correct?

At year 7 in HS, they normally start to teach division of fraction. This happens at the same time when algebra is introduced. For gifted kids, they normally learn simple fraction at grade 2, fraction division at end of grade 3 and basic algebra at grade 4. The rule about division of fraction can be derived by using basic algebraic skills. I reckon it's a crime that teachers do not lead kids through this exercise by end of year 7 or early year 8. I think if kids do not completely master something, it's a learning gap. As they get older there is no time to bridge these gaps.

Mathematics is the only area of study that people can strive to be completely correct. It looks like this spirit isn't popular any more?

No, it's not. Probably never was. To be quite frank, the average student won't give a f*ck where the formula came from as long as they get the right answer and pass. Maybe you haven't noticed, but maths isn't a particularly popular subject among the masses in school.

At the moment, I think it's more important that the kids are taught correct grammar and spelling (seeing many teachers fail dismally at this).

tommykins · Jan 23, 2009

I did 4unit so I have abit of a clue on how to derive equations, but mate not everyone is able to share your passion of mathematics. (which is a shame really, maths is bloody awesome)

Kids already hate it as it is, if you make it anymore complex there will be a huge drop in numbers.

Schoey93 · Jan 23, 2009

oasfree said:
Oh gosh, is that how you guys at HS think mathematics should be taught or learned? That way is simply the applying of a rule that teachers did not care to fully explore and explain how that rule was derived during the last 200 years. Wait until you guys get into University and if you guys ever enroll to learn pure mathematics where they start asking you to explore the foundation of mathematics, mathematical logic, number theory, set theory and the idea that mathematics is the closest to perfection among the fallible world of knowledge we acquire through the senses. Suddenly you start to doubt everything that you have ever learned before. I know perhaps I am asking too much of teachers. Many primary school teachers do not even learn mathematics at University (or even 3-4 unit maths when they were at HS).

When I read the NSW mathematics syllabus, it stated that kids at stage 2-3 in middle primary school should start to develop their own methods along with methods taught by teachers. I thought if kids are really like that, they would all become PhDs by the time they reach University. Perhaps that's the ideal world that never exists. I guided my own kid in this spirit, and the kid started to correct primary school teachers when they made silly mistakes (since middle of grade 3). Fortunately the teachers had been kind to my kid.

Honestly, any guy here between year 10-12 who have been taught to divide fraction that way (and have not been re-taught with a complete algebraic logical proof to derive the rule) can really trust that this method really produces correct results? Have you ever ask how they came up with such a rule? Have you guys tried to go back and derive the rule yourself to be completely sure that it's absolutely correct?

At year 7 in HS, they normally start to teach division of fraction. This happens at the same time when algebra is introduced. For gifted kids, they normally learn simple fraction at grade 2, fraction division at end of grade 3 and basic algebra at grade 4. The rule about division of fraction can be derived by using basic algebraic skills. I reckon it's a crime that teachers do not lead kids through this exercise by end of year 7 or early year 8. I think if kids do not completely master something, it's a learning gap. As they get older there is no time to bridge these gaps.

Mathematics is the only area of study that people can strive to be completely correct. It looks like this spirit isn't popular any more?

Lockhart's Lament

wixxy2348 · Jan 23, 2009

black_kat_meow said:
That's how I learnt... and everyone I know.

Ditto.

If they'd tried to teach me any other way I would have stopped listening.

oasfree · Jan 23, 2009

Thanks a lot guys, you guys are very honest. I fully appreciate the enormity of the problem right now. I fully agree that language, spelling, grammar are all important things but mathematics is very important too for a nation that wants to become smart. The Russian, US and Europeans did not rely on languages alone to build technologies and explore space or deter aggression by building sophisticated weapons. Now we have the rise of the sleeping giants like India and China to reckon with too. I have the benefit of University mathematics and philosophy to back my passion for truth. To be aware of what is going out there in schools really help me to see how kid will eventually fit into this education system.

I wonder if selective school kids feel same way about mathematics! If they also feel same way, I wonder if they should go to selective schools at all! if the teachers there teach the same way, they should not be there at all! I would appreciate if some selective school students give me comments on this issue.

When I see math education at primary school being TOO basic, I started to teach my own kid at home. I think if teachers see how I teach my kid, they would wonder if I want my kid to become another Pythagoras or some character like that. But honestly we treat math as a fun subject, more of a philosophy and history telling in the old fashion way. But it is always about the credibility of an idea and how to prove that the idea is mathematically correct. We raise all silly questions like "Why 1 + 1 = 2?" or "When you chop an orange into 2 parts, why is it wrong to say now we have 2 > 1? Why it looks like the sum of the parts are bigger than the whole?". We ask questions like "When does it make sense to divide something? Would division change the nature of things to render the operation meaningless?". So by grade 3, the poor teacher at primary got pushed by the kid mathematically. Any silly mistake bounced back to her face quickly when the kid politely suggested that it was wrong and offered a logical reasoning to lead to what was right. This was duly noted in the school report and a frank admission by the teacher at parent & teacher interview. It's not just mathematics. The kid is well rounded in all other subjects too.

Hey guys, I'd like to offer a suggestion. If you guys take a bit of trouble to go back to the very basic and attempt to logically prove that some of those simple mathematical rules are COMPLETELY correct, you will lift your math performance dramatically. You will find that your 3u and 4u math extensions become interesting and easier to learn. I know some will feel "I don't give a damn". But you guys are the future of the country. Old farts like me and your parents will eventually become much older and disappeared into the ground. But as long as we are paying taxes so that all kids can have good education, I wish for the best return for the investment for the country. Going at this rate, Australia will dig all the minerals and sell until we have lots of holes in the country and the future kids will be very poor.

TearsOfFire · Jan 23, 2009

oasfree said:
I wonder if selective school kids feel same way about mathematics! If they also feel same way, I wonder if they should go to selective schools at all! if the teachers there teach the same way, they should not be there at all! I would appreciate if some selective school students give me comments on this issue.

If you go to a selective school, it doesn't mean you have good teachers.

Trebla · Jan 23, 2009

oasfree said:
Oh gosh, is that how you guys at HS think mathematics should be taught or learned? That way is simply the applying of a rule that teachers did not care to fully explore and explain how that rule was derived during the last 200 years. Wait until you guys get into University and if you guys ever enroll to learn pure mathematics where they start asking you to explore the foundation of mathematics, mathematical logic, number theory, set theory and the idea that mathematics is the closest to perfection among the fallible world of knowledge we acquire through the senses. Suddenly you start to doubt everything that you have ever learned before. I know perhaps I am asking too much of teachers. Many primary school teachers do not even learn mathematics at University (or even 3-4 unit maths when they were at HS).

First of all, the mind of the average primary school kid would more than likely fail to grasp the derivation with high mathematical rigour. Mathematics has to be built on intuition at primary level. Simple operations like addition (e.g. counting objects) and multiplication (e.g. rows/columns) are taught based on intuition.

oasfree said:
When I read the NSW mathematics syllabus, it stated that kids at stage 2-3 in middle primary school should start to develop their own methods along with methods taught by teachers. I thought if kids are really like that, they would all become PhDs by the time they reach University. Perhaps that's the ideal world that never exists. I guided my own kid in this spirit, and the kid started to correct primary school teachers when they made silly mistakes (since middle of grade 3). Fortunately the teachers had been kind to my kid.

The intentions of the syllabus are not necessarily outcomes that are met. It assumes the student is fully engaged with syllabus when in reality this is not always the case. However, those who do engage with the syllabus do develop their own methods.

oasfree said:
Honestly, any guy here between year 10-12 who have been taught to divide fraction that way (and have not been re-taught with a complete algebraic logical proof to derive the rule) can really trust that this method really produces correct results? Have you ever ask how they came up with such a rule? Have you guys tried to go back and derive the rule yourself to be completely sure that it's absolutely correct?

I'm not quite sure what you mean. I was taught these concepts based on intuitive examples way back in primary school. Most good textbooks also use intuition to explain them (e.g. taking half of a semi-circle leads to a quarter of a circle).

oasfree said:
At year 7 in HS, they normally start to teach division of fraction. This happens at the same time when algebra is introduced. For gifted kids, they normally learn simple fraction at grade 2, fraction division at end of grade 3 and basic algebra at grade 4. The rule about division of fraction can be derived by using basic algebraic skills. I reckon it's a crime that teachers do not lead kids through this exercise by end of year 7 or early year 8. I think if kids do not completely master something, it's a learning gap. As they get older there is no time to bridge these gaps.

I think that is a problem which is isolated within individual schools, not necessarily across the board.

That being said, I do think there is a problem in terms of the way senior levels of high school mathematics is being taught. Many teachers simply write down a formula with little derivation and expect students to apply it. I was lucky enough to have teachers who basically derived most formulae in Year 11and 12 to show where they came from but most teachers don't do this despite the syllabus explicitly stating that the proof of some formulae are required, because they are not examinable (and some students also pressure them to stick to examinable material only). Another problem is that only the best of the cohort would truly understand and appreciate these derivations, but the majority would hardly care.

oasfree · Jan 23, 2009

Trebla, thanks for your comments. One of the most powerful and most abused tool of the human mind is the ability to generalise. In mathematics, this is a formal concept called generalisation which, in many ways, simplifes and extracts the essentials to produce rules that could qualify over as many instances of experience as possible. For example, when it's about dividing fraction, one would like to see this form

For all a, b, c and d as members of the set of real numbers, this rule is valid
<table class="tborder" style="border-top-width: 0px;" width="100%" align="center" border="0" cellpadding="3" cellspacing="1"><tbody><tr title="Post 4066942" valign="top"><td class="alt1">a/b <link rel="File-List" href="file:///C:%5CDOCUME%7E1%5CDAO%7E1.OYS%5CLOCALS%7E1%5CTemp%%20%205Cmsohtml1%5C01%5Cclip_filelist.xml"><!--[if gte mso 9]><xml> <w:WordDocument> <w:View>Normal</w:View> <w:Zoom>0</w:Zoom> <w

unctuationKerning/> <w:ValidateAgainstSchemas/> <w:SaveIfXMLInvalid>false</w:SaveIfXMLInvalid> <w:IgnoreMixedContent>false</w:IgnoreMixedContent> <w:AlwaysShowPlaceholderText>false</w:AlwaysShowPlaceholderText> <w:Compatibility> <w:BreakWrappedTables/> <w:SnapToGridInCell/> <w:WrapTextWithPunct/> <w:UseAsianBreakRules/> <w

ontGrowAutofit/> </w:Compatibility> <w:BrowserLevel>MicrosoftInternetExplorer4</w:BrowserLevel> </w:WordDocument> </xml><![endif]--><style>  </style> ÷ c/d = a/b x d/c</td></tr></tbody></table>
The rule is a generalised form of all instances of things like this

<table class="tborder" style="border-top-width: 0px;" width="100%" align="center" border="0" cellpadding="3" cellspacing="1"><tbody><tr title="Post 4066942" valign="top"><td class="alt1"> 2/3<link rel="File-List" href="file:///C:%5CDOCUME%7E1%5CDAO%7E1.OYS%5CLOCALS%7E1%5CTemp%%20%205Cmsohtml1%5C01%5Cclip_filelist.xml"><!--[if gte mso 9]><xml> <w:WordDocument> <w:View>Normal</w:View> <w:Zoom>0</w:Zoom> <w

unctuationKerning/> <w:ValidateAgainstSchemas/> <w:SaveIfXMLInvalid>false</w:SaveIfXMLInvalid> <w:IgnoreMixedContent>false</w:IgnoreMixedContent> <w:AlwaysShowPlaceholderText>false</w:AlwaysShowPlaceholderText> <w:Compatibility> <w:BreakWrappedTables/> <w:SnapToGridInCell/> <w:WrapTextWithPunct/> <w:UseAsianBreakRules/> <w

ontGrowAutofit/> </w:Compatibility> <w:BrowserLevel>MicrosoftInternetExplorer4</w:BrowserLevel> </w:WordDocument> </xml><![endif]--><style>  </style> ÷ 3/4 = 2/3 x 4/3</td></tr></tbody></table>
To be able to state that this rule works over ALL real numbers is a big step that no one can state that without formal proof.

While I don't expect that kids in primary school can easily handle this rigour, but I think year 7-8 should be able to handle it. Infact I learnt it when I was very young in the olden school teaching style. To derive this, one only need to use very basic basic algebraic transofrmations. I have found it quite possible to teach young kids the idea by taking some of the fancier stuff out of it. At primary schools kids are shown pictures to learn fraction. But when they hit division of or multiplication of fractions, pictures are no longer adequate. And this is where basic algebra steps in. I personally think this is essential for HS students to really build the ability to tackle 3u and 4u extensions. But students here don't seem to recall any teacher that taught them to derive even basic rules. That's my worry for future generations.

tommykins · Jan 23, 2009

oasfree said:
I wonder if selective school kids feel same way about mathematics! If they also feel same way, I wonder if they should go to selective schools at all! if the teachers there teach the same way, they should not be there at all! I would appreciate if some selective school students give me comments on this issue.

I went to a selective school and luckily enough I had great teachers for prelim and hsc 4unit math.

We derived most of the rules we had and it was pretty great. We also had to prove a few of them as well.

Although fustrating, the high you get from being mentally fustrated and leading to a correct answer is the best.

PS. I'm planning to pursue mathematics even when I'm older, I enjoy reading books on discoveries (game theory had me VERY interested, abit of psychology mixed with math) and therefore teach my kid as well.

But I hope you realise that general society is depreciating, the course is getting dumber - general society cares not for those who enable the wolrd to function (ie. mathematicians/mathematics), all they care is the world is working.

During my 4unit hsc study period I look at pre 1980 exams and was simply amazed at the amount of questions I couldn't do bar the topics that have been taken out.

oasfree · Jan 24, 2009

tommykins said:
I went to a selective school and luckily enough I had great teachers for prelim and hsc 4unit math.

We derived most of the rules we had and it was pretty great. We also had to prove a few of them as well.
...

Thanks for the info. So it's not dead. It's again up to the teacher to teach. I guess soon you will start University where you will find that the ability to derive formulas is a must. This is particularly true if you take all strands of mathematics including applied math and pure math. In pure math you will deal with the issues at the foundation of mathematics and face questions that the masters of the old time face in logic, number theory, set theory, ... You will investigate questions whether anything is valid at all. If you also combine math with philosophy, you will start to deal with epistemology which is the study about how on Earth humans acquire knowledge and truth. You will heavily use inductive and deductive methods of reasoning. All this is very interesting. However I suppose you need to study something that will get a good job and money as well. Well, that was what I did any way. One major for making money and one major for pure enjoyment of scholarly knowledge. Best of luck!

tommykins · Jan 24, 2009

I'm planning to do mathematics after my engineering degree as a post grad.

DownInFlames · Jan 24, 2009

oasfree said:
Oh gosh, is that how you guys at HS think mathematics should be taught or learned? That way is simply the applying of a rule that teachers did not care to fully explore and explain how that rule was derived during the last 200 years. Wait until you guys get into University and if you guys ever enroll to learn pure mathematics where they start asking you to explore the foundation of mathematics, mathematical logic, number theory, set theory and the idea that mathematics is the closest to perfection among the fallible world of knowledge we acquire through the senses. Suddenly you start to doubt everything that you have ever learned before. I know perhaps I am asking too much of teachers. Many primary school teachers do not even learn mathematics at University (or even 3-4 unit maths when they were at HS).

When I read the NSW mathematics syllabus, it stated that kids at stage 2-3 in middle primary school should start to develop their own methods along with methods taught by teachers. I thought if kids are really like that, they would all become PhDs by the time they reach University. Perhaps that's the ideal world that never exists. I guided my own kid in this spirit, and the kid started to correct primary school teachers when they made silly mistakes (since middle of grade 3). Fortunately the teachers had been kind to my kid.

Honestly, any guy here between year 10-12 who have been taught to divide fraction that way (and have not been re-taught with a complete algebraic logical proof to derive the rule) can really trust that this method really produces correct results? Have you ever ask how they came up with such a rule? Have you guys tried to go back and derive the rule yourself to be completely sure that it's absolutely correct?

At year 7 in HS, they normally start to teach division of fraction. This happens at the same time when algebra is introduced. For gifted kids, they normally learn simple fraction at grade 2, fraction division at end of grade 3 and basic algebra at grade 4. The rule about division of fraction can be derived by using basic algebraic skills. I reckon it's a crime that teachers do not lead kids through this exercise by end of year 7 or early year 8. I think if kids do not completely master something, it's a learning gap. As they get older there is no time to bridge these gaps.

Mathematics is the only area of study that people can strive to be completely correct. It looks like this spirit isn't popular any more?

Holy shit, batman! You in love with the mathematics!

Well, for most people, knowing the rule and how to apply it is all they will ever need to know. Not everyone is going to become a mathemetician, and therefore the most practical (and easy-to-remember) way of learning is "Invert and multiply" or "divide by 100 then times by 40 to get 40%"

I agree with you in principle, learning to derive your own methods is certainly the best way to understand what exactly you're doing. But kids who are actually interested will pick up on this anyway, by playing around with the numbers.

For teaching the "invert and then multiply" method, I think it's pretty standard to go through a few examples in class of how it basically works, but thats as far as most people need to go, and they don't really need to be able to understand it. People who are going on to study math at uni will tend to be the people who innately understand these things.

Also, what kind of primary teacher needs 3 unit math? They need to be able to explain primary school math WELL (and be able to cover a vast range of topics of which math is not the most important.)

Will Shakespear · Jan 24, 2009

DownInFlames said:
Well, for most people, knowing the rule and how to apply it is all they will ever need to know. Not everyone is going to become a mathemetician, and therefore the most practical (and easy-to-remember) way of learning is "Invert and multiply" or "divide by 100 then times by 40 to get 40%"

not rly

for most people, the rules they "learn" at high school are absolutely useless

by that i mean high school maths is completely useless for just about everyone

you don't even need to apply the rule, you just pick up a calculator

the brain-expansion you get from learning where the rule comes from is probably more beneficial than learning formulas that won't be retained after the exam

Dividing fraction - how do they teach? (2 Viewers)

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