I had (have?) a problem where I needed to deeply understand a topic to be happy with it. I'd do every question in the back of the textbook. Build little excel models. Ask the profs weird/"obvious" questions.
It takes too goddam long though. At university, there's often not enough time to deeply understand something before the exam. And your competition is just learning the tricks. And it's ego crushing to do poorly on an exam.
I did okay though, despite. I ended up teaching at a good school and being published, but my grades would have been 5-10% higher had I just learned the tricks.
Teaching is frustrating too. Watching the kids that only care about the tricks ace the exam, and the ones with curious minds screw up a few mechanical things.
> Teaching is frustrating too. Watching the kids that only care about the tricks ace the exam, and the ones with curious minds screw up a few mechanical things.
Or worse, the majority of kids that don't even realise the subject is more than just the tricks themselves.
That's because these tricks is how you perform the operation quickly and efficiently. I'm not sure if you can even meaningfully derive the algebraic process and meaning from first principles. You should nix the cute mnemonic stories, since those probably serve to confuse people, but the methods themselves are absolutely solid. And the one shown right on the front page (for me) cross multiply is how you divide by fractions. a/b / c/d = a/b * d/c. The fact that 1/a is the reciprocal of a, whatever a is and that 1/(a/b) = b/a should be the first thing you tell people after you tell them about fractions. Then you show why cross multiply works. That shouldn't take more than one lesson.
In the name of getting rid of "mechanical" learning, most schoolkids these days don't even get to learn multiplication tables. While learning the tables is a pain, they pay rich dividends.
As an example, an 8th-grade kid who I help with math was struggling to find the square root of 529. He tried dividing it by his usual set of numbers, and pretty much hit a brick wall. I tried to get him to see that 529 was between 400 and 625, so its square root had to lie between 20 and 25. He did see it, but it's going to take him a while to figure out that he can do this with any number, if he can remember (or quickly work out) a few squares.
Now, one /could/ argue that computing the square root of a number is in itself mechanical ... but just a nodding familiarity with numbers can make even that a breeze.
And yes, the root cause of the problem is that math is taught as a bag of complicated tricks which need to be remembered, and not understood :/
Thanks to all of you for the interesting comments on this online book, which came to this thread overnight in my time zone. I am a mathematics teacher, and this link was recommended to me by a parent just more than a day ago.
To answer one frequently asked question in the previous comments, no, I don't think any school anywhere includes teachers who use all these tricks in their teaching, and, yes, I think most of these classroom teacher tricks are specific to the United States. In the United States, the great majority of elementary school teachers are generalists, teaching all school subjects to their pupils, and their higher education does not prepare them well for teaching mathematics.[1] By contrast, elementary teachers in many countries where students learn more mathematics more thoroughly are subject-matter specialists, with mathematics teachers teaching only elementary mathematics, and other teachers teaching elementary pupils other subjects.
My teaching is in two contexts: since 2007, I have taught voluntary-participation, extracurricular courses in prealgebra mathematics with additional advanced topics to middle-elementary age (mostly fourth grade) pupils in weekend supplementary classes. Many of those students are quite advanced intellectually for their age. They come to my classes (mostly through word-of-mouth recommendations from their parents' friends) from a ten-county expanse of Minnesota. Just this school year (that is, just since August) I am also on the faculty of an independent school, teaching all sixth grade pupils and "honors" seventh grade students mathematics at about the same level, although it is my intention this school year to move the seventh graders along into topics that can properly be called "beginning algebra" topics (to solving quadratic equations and graphing systems of two linear equations on the Cartesian plane).
The school where I teach is reforming its mathematics curriculum with advice from a nonprofit consulting organization.[2] The reform program at my school is informed by international best practice in primary and secondary education and by what mathematics background is necessary to succeed in higher education in universities like MIT. (The founder of the consulting organization is an MIT alumna.) As the submitted ebook says, the hardest thing for a teacher to do is to encourage students to think rather than just rely on a mindless trick. This year I will have to set homework and tests that I write myself to probe for deep understanding of mathematical concepts and relentlessly try to find out how (and even whether) the learners in my care think about mathematics. Most of the rest of today on my weekend schedule is slated for writing a major unit test for my seventh graders, who use an excellent textbook[3] that is part of a textbook series that doesn't provide teachers with ready-made unit tests. The textbook is based on problem-solving and explaining mathematics from first principles (Chapter 1 takes the field properties of the real numbers as an axiom system to explain many principles of arithmetic) and is the best textbook for its topic available in English.
I agree. My father was working in thermodynamics, and he quietly steered me from tricks to being able derive things myself. For example, in cross-multiplication, percents, unit conversion, long division.
I remember him saying: "Forget all rules about percentages. All you need to know it's like a number multiplied by 100."
He also wasn't fond of mole as unit, he felt that it obscures things especially for students.
Later I applied the same method myself on algebra (deriving binomial expressions), trigonometric functions (deriving constants from unit circle, other relations from the basic relations of sin(x+y)) and so on.
I guess this actually applies to any field. Do not start with tricks and shortcuts until you understand and can do the deed with basic methods.
It's still a complicated approach without proper material or teachers to orient you. There are "obvious" derivations that may be possible to produce by most people by putting a couple of hours on it, and then there are other "obvious" derivations that took hundreds of years and thousands of man-hours, and when you are learning you aren't on a position to tell them apart. Others are a bit in the middle and can be turned into the "easy" kind by making a few not-so-obvious assumptions. But that still requires a proper guidance.
Even derivations itself probably contain plenty of not so obvious "tricks" that should be known to develop them. It's not so clear cut what is a trick and what are obvious assumptions from the real world. I suspect that the regular increase of IQ every generation may be related to all those hidden tricks and skills that everybody acquires at school or from general culture, but that at some point they become invisible obvious assumptions and common sense.
Another easier approach may be to start with the tricks and work backwards from there to more general premises.
> I remember him saying: "Forget all rules about percentages. All you need to know it's like a number multiplied by 100."
I don't remember what rules or tricks I was taught about percentages, but I do remember realising that "%" meant "/ 100" and from there everything else followed.
Ooh I'd be excited to read this. I recently found "Linear Algebra Done Right." How, you ask? They defer using determinants until the end of the book. Determinants are hard computationally, hard to define, and not obviously useful. Without them (and I haven't read the book but I will), one has to make more general, less computational proofs.
The determinant of a linear map is its volume scale factor: any shape of volume V in the input space gets transformed to a shape of volume det(L)V in the output space. This makes a bunch of things easy to understand:
- Why you can calculate the volume of a slanted box with a determinant
- Why a matrix is singular if its determinant is zero
- Why the determinant is the product of the eigenvalues
- Why change of variables in integration gets a det(J) factor
Sadly, most linear algbra texts introduce the determinant as some random summation formula or with a series of unmotivated axioms. This is a general problem with mathematics: symbols over geometry, and formal proofs over intuitive understanding when it should have been the other way around.
For the opposing view see Street Fighting Mathematics which encourages not only using tricks but using them well and often.
The problem is if you only learn the trick without the reasoning behind it. The solution isn't to not learn the trick, it's to learn why it works.
I think it's a disservice to kids who will go into science and engineering if they've never been allowed to use heuristics before. Too often you learn a long-form solution method and then the "trick" to solve it quickly and you need to be able to do both.
The linked website seems to confuse mnemonics (like FOIL) with 'tricks' like cross-multiply (which I thought was for ratios: x/5 = 3/15- I learned 'invert and multiply' for dividing fractions).
I think that tricks that allow you to do things quickly in your head are great- they help build numeracy.
That's horrifying. Not the book, but the "tricks". All of those are about memorizing the superficial (graphical things) without understanding what's happening, weakening the connection between graphical representation and performed operation. I'm glad I was never exposed to those "tricks" and I feel I'd have much trouble trying to remember them because I understand what's actually happening.
I agree with the premise that it is destructive for students to merely learn tricks in order to pass classes.
However, the existence of tricks is enormously useful once understanding the underlying mechanism of a particular tool is not the focus of a problem at hand. Abstraction is a fundamental human cognitive faculty. For instance, if a student understands why the cross-multiplication 'trick' works, then they should be free to use it as they please, provided they can actually explain why it works if prompted to do so. The notion that there was a 'right' way to do something (like use common denominators) was stifling and frustrating during my school years. If I can explain and justify the trick - then let me use it. On the other hand, being boxed into doing things the instructor-sanctioned way can lead to equally vacuous understanding: "Teacher says find a common denominator so that's what I'll do even though I don't know why".
Additionally, I will argue that all methods for doing computations with fractions are 'tricks' at some level. After all, they are just theorems on the field of quotients of the integers embedded in the reals. One should not be precluded from using a 'trick' because one of these theorems ('common denominator method good - your trick bad') is more familiar to an instructor. Replacing one 'trick' with what is actually just another does not facilitate understanding.
Of course, this is predicated on actually understanding the tool in the first place.
I take an opposite viewpoint from the author(s). Students should be encouraged to develop and use tools. The utility of hiding complexity [0] with tooling is part of the very essence of what it means to be a hacker. It is also very useful in other fields. For example, a physicist solving for the flow of some fluid does not need to think about why a particular fraction trick works. This would draw precious cognitive resources better served elsewhere.
Once a concept is understood, tricks become useful tools. In a field such as building construction, short-cuts are often expensive in the long run because the benefit of making some compromise (e.g. use cheap plaster) is outweighed by its consequences (e.g. need to re-plaster after short amount of time). This mode of thinking does not apply to mathematics.
Tricks are not 'bad' and should not be nixed. They should be embraced and presented as tools of great utility.
Many of the "tricks" this book indicts aren't good hacks and aren't easily formulated as theorems. They are terrible hacks that are ridiculously inefficient, distracting, and only necessary if you fundamentally don't understand what's actually going on.
See "Butterfly Method, Jesus Fish" or "Backflip and Cartwheel":
They're both super inefficient tricks that are totally unnecessary if you know and understand the theorem they embody.
> If I can explain and justify the trick - then let me use it.
This book is a guide for teachers, not a rule book for students.
For the tricks that are arguably good hacks, the authors provide a simple argument: the time investing in teaching the trick is not worth it, and is better spent somewhere else (see the Jim Doherty quote just before the TOC).
The argument isn't that students should not be allowed to use certain theorems if they understand those theorems. Rather, the argument is that teachers should invest their finite teaching resources explaining other theorems instead.
Often, the authors are arguing that there is an equivalent and equally useful formulation of the same theorem (or a similar one) that's easier to derive and understand. Which is the sort of justification any working mathematician should be on board with (they don't need to agree with the conclusion, but the form of the justification is at least reasonable).
> This mode of thinking does not apply to mathematics.
But this does apply in education writ large. E.g. sacrificing understanding for good performance on a standardized test.
This is not a book telling you how you should do mathematics in your work day. It's a book advocating for certain ways of allocating classroom teaching time.
I fully support the premise, but I started reading and have no idea what half of the tricks actually are. I think the book could benefit from explaining the tricks for those of us who have not had the misfortune to have been taught them.
Cross multiplication is a trick to multiply by the 'reciprocal' of the fraction.
The failing here is that you don't understand why you get the right answer here.
Why is (1/2) / (1/4) = 2? It's because there are 2 quarters in every half, it's not because (1/2) * (4/1) = 2 that's just a 'convenient trick' to get the answer quickly. This is a great idea if your goal is to pass some maths exams which have a fixed format in the near future. It's a terrible idea if you want to be able to apply mathematics to anything.
Just quickly skimmed a bit of the book, but I have never seen or heard any of those tricks (German here) with the exception of the formula triangle (still have that in mind for Ohm's law). That being said, I mainly need it for remembering the relationship and have no trouble solving the formula for a different quantity.
I'm struggling to remember how exactly I was taught, but cross-multiplying was the method I've used for as long as I can remember. I really believe it was how I was taught when we first encountered fractional division.
Though it is higher level than this book appears to target, i had a similar issue with the obfuscation of the trig functions. Year after year i would ask my math teachers how to calculate sin and cos, to no avail.
I remember sitting in a college math course as we deduced the geometric series that represents each positively giddy.
Interesting. I have gotten very far in life just with many of the computational tricks mentioned in the book. If you do enough multiplication and addition you will think a lot more about what the tricks mean.
I think, I don't see there the "proving backwards" trick described. The technique goes as follows: a student starts with something (for example, an equation) they need to prove, and transforms it until they get some trivial identity like 1=1.
If they succeed in this procedure, they claim that the original equation they started with was correct too.
Everyone does it, and it takes ages to make them unlearn it.
If somebody is wondering why the technique is incorrect: starting from a false premise, any statement can be proven, including things like 1=1 or 1=0.
As long as your steps are implications in the reverse direction (from the later to the earlier statement) this is perfectly correct. In most cases I've seen people do that the transformations are actually equivalences, which muddies the waters somewhat.
What is a problem is when people do use reverse-implications that are not equivalences, arrive at something false, and claim that the original equation is false.
I've responded to a few comments here already, but I'm a math/CS teacher who is a friend and research partner of the author, and I contributed a very small amount to Nix the Tricks. I'd be more than happy to take any questions and/or forward them on.
A possible solution to fix this in schools may be to include questions that cannot be solved by applying tricks blindly. Adding these questions will allow teachers to identify which students are simply using tricks and which students deeply understand the material.
FYI, the common core mathematics standards are largely a backlash to the sorts of tricks this book is complaining about. And a member of the group that coauthored the book (MTBoS) has also published a book that speak favorably about common core.
That can bring difficulties, as far as I know the commutative property is a fundamental property in many algebra systems. So it's one of the few assumptions from where everything else builds up, not sure how would you go about explaining that.
You can show a few pictures of examples of how it works in real life, but axioms are the basis from which you build everything else. You just accept them as being useful and prove the rest.
I was never even taught this "trick." The language used is shitty.
The way I was taught, dividing a fraction by a fraction is equivalent to multiplying the first fraction by the second fraction's reciprocal. It's easy to check this yourself. "Just cross multiply!" is ridiculous...
You are definitely judging the book by its cover. If you follow the link, the full book has a lot more "tricks" and "solutions". The cross-multiply one is just an example. Of course many of them are culture/location dependent, but I'm sure you will find ones that you are familiar with.
I guess it is saying that people are going straight to the "trick" (multiply the reciprocal) instead of understanding that they are applying the fact that:
a / (b/c) = a * (c / b) (a in Q, b,c in N and b <> 0)
And that they therefore have a less rich understanding of when the rule is applicable and when it isn't appropriate. (Such as a situation where b is actually zero, perhaps if it's "hidden" in another form like x-y.
[+] [-] elchief|10 years ago|reply
It takes too goddam long though. At university, there's often not enough time to deeply understand something before the exam. And your competition is just learning the tricks. And it's ego crushing to do poorly on an exam.
I did okay though, despite. I ended up teaching at a good school and being published, but my grades would have been 5-10% higher had I just learned the tricks.
Teaching is frustrating too. Watching the kids that only care about the tricks ace the exam, and the ones with curious minds screw up a few mechanical things.
[+] [-] antod|10 years ago|reply
Or worse, the majority of kids that don't even realise the subject is more than just the tricks themselves.
[+] [-] anon4|10 years ago|reply
[+] [-] mangamadaiyan|10 years ago|reply
In the name of getting rid of "mechanical" learning, most schoolkids these days don't even get to learn multiplication tables. While learning the tables is a pain, they pay rich dividends.
As an example, an 8th-grade kid who I help with math was struggling to find the square root of 529. He tried dividing it by his usual set of numbers, and pretty much hit a brick wall. I tried to get him to see that 529 was between 400 and 625, so its square root had to lie between 20 and 25. He did see it, but it's going to take him a while to figure out that he can do this with any number, if he can remember (or quickly work out) a few squares.
Now, one /could/ argue that computing the square root of a number is in itself mechanical ... but just a nodding familiarity with numbers can make even that a breeze.
And yes, the root cause of the problem is that math is taught as a bag of complicated tricks which need to be remembered, and not understood :/
[+] [-] tokenadult|10 years ago|reply
To answer one frequently asked question in the previous comments, no, I don't think any school anywhere includes teachers who use all these tricks in their teaching, and, yes, I think most of these classroom teacher tricks are specific to the United States. In the United States, the great majority of elementary school teachers are generalists, teaching all school subjects to their pupils, and their higher education does not prepare them well for teaching mathematics.[1] By contrast, elementary teachers in many countries where students learn more mathematics more thoroughly are subject-matter specialists, with mathematics teachers teaching only elementary mathematics, and other teachers teaching elementary pupils other subjects.
My teaching is in two contexts: since 2007, I have taught voluntary-participation, extracurricular courses in prealgebra mathematics with additional advanced topics to middle-elementary age (mostly fourth grade) pupils in weekend supplementary classes. Many of those students are quite advanced intellectually for their age. They come to my classes (mostly through word-of-mouth recommendations from their parents' friends) from a ten-county expanse of Minnesota. Just this school year (that is, just since August) I am also on the faculty of an independent school, teaching all sixth grade pupils and "honors" seventh grade students mathematics at about the same level, although it is my intention this school year to move the seventh graders along into topics that can properly be called "beginning algebra" topics (to solving quadratic equations and graphing systems of two linear equations on the Cartesian plane).
The school where I teach is reforming its mathematics curriculum with advice from a nonprofit consulting organization.[2] The reform program at my school is informed by international best practice in primary and secondary education and by what mathematics background is necessary to succeed in higher education in universities like MIT. (The founder of the consulting organization is an MIT alumna.) As the submitted ebook says, the hardest thing for a teacher to do is to encourage students to think rather than just rely on a mindless trick. This year I will have to set homework and tests that I write myself to probe for deep understanding of mathematical concepts and relentlessly try to find out how (and even whether) the learners in my care think about mathematics. Most of the rest of today on my weekend schedule is slated for writing a major unit test for my seventh graders, who use an excellent textbook[3] that is part of a textbook series that doesn't provide teachers with ready-made unit tests. The textbook is based on problem-solving and explaining mathematics from first principles (Chapter 1 takes the field properties of the real numbers as an axiom system to explain many principles of arithmetic) and is the best textbook for its topic available in English.
[1] http://condor.depaul.edu/sepp/mat660/Askey.pdf
http://www.ams.org/notices/199908/rev-howe.pdf
http://www.amazon.com/Knowing-Teaching-Elementary-Mathematic...
https://math.berkeley.edu/~wu/Stony_Brook_2014.pdf
https://math.berkeley.edu/~wu/MSRI_2014_1.pdf
[2] http://www.mathwalk.org/
(I liked the old website of this organization better than the new website.)
[3] http://www.artofproblemsolving.com/store/item/prealgebra
The Art of Problem Solving website is a treasure trove of interesting mathematics education resources for learners of all ages.
[+] [-] asgard1024|10 years ago|reply
I remember him saying: "Forget all rules about percentages. All you need to know it's like a number multiplied by 100."
He also wasn't fond of mole as unit, he felt that it obscures things especially for students.
Later I applied the same method myself on algebra (deriving binomial expressions), trigonometric functions (deriving constants from unit circle, other relations from the basic relations of sin(x+y)) and so on.
I guess this actually applies to any field. Do not start with tricks and shortcuts until you understand and can do the deed with basic methods.
[+] [-] carapat_virulat|10 years ago|reply
Even derivations itself probably contain plenty of not so obvious "tricks" that should be known to develop them. It's not so clear cut what is a trick and what are obvious assumptions from the real world. I suspect that the regular increase of IQ every generation may be related to all those hidden tricks and skills that everybody acquires at school or from general culture, but that at some point they become invisible obvious assumptions and common sense.
Another easier approach may be to start with the tricks and work backwards from there to more general premises.
[+] [-] tome|10 years ago|reply
I don't remember what rules or tricks I was taught about percentages, but I do remember realising that "%" meant "/ 100" and from there everything else followed.
[+] [-] delish|10 years ago|reply
[+] [-] jules|10 years ago|reply
- Why you can calculate the volume of a slanted box with a determinant
- Why a matrix is singular if its determinant is zero
- Why the determinant is the product of the eigenvalues
- Why det(I) = 1, why det(AB) = det(A)det(B), and why det(A^-1) = 1/det(A)
- Why change of variables in integration gets a det(J) factor
Sadly, most linear algbra texts introduce the determinant as some random summation formula or with a series of unmotivated axioms. This is a general problem with mathematics: symbols over geometry, and formal proofs over intuitive understanding when it should have been the other way around.
[+] [-] stsp|10 years ago|reply
[+] [-] nerd_stuff|10 years ago|reply
The problem is if you only learn the trick without the reasoning behind it. The solution isn't to not learn the trick, it's to learn why it works.
I think it's a disservice to kids who will go into science and engineering if they've never been allowed to use heuristics before. Too often you learn a long-form solution method and then the "trick" to solve it quickly and you need to be able to do both.
https://www.edx.org/course/street-fighting-math-mitx-6-sfmx
[+] [-] dhimes|10 years ago|reply
I think that tricks that allow you to do things quickly in your head are great- they help build numeracy.
[+] [-] scotty79|10 years ago|reply
[+] [-] Camillo|10 years ago|reply
[+] [-] daniel-levin|10 years ago|reply
However, the existence of tricks is enormously useful once understanding the underlying mechanism of a particular tool is not the focus of a problem at hand. Abstraction is a fundamental human cognitive faculty. For instance, if a student understands why the cross-multiplication 'trick' works, then they should be free to use it as they please, provided they can actually explain why it works if prompted to do so. The notion that there was a 'right' way to do something (like use common denominators) was stifling and frustrating during my school years. If I can explain and justify the trick - then let me use it. On the other hand, being boxed into doing things the instructor-sanctioned way can lead to equally vacuous understanding: "Teacher says find a common denominator so that's what I'll do even though I don't know why".
Additionally, I will argue that all methods for doing computations with fractions are 'tricks' at some level. After all, they are just theorems on the field of quotients of the integers embedded in the reals. One should not be precluded from using a 'trick' because one of these theorems ('common denominator method good - your trick bad') is more familiar to an instructor. Replacing one 'trick' with what is actually just another does not facilitate understanding.
Of course, this is predicated on actually understanding the tool in the first place.
I take an opposite viewpoint from the author(s). Students should be encouraged to develop and use tools. The utility of hiding complexity [0] with tooling is part of the very essence of what it means to be a hacker. It is also very useful in other fields. For example, a physicist solving for the flow of some fluid does not need to think about why a particular fraction trick works. This would draw precious cognitive resources better served elsewhere.
Once a concept is understood, tricks become useful tools. In a field such as building construction, short-cuts are often expensive in the long run because the benefit of making some compromise (e.g. use cheap plaster) is outweighed by its consequences (e.g. need to re-plaster after short amount of time). This mode of thinking does not apply to mathematics.
Tricks are not 'bad' and should not be nixed. They should be embraced and presented as tools of great utility.
[0] Such as how fractions work
edit: spelling
[+] [-] nmrm2|10 years ago|reply
See "Butterfly Method, Jesus Fish" or "Backflip and Cartwheel":
http://www.nixthetricks.com/NixTheTricks2.pdf
They're both super inefficient tricks that are totally unnecessary if you know and understand the theorem they embody.
> If I can explain and justify the trick - then let me use it.
This book is a guide for teachers, not a rule book for students.
For the tricks that are arguably good hacks, the authors provide a simple argument: the time investing in teaching the trick is not worth it, and is better spent somewhere else (see the Jim Doherty quote just before the TOC).
The argument isn't that students should not be allowed to use certain theorems if they understand those theorems. Rather, the argument is that teachers should invest their finite teaching resources explaining other theorems instead.
Often, the authors are arguing that there is an equivalent and equally useful formulation of the same theorem (or a similar one) that's easier to derive and understand. Which is the sort of justification any working mathematician should be on board with (they don't need to agree with the conclusion, but the form of the justification is at least reasonable).
> This mode of thinking does not apply to mathematics.
But this does apply in education writ large. E.g. sacrificing understanding for good performance on a standardized test.
This is not a book telling you how you should do mathematics in your work day. It's a book advocating for certain ways of allocating classroom teaching time.
[+] [-] jstanley|10 years ago|reply
[+] [-] iamflimflam1|10 years ago|reply
[+] [-] aptwebapps|10 years ago|reply
[+] [-] sleepychu|10 years ago|reply
The failing here is that you don't understand why you get the right answer here.
Why is (1/2) / (1/4) = 2? It's because there are 2 quarters in every half, it's not because (1/2) * (4/1) = 2 that's just a 'convenient trick' to get the answer quickly. This is a great idea if your goal is to pass some maths exams which have a fixed format in the near future. It's a terrible idea if you want to be able to apply mathematics to anything.
[+] [-] ygra|10 years ago|reply
[+] [-] lacksconfidence|10 years ago|reply
[+] [-] enjo|10 years ago|reply
[+] [-] liquidise|10 years ago|reply
I remember sitting in a college math course as we deduced the geometric series that represents each positively giddy.
[+] [-] skaevola|10 years ago|reply
https://www.youtube.com/watch?v=1hcKERTnNi0
What an awful way to teach.
[+] [-] hammerbrostime|10 years ago|reply
[+] [-] msie|10 years ago|reply
[+] [-] a-nikolaev|10 years ago|reply
If they succeed in this procedure, they claim that the original equation they started with was correct too.
Everyone does it, and it takes ages to make them unlearn it.
If somebody is wondering why the technique is incorrect: starting from a false premise, any statement can be proven, including things like 1=1 or 1=0.
[+] [-] robryk|10 years ago|reply
What is a problem is when people do use reverse-implications that are not equivalences, arrive at something false, and claim that the original equation is false.
[+] [-] arielby|10 years ago|reply
[+] [-] pflats|10 years ago|reply
[+] [-] j7ake|10 years ago|reply
[+] [-] phanimahesh|10 years ago|reply
[+] [-] droopybuns|10 years ago|reply
[+] [-] nmrm2|10 years ago|reply
[+] [-] amelius|10 years ago|reply
[+] [-] Jabbles|10 years ago|reply
https://www.dpmms.cam.ac.uk/~wtg10/commutative.html
[+] [-] carapat_virulat|10 years ago|reply
You can show a few pictures of examples of how it works in real life, but axioms are the basis from which you build everything else. You just accept them as being useful and prove the rest.
[+] [-] elevenfist|10 years ago|reply
The way I was taught, dividing a fraction by a fraction is equivalent to multiplying the first fraction by the second fraction's reciprocal. It's easy to check this yourself. "Just cross multiply!" is ridiculous...
[+] [-] xtreme|10 years ago|reply
[+] [-] eterm|10 years ago|reply
a / (b/c) = a * (c / b) (a in Q, b,c in N and b <> 0)
And that they therefore have a less rich understanding of when the rule is applicable and when it isn't appropriate. (Such as a situation where b is actually zero, perhaps if it's "hidden" in another form like x-y.