Nix the Tricks: Math tricks defeat understanding(nixthetricks.com)
nixthetricks.com
Nix the Tricks: Math tricks defeat understanding
http://www.nixthetricks.com/#hn-repost
84 comments
> 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.
Or worse, the majority of kids that don't even realise the subject is more than just the tricks themselves.
And who's fault is that?
It's the system that's at fault. The kids want to cram for the test because the GPA follows them, like the Akashic Records, and that's why you get textbooks that promote un-understood shortcuts. You will have encountered the problem if you have taught university or highschool, and you will have identified it as a hard problem.
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.
P.S. On a tangential note, it would help if the names for mathematical terms were in English, rather than some weird amalgamation of Greek, Latin, French and German. In my language, all n-gons are called n-anguloids. So you have threeanguloid, fouranguloid, fiveanguloid, "a proper N-anguloid", etc... And we use the normal names for numbers, rather than Greek ones. And a rectangle is "right-angle-oid" (granted, that's the name in English too, if you squint a bit and know what phonetic shifts to look for and your German is good). Square is quadrat, though. Then again, it's still an everyday word, even if we borrowed it from Greek millennia ago. And the word for "perimeter" also means "a walk around" and "a lap". You literally tell people "the length of the lap of a rightanguloid is the sum of the four sides", which is kind of obvious if you imagine walking the perimeter. And the word for "angle" is the same as "corner". Sadly, that ship has sailed long ago.
I am 99% sure you're talking about Dutch, and in that case: we owe a lot of those non-borrowed terms, including the very term 'wiskunde' (mathematics) to Simon Stevin[1].
[1] https://en.wikipedia.org/wiki/Simon_Stevin
[1] https://en.wikipedia.org/wiki/Simon_Stevin
Just for clarity on all my responses in these threads, I'm a (minor) contributor to Nix the Tricks and a personal friend and research partner of the author.
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 x d/c.
Exactly! And that's exactly what the book recommends. Remember, the target audience of the book is teachers. If a teacher tells them to just draw a butterfly and multiply the matching parts, the students are not actually learning the mathematics. You can use whatever method you want once you understand the mechanics.
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 x d/c.
Exactly! And that's exactly what the book recommends. Remember, the target audience of the book is teachers. If a teacher tells them to just draw a butterfly and multiply the matching parts, the students are not actually learning the mathematics. You can use whatever method you want once you understand the mechanics.
On a slightly tangential note -
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 :/
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 :/
And since it ends in a 9... :)
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.
[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.
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.
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.
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.
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.
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.
But it doesn't mean / 100, it means "per 100"
as described here in "The history of the percent sign"
http://www.shadycharacters.co.uk/2015/03/percent-sign/
as described here in "The history of the percent sign"
http://www.shadycharacters.co.uk/2015/03/percent-sign/
I went to elementary school in Czechoslovakia, late 80s, and we learned some rules about percentages (which I believe was just a modified formula with the multiplication by 100) as a part of what you call cross-multiplication.
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 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.
- 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.
Another application of determinants:
- Compute a vector perpendicular to other two vectors.
- Compute the rotational of a vector field
- Determine if three vector are linearly dependent
- If you consider a complex number z = a + bi, then there is a linear transformation from C to C given by x -> xz, than you can see as a linear transformation from R^2 to R^2, the determinant of that matrix is the amplifying factor for the area of figures.
At least in 3-space, they also have an interesting relationship with signed tetrahedral volumes, the scalar triple product, and Pluecker coordinates. I once got a published paper out of those connections (http://www.cs.utah.edu/~aek/research/triangle.pdf).
Nice, and that generalizes neatly to n-space with exterior algebra / Grassmann coordinates.
Seriously, thank you for this. This makes so much of what I've been learning make sense and helps with the intuition around determinants. Why isn't this the first thing they teach about determinants?
I asked a friend, and he said "but isn’t it obvious that if you sum up this and this and...". Yes. To many mathematicians, it is just obvious. Maybe that’s the issue.
Understanding what is and what isn't obvious to other people is a good chunk of what makes the difference between a good teacher and a bad one--possibly the single most important factor. Yet it has nothing to do with what makes a good mathematician.
Yes. But obviously, at a university, most professors are selected for their research. Which does not, unless they research teaching, necessarily mean they are good at teaching.
"Linear Algebra Done Right" helped me with getting through college :)
I can highly recommend it.
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.
https://www.edx.org/course/street-fighting-math-mitx-6-sfmx
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
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.
I think that tricks that allow you to do things quickly in your head are great- they help build numeracy.
FOIL is the worst offender in my experience tutoring people in math. I recall several instances of people being completely stumped when trying to multiply a trinomial and a binomial, because they couldn't figure out how to apply FOIL.
Then come up with a better mnemonic or trick that captures the essence of the general case.
Like Fanana: First times all of them, next times all of them, next times all of them... it's not 100% clear but it took 10 seconds to come up with and it works with trinomials and beyond.
Or FettuchENE: First times each of them, plus next times each of them.... Ok, that's kind of bad, but you get the point. FOIL sticks around not becuase it's the best but because it's the most memorable, it's a meme. We should introduce better memes to compete with it.
Like Fanana: First times all of them, next times all of them, next times all of them... it's not 100% clear but it took 10 seconds to come up with and it works with trinomials and beyond.
Or FettuchENE: First times each of them, plus next times each of them.... Ok, that's kind of bad, but you get the point. FOIL sticks around not becuase it's the best but because it's the most memorable, it's a meme. We should introduce better memes to compete with it.
Why do we need a mnemonic or a meme for this? Is understanding distribution really that much harder than memorizing FOIL?
At the same time, understanding distribution is so much more powerful because it generalizes to many other uses and later concepts.
I think FOIL sticks around primarily because it is what people are taught. Even when people actually do understand distribution, they may not realize that FOIL is just a special case of distributing.
At the same time, understanding distribution is so much more powerful because it generalizes to many other uses and later concepts.
I think FOIL sticks around primarily because it is what people are taught. Even when people actually do understand distribution, they may not realize that FOIL is just a special case of distributing.
Is it harder to understand distribution than memorize FOIL? Probably. It's memorizing a process vs. understanding a concept. FOILing won't help you master algebra but if you're in a science class and your teacher needs to get you up to speed on multiplying pairs of binomials then they'll likely just show you FOIL.
I agree with you that FOILing is stupid and might make the student worse off in the long run, but I think depriving them of getting practive with heuristics might be worse. Ideally a student would notice that FOIL isn't a good trick and stop using it, but there are other "tricks" that are extremely useful.
I agree with you that FOILing is stupid and might make the student worse off in the long run, but I think depriving them of getting practive with heuristics might be worse. Ideally a student would notice that FOIL isn't a good trick and stop using it, but there are other "tricks" that are extremely useful.
If you're using FOIL to replace the distributive property then you're doing it wrong. It's simply a bookkeeping device, like the little ditties that people have to "memorize" the names of the planets or the Great Lakes.
What's wrong with the sum of all ways of picking a term from each factor?
I both tutored and taught math. Never had a problem with FOIL.
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 have to ask, do those "tricks" represent the way math is usually taught in America? It looks like a bad joke to me.
> do those "tricks" represent the way math is usually taught in America?
Anecdotally, yes -- and damn proud of it! A lot of the backlash against Common Core Mathematics is from people who want Mathematics education to stay that way.
The reason is that a huge amount of mathematics educators in American secondary schools aren't properly trained in even elementary Mathematics.
At the lower grades (ages < 15 or 16) it's mostly because you can go through all of high school and university in America without understanding anything about Mathematics. For example, most people at the university I attended never took a Calculus course. Everyone was required to take a Mathematics course, but there are math courses that are essentially repeats of what you would expect a non-exceptional sixth grader to understand -- matching linear functions to their graphs and the like. These courses even get recommended by teaching colleges because even Calculus I was perceived as "hard" or "advanced" "math". And Calculus I itself was entirely superficial and mostly trick-based (because calculus isn't proof based except at a tiny handful of elite universities).
Result: most Mathematics educators a typical student encounters before the age of 15 have never written a proof (with practical consequences; e.g., they don't know what induction is). Even worse, these educators would probably have a hard time defining a function in terms that a working mathematician would find meaningful (educators now have their own cutesy but non-rigorous definitions for things like "number" and "function").
At the upper ages (~15 until university), I'm convinced it's because teachers are not properly compensated and so teaching doesn't appeal to the best students. Many high school teachers that teach math at least complete an undergraduate degree in Mathematics, although even that is not actually required in many states. However, based upon my anecdotal experience tutoring and interacting with peers while in university, Education students are typically in the bottom half of Mathematics students and often rely on (and prefer :-( ) "trick"-based "mathematics", even in proof-based courses (memorizing formats of theorems and corresponding proofs, then figuring out what symbols/numbers to plug in. But with zero understanding of the underlying argument).
Anecdotally, yes -- and damn proud of it! A lot of the backlash against Common Core Mathematics is from people who want Mathematics education to stay that way.
The reason is that a huge amount of mathematics educators in American secondary schools aren't properly trained in even elementary Mathematics.
At the lower grades (ages < 15 or 16) it's mostly because you can go through all of high school and university in America without understanding anything about Mathematics. For example, most people at the university I attended never took a Calculus course. Everyone was required to take a Mathematics course, but there are math courses that are essentially repeats of what you would expect a non-exceptional sixth grader to understand -- matching linear functions to their graphs and the like. These courses even get recommended by teaching colleges because even Calculus I was perceived as "hard" or "advanced" "math". And Calculus I itself was entirely superficial and mostly trick-based (because calculus isn't proof based except at a tiny handful of elite universities).
Result: most Mathematics educators a typical student encounters before the age of 15 have never written a proof (with practical consequences; e.g., they don't know what induction is). Even worse, these educators would probably have a hard time defining a function in terms that a working mathematician would find meaningful (educators now have their own cutesy but non-rigorous definitions for things like "number" and "function").
At the upper ages (~15 until university), I'm convinced it's because teachers are not properly compensated and so teaching doesn't appeal to the best students. Many high school teachers that teach math at least complete an undergraduate degree in Mathematics, although even that is not actually required in many states. However, based upon my anecdotal experience tutoring and interacting with peers while in university, Education students are typically in the bottom half of Mathematics students and often rely on (and prefer :-( ) "trick"-based "mathematics", even in proof-based courses (memorizing formats of theorems and corresponding proofs, then figuring out what symbols/numbers to plug in. But with zero understanding of the underlying argument).
A lot of the backlash against Common Core Mathematics is from people who want Mathematics education to stay that way.
You've got it backwards here; the Common Core curriculum is basically all "tricks" without understanding.
You've got it backwards here; the Common Core curriculum is basically all "tricks" without understanding.
The explicit, stated strategy of Common Core Mathematics is to emphasize understanding and conceptual learning over rote execution. And it does this, in practice, by teaching multiple different strategies for solving the same problem, and teaching why they all work the same.
This, of course, leads to the hyperbolic frothing at the mouth on mommy-blogs and the like when a take-home sheet featuring just one of the many strategies taught is not the vaunted One True Trick that the parent in question learned as a child. Cue blogs titled "I'm a certified public accountant and even I can't do math this way!", which of course proves the very point Common Core is trying to make -- that older generations don't actually even know Math at all, but rather a bag full of tricks without a hint of understanding.
This, of course, leads to the hyperbolic frothing at the mouth on mommy-blogs and the like when a take-home sheet featuring just one of the many strategies taught is not the vaunted One True Trick that the parent in question learned as a child. Cue blogs titled "I'm a certified public accountant and even I can't do math this way!", which of course proves the very point Common Core is trying to make -- that older generations don't actually even know Math at all, but rather a bag full of tricks without a hint of understanding.
This is an utter myth promoted by Glen Beck fanbloggers. Every time one of these attempted potshots goes viral, a sane person like Friendly Atheist patiently explains how ignorant the ranter is.
Just for clarity on all my responses in these threads, I'm a (minor) contributor to Nix the Tricks and a personal friend and research partner of the author.
Yes, all of the tricks were submitted by math teachers that we have seen used in practice. I don't think that we are all Americans, but a most of us are.
I personally haven't seen too many of them, especially the weirder ones, but I know they were not invented whole-cloth for the book.
Yes, all of the tricks were submitted by math teachers that we have seen used in practice. I don't think that we are all Americans, but a most of us are.
I personally haven't seen too many of them, especially the weirder ones, but I know they were not invented whole-cloth for the book.
I'm in my late 20s. I think a lot of these tricks were a result of George Bush II's "No Child Left Behind" testing mandate (2002? around there), where test results directly correlated to the amount of federal funding you'd receive. In response, teachers adopted quick tricks in order to optimize for test performance (i.e. 'teach for the test'). I'd wager these "tricks" are what teachers are forced to use to cater to effectively get the average student to perform optimally on those standardized tests. People just now entering their undergraduate programs, I speculate, will be familiar with these shortcuts.
Math is taught in a wide variety of ways, depending on where you live, what type of schools you're going to (private east-coast Andover/Exeter vs inner-city Chicago), how rich your city is (higher taxes => more money for schools => safer/better environments => fosters a better learning experience & a life at home that values rigorous grasp of academic concepts), etc.
I was one of those lucky guys to grow up in an environment that had a heavy emphasis on academia at the 'publish or perish (then tenure)' level and my school systems emphasized concepts over rote memorization (i.e., a course on geometric proof formulation where you are given a set of axioms from which you must derive all consequential Lemmas, which is basically the opposite of 'heres a few tricks'),
If your class size is 35, and you have even one or two disruptive kids in class I'd imagine even the most dedicated teachers would start 'teaching by the tricks, for the test'.
To answer your question: America is a big place. The difference between most Euro countries is culturally less than the difference between NYC and Harlan, Kentucky.
An interesting corollary of this is (let's choose to ignore ethical implications here -- I believe everyone has the right to free, high quality education, housing, and food), but purely from the POV of workplace-competition-over-jobs, the more schools continue to teach poor analytic skills, the less competent my competitors are. I'm no Perelman or Feynman, but I do pretty fine
Math is taught in a wide variety of ways, depending on where you live, what type of schools you're going to (private east-coast Andover/Exeter vs inner-city Chicago), how rich your city is (higher taxes => more money for schools => safer/better environments => fosters a better learning experience & a life at home that values rigorous grasp of academic concepts), etc.
I was one of those lucky guys to grow up in an environment that had a heavy emphasis on academia at the 'publish or perish (then tenure)' level and my school systems emphasized concepts over rote memorization (i.e., a course on geometric proof formulation where you are given a set of axioms from which you must derive all consequential Lemmas, which is basically the opposite of 'heres a few tricks'),
If your class size is 35, and you have even one or two disruptive kids in class I'd imagine even the most dedicated teachers would start 'teaching by the tricks, for the test'.
To answer your question: America is a big place. The difference between most Euro countries is culturally less than the difference between NYC and Harlan, Kentucky.
An interesting corollary of this is (let's choose to ignore ethical implications here -- I believe everyone has the right to free, high quality education, housing, and food), but purely from the POV of workplace-competition-over-jobs, the more schools continue to teach poor analytic skills, the less competent my competitors are. I'm no Perelman or Feynman, but I do pretty fine
OMG. Bush has been out of office for 6 years already. “It’s Bush’s fault” is not a legitimate response, and almost never was. I’m not saying he was good for the country, oh no not at all, but you can’t blame this on him. This partisanship is a disease on the nation.
Many of the tricks are there because math in the US has traditionally been taught for repetitive manual use. You need your accounts calculated, you get somebody to add all those numbers and put the result on paper. I was just looking at a math book from the 1940s, and the first couple chapters are all shortcuts to doing multi-digit multiplication and stuff. Math education has not caught up to a reality filled with abundant accurate computing power.
This horrible way of teaching math really is the most common way it is taught in America, and it has been so ever since schools were started in America. Not all those tricks are common, especially things like horses and Jesus fish. Those stupid drawings are only a desperate attempt to use a mnemonic to memorize a rote procedure.
Every previous effort I’ve noticed has tried to eliminate this problem and failed. New Math in the 1960s. Reform Math in the 1990s. Everyday Math in the 2000s. The problem is not in the exact curriculum used. A good teacher with enough autonomy can teach well with any of these curricula. Obviously, there is a problem somewhere between concept and implementation.
Many of the tricks are there because math in the US has traditionally been taught for repetitive manual use. You need your accounts calculated, you get somebody to add all those numbers and put the result on paper. I was just looking at a math book from the 1940s, and the first couple chapters are all shortcuts to doing multi-digit multiplication and stuff. Math education has not caught up to a reality filled with abundant accurate computing power.
This horrible way of teaching math really is the most common way it is taught in America, and it has been so ever since schools were started in America. Not all those tricks are common, especially things like horses and Jesus fish. Those stupid drawings are only a desperate attempt to use a mnemonic to memorize a rote procedure.
Every previous effort I’ve noticed has tried to eliminate this problem and failed. New Math in the 1960s. Reform Math in the 1990s. Everyday Math in the 2000s. The problem is not in the exact curriculum used. A good teacher with enough autonomy can teach well with any of these curricula. Obviously, there is a problem somewhere between concept and implementation.
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.
[0] Such as how fractions work
edit: spelling
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
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":
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.
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.
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.
Same here. I read through the first 10 or so and hadn't heard or been taught any of them. Is it a particular US thing? (British here for reference)
Mostly North American.
If you're interested, these videos explain and discuss each "trick" in more detail:
https://www.youtube.com/watch?v=1gxLwLBl4mM&index=4&list=PLA...
It's a work in progress...
If you're interested, these videos explain and discuss each "trick" in more detail:
https://www.youtube.com/watch?v=1gxLwLBl4mM&index=4&list=PLA...
It's a work in progress...
Wrt the example of fraction division, isn't this the standard way? Was anyone here taught to cross-multiply for division?
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.
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.
> Why is (1/2) / (1/4) = 2? It's because there are 2 quarters in every half
That sounds like looking for a "trick" to me, rather than the much simpler generic rearrangement. Trying to cram that phrase into more complex ones I'd start thinking "Wait, how many five eighths are there in seventy sixteenths?"
If the question is "what is (1/2) / (1/4)" and you can't see it, then the idea of rearranging the equation and why rearranging it is OK is pretty important and general.
(5/8) / (70/16) = x
5/8 = 70x / 16
5 . 16/8 = 70x
5 . 2 = 70x
10 = 70x
1/7 = x
That seems simpler to me than "There is one seventh of seventy sixteenths in five eighths" and it requires knowing how basic multiplication and division works, and that the equals sign means that the things on both sides are equal, so if we multiply both sides by the same number or divide by the same number then that's OK. That can then lead into why you need to be careful about values being 0 or why sometimes you have to add ± to an answer.
You can also just generally explore rearrangements, try out different modifications and see if it makes things simpler, go back and try others.
That sounds like looking for a "trick" to me, rather than the much simpler generic rearrangement. Trying to cram that phrase into more complex ones I'd start thinking "Wait, how many five eighths are there in seventy sixteenths?"
If the question is "what is (1/2) / (1/4)" and you can't see it, then the idea of rearranging the equation and why rearranging it is OK is pretty important and general.
(5/8) / (70/16) = x
5/8 = 70x / 16
5 . 16/8 = 70x
5 . 2 = 70x
10 = 70x
1/7 = x
That seems simpler to me than "There is one seventh of seventy sixteenths in five eighths" and it requires knowing how basic multiplication and division works, and that the equals sign means that the things on both sides are equal, so if we multiply both sides by the same number or divide by the same number then that's OK. That can then lead into why you need to be careful about values being 0 or why sometimes you have to add ± to an answer.
You can also just generally explore rearrangements, try out different modifications and see if it makes things simpler, go back and try others.
Oh, I see. I thought cross multiplying was referring to what you do in order to add or subtract two fractions with different denominators. Should have just looked it up.
The failing here is that you don't understand why you get the right answer here.
By 'you' you mean the putative student mislead by shortcuts, right? I understand perfectly well what's happening with reciprocal multiplication and why it works. ;)
The failing here is that you don't understand why you get the right answer here.
By 'you' you mean the putative student mislead by shortcuts, right? I understand perfectly well what's happening with reciprocal multiplication and why it works. ;)
Just for clarity on all my responses in these threads, I'm a (minor) contributor to Nix the Tricks and a personal friend and research partner of the author.
Why is (1/2) / (1/4) = 2? It's because there are 2 quarters in every half, it's not because (1/2) x (4/1) = 2 that's just a 'convenient trick' to get the answer quickly.
No, both of these are true. Your logical answer is correct: 1/4 goes into 1/2 twice because there are two quarters in every half. This is how our ancestors conceived division and fractions, and that's how physically dividing things works in the real world. But division is not an independent operation; it's the inverse of multiplication. As plonh mentions downthread, division is, quite literally, defined on a field (like Q) as multiplying by the reciprocal. (1/2) / (1/4) = 2 because (4/1) = (1/4)^-1 and (1/2) * (4/1) = 2. Any other mathematical explanation is simply justifying this fact in more palatable terms.
Why is (1/2) / (1/4) = 2? It's because there are 2 quarters in every half, it's not because (1/2) x (4/1) = 2 that's just a 'convenient trick' to get the answer quickly.
No, both of these are true. Your logical answer is correct: 1/4 goes into 1/2 twice because there are two quarters in every half. This is how our ancestors conceived division and fractions, and that's how physically dividing things works in the real world. But division is not an independent operation; it's the inverse of multiplication. As plonh mentions downthread, division is, quite literally, defined on a field (like Q) as multiplying by the reciprocal. (1/2) / (1/4) = 2 because (4/1) = (1/4)^-1 and (1/2) * (4/1) = 2. Any other mathematical explanation is simply justifying this fact in more palatable terms.
I've never heard of most of the tricks mentioned and agree they seem like terrible things to teach, but calling cross multiplication a trick seems bizarre.
Cross multiplication is multiplying by the reciprocal. In neither case do you have to understand what you're doing. You can make this complaint about every arithmetic operation.
Whaaa? 1/(1/x))=x is absolutely true for all x!=0 in a field.
How does your method explain how to solve "1/2 / 3/4" ? How does talking about how many 4/3 are in 1/2 help?
How does your method explain how to solve "1/2 / 3/4" ? How does talking about how many 4/3 are in 1/2 help?
How does it not? There's 2/3 of a 3/4 in a 1/2
(I'm assuming that last sentence was typoed and "talking about how many 3/4 in 1/2" was intended.)
(I'm assuming that last sentence was typoed and "talking about how many 3/4 in 1/2" was intended.)
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.
fwiw, I was taught in the mid 90's about how (a/b) / (c/d) = (a/b) * (d/c) = (ad)/(bc)
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.
I remember sitting in a college math course as we deduced the geometric series that represents each positively giddy.
I hadn't heard of the turtle head method before so I searched and found this video:
https://www.youtube.com/watch?v=1hcKERTnNi0
What an awful way to teach.
https://www.youtube.com/watch?v=1hcKERTnNi0
What an awful way to teach.
The link to the table of contents is broken, here is the proper link:
http://www.nixthetricks.com/Contents.html
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.
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.
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.
Yes, I know ) But then this whole derivation now lives in a "special world", where you cannot do all math, but only certain subset.
Your example is cool. I have not seen many exmaples of such logic, but people probably do it too.. )
Your example is cool. I have not seen many exmaples of such logic, but people probably do it too.. )
The most common example of such logic is multiplying by maybe-zero (transforming a=b into ac=bc) and factoring out a possibly-zero factor (transforming ac!=bc into a!=b).
That's why you make sure all operations you do are bidirectional.
If you are careful and each step gives an equivalent statement, this is correct. For example, if you read math books/papers, whenever such transformation is made, the author usually states that it is equivalent.
However, the problem is that students apply this technique blindly manipulating the formula in any way they can/want. Such carelessness is the real problem.
However, the problem is that students apply this technique blindly manipulating the formula in any way they can/want. Such carelessness is the real problem.
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.
This book was a horrifying read. I never came across any of the "tricks", besides cross-multiplication for dividing fractions.
I'm a parent. I'm finding this a valuable read for checking in on my kids' common core math.
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.
Does it explain why ab = ba?
Timothy Gowers has a few thoughts on this:
https://www.dpmms.cam.ac.uk/~wtg10/commutative.html
https://www.dpmms.cam.ac.uk/~wtg10/commutative.html
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.
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.
ab=ba is not an axiom for integers (where multiplication is defined as repeated addition), or rationals (where multiplication is defined in terms of integer multiplication). I suspect that it is similarly not an axiom in the real numbers (and, by extension, the complex numbers), but it has been a while since I looked of how the reals were defined.
If we consider fields, then the field operation commonanly referred to as multiplication, is defined to be commutative.
If we consider fields, then the field operation commonanly referred to as multiplication, is defined to be commutative.
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...
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.
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.
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.