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.
Or worse, the majority of kids that don't even realise the subject is more than just the tricks themselves.
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.
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 :/
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. 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. 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 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 liked the old website of this organization better than the new website.)
The Art of Problem Solving website is a treasure trove of interesting mathematics education resources for learners of all ages.
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.
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 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.
as described here in "The history of the percent sign"
- 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.
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.
I think that tricks that allow you to do things quickly in your head are great- they help build numeracy.
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.
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.
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.
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).
You've got it backwards here; the Common Core curriculum is basically all "tricks" without 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.
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.
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
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.
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  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.
 Such as how fractions work
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.
If you're interested, these videos explain and discuss each "trick" in more detail:
It's a work in progress...
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.
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.
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. ;)
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.
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?
(I'm assuming that last sentence was typoed and "talking about how many 3/4 in 1/2" was intended.)
I remember sitting in a college math course as we deduced the geometric series that represents each positively giddy.
What an awful way to teach.
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.
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.
Your example is cool. I have not seen many exmaples of such logic, but people probably do it too.. )
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.
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.
If we consider fields, then the field operation commonanly referred to as multiplication, is defined to be commutative.
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...
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.