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Nix the Tricks: Math tricks defeat understanding

139 points by tokenadult 11y ago

droopybuns 11y ago
I'm a parent. I'm finding this a valuable read for checking in on my kids' common core math.
- nmrm2 11y ago
  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.
delish 11y ago
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.
- stsp 11y ago
  "Linear Algebra Done Right" helped me with getting through college :) I can highly recommend it.
- jules 11y ago
  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.
  - torrance 11y ago
    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?
    - kuschku 11y ago
      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.
      - technomancy 11y ago
        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.
        kuschku 11y ago
        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.
  - a_e_k 11y ago
    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 http://www.cs.utah.edu/~aek/research/triangle.pdf).
    - jules 11y ago
      Nice, and that generalizes neatly to n-space with exterior algebra / Grassmann coordinates.
aptwebapps 11y ago
Wrt the example of fraction division, isn't this the standard way? Was anyone here taught to cross-multiply for division?
- lacksconfidence 11y ago
  fwiw, I was taught in the mid 90's about how (a/b) / (c/d) = (a/b) * (d/c) = (ad)/(bc)
- enjo 11y ago
  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.
- sleepychu 11y ago
  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.
  - aptwebapps 11y ago
    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. ;)
  - IanCal 11y ago
    > 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.
liquidise 11y ago
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.
msie 11y ago
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.
elchief 11y ago
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.
- antod 11y ago
  > 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.
  - fleshshelf 11y ago
    And who's fault is that?
    - HarryHirsch 11y ago
      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.
- mangamadaiyan 11y ago
  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 :/
jstanley 11y ago
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.
- iamflimflam1 11y ago
  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)
  - OTmath 11y ago
    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=PLAmlm9_MPQMSJ4kRHofj7xkJN1aqmGuQ8 https://www.youtube.com/watch?v=1gxLwLBl4mM&index=4&list=PLA... It's a work in progress...
asgard1024 11y ago
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.
- carapat_virulat 11y ago
  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.
- tome 11y ago
  > 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.
  - 11y ago
scotty79 11y ago
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.
- Camillo 11y ago
  I have to ask, do those "tricks" represent the way math is usually taught in America? It looks like a bad joke to me.
  - nmrm2 11y ago
    > 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).
amelius 11y ago
Does it explain why ab = ba?
- carapat_virulat 11y ago
  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.
  - gizmo686 11y ago
    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.
- Jabbles 11y ago
  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
j7ake 11y ago
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.
a-nikolaev 11y ago
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.
- arielby 11y ago
  That's why you make sure all operations you do are bidirectional.
  - a-nikolaev 11y ago
    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.
- robryk 11y ago
  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.
  - a-nikolaev 11y ago
    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.. )
daniel-levin 11y ago
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
- nmrm2 11y ago
  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 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.
tokenadult 11y ago
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://condor.depaul.edu/sepp/mat660/Askey.pdf http://www.ams.org/notices/199908/rev-howe.pdf http://www.ams.org/notices/199908/rev-howe.pdf http://www.amazon.com/Knowing-Teaching-Elementary-Mathematics-Understanding/dp/0415873843 http://www.amazon.com/Knowing-Teaching-Elementary-Mathematic... https://math.berkeley.edu/~wu/Stony_Brook_2014.pdf https://math.berkeley.edu/~wu/Stony_Brook_2014.pdf https://math.berkeley.edu/~wu/MSRI_2014_1.pdf https://math.berkeley.edu/~wu/MSRI_2014_1.pdf [2] http://www.mathwalk.org/ 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 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.
hammerbrostime 11y ago
The link to the table of contents is broken, here is the proper link: http://www.nixthetricks.com/Contents.html http://www.nixthetricks.com/Contents.html
skaevola 11y ago
I hadn't heard of the turtle head method before so I searched and found this video: https://www.youtube.com/watch?v=1hcKERTnNi0 https://www.youtube.com/watch?v=1hcKERTnNi0 What an awful way to teach.
nerd_stuff 11y ago
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 https://www.edx.org/course/street-fighting-math-mitx-6-sfmx
- dhimes 11y ago
  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.
  - dsharlet 11y ago
    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.
    - nerd_stuff 11y ago
      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.
      - 11y ago
elevenfist 11y ago
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...
- xtreme 11y ago
  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.
- eterm 11y ago
  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.
pflats 11y ago
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.
phanimahesh 11y ago
This book was a horrifying read. I never came across any of the "tricks", besides cross-multiplication for dividing fractions.

139 points by tokenadult 11y ago

droopybuns 11y ago

I'm a parent. I'm finding this a valuable read for checking in on my kids' common core math.

nmrm2 11y ago
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.

delish 11y ago

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.

stsp 11y ago
"Linear Algebra Done Right" helped me with getting through college :) I can highly recommend it.
jules 11y ago
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.
- torrance 11y ago
  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?
  - kuschku 11y ago
    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.
    - technomancy 11y ago
      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.
      - kuschku 11y ago
        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.
- a_e_k 11y ago
  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 http://www.cs.utah.edu/~aek/research/triangle.pdf).
  - jules 11y ago
    Nice, and that generalizes neatly to n-space with exterior algebra / Grassmann coordinates.

aptwebapps 11y ago

Wrt the example of fraction division, isn't this the standard way? Was anyone here taught to cross-multiply for division?

lacksconfidence 11y ago
fwiw, I was taught in the mid 90's about how (a/b) / (c/d) = (a/b) * (d/c) = (ad)/(bc)
enjo 11y ago
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.
sleepychu 11y ago
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.
- aptwebapps 11y ago
  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. ;)
- IanCal 11y ago
  > 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.

liquidise 11y ago

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.

msie 11y ago

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.

elchief 11y ago

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.

antod 11y ago
> 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.
- fleshshelf 11y ago
  And who's fault is that?
  - HarryHirsch 11y ago
    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.
mangamadaiyan 11y ago
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 :/

jstanley 11y ago

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.

iamflimflam1 11y ago
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)
- OTmath 11y ago
  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=PLAmlm9_MPQMSJ4kRHofj7xkJN1aqmGuQ8 https://www.youtube.com/watch?v=1gxLwLBl4mM&index=4&list=PLA... It's a work in progress...

asgard1024 11y ago

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.

carapat_virulat 11y ago
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.
tome 11y ago
> 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.
- 11y ago

scotty79 11y ago

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.

Camillo 11y ago
I have to ask, do those "tricks" represent the way math is usually taught in America? It looks like a bad joke to me.
- nmrm2 11y ago
  > 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).

amelius 11y ago

Does it explain why ab = ba?

carapat_virulat 11y ago
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.
- gizmo686 11y ago
  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.
Jabbles 11y ago
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

j7ake 11y ago

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.

a-nikolaev 11y ago

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.

arielby 11y ago
That's why you make sure all operations you do are bidirectional.
- a-nikolaev 11y ago
  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.
robryk 11y ago
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.
- a-nikolaev 11y ago
  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.. )

daniel-levin 11y ago

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

nmrm2 11y ago
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 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.

tokenadult 11y ago

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://condor.depaul.edu/sepp/mat660/Askey.pdf http://www.ams.org/notices/199908/rev-howe.pdf http://www.ams.org/notices/199908/rev-howe.pdf http://www.amazon.com/Knowing-Teaching-Elementary-Mathematics-Understanding/dp/0415873843 http://www.amazon.com/Knowing-Teaching-Elementary-Mathematic... https://math.berkeley.edu/~wu/Stony_Brook_2014.pdf https://math.berkeley.edu/~wu/Stony_Brook_2014.pdf https://math.berkeley.edu/~wu/MSRI_2014_1.pdf https://math.berkeley.edu/~wu/MSRI_2014_1.pdf [2] http://www.mathwalk.org/ 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 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.

hammerbrostime 11y ago

The link to the table of contents is broken, here is the proper link: http://www.nixthetricks.com/Contents.html http://www.nixthetricks.com/Contents.html

skaevola 11y ago

I hadn't heard of the turtle head method before so I searched and found this video: https://www.youtube.com/watch?v=1hcKERTnNi0 https://www.youtube.com/watch?v=1hcKERTnNi0 What an awful way to teach.

nerd_stuff 11y ago

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 https://www.edx.org/course/street-fighting-math-mitx-6-sfmx

dhimes 11y ago
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.
- dsharlet 11y ago
  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.
  - nerd_stuff 11y ago
    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.
    - 11y ago

elevenfist 11y ago

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...

xtreme 11y ago
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.
eterm 11y ago
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.

pflats 11y ago

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.

phanimahesh 11y ago

This book was a horrifying read. I never came across any of the "tricks", besides cross-multiplication for dividing fractions.