
Nix the Tricks: Math tricks defeat understanding - tokenadult
http://www.nixthetricks.com/#hn-repost
======
elchief
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.

~~~
anon4
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.

~~~
anon4
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.

~~~
anvandare
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](https://en.wikipedia.org/wiki/Simon_Stevin)

------
tokenadult
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-
Mathematic...](http://www.amazon.com/Knowing-Teaching-Elementary-Mathematics-
Understanding/dp/0415873843)

[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.

------
asgard1024
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.

~~~
tome
> 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.

~~~
SixSigma
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/](http://www.shadycharacters.co.uk/2015/03/percent-sign/)

~~~
plonh
/ means "per"

~~~
Retra
and × means "of"

------
delish
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.

~~~
jules
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.

~~~
a_e_k
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
Nice, and that generalizes neatly to n-space with exterior algebra / Grassmann
coordinates.

------
nerd_stuff
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
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
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
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.

~~~
dsharlet
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.

~~~
nerd_stuff
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.

~~~
dhimes
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.

------
scotty79
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
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
_> 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).

~~~
acheron
_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.

~~~
21echoes
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.

------
daniel-levin
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
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.

------
jstanley
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
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
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...](https://www.youtube.com/watch?v=1gxLwLBl4mM&index=4&list=PLAmlm9_MPQMSJ4kRHofj7xkJN1aqmGuQ8)

It's a work in progress...

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

~~~
sleepychu
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.

~~~
plonh
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?

~~~
ashark
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.)

------
liquidise
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.

------
skaevola
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.

------
hammerbrostime
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)

------
msie
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.

------
a-nikolaev
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.

~~~
robryk
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
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.. )

~~~
robryk
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).

------
pflats
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.

------
j7ake
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.

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

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

~~~
nmrm2
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.

------
amelius
Does it explain why a _b = b_ a?

~~~
carapat_virulat
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
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.

------
elevenfist
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
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.

