
Common core long division. - bane
https://www.facebook.com/dan.bongino/photos/a.517057181720381.1073741827.101043269988443/620433314716100/?type=1
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philh
I find the comments that people can't understand this method to be kinda
bizarre.

I haven't heard of common core, but after a few moments of looking at this,
the algorithm becomes clear: 432 is between 80 and 800, so repeatedly subtract
80 until you get below 80. Then repeatedly subtract 8. Keep track of the
subtractions you're making, and add them up.

Slower and more verbose than the method I learned, and I'm not defending (or
condemming) it, but I kind of wonder if some people have a reaction like "this
is not what I would have expected to see, and I can't immediately understand
it, so it sucks".

~~~
jrockway
I think the problem is that the parents were taught how to do things, not how
to think about things.

Nowadays, you don't learn long division so you can divide two numbers given a
large amount of time and a big sheet of paper. Your computer will do that for
you. You learn it as a precursor to algebra and number theory and as a way to
apply an algorithm practically. That's what the Facebook comments don't
understand.

(I don't really see the benefit in this case, but I'm tired of the knee-jerk
reactions about everything new.)

~~~
betterunix
Ironically, the previous generation was also supposed to be taught how to
think rather than just how to do math:

[https://en.wikipedia.org/wiki/New_Math](https://en.wikipedia.org/wiki/New_Math)

~~~
jrockway
Good point. We should have also taught them to remember to think before
posting to Facebook.

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ahsanhilal
The common core standards which talk about division are described here
[http://www.corestandards.org/Math/Content/7/NS](http://www.corestandards.org/Math/Content/7/NS)

At no point do the standards direct you how to teach long division. Rather
they try make sure that kids who go through 5,6,7 grades have a basis in
division and consequently long division. Claiming that common core directs
teachers to teach division in a certain manner is false. The fact that the kid
is learning how to do division in a weird manner is probably the fault of the
teacher who does not understand it herself.

Common Core is a set of guidelines regarding what should be taught in a
curriculum to provide some structure to the curriculum variability across
classes, schools, districts and states. Common Core definitely has some
problems and most of them pertaining to implementation, teacher training, and
the fact that due to standards some students are being set up to fail.
However, it definitely does not direct anyone how to teach, but rather what
they should be teaching at the minimum.

~~~
blakesterz
" However, it definitely does not direct anyone how to teach" I'm a parent of
3 kids in a NY elementary school. Several members of my family are teachers or
administrators in schools. As a parent I disagree with that, and every single
teacher/administrator would disagree with that. The entire point of COMMON
core is to have everyone doing the same thing. This is one of the many reasons
teachers hate it.

~~~
burkemw3
Deidre Austen responded [0] to the original post that the standard being
assessed is "find whole number quotients of whole numbers with up to 4-digit
dividends and 2-digit divisors using strategies based on place value, the
properties of operations, and/or the relationship between multiplication and
division. Illustrate and explain the calculation by using equations,
rectangular arrays, and/or area models."

Considering that this school has has chosen a reform (or non-standard)
division approach to achieve this standard, instead of using regular long
division, shows to me that curriculum managers do have options for ways to
achieve the standards.

(Hopefully, this Facebook link works correctly)

[0]
[https://www.facebook.com/dan.bongino/photos/a.51705718172038...](https://www.facebook.com/dan.bongino/photos/a.517057181720381.1073741827.101043269988443/620433314716100/?type=1&comment_id=1858450&offset=0&total_comments=312)

~~~
ahsanhilal
Exactly teachers have a lot of leeway in how they want to teach something. The
most common reasons for failure are that the teacher does not understand
exactly what she is teaching. That is a failure of the district and school in
not recognizing that the teacher is not well-trained to teach the subject
matter.

------
jedberg
I don't think people realize that the purpose of elementary education is not
to teach the kids how to do something -- it's to teach them how to _think_
about doing something.

This method is longer but it helps them grasp the concept of what division is
so that they can then extend that thinking.

~~~
ishener
is this system really needed in order to grasp the concept of division? isn't
saying "divide these 30 oranges to 6 boxes" enough? what does this system add
to grasping the concept beyond this sentence?

actually i think the two methods of division are entirely pointless to teach
youth. they are both mechanical algorithms that contribute very very little to
the logical sophistication of kids, especially when you consider the other
things you could teach them instead (set theory).

in other words, one method is short but easily forgotten, the other is long
and remembered, but actually in real life you don't need to remember how to
divide numbers and you also don't need to do it fast.

~~~
wavesounds
"isn't saying "divide these 30 oranges to 6 boxes" enough? "

So your plan is to teach division by telling someone to 'divide'?

~~~
ishener
why not... i'm not teaching the dictionary meaning of the word, i'm teaching
the concept...

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ef4
I'm getting really tired of the complaints from teachers and parents about
these education reforms. Not because the reforms are any good, but because
they're so tiny and insignificant.

"School math" is already an absurd parody of the real thing (rather than
repeat the argument, see Paul Lockhart "A Mathematician's Lament").

And I'm supposed to get up in arms about switching to a slightly different
long-division algorithm? Seriously? These parents and teachers are so ignorant
of mathematics that they can't even form coherent arguments about what they
don't like beyond "it's different from the steps I learned to mindlessly
repeat as a child".

~~~
stefan_kendall3
I would agree with you, but I've actually seen the homework assignments and
only acceptable correct answers to common core work, and it's disgusting.

Trying to solve 72/9? Better show your work by drawing 72 circles, or you get
no credit.

~~~
zecho
Depends on what the outcome of learning is supposed to be. If it is to simply
recite 8, then what have you learned other than a factoid? 72 circles in 9
columns is 8 rows of circles and now you see the relationship clearly. If you
rotate your grid, you have 8 columns of 9 circles. If the next question is
72/12 or 72/3 those relationships become even more apparent as the shape of
the grid changes.

I see no problem taking math out of abstraction in this way, especially as one
moves onto visualizing more complex math. Seeing how the symbols interact with
their meaning is a good thing.

I recently introduced my 5 year old to Dragonbox. She picked up algebra
quickly, without the formalities, of course. I was pretty amazed by how
quickly she understood it, too.

Playing with numbers is a really a great way to learn the meaning of formulas.
There are entire schools of thought, such as Montessori, that believe numbers
should be taught in this tactile way. Seeing it first hand, and comparing it
to my own struggles as a child, I'm inclined to see great value in drawing 72
little circles out.

------
joliv
This actually looks like a much better way to learn how division works—taking
a number out of another number some number of times. While it may be less
efficient than the other method, if we really cared about efficiency we would
just teach division with calculators.

EDIT: Ooph, he's running for the House of Reps in my state
([http://www.bongino.com/](http://www.bongino.com/))

~~~
RogerL
I find the Facebook comments interesting - lot's of "I can't figure this out".
I suspect, strongly, that a lot of people learned the traditional manipulation
but never understood why they were doing it. This just make it explicit: 4
ten's, and so on.

Now, if children are being forced to use this technique throughout their
education something is wrong. If this is a step on the way to the standard
technique, then so much the better. I recall pre-calc: we were forced to do a
bunch of cumbersome math to compute limits before being taught the 'easy' way,
because the easy way, being symbolic manipulation, could obscure what is going
on and keep you from learning the concepts behind limits. Same thing. I rarely
do any math the way it is initially taught. When is the last time you wrote -3
= -3 on both sides of an algebraic system to move a 3 to the other side of an
equation?

~~~
wolfgke
> I find the Facebook comments interesting - lot's of "I can't figure this
> out". I suspect, strongly, that a lot of people learned the traditional
> manipulation but never understood why they were doing it.

If you really want students to understand _why_ they are doing it, I see no
other way than teaching methods for proving the correctness of algorithms, i.
e. Hoare logic:
[https://en.wikipedia.org/wiki/Hoare_logic](https://en.wikipedia.org/wiki/Hoare_logic)

------
Someone
This example is overly conservative, but the approach is sound.

Yes, one should expect students to know that you can fit eight five times into
43, but things start being different fast when doing trickier divisions. For
example take 903/13\. I don't have to think to know 6 times 13 equals 78 and 7
times 13 equals 91, but the average student? Forget it.

I would be happy to see the average student take out a 5 first (903 = 50 * 13
+ 253), then another one (903 = 50 * 13 + 10 * 13 + 123). From there, I would
have to think a bit to realize that one can take out a 9 (123 is very close to
113). The typical student? Forget it.

And that gets increasingly difficult when the divisor gets larger. What's the
first digit of 9230/131?

So, yes, this strategy isn't optimal from a # of operations viewpoint, but for
many students, it probably is the best way to get an answer. What would you
prefer, a student who doesn't get the answer, or one that gets the right
answer in a suboptimal way?

For me, the main question is what the teacher taught: did (s)he only tell
students about this slowest, sure-fire way, or did she teach them the 'right'
way and told them "if you aren't sure what digit to take next, pick one that
you are sure about"?

~~~
noobermin
For the reasons you said, I don't think that this is a general algorithm they
are teaching. May be it is a stepping stone to get to understanding of what
division means first and then they'll teach them long division. Clearly this
is an inefficient method for larger numbers, so I don't see how this can be
expected to be useful as a go-to algorithm for calculating division.

~~~
ronaldx
I doubt she has been taught to repeatedly take off 10x8s.

I find it more likely she been taught that it doesn't matter what she takes
off: as long as she gets to the end without error, her answer will be correct.

I imagine that she has reasoned that taking off 50x8 is difficult because she
doesn't yet have instant access to 5x8 (which would be rare for her age group)
but she can confidently subtract 8s without error (also rare). Perhaps she
reasoned this earlier and has been consistently using an inefficient version
of the method when she doesn't need to. In that case, she will likely be able
to fix this herself when she realises.

[ _Edited to add:_ I interpolate this because I am a schoolteacher who teaches
math. It's a big part of my job to recognise what support a student needs:
here, most likely she needs 8-times-table support.

The method that I describe is well-enough established in the school system
that university entrants use it. Of course, it's possible that her teacher has
taught her an inefficient method, but that would be unusual. There _are_ bad
math teachers in elementary school, but that's not a typical teaching error.]

~~~
noobermin
That's a bit of interpolation there. I will admit that we are given very
little to work on, just a piece of paper with the algo partially done on it by
a child... I'm assuming you think that because of the "10x8" on the side and
it could be easily "20x8". It looks like they are teaching it by places, and
the 10x_ represents the order of magnitude that is currently being operated,
and she was to see how many "8"'s in that magnitude are in the number. That
was my assumption, at least.

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gamegoblin
I have tutored a lot of math in university. Mainly college algebra, trig, the
basics that you probably should have learned in highschool but didn't care to,
wasn't offered, or was taught poorly.

Out of maybe 30 students I've tutored, only 1 could do long division. It seems
that students learn it, forget it, briefly relearn it when learning to do
polynomial division, then forget it again.

~~~
gnoway
So in your opinion, would replacing long division by iterative subtraction
(what I'm calling this - I don't know what it's called) improve the situation?

In this case it looks like you still need to understand multiplication and
subtraction to complete the exercise, so I am not really sure what's gained by
switching to this method.

~~~
ronaldx
Despite claiming you don't understand the gain, you just said it: students
gain a more complete understanding of how multiplication and subtraction are
related to division, which is the primary goal of this stage of math
education.

~~~
wolfgke
If this is your goal you better teach the student about how to define addition
and multiplication using the the formalization from the Peano axioms, i. e.

    
    
      +(0, y) = y for all y in N
      +(S(x), y) = S(+(x, y)) for all x, y in N
      *(0, y) = 0 for all y in N
      *(S(x), y) = +(y, *(x, y)) for all x, y in N
    

I know no way that states more clearly how addition and multiplication are
related. Division is than simply stated as: for given a, b in N find x such
that

    
    
      *(b, x) = a.
    

This states clearly how multiplication (obvious) and addition (by definition
of multiplication) is related to division.

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jliechti1
In hindsight, I think it would have been helpful if the schools I went to
introduced the concept of an _algorithm_. I remember we spent a lot of time
learning Lattice multiplication
([http://en.wikipedia.org/wiki/Lattice_multiplication](http://en.wikipedia.org/wiki/Lattice_multiplication))
and I would get frustrated having to draw all the little boxes when the
"traditional" way was much faster for me. It would have been beneficial if
they stressed the concept of algorithms and showed me there are, in fact, many
more ways to multiply numbers together.

~~~
NAFV_P
Lattice, I never heard of it when I was in school. When I first came across
the term about two years ago, I realised I had already worked out the
algorithm a few months beforehand.

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ronaldx
The daughter is correctly conceptualising number by following this method
(and, although I cannot see the answer, her method is executed correctly up to
the point I can see). She doesn't seem confused.

Her method is not perfect but is very well developed if we compare her
handwriting. If the father genuinely thinks that this is an easy problem for
her age group, then her handwriting is behind where it should be.

So, Daniel Bongino, what's your problem with this? Your child is developing
her number skills just in the same way that she is developing her fine motor
skills. That's great.

------
dworin
I find it interesting how all of the people who complained about how terrible
math education was when they were kids turn into grownups who wish they would
just go back to the traditional way of teaching math.

My sister is an elementary school teacher, and she's posted some videos on her
Facebook page explaining how common core teaches - and they look so much
BETTER than the way I learned to do math.

As far as I can tell, the old system was some combination of "memorize times
tables" and "structured brute force the answer." The new system tries to teach
kids ways to think about numbers and simplify them so that they can use mental
math.

The people who developed the standards came up with them by studying which
approaches to teaching actually got kids to learn math. I don't think "we did
things a certain way for hundreds of years before we started looking at it
scientifically" is a strong refutation of that approach.

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thewopr
The problem here is not the method the students are learning. The problem here
is that they are learning long division at all. Does ANYBODY care the method
by which children learn to divide large numbers?

I would be willing to bet that behind the vast majority of those outraged
comments is an adult that hasn't used long division in decades.

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Aardwolf
Learning to sum, subtract, multiply and divide in your head is pretty
important, e.g. to be able to work with money, so that's a super useful thing
to learn in primary school. But doing long division on paper is not useful
imho. It's not even useful for high level math, or programming (except for CPU
design). Why do they teach it in primary school? I always wondered that, since
I never needed it (at least not in decimal).

~~~
betterunix
It absolutely is important to learn the long division _algorithm_ for
programming. How else do you deal with integer division?

(Yes, _integer_ division, which people refer to as "big number" division for
some bizarre reason.)

~~~
redblacktree
$ python

>>> 523 / 6

87

>>>

I can't remember any time when I've been called upon to write an algorithm for
integer division.

~~~
betterunix
1\. Perhaps you should start now, since your Python interpreter gave you the
wrong answer. It should be obvious that 523/6 is not an integer.

2\. That is kind of like saying, "I can't remember any time when I have been
called upon to implement merge sort, therefore there is no point in teaching
the algorithm!!"

~~~
redblacktree
It's not _wrong_ it's just the result of integer division where the remainder
has been dropped. What makes you say it is "wrong?" Were you expecting a
floating-point result?

To your point #2, you didn't argue that teaching these concepts aids in the
general understanding of how to create algorithms, you said, "How else do you
deal with integer division?" To which my answer is: You don't. It's a solved
problem.

~~~
betterunix
I would expect either a pair of integers, either a rational number or a
quotient and remainder. 523/6 is not 87.

As for "solved problems," it is not at all unusual to be asked to re-implement
an algorithm that is already implemented elsewhere. Maybe there is no
implementation for your platform. Maybe you are writing a new language. You
could find yourself implementing integer division just as much as you could
find yourself implementing a sorting algorithm, a hash table, etc.

[Edit: Also worth noting is that in the context of education it is common to
ask students to re-implement known algorithms and build up systems on their
own.]

~~~
redblacktree
In the context of education, I agree, but you strech credulity with this:

> Maybe there is no implementation for your platform.

I challenge you to find me a single example of a platform that does not
already have an implementation for integer division. This functionality is
_basic_.

Edit: Also, you seem to have strange expectations with respect to results of
integer division. Many languages in common use will give the result above. If
you want the remainder, you're expected to use modulus.

For example:

>>> '%s %s/%s' % (523 / 6, 523 % 6, 523)

'87 1/523'

>>>

