Common core long division.

https://www.facebook.com/dan.bongino/photos/a.517057181720381.1073741827.101043269988443/620433314716100/?type=1

by bane • 12 years ago

27 59 12 years ago

joliv• 12 years ago

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/)

RogerL• 12 years ago

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?

wolfgke2• 12 years ago

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

gamegoblin2• 12 years ago

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.

gnoway2• 12 years ago

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.

ronaldx5• 12 years ago

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.

gamegoblin2• 12 years ago

I am suspicious of it because it seems to mire the situation with division in which there isn't an integer answer, whereas the traditional long division method provides an intuitive method for extracting digits beyond the decimal.

But the method presented is simple and easy to remember. Seems like a simplicity vs. efficiency debate.

jliechti1• 12 years ago

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) 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• 12 years ago

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.

philh2• 12 years ago

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".

jrockway2• 12 years ago

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.)

betterunix5• 12 years ago

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

lnanek2• 12 years ago

It's basically just subtracting whatever you can think of until you finish, it isn't really teaching division at all. The main idea is just to prevent having to teach kids the times tables so they can be promoted out of the system easier.

DougBTX• 12 years ago

Multiplication is a way to do adding faster, and division is a way to do subtraction faster. Pretty cool if kids are being encouraged to realise that relationship.

It was only when I watched videos of those mechanical "fire control" computers on YouTube that I understood that integration is just a way of adding. Multiplication is adding the same number again and again, which is why integrating a constant (that number) is the same as multiplication. Cute.

zecho2• 12 years ago

I think it's a clever way to introduce the properties of numbers without forcing memorization of a multiplication table. It's a very brute force technique. Of course memorization is useful, but so will understanding these properties as the kids enter algebra.

jedberg• 12 years ago

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.

ishener2• 12 years ago

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• 12 years ago

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

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

thewopr4• 12 years ago

I absolutely agree. There is a huge, amazing array of concepts that never gets touched upon because it is important that students learn to divide large numbers by hand. They were designed for an era where "computers" were people and companies hired people to keep, literally "the books" in order. That world disappeared decades ago, but this relic remains.

ronaldx5• 12 years ago

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.

Someone2• 12 years ago

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"?

blakesterz2• 12 years ago

"or did she teach them the 'right' way and told them" ... there is ONLY the right way to do it. If the kids do it "the wrong way" they don't get credit.

watwut2• 12 years ago

That is definition of a bad math teacher.

thewopr4• 12 years ago

I find this indicative of poor teaching in general, not what standards they are trying to teach. Common core doesn't hurt or help with this issue. Being rigid with methods and not aiming for understanding is the problem.

noobermin3• 12 years ago

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.

ronaldx5• 12 years ago

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.]

Someone2• 12 years ago

On the contrary: for larger numbers, it is an efficient method, in the sense that one can expect students to do it.

For example, what's the first digit of 1019794644/1457? I would spend a few seconds before taking a guess at 7, compute 71457 to get 10199, conclude "oh, it must be 6", compute 61457, subtract it, etc.

And I can make a good educated guess to get that initial 7 because I know 14 = 1/7 (in the same sense that 27 = 1/37 and 91 = 1/11. It helps to know some factorisations of integers close to 10^n). You can't expect all students to do that, and you don't have to. It turns out that, if they just guess a low end 5, they do better than me, because they only have to compute 51457 (for an expert, a bit easier than either 6 or 7, but for them, likely equally difficult), and, from there, already will have simplified the problem. Their next multiplication will be 11457. After 'my' 71457, I still have to do a more difficult 61457.

Also, looking at it objectively, it turns out that I do the exact same thing as that student that "isn't that good at long division". The only difference it that that student is honest and that I lie, by writing the answer in a way that suggests that that '6' came to me by some divine intervention, and not by correcting an initially incorrect guess.

ahsanhilal3• 12 years ago

The common core standards which talk about division are described here 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.

blakesterz2• 12 years ago

" 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.

ahsanhilal3• 12 years ago

I work with over 30 schools in Bay Area, Chicago and New York as well, and have been working with teachers, districts etc on common core for the past two years. The entire point of CCSS is to make sure that WHAT is taught in a grade level does not vary. HOW varies so much from district to district, as each school/classroom contains so much demographic variability. My point was that it does not direct you "how" to teach kids division. Rather it provides guidelines that they should be taught division. Please go read the entire standards list and tell me if they contain anything more than suggestion on how to teach any mathematical curriculum. I agree teachers hate it, but its mostly because: 1. They are being forced to adapt to something without adequate amount of training 2. They have children with a whole range of skill sets which do not map to the common core for the grade levels they teach. So some children might be ahead and some might be behind for the particular grade level, so its hard to make sure that everyone in a grade level is mapping on the standards

watwut2• 12 years ago

Same thing everybody is doing in this case is long division. Everybody gotta get the result when faced with long division exercise.

If some administrators in schools or school district decided that there is only one way to do long division, then it is on them.

burkemw3• 12 years ago

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...

noobermin3• 12 years ago

Read the GP's link. You'll see that the COMMON core (despite its name) does not quite do that. It is up to the states and I think even local districts to decide how to implement it. The idea is that a common core is to set up ideas that every student should learn (that's the COMMON part), not _how_ they should learn it.

thewopr4• 12 years ago

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.

ef42• 12 years ago

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".

thewopr4• 12 years ago

A Mathematician's Lament is a good document to read when thinking about this. Link for the lazy

http://www.maa.org/sites/default/files/pdf/devlin/LockhartsL...

stefan_kendall3• 12 years ago

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.

zecho2• 12 years ago

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.

ef42• 12 years ago

Sure, but that kind of absurdity is nothing new. Most people just take the older absurdities for granted because it's all they know.

saticmotion• 12 years ago

I finally created an account on HN, just to thank you for telling me about Paul Lockart's essay. Only now do I realize why I don't like, nor properly understand mathematics - at least not mathematics as it's presented in schools.

Things like Conway's Game of Life, fractals, etc are all so insanely interesting and their results so beautiful. But in school they have you calculating boring functions that no one really knows what they're about.

Aardwolf• 12 years ago

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).

betterunix5• 12 years ago

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.)

redblacktree4• 12 years ago

$ python

>> 523 / 6

>>

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

dworin• 12 years ago

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.

Common core long division.

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