Common core long division.
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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".
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.)
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.
I would also point out that the repeated addition algorithm makes a lot of sense in base 1, but that shift and add makes far more sense in base 2 or higher.
As for integration, I would say it is better to think of it as a limit involving both addition and multiplication.
Can you elaborate? If you need to do 13 + 13 + 13 + 13, you can do that as 13 x 4, which I guess is what you mean by the first part; but what analagous subtraction could you replace with a division?
52/13:
52 - 13 = 39
39 - 13 = 26
26 - 13 = 13
13 - 13 = 0
count the subtractions and you get 4.
As long as we're sticking to integer division, this works out fairly well. Here, a calculation with a remainder:
54/13:
54 - 13 = 41
41 - 13 = 28
28 - 13 = 15
15 - 13 = 2 (smaller than 13, so this is our remainder)
54/13 = 4 with 2 reaminder
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.
If some administrators in schools or school district decided that there is only one way to do long division, then it is on them.
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...
This method is longer but it helps them grasp the concept of what division is so that they can then extend that thinking.
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.
So your plan is to teach division by telling someone to 'divide'?
"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".
http://www.maa.org/sites/default/files/pdf/devlin/LockhartsL...
Trying to solve 72/9? Better show your work by drawing 72 circles, or you get no credit.
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.
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.
EDIT: Ooph, he's running for the House of Reps in my state (http://www.bongino.com/)
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?
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
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"?
Question: "Given that the new primary maths curriculum no longer includes the chunking method in division calculations ... assessments in maths will give credit for a pupil’s working?
Answer: "...chunking will not be rewarded in method marks — long division will."
*"Chunking" here refers to the method of the parent article. Chunking has been removed from the curriculum and only the traditional method will receive credit. That's the definition of bad math teaching, from the UK government.
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.]
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.
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.
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.
+(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
*(b, x) = a.
That said, I don't have a big problem with this as a method for learning division. I think my reaction is related more to the memory of 'show your work!' and the thought that this would mean more work to show.
But the method presented is simple and easy to remember. Seems like a simplicity vs. efficiency debate.
Student probably hasn't studied decimal numbers to an extent that this would be relevant at this stage.
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.
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.
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.
(Yes, integer division, which people refer to as "big number" division for some bizarre reason.)
>>> 523 / 6
87
>>>
I can't remember any time when I've been called upon to write an algorithm for integer division.
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!!"
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.
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.]
> 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'
>>>