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tl;dr I'm interested in opinions on short division. Below I discuss my experience dealing with young children and long division versus short division. For those that don't know of it, wikiHow has a nice explanation on short division.


Recently I've been asked to help some children around the age of 10 with math, and almost immediately I find that they struggle with division. They take a long time to divide numbers and they always use long division, or some variant of it. I've even seen one person divide as follows:

\begin{array} {r|l} \require{enclose} 3 \enclose{longdiv}{1~8~6} \\ \underline{{}-~1~5~0} & 5~0 \\ 3~6 \\ \underline{{}-~3~0} & 1~0 \\ 6 \\ \underline{{}-~6} & \underline{\hphantom{5~}2} \\ 0 & 6~2 \end{array}

which just frustrates me. Excuse my rant. But I would expect people to know that one can directly see that the answer is $62$ by seeing that $18\div3=6$ and $6\div3=2$. Furthermore I ask them why they use $50$ on the first step instead of $60$, and why not just not have the zero and do it digit-by-digit and they say they noticed $50$ "worked" and that they've been taught this way at school.

Poor dividing aside, long division inherently takes a long time to do, as the name implies, and when given a limited amount of time to do an assignment or assessment in class, time consuming division can really hurt. Likewise, there is what is known as short division, which generally takes much less time but requires more mental calculations. But it would appear that no-one knows about short division explicitly, at least out of everyone I've talked to. I myself actually "figured out" short division at a young age from examples like $186\div3$ by simply skipping steps, but didn't even know it was called this before writing this question.

Out of all of the children I've helped with division, the only ones that struggle now are the ones that don't have their multiplication tables memorized. Dividing by multi-digit numbers is still a process, but it's not the kind of thing I would expect someone to be able to do easily. Dividing by single-digit numbers occurs often though, for example reducing fractions. Some of the kids with stronger mental arithmetic skills that I've shown this to can now do this process digit-by-digit mentally, so that they don't write anything out when doing $\frac{38}{56}=\frac{19}{28}$ for example.

Considering the usefulness of short division, it strikes me as surprising that this isn't well-known and taught more! Does anyone have any experience with teaching short division?


Edit:

I may've glossed over why, exactly, I find short division not only less space consuming, but also easier and faster. Firstly, this does not require a huge amount of mental arithmetic. In fact it only requires being able to compute very small divisions several times over.

The essence of short division is that we are just repeatedly finding what goes into something and what the remainder is, which most of the kids I've dealt with are capable of doing mentally, as far as what's required for short division. Note that I'm not talking about problems with divisors larger than say 11.

Take, for example, $847\div3$. Most kids I've worked with would know right off the bat that $8\div3=2\rm~R~2$. So they write a 2 on top and 2 by the 8. Then they know that $24\div3=8\rm~R~0$. So they write an 8 on top and 0 by the 4. Finally they know $7\div3=2\rm~R~1$. This gives the final answer: $847\div3=282\rm~R~1$.

These steps are not difficult at all as long as the divisor is at most 11 or 12. In fact most of the kids find this easy to do. For larger divisors I wouldn't expect someone to do this mentally with short division. I would expect them to use long division as usual for something such as $847\div13$.

It also helps with situations such as the aforementioned fraction reduction such as $\frac{38}{56}=\frac{19}{28}$. It's not too hard to see that 2 goes in, but with long division writing this out can take a while. Hence why this is also faster in such cases.

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    $\begingroup$ Short division and long division take the same amount of time - indeed, they use the exact same processes. The difference is the amount of space on the page that they take to express as half of the steps are done mentally and not written down. $\endgroup$ – Matthew Daly Jan 6 at 12:17
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    $\begingroup$ The page you linked says, "It is an abbreviated form of long division — whereby the products are omitted and the partial remainders are notated as superscripts." I don't find the form with superscripts simpler or quicker than long division. It seems your main gripe is why the kids chose 50 instead of 60, or rather why did not they choose 0, then 6. Suppose you had 176 instead of 186 and you chose 6, it would be too much, so you go one down and choose 5, then continue as usual. What they do with 50 is the same trial and error approach. The beauty of the algorithm is that it still works. $\endgroup$ – Rusty Core Jan 6 at 18:40
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    $\begingroup$ As a person experienced with numbers, you have much better chunking available to you, and can hold a remainder in one of your 7±2 available mental slots while you do some other calculation. Many students will not be able to do the required mental calculations yet, the same way you can't hold three new phone numbers in your head at once. $\endgroup$ – Ryan Cavanaugh Jan 6 at 22:48
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    $\begingroup$ I remember I was taught short division some time in primary school, I'm not sure if long division was also taught, but if it was I don't remember it. I certainly find short division more understandable than long division. It wasn't until I was towards the end of highschool and was learning polynomial division that I worked out how to do long division. $\endgroup$ – Potato44 Jan 6 at 23:01
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    $\begingroup$ As a long time teacher in English primary schools, I found that division (of any sort) was always the last of the 4 to be taught - to the extent that add, subtract and times left little room for it, sadly. Also times tables were often thought to be old fashioned, therefore not concentrated upon. Without times tables, division is a mystery. In fact, I worked with teachers who didn't know their full times tables! So, not really surprised by the contents of this question (inc. rant!). $\endgroup$ – Tim Jan 7 at 8:13
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It's pretty rare these days for anyone to actually do division by hand. Most people reach for a calculator. Given those realities, I would question whether it even makes sense that we spend such a vast amount of time teaching young children even one algorithm for division. Maybe we should postpone it, deemphasize it, or replace long division with a slower or more space-consuming algorithm that's conceptually simpler and easier to learn. For example, we could teach kids to make an initial estimate of the result, multiply, find the error (which may be positive or negative), and then either decide whether their estimate is good enough or needs to be refined. Or we could teach them to do a binary search. We're no longer living in 18th century London, where human computation is the only option, so it has to be efficient, and every little kid has to be prepared for a life as a clerk or a bookkeeper who uses a quill pen.

I myself actually "figured out" short division at a young age from examples like 186÷3 by simply skipping steps, but didn't even know it was called this before writing this question.

This is typical of the type of person who wants to understand why math works, is comfortable with numbers, and has good mental habits involving sense-making and pre-estimation. Most kids never develop these habits and attitudes. Our main goal should be to try to inculcate these in kids. Where I live (California), my kids actually did get explicit instruction in pre-estimating the results of arithmetic problems. They had worksheets where they were forced to do it. They grumbled about its being extra work, but I thought it was great.

If you're in the habit of pre-estimating, and also of checking your results afterward, then it's almost inevitable that you'll start to notice that some examples like 186/3 can be done without setting pencil to paper.

Out of all of the children I've helped with division, the only ones that struggle now are the ones that don't have their multiplication tables memorized.

This is unfortunately a deeper problem. For many kids, being forced to memorize their multiplication tables represents the point in there lives where math goes from being interesting to being horrible drudgery. This is true even for many kids who are smart and good at math. There is the danger that you win the battle but lose the war. The kid becomes proficient at arithmetic but learns to hate math. Again, I wonder whether the problem isn't just that we spend too much time on it at too early a stage. Maybe kids could learn the basic algorithms of arithmetic while having access to a wall chart of the times tables, or a copy on their desk. After a year or two of this, they would have learned most of the facts by osmosis. Personally, I'm 54 years old, and I still do 6x8 by breaking it down into 2x3x8.

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    $\begingroup$ There's calculators so nobody needs to know math .. hilarious - but totally wrong view. I went to school starting 1981 - and we used "written" math until grade 7 - and I find it mandatory to know at least the foundation without calculators. What is the calculator good for, if you don't even know the basic rules? Of course as OP mentioned part of those basics are multiplication tables - we used to learn them - and I assume its not much different now $\endgroup$ – eagle275 Jan 6 at 14:54
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    $\begingroup$ @eagle275: There's calculators so nobody needs to know math .. hilarious - but totally wrong view. I agree that that would be ridiculous. And it's not what I said. $\endgroup$ – Ben Crowell Jan 6 at 14:56
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    $\begingroup$ Joined to add this - I was one of those kids that failed at Memorizing multiplication tables, but excelled at higher level math - math only really started to flow for me when we moved from 'this is the answer, memorize it' to adding letters to math problems. I usually do my own breakdown of problems into smaller parts, and add them together to get solutions as well. .. Bravo @BenCrowell $\endgroup$ – Cinderhaze Jan 6 at 18:09
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    $\begingroup$ "We could teach kids to make an initial estimate of the result, multiply, ... and then either decide whether their estimate is good enough or needs to be refined." — This is exactly how long division works. If I need to divide, say, 847 by 13 I can see that 13 does not "go into" 8, so I estimate how many 13 "goes into" 83. Say, I estimated 5. Then I do 13*5 = 65. Not enough. I can fiddle with another estimation or just use 5, and then the algorithm will ensure my calculation is still correct. Kids should be taught how the algorithm works. Calculators will never teach estimation. $\endgroup$ – Rusty Core Jan 6 at 18:50
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    $\begingroup$ There's far more interesting things in math than [short/long] division. $\endgroup$ – Mateen Ulhaq Jan 7 at 11:19
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If you are tutoring, it's important to value whatever algorithms work. Your frustration with the (new to me) lovely algorithm you show concerns me. It shows why each step makes sense, which is much better than the form of long division I learned.

"In fact a lot of kids find it [short division] amazing to learn."

If you show an individual a new process at just the right moment (when it really makes their work easier), it will be more exciting for them, and easier to remember. A tutor who pays attention to their students has more flexibility that way than a teacher who has to give lessons to maybe 30 kids at once.

There is a sensible sequence: after kids get passably good at multiplication, they learn small division problems, then long division, done either as you show above, or stacked, showing all of the steps in a meaningful way. Once kids really understand that, then you show them the beauty of short division.

To get to your main question, though, it could be that short division isn't taught, because the makers of the "Common Core State Standards" may have left it out. [I have mixed opinions about these standards. The 8 standards for mathematical practice are lovely. The individual content standards have many problems. The name seems like an oxmoron to me - if they are common across the nation, then they can't really be state standards, can they?]

I looked at this document, and found this for 6th grade: CCSS.Math.Content.6.NS.B.2 Fluently divide multi-digit numbers using the standard algorithm.

The implication is that there is one standard algorithm. I'd say there are about 3 standard algorithms (or more), counting all of the way to do long division, and then the one way to do short division. This statement, along with a failed search, made me think that perhaps short division is not taught.

Can any long-term elementary teachers here confirm my suspicion?

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    $\begingroup$ I can answer your side question about CCSS being common but state-based. There are federal standards with some different acronym (that I currently forget). Then the each state creates their own CCSS that is at least 95% compliant with those federal standards. So it gives enough flexibility that 46 different stakeholders didn't have to unanimously agree to every facet of mathematical education but enough commonality that New York could accept an 8th grade education taught in Florida at face value. $\endgroup$ – Matthew Daly Jan 6 at 17:50
  • $\begingroup$ They are common because Coleman and Gates wanted them to be. They are state standards, because by law USED cannot impose unified national curriculum onto presumably independent states. The law needs to be changed to do that, but CC became too toxic for the politicians to touch it with a ten-foot pole. OTOH, CC helped bringing back into the limelight 25-year old math programs, once rejected for their watered-down, calculator-heavy group-based approach, but now "aligned with" CC (basically, only the covers of the textbooks were changed). So at least this goal has been achieved. $\endgroup$ – Rusty Core Jan 6 at 18:58
  • $\begingroup$ "Your frustration with the (new to me) lovely algorithm you show concerns me. It shows why each step makes sense, which is much better than the form of long division I learned." Yes! And that's why I love it so much as well. But it appears to get hardly any love around. $\ddot\frown$ $\endgroup$ – Simply Beautiful Art Jan 7 at 2:16
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Not a teacher here, but I noticed when my kids went to school there was far less emphasis than I remember on techniques that require above average insight or intuition. I think there's more pressure these days toward making sure most students achieve a predetermined minimum performance, and less on helping high-performing students stretch their capabilities. A couple of my kids' teachers complained to me that I was teaching them things outside their curriculum. Just anecdotal evidence, but there could be a grain of truth.

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  • $\begingroup$ Hello and welcome to the community! I too feel like this is the case, even though I am still a young one! But even so, I feel like everyone here is overestimating the capability requirements to do short division. Quite literally it follows through the same steps as long division, except it's expected that you can do things like $8\div3=2\rm~R~2$ mentally, which most kids go through before reaching long division anyways. Then you simply have to write those values down. It's not something incredibly convoluted at all. $\endgroup$ – Simply Beautiful Art Jan 7 at 1:35
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Both of my sons are learning division right now (or rather, just finished the section), one in a public school using Eureka Math (in 4th grade level math), one in a Montessori (in Primary, at 1st grade age).

The public school focused on long division certainly, but (either because of the curriculum, or more likely because of the teacher, as I don't remember seeing this in Eureka's curriculum online) they definitely taught something akin to short division as well. In fact, they taught about four different ways to do division - which turned out helpful for my son, who's extremely gifted at math but for some reason struggled with long division. None of them were explicitly short division, certainly.

One method he was taught was a kind of estimation, that seems sort of like what you showed above (and disliked) - but it was extremely helpful for him. Showing a slightly more complex example:

2954 ÷ 16: 16 x 200 = 3200 (200 16's is close to the number) - 320 = 2880 (20 16's less, gets you closer, now a little below) + 64 = 2944 (now we're right there)


200 - 20 + 4 = 184 (r10)

He would do these in a few seconds, and instantly realize the right estimation to make, often faster than I could do the division (and I'm very gifted at mental math myself). Again, he's highly gifted at mental math, so I don't know if this is common for other students or if other students were even taught this (as opposed to him figuring it out), but it works very well for him. It's also not that far from short division in some ways, either, though certainly not the same.

The Montessori of course still uses the "racks and tubes" method, which is basically long division with beads. A great way to teach children I think who don't intuit long division immediately.

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Doing a quick google search for "why do students struggle with division?" made me realize myself why long division is difficult for a lot of students. Common reasons claimed include:

  • The long division algorithm is long. Remembering the steps can be difficult.

  • The long division algorithm may not be intuitive. We can tell them that it's just "how many groups of 5 go into 260" but they don't understand what this actually means, especially with more digits involved.

  • We get lost focusing on the procedure too much. Again, they know we want groups of 5 that go into 260, but they don't understand how the steps get them there.

One site suggested trying out short division, and another suggested trying out estimation. For short division, they claimed that long division can make simple problems become overly complex, such as the $186\div3$ example. The advocate for estimation said that estimation compels the student to think on their own and build their own number sense and intuition.

I also think that if they're taught to multiply 5 by 50 and then subtract the result from 260, they might get overwhelmed by the sizes of the numbers. If they're taught to look at the first digits only, divide, multiply, subtract, and drop down the next digit, the numbers become disconnected and lose meaning.

Estimating helps in the first situation above. It let's students figure out how to avoid getting overwhelmed by all the numbers as well as how to handle larger divisors. It can also involve something similar to the method of adding/subtracting of bridging 10s. A nice example I found was dividing 700 by 20. By first recognizing how many times 20 went into 100 first, the problem can become more intuitive, and this is applicable in life. When pulling out money to pay for something, we often think in larger groups that are more easily managed.

Short division helps in the second situation above. It avoids dropping all the digits down by instead moving the remainders up, and it simplifies the "divide multiply subtract" step down to make students realize that it's just smaller division problems that they can do easily. By emphasizing these steps as one whole step, the process can begin to make more sense.

In my opinion, I also find short division intuitive as well. It gives meaning to the usage of remainders on small numbers when going to larger numbers, since we're taking out large groups, and then splitting up the remainders. A nice example I found online was splitting 103 skittles between 4 people. Have students try to do this, and then step in and suggest sharing groups of 10 skittles at a time. How many skittles are left after all the groups of 10 are shared? This is generally what all division algorithms boil down to.

Is short division faster? Maybe. In my experience, yes for most. A lot of students are capable of thinking in the larger step of "how many times does it go in and what's the remainder?", when the quotient and divisor are less than 10. If they can do that, then multiplying it out and subtracting no longer becomes necessary to explicitly write out. And it certainly saves space, but that shouldn't be the concern to begin with.

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    $\begingroup$ The problem is that many people only manage to think up to the tip of their nose. I was shopping a while back, when the cash register "broke" - the customer in front had 37 bottles he wanted to return for his deposit (0,25 per bottle ) and both customer and shop clerk were unable to proceed from there. After a short time I asked what the problem was in giving the customer his 9.25 deposit back. Now both asked how I calculated that and I replied "you can go both ways: 36 plus 1 bottle - or 40 minus 3 bottles" - then both "That way it indeed is simple" But most people are unable to use the brain $\endgroup$ – eagle275 Jan 7 at 15:23
  • $\begingroup$ That is exactly what we are trying to get at. A lengthy division algorithm presented to an 8 year old is anything other than simple to them. The point is that hopefully by presenting it differently they may actually understand how to divide, instead of mindlessly following a procedure. $\endgroup$ – Simply Beautiful Art Jan 7 at 15:45
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Long division is a useful thing to teach because at some point, polynomial long division is going to be something that someone is going to want to teach them, and it helps if they've seen something similar before. Thus, we don't want to replace it with anything.

The question, then, becomes whether we want to add a second algorithm, and I can't see any particularly good reason to do so (and it needs to be a particularly good reason, given the limited availability of time), and indeed several reasons not to:

  1. As you yourself demonstrate, capable students with a strong number sense will pick it up naturally by just skipping steps, so will benefit little from being taught it explicitly, whereas instead investing that time into developing that number sense will place more people into this category.
  2. For those students without such a strong number sense, teaching them two different algorithms won't magically give them the strong number sense needed for short division to actually be practically useful. The student from your first example would still not see that 60 and 3 are the relevant factors, regardless of the algorithm in question.
  3. If we did want to invest time into teaching fast and efficient mental arithmetic algorithms, there are many higher-priority algorithms to teach before short division (addition/subtraction/multiplication is more common, anzan-based algorithms are faster).
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