Is short division taught these days and if not, why not?

Question

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.

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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?

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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.

Answer 1

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.

Answer 2

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 ways 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?

Answer 3

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.

Answer 4

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).

Answer 5

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)

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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.

Answer 6

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.

Answer 7

I know I am late to the party, but I absolutely hate short division. It works for simple division problems, for those that you can almost do in your head. But try dividing 35/43 or 692/37. This is where most students either slow WAY down or get confused and frustrated. A really good example is when you are trying to convert 1/8 into a fraction or percentage. This is what long division is really great for. It takes all of the math that you have to do in your head and puts it onto the paper, making it easier to understand what is going on and what needs to happen next. I am all for teaching short division as a stepping stone for long division, but long division needs to be a priority. I have lost count of how many students I have taught long division to because they got stuck with short division and a more complicated division problem.

Answer 8

One of the benefits of the partial quotients division method as illustrated above with 186 / 3 is NOT in dividing by single digit divisors but in multi-digit divisors. Students are generally taught the bus stop method when dividing by a single digit (which makes sense). But when you start dealing with multiple digit divisors this approach can help students to see that division is really just a process of subtractions. They do not need to be accurate in the multiples they use (as required by the traditional long division method) as this approach is more flexible and can limit the multiples to just 1x 2x & 5x (as these are the factors of 10) and focus on reducing the remainder. Students should be taught that fractions & division are very similar as it may be possible to simplify the division problem by reducing both dividend & divisor by the same number just like simplifying a fraction. As long division is fraught with difficulty it puzzles me why student are not taught to check their long division answers by using Sum of the Digits: (divisor x quotient) + remainder = dividend.