Long division: why fourth graders fall apart, and how to teach it at home

Long division is where many children's confidence in math breaks, and the reason is simple: the written method is a ritual (divide, multiply, subtract, bring down) stacked on top of three things the child has to own first: what division means, the multiplication facts, and a sense of how big the answer should be. A fourth grader who is missing any one of those can't learn the ritual, and a fourth grader who learns the ritual without them forgets it by June. The standards most states use know this; they don't ask for the standard long-division algorithm until sixth grade. Fourth and fifth graders are meant to divide with place-value strategies, and the best of those, partial quotients, is the one to teach at home.

When are kids supposed to learn long division?

Later than most parents think. The standards most states use set it out in three steps. Third graders learn what division is: 56 ÷ 8 is "how many in each of 8 equal shares," or "how many shares of 8," and division is an unknown-factor problem (8 × ? = 56). Fourth graders divide up to four-digit numbers by one-digit numbers "using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division," and illustrate the calculation with equations, arrays or area models. Fifth graders do the same with two-digit divisors. The standard algorithm, the one you learned, appears in sixth grade: "fluently divide multi-digit numbers using the standard algorithm."

So a fourth grader who can't do the vertical method hasn't missed a fourth-grade target. A fourth grader who can't say what 56 ÷ 8 means, or who has no way to divide 96 by 4, has, and those are the gaps to fill.

Why does long division matter so much?

Because it's the point where arithmetic becomes the math that comes later. In a 2012 study that followed 599 American children from ages 10–12 to 15–17 and 3,677 British children from 10 to 16, how well children understood fractions and division at age 10 predicted their algebra and overall math achievement in high school, after accounting for parents' education and income, IQ, reading, working memory and whole-number addition, subtraction and multiplication. The study's lead author, Robert Siegler, said the message was that "we need to improve instruction in long division and fractions." Division is the operation that produces fractions, ratios, rates and decimals; a child who can divide with understanding has a foothold on all of them.

Why does my child fall apart at it?

Four causes, usually in combination. Find yours before you teach anything.

The facts aren't automatic. Every step of dividing 672 by 8 asks "how many 8s in this?" A child who has to count up by 8s to answer runs out of attention before the second step. Division is multiplication facts, used backwards, under load. If 7 × 8 takes more than a couple of seconds, start with the times-tables routine and come back; it's the most common cause and the easiest to fix.

Division doesn't mean anything yet. A child who sees 56 ÷ 8 as a symbol to react to, rather than as 56 things shared among 8 or 56 counted in 8s, has nothing to check an answer against. The third-grade meaning has to be there.

No sense of size. "672 ÷ 8 is about 80, because 8 × 80 is 640." A child who can say that catches a wrong answer of 9 or 840 instantly. A child who can't will write whatever the ritual produces.

The ritual itself. Divide-multiply-subtract-bring-down is a sequence of four unexplained moves, done in a specific layout, with the first digit treated differently from the rest. Taught first, it's memorized; memorized, it's fragile. The standards delay it for exactly this reason.

How do I teach division at home?

Five steps. Each builds on the last; don't skip to the written method because it's familiar.

11. Make it mean something. Share real things. 24 crackers for 4 people: deal them out, one each, around and around. "How many each?" Then the other meaning: 24 crackers in bags of 4. "How many bags?" Say both out loud. Write 24 ÷ 4 = 6 under each, and 4 × 6 = 24 next to it; division is the question "4 times what?"

22. Connect it to the facts. For a few days, every division question is a multiplication question asked backwards: "56 ÷ 8? Eight times what is 56?" This is the third-grade standard, and it is the whole of single-digit division once the facts are there.

33. Estimate first, every time. Before any method: "About how many?" For 672 ÷ 8: "8 times 50 is 400, 8 times 100 is 800, so somewhere between 50 and 100, closer to 100." A child with this habit has a check for every answer the rest of their life.

44. Teach partial quotients (the "box" method). Write 672 with 8 on the outside, as usual, but instead of working digit by digit, take easy chunks of the answer. "How many 8s can I take out of 672 easily? 50 of them; that's 400, leaving 272. Another 30? That's 240, leaving 32. Four more 8s is 32, leaving 0." Write 50, 30 and 4 down the side and add them: 84. The chunks can be any size; a child who takes 10 at a time gets the same answer with more steps, which is fine. This is the method the fourth-grade standards describe: strategies based on place value and the link to multiplication, shown with equations or a model. It never breaks, it uses facts the child has (8 × 50, 8 × 10), and the child can see what each number means.

55. Then show the standard method as a shortcut. Once partial quotients are easy, show the vertical method as the same thing with the chunks forced to be the biggest possible at each place: 8 into 6 doesn't go, 8 into 67 goes 8 times (that's the 80), and so on. Children who've done partial quotients first see the standard method as a compression of something they understand; children who haven't see a ritual. The standards give until sixth grade to be fluent with it. Use the time.

Here's what step 4 sounds like with a nine-year-old on 96 ÷ 4, after step 3's estimate of "more than 20, because 4 × 20 is 80":

• Parent"How many 4s can you take out of 96 easily?"
• Child"Twenty. That's 80."
• Parent"What's left?"
• Child"16."
• Parent"How many 4s in 16?"
• Child"Four."
• Parent"So how many 4s altogether?"
• Child"Twenty and four. 24."
• Parent"Check it."
• Child"4 times 24... 80 and 16. 96."

No bring-down, no mystery, and the check is built in.

If you'd like the meaning step to come with pictures, Mathwell's division lessons build it from rectangles: 24 ÷ 4 as a rectangle with 24 squares and 4 rows, so the quotient is something the child can see, and when the child is stuck, the Tutor offers the smaller step ("how many 4s make 80?") rather than the answer. Mathwell is a browser-based K–5 math tutor that reads every problem aloud and teaches with hints instead of answers; anyone can try the demo lesson without signing up, and after that there's a 7-day free trial with a card on file, then $19 a month, $99 a year, or $169 for a three-year K–5 pass. It's for kindergarten through fifth grade only, and the child answers by tapping or typing, not by speaking. The crackers come first.

What about remainders?

Teach them with the crackers too: 25 crackers for 4 people is 6 each with 1 left over, and what happens to the 1 depends on the story (cut it up, leave it, give it to the dog). Children who meet remainders as "what's left after sharing" handle "R1" on paper without trouble, and are ready, in fifth grade, for the idea that the leftover can be written as a fraction (6 and 1/4). A child taught "R1" as a rule first often can't say what the 1 is.

When should I worry?

In fourth grade, trouble with the vertical method is not a worry; the standards don't ask for it. Trouble with the meaning (can't say what 20 ÷ 5 asks), with the facts (counts up by 5s for every fact), or with size ("20 ÷ 5 is 100" and no flinch) is worth telling the teacher, in those words, so the help goes to the right place. In fifth grade, a child who still can't divide 96 by 4 any way at all needs steps 1 to 4 above, two or three weeks of ten-minute sessions, and usually the times tables first.