My child subtracts the small number from the big one, no matter where it is. How to fix subtraction with regrouping

When a child writes 42 − 17 = 35, they are not being careless. They are following a rule, "you can't take 7 from 2, so take 2 from 7," and that rule has worked on every subtraction problem they met before this one. Researchers who studied thousands of children's subtraction errors found that mistakes like this are systematic: the child's procedure hits a wall, and the child patches it with something sensible. Telling them to "remember to borrow" doesn't fix a patch; it adds a second rule they don't understand to the first one. What fixes it is making a ten real, with a bundle of ten straws or ten pennies traded for a dime, so that 42 can be seen as 3 tens and 12 ones, and "borrowing" becomes "trading," which a child can see the reason for.

Why does my child subtract the small number from the big one?

Because until regrouping arrived, it was always right. In 48 − 23, 59 − 14, and every problem in first grade, the bottom digit was smaller than the top one in each column. A child who learned "subtract the digits" learned a procedure that never failed. Then 42 − 17 shows up, the procedure says "2 take away 7," and the child hits what the researchers John Seely Brown and Kurt VanLehn called an impasse: a step the child believes can't be done. Their theory, built from large collections of children's subtraction work, is that a child at an impasse doesn't stop; they repair the procedure with the nearest sensible move, and one of the nearest is to swap the digits and subtract anyway. The result looks like a mistake and behaves like a rule: it produces the same wrong answers every time, which is how you can tell it from a slip.

Two things follow. First, the child is reasoning, which is good news, because a child who reasons can be shown a better rule. Second, the error won't go away with reminders, because it isn't forgetting. It goes away when the child understands why 42 can be rewritten as 30 and 12, and that understanding has a name: place value.

What are second and third graders expected to do?

The standards most states use are specific here, and worth reading before you teach, because they don't say what many of us learned. Second graders are expected to subtract within 100 fluently "using strategies based on place value," and to subtract within 1,000 "using concrete models or drawings," understanding that "sometimes it is necessary to compose or decompose tens or hundreds." They're also expected to "explain why addition and subtraction strategies work." The standard written method, with the crossed-out digits, isn't expected to be fluent until fourth grade.

So a second grader who can't do the crossed-out-digit routine isn't behind. A second grader who doesn't know that 42 is four tens and two ones is missing the thing the routine stands on, and that's where the work is.

How do I fix it at home?

Seven steps. Most families get through them in two or three weeks of ten-minute sessions.

11. Check place value first. Ask: "What's 42? How many tens, how many ones?" If the answer is quick ("four tens and two"), move on. If it isn't, stop here and build it: bundles of ten straws (or coffee stirrers, or craft sticks) with a rubber band, and loose ones. 42 is four bundles and two loose. Build 36, 50, 17. A child who can't see 42 as four tens and two will never see why it can also be three tens and twelve; how the teen numbers teach place value is the kindergarten version of the same idea, and sometimes that's where the gap is.

22. Make the ten real. The federal What Works Clearinghouse guide for elementary students who struggle with math makes concrete representations one of its six strongly supported recommendations, and this is the topic they were made for. Bundles of ten and loose ones are enough. Dimes and pennies work too, with the extra benefit that a dime is obviously a ten and obviously not ten things.

33. Do 42 − 17 with the stuff, slowly. Lay out four bundles and two loose. "Take away 7 loose ones. Can we?" (No, there are only 2.) "What can we do?" Let the child find it: open a bundle. Now there are three bundles and twelve loose. "Now take 7." Five loose left. "Take away 1 bundle." Two bundles left. "What's left?" Two tens and five: 25. Say the whole story back: "42 is the same as 30 and 12, and 30 and 12 take away 17 is 25."

44. Say "trade," not "borrow." Nobody gives the ten back. The child traded one ten for ten ones, and the number didn't change. "Regroup" is the school word; "trade" is the one children understand first. The word matters because "borrow" makes the crossed-out digits look like magic, and "trade" makes them bookkeeping.

55. Write it while you do it. The second time through, keep the paper next to the straws. When the child opens a bundle, cross out the 4 and write 3 above it, and write the little 1 next to the 2 to make 12, and say what each mark means: "that's the bundle we opened; that's where its ten ones went." The notation stops being a ritual the day the child sees that each mark records something they just did with their hands.

66. Estimate first, every time. "42 take away 17: that's about 40 take away 20, so about 20." Then solve. A child who has estimated 20 and gets 35 knows something went wrong, which is the habit that catches the small-from-large rule on its own. Add it back to check: 25 + 17 should be 42.

77. Give them a second route. Subtraction is also distance. On a number line, from 17 up to 20 is 3, 20 to 40 is 20, 40 to 42 is 2: 25. The same guide puts the number line among its strong recommendations, and the standards expect children to use "the relationship between addition and subtraction." A child with two routes can check one against the other, and some children find counting up far easier than regrouping for years.

Here's what step 3 sounds like with an eight-year-old who has been writing 35:

• Parent"Here's 42: four bundles, two loose. Take away 7 loose ones."
• Child"There's only 2."
• Parent"So what can we do?"
• Child"Take 2 from the 7?"
• Parent"Where would the other 5 come from? Look at the bundles."
• Child"Open one?" (Takes the band off.) "Now there's... 12 loose."
• Parent"And how many bundles?"
• Child"Three. Take 7 from 12 is 5. Take one bundle is... 2 bundles. 25."
• Parent"Is 42 still 42 when it's three bundles and twelve?"
• Child"Yeah. It's just opened."

That last answer is the whole lesson.

This is also the picture Mathwell uses for regrouping: ones that gather into a ten you can see, rather than a small 1 that appears on the page by rule, and when a child is stuck, the Tutor offers the smaller step ("can you take 7 from 2? what could you trade?") 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. A rubber band and ten straws remain the best version of the lesson.

What about subtracting across a zero?

40 − 17 and 302 − 145 produce their own family of bugs, and the same researchers catalogued them: children who change the 0 to 10 without taking a ten from the next column, children who stop at the zero, children who skip it. The cause is the same impasse, one level up: "I need to open a ten, but there are no tens." With the stuff, the answer is visible. For 40, there are four bundles and no loose ones; to take 7, you open a bundle, and now there are three bundles and ten loose. For 302, you'd have to open a hundred to get tens before you can open a ten to get ones, and a child who has done it with bundles of a hundred (ten bundles of ten in a bag) does it on paper without the usual confusion. Don't introduce zero problems until the plain regrouping is solid.

Should I teach it the way I learned it?

If the way you learned it was "borrow one, cross it out, put a one next to the two," teach the trade with straws first and the notation second, and you'll be teaching the same method with the reason attached.

The crossed-out digits are the standard method, and your child will use them; the standards simply put understanding first and fluency with the written form in fourth grade. What to avoid is the sentence "you can't take a bigger number from a smaller one," which is false (they'll learn negative numbers later) and which is, incidentally, the exact belief that produces the small-from-large bug.

If the problem arrives as a story ("Maya had 42 stickers and gave away 17"), the story may be the obstacle rather than the arithmetic; the retell-and-act-it-out routine for word problems sorts out which.

When should I worry?

In second grade, not at all; this is the year the error is supposed to appear and be fixed. In third grade, a child who still swaps digits after a few weeks of work with bundles usually has a place-value gap underneath, and that's what to tell the teacher: "she's not sure 42 is four tens and two." In fourth grade, when the written method is expected to be fluent, a child still at the impasse needs the bundles again, not more worksheets, and probably a conversation about whether the facts within 20 are automatic, since a child who has to count 12 − 7 on their fingers has no attention left for the regrouping.