Last week, I attended Carole Fullerton‘s parent presentation. She discussed strategies students have for adding two-digit numbers. Carole’s timing was great since I’ve been having similar discussions with teachers in recent weeks.

How many ways can you add 59 + 37?

The most common strategy that I see students use is to add the tens, add the ones, and then combine. Students working with ten frames naturally begin by grouping the 10’s, not the 1’s, together.

Students find other strategies. For example,

- Add 1 to 59 to make 60. Take 1 away from 37 to make 36. 60 and 36 is 96. (make ten)
- Add 1 to 59 to make 60. Add 3 to 37 to make 40. 60 and 40 is 100. Take the extra 4 away. (friendly numbers and compensation)
- 30 more than 59 is 89. 7 more than 89 is 96. (add on)

These are the strategies I use to compute mentally. On paper, I fall back to the traditional right-to-left digit algorithm. It’s the result of performing thousands of such calculations in elementary school.

Students should be encouraged to write their mental math strategies down on paper. Some students will have to.

Teachers and parents appreciate these strategies. They make sense. Teachers and parents want mathematics to make sense to their kids. But at some point they always ask *the* question: “When should they learn the traditional/regular/real way?” They ask this because they are concerned their kids will not be prepared. “But do these strategies work for three-digit addition?” Yes.

“Relax. This will look familiar,” Carole joked. The same natural left-to-right strategy described above can be written vertically. We start with 50 and 30 is 80. Already, we know the sum is greater than 80.

Compare that with the traditional algorithm. We start with 9 and 7 is 16. We know the sum will have a 6 in the one’s place.

Which piece of information is more important? Carole made the point that accuracy is important. Always was, always will be. But it’s not where we should start. Strategies should be built on conceptual understanding. The emphasis of a left-to-right strategy is on number whereas the emphasis of a right-to-left strategy is on digits.

In her new blog, Amy Newman also writes about this. As well, she shares Carole’s key messages for parents helping children at home.

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OK. Dig this. The choice between (1) continuing to go left to right the way you do when you’re using mental math strategies, and (2) abstract efficiency, is a false one. There exists an efficient left-to-right algorithm!

I use it to challenge my preservice elementary teachers on their assumptions about what it means to understand an operation or an algorithm. They tend to claim that they understand the standard algorithm because they can do it, but then they claim that the crossing out in the sum (that you see in the videos I linked to) “doesn’t make any sense” and “will confuse kids”. But that crossing out step is no less sensical than carrying. It’s just less familiar.

Thanks for sharing this. It’s great to have ways to challenge the belief that “I can do it = I understand it”. As you said, it’s possible to think about place value while doing the standard algorithm, but typically we don’t. Ask most to explain why we “carry the one” and few will think of it as representing ten tens in the hundreds column.

Really like this post, Chris. Thank you for sharing.

Most natural way, IMO, is context dependent. In the given example, 59+37 = 60+37-1 = 97-1 = 96. Ta da!!

Another nice multiplication example, a favorite of Gauss, is 13 x 19 = 13 x 20 – 13 = 260 – 10 – 3 = 250 -3 = 247.

Subtraction is even easier: 62 – 29 -> 1 step to 30, 30 more to 60 and 2 more = 33 steps.

500 – 45 -> 5 steps to 50, 50 more to 100 , 400 more to 500 = 455 (try doing 500 – 45 with kids by “borrowing” ).

“Most natural way, IMO, is context dependent.”

Agreed. Flexibility is key. Carole talked about the need to have a second strategy in the back pocket. Your strategy is similar to the first two in the bulleted list. It’s probably how I would naturally go about it as well. There are others, but can you think of any contexts where adding the units first is natural? I can’t.

Carole also talked about subtraction and presented examples similar to yours. Thanks for including it in your comment. It makes much more sense than borrowing, doesn’t it? While this strategy wasn’t new to me, Carole’s take on it was. She said, “The brain craves forward motion.” (Carole briefly commented on this on Amy’s blog.)

The shift in how students learn mathematics is happening. You make several important connections. Thanks for taking the time to write and share these important ideas.

I’ve been collecting fast (or mental) arithmetic tricks here

http://www.cut-the-knot.org/arithmetic/rapid/rapid.shtml

I think a huge advantage of of the left-right algorithm is there is less “magic” happening, and it makes for a smoother transition to having students re-grouping brackets using the associative law when they’re introduced to algebra.