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I am tutoring a Grade 2 girl in arithmetic. She has demonstrated an ability to add two-digit numbers with carrying. For example:

$$\;\;14\\ +27\\ =41$$

I asked her to write this out horizontally, and this is what she produced.

$$12+47=41$$

She evidently is failing to see the numbers and is confounding the vertical addition of the digits with the horizontal reading of the numbers.

With practice, and prompting, she is able to get this right, but it seems as though she sees the sum as a matrix of four digits, and is missing the numbers.

Any insights on how to help her?

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    $\begingroup$ Start her by adding the words "tens" and "ones" near every appropriate digit. So her rendering can be read as one "tens" and two "tens" plus 4 "ones" and 7 "ones" equals (or gives) 4 "tens" and 1 "ones". Then ask what people might mean by 78? 7 "ones" and 8 "tens"? If she can handle the mechanics of carry, she might get the abstraction of left-to-right numerical representation. Your "evidently" sentence is one I question. Gerhard "Failure Backwards is Success Forwards?" Paseman, 2015.10.01 $\endgroup$ – Gerhard Paseman Oct 1 '15 at 17:33
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    $\begingroup$ Have you listened to her count? For example: Try asking her to count up to 25. If she is chanting the correct number names: Can she write them down, too? Ask her if she can write down the number that you say. Then say, e.g., "12." Can she write it? If she can, tell her you are going to say two numbers for her to write down. Then say, e.g., "12 and 47." Can she write them both? What about "12 plus 47"? Can she write this down? Can she write it down horizontally first and vertically second? $\endgroup$ – Benjamin Dickman Oct 2 '15 at 2:18
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    $\begingroup$ One more thought [for now]: Can she vertically compute $14 + 9$? If so: How would she write this ($14 + 9 = 23$) horizontally? $\endgroup$ – Benjamin Dickman Oct 2 '15 at 2:33
  • $\begingroup$ I don't think I'm understanding this. In my opinion the student is demonstrating a lack of understanding of place value. The fact they can add in columns is a red herring as a full understanding of place value is unecesary for the algorithm. $\endgroup$ – Karl Oct 2 '15 at 20:14
  • $\begingroup$ She definitely doesn't understand place value yet. She's learned the mechanics of addition. $\endgroup$ – Adam Hrankowski Oct 2 '15 at 20:45
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Does she have difficulty with reading or other taking in other visual information? Her problem might have nothing to do with understanding numbers and everything to do with a learning difference in how she perceives visual information. I've had students who could answer questions but couldn't handle a worksheet (with the same questions) if it was organized in an unusual way. You might check with the child's teacher to see if this is an issue and/or if she has an IEP (Individualised Education Plan).

Note that the reason that I suspect this, is that she seems to understand the numbers when you question her. I suggest you rule out learning issues and continue to reinforce the different formats using one and tens as Gerhard suggested in his comment.

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  • $\begingroup$ I wondered about that as my son had a similar problem learning how to read. He was tested by an optometrist and he was shown to have problems tracking the printed letters. We treated it with eye exercises. This little girl seems to be reading fine. I'm wondering if she's young for her grade level. She's shown some improvement, and maybe it's all a little much for her. $\endgroup$ – Adam Hrankowski Oct 2 '15 at 1:48
  • $\begingroup$ I've known students who read fine, but still have trouble with information that's organized in a different visual way, so it's still worth thinking about. $\endgroup$ – Amy B Oct 2 '15 at 11:13
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    $\begingroup$ If Grade 2 is like second grade in U.S., AND she is not yet 7, I would say you have someone ahead of the curve, and that her perspective should not be "forced" into seeing your perspective, so much as help her understand why others have your perspective. Not that I am accusing the original poster of bad form or method, just presumption. I have a gut feeling that your and my understanding of numbers isn't necessarily better than hers. Gerhard "What Else Does Horizontal Mean?" Paseman, 2015.10.02 $\endgroup$ – Gerhard Paseman Oct 2 '15 at 15:59
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She evidently is failing to see the numbers and is confounding the vertical addition of the digits with the horizontal reading of the numbers.

With practice, and prompting, she is able to get this right, but it seems as though she sees the sum as a matrix of four digits, and is missing the numbers.

That may indeed be what she is doing, but there is an alternative hypothesis to consider, which is that she sees and interprets the arrangement of the digits perfectly well, but does not understand what you mean when you ask her to write the problem horizontally.

A good way to probe what is going on is to ask her to read aloud the original problem. If she reads it as "fourteen plus twenty-seven equals forty-one" then that means she is correctly parsing the vertically-written sum, and suggests that she just doesn't understand what you are asking her to do. In that case, try revoicing her answer back to her, and ask something like "Could you write fourteen plus twenty-seven equals forty-one on one line?"

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I agree that you should ask her to read the orginal problem aloud. Does she understand that the problem reads "fourteen plus twenty-seven?" If you write a problem both "horizontally" and "vertically" and then ask her to read both versions aloud, does she read them the same? Can she demonstrate that problem with unifex cubes or on a number line? Can she explain in some other way why 14 + 27 = 41?

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