My son's new way of doing addition

Beth,

I wouldn’t have him change a thing, as long as he knows how to do it from right to left for the sake of a strict teacher. Actually, many people who are particularly adept in math tend to add from left to right, especially when they are doing it in their heads. A lot of math books do teach it that way, as an alternative, when they teach doing it in their heads. It’s actually a much quicker method - try it!

Wow. The first thing that needs to be said is your son is bright to have figured that out. I’m impressed.

Here’s what I’d do. I’d heap praise on this wonderful child for his ingenuity but then tell him,
“you know what? Even though you get the right answer this way now, it’s not always going to work. The numbers will get bigger and bigger! (Show him some big numbers) and doing in the way you’re doing it, will make it take toooo long when you’re adding those really big numbers.

So, let’s try it this way. You’ll still get the right answer and it’s a shortcut way to do the adding even if it doesn’t feel like a shortcut at first.”

Then show him the right way. Since he has visual spatial problems you might lay a piece of paper over the 10s column at first so he has to look down the 1s column.

What a clever child!

Susan is right- what he is doing is an alternative algorythm and it is perfectly legitimate. It will be a bit cumbersome as he gets into larger numbers, but not much really. Sounds to me like he might be working his way around traditional regrouping:) As long as he has a clear understanding of place value- and this is really the essential concept- he should be fine.

There are actually several multiplication algorythms that work the same way, so I don’t know that I would worry too much about that either. The one that corresponds directly to what he is doing with addition can take up a lot of space on the paper which is a disadvantage, but it does work and it keeps kids from forgetting the place value as they work through the problem. He can also use something called a lattice, which is based on Napiers Bones- one of those “out there” things in math that always works, is fairly efficient and is only beginning to be widely taught. It is harder to explain than it is to do- but I will try if you like- let me know.

I would be more interested to see what what he does with subtraction:) especially when he has to regroup. While there is nothing inherently wrong with working from left to right in a math problem as long as his conceptual understanding is good- including the all important place value- it is hard for adults who have been taught traditional methods to look at. However, the fact that they are traditional and comfortable for us doesn’t make them the only way- or even the most effective way.What does his teacher say about it?

Robin

Thanks for your responses. I have to admit my reaction wasn’t “a clever child” but rather “Oh no, something else we have to work on.” So you collectively helped reorient me. I think my son has more math ability than I and it never occurred to me that what he was doing was anything but “the wrong way”, even though he was getting the right answer. He does do amazing things in his head and I think you are right about him finding a way to do what he does in his head on paper.

I know this is not the way his teacher taught him, but I don’t know what he does at school. His sp. ed teacher is pretty dogmatic and doesn’t communicate with us except to show up at IEP meetings. (Our goal is to improve his reading and visual-spatial abilities enough that we can get him out of resource room—his conceputal skills are several years above grade level). Anyway, I am sure she would not be impressed. I was wondering about bringing it to her attention but think I will just experiment and see what he can do in terms of larger numbers and subtraction with regrouping.

Robin,

I would love to hear more about this “lattice” and the alternative multiplication algorithms. My gt/ld son tests as very superior in math concepts but only low average in calculation. He often has difficulty employing the usual methods of calculating, does everything in his head and frequently is unable to show his work or explain how he arrived at an answer. He tends to invent his own ways of calculating, some of which work and some of which don’t. I’d be very interested in exploring alternatives that might work for him. Thanks!

Andrea

The lattice is actually quite cool because it looks the same for every problem and the organization is the same- you don’t have to remember to add zeroes with every place value switch like you do with more standard methods. It looks a bit like a multiplication table. Here goes:

45 * 64

you have two, two digit numbers to multiply. Draw a 2*2 grid. ( I can’t do that here:) with diagonals drawn from the right corner to the left corner of each box and extending out a bit from the bottom left corner of the outside left edges. We use 1/2 in graph paper while kis are learning how to draw the grids.

Across the top of the grid, one digit over each box, write 45. Down the right hand side, one digit beside a box, put 64. You should now have what looks like a multiplication grid with diagonals. (imagine the grid here) Remember- the digits are outside the box.

4 5
6
4

Then you just multiply, placing the tens digit above the diagonal in each box and the ones digit below. 5*6= 30 - 3 above and 0 below, 5*4= 20- 2 above and 0 below, 4*6 = 24- 2 above and 4 below, 4*4= 16, 1 above and 6 below.

The numbers in each diagonal are added- that’s why you extend the lines. Start from the lower right and work upwards toward the top left. So- under the lower right corner is a zero. Write that outside the grid.

The next diagonal has a 0+2+6= 8 (one the first line from the right)
Next 3+4+1=8

2

it looks kind of like this 2
8
8 0

Then you read around from top right to bottom left 2880. That is the answer.

This is really hard to explain without being able to draw:( Hope I wasn’t too confusing.

For larger multiples, you change the dimensions of the grid. So- 3 digits * 2 digits would have a three by two grid, etc.If you have to regroup when you are adding- you carry it up to the next diagonal.

If this is confusing- let me know and I will email you an explanation with pictures:) It is quite easy but it is so visual that words are hard.

Robin

Robin,

I think I need a picture :)

Andrea

Dr. Math not only has a wonderful explanation of how the Lattice method works, but also tells you WHY it works!

Dr. Math is the kind of site you want to bookmark and consult regularly. Her archives are a goldmine!

Jenny

http://navigation.helper.realnames.com/framer/1/113/default.asp?realname=Dr%2E+Math&url=http%3A%2F%2Fforum%2Eswarthmore%2Eedu%2Fdr%2Emath&frameid=1&providerid=113&uid=30008485

If this URL doesn’t work properly, try a search for Dr. Math, and then search the site for “lattice multiplication”. It’s worth the effort.

BTW, whatever happened to the ability to paste in hotlinks on this site? :-(

Thanks, Jenny! What a great site! My kids always want to know why things work the way they do and this should help me answer their questions.

Andrea

I don’t need a picture, your words described it for me very well, and I thank you for your effort. In fact I am going to show this method of multiplication to my fourteen yr old son who is still having difficulty with math in general, especially the fundamentals. He is considered to be nonverbal LD, and may need a different approach to multiplication.

Ann

You are very welcome- I hope it “takes” for him!

Robin

Robin
Please email off the discussion board with a visual regarding the lattice. I tutor many students that could use this, but with just word explanations it appaers to be quite confusing for them.
I am always interested in learning new math techniques. So many of my kids have such problenms in math!
Thanks Judy

email to [email protected]

Judy,

The Dr. Math site also provides visuals along with the explanations if you find you are having trouble getting them via e-mail from Robin.

Jenny

I add this way, left to right, mentally myself, and it’s a good way to do things mentally, and mathematically absolutely correct. In fact it’s better for estimation, because the most important digits come first. He should keep this method as a tool for quick mental arithmetic.

Two things do worry me. You say he still counts the ones on his fingers. Counting on fingers is slow and cumbersome and a real nuisance when you run out of fingers (7 + 8 anyone?). Finger-counting tends to be very error-prone as you forget which finger is counted and which the counter. Finger-counters hit a brick wall when numbers get large (the same kind of problem as word-memorizers in upper elementary reading) and they have to do an awful lot of backtracking with the usual fights and frustration.
The other is the person replying who mentioned the pattern of “does wonderfully in his head but doesn’t like to put steps on paper”. Math teachers get very familiar with this, and we react to it the same way as reading teachers do to the kid who can “read” very well in his own book but somehow can’t read any other books — we try very hard to explain that there is a *vital process* missing here, and the sooner you do the real work, the sooner you will get somewhere; the longer you put it off and refuse to try something different, the longer you will stay stalled in the same place and the worse your failure will get.

Your son needs to learn a systematic way of dealing with numbers. The standard algorithms became standard because they are both correct and efficient.

The lattice system is perfectly logical and correct, but it is a problem to have to explain it to his new teacher every year, especially if his teacher is one of the mathematically untalented who works by rote, whose minds are therefore set in concrete — and he’s guaranteed to get a couple of those in his school career.

Invented algorithms tend to either be slow or error-full or often both.

Have a care with judging mathematical correctness by the right answer. Yes, in Grade 1, there is only one step and the answer is right or not. Later, we try to teach higher-level thinking skills, and the purpose of an exercise is often to illustrate a particular problem-solving method, NOT to get the answer. Heck, the answers are in the back of the book, and if we just want answers, we can photocopy the page faster and more neatly than handwriting. Many a middle-school algebra student has been set on the road to failure by having been trained to be too focused on answers, and not understanding the question.

What your son is doing in inventing his own logic is actually a positive step towards this higher-order problem-solving. But he still has to be able to write his logic down and communicate it to someone else in a language they understand, and that is where standard systems — like standard handwriting and spelling — become important.