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I’ve been teaching my kids addition tables (1+3=4, 2+3=5, 3+3=6, etc.)

I only just found out about number bonds (1+4=5, 2+3=5, 4+1=5). This seems a better method because it’s mastering all the components of a single number, rather than mastering what a single number like 2 will do to all the other numbers. Am I right? Or should I teach rote memorisation of both?

My 5 year old seems pretty good at problem solving but I’d like him to improve his rote memory. He actually enjoys it so I’m not taking time away from focusing on understanding.

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    $\begingroup$ To avoid being too opinion-based, you might want to rephrase your main question to something like "Are there any studies that show that, when teaching addition, it is more effective to use the concept of number bonds rather than to use rote memorization?" $\endgroup$ – Joel Reyes Noche Jul 3 at 11:01
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    $\begingroup$ Mat I ask what country you are from? It appears this phrase is used more commonly in the UK and Singapore. $\endgroup$ – JoeTaxpayer Jul 3 at 13:09
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    $\begingroup$ I think number bonds complement addition tables but, alone, they are nearly useless. For example to find $14+17$, you could add $10+10=20$ and $4+7=11$ then $20+11=31$. Or we could use number bonds: the $4$ in $14$ needs a $6$ to become $20$; take this $6$ from $17$ we get $11$; add it to our $20$ we get $31$. $\endgroup$ – Paracosmiste Jul 3 at 14:44
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    $\begingroup$ It is also useful in subtraction. To find $31-14$, I remember that $14$ needs a $6$ to become $20$ so I remove $20$ from $31$ and then add $6$. $\endgroup$ – Paracosmiste Jul 3 at 14:46
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    $\begingroup$ I have seen the number bonds for a sum of 10 more often, and these are very useful. $\endgroup$ – Sue VanHattum Jul 4 at 15:43
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My sense is that the formalism of "number bonds" is a bad idea. Here are some reasons.

  1. At this level there is no need for new jargon and little good is accomplished by introducing it. It only creates potential new confusion. "Number bond" is particularly terrible terminology, as children are very unlikely to be familiar with the word "bond", whose usual meanings have, moreover, little to do with addition. Additionally, there is already well established terminology for these "bonds" - they are (ordered) partitions of the integer into two parts. It is never a good idea to invent new terminology for well established notions. "Partition" is better terminology because it reflects the (easily explained to a child) idea of representing the number by a linear series of dots and dividing the series somewhere.

  2. While it's quite reasonable to pose, as an exercise, the problem of finding all (ordered or unordered - two different problems) pairs (implicitly of positive integers) that sum to a given integer (e.g. $5$), using such an exercise as the basis for teaching arithmetic seems misguided, because on the one hand no one can answer it that has not understood how to sum natural numbers, and, on the other hand, there is no general reason to link pairs whose sums are equal (unless one is studying partitions of integers, a topic that is not generally broached at the elementary school level).

  3. I don't see how "number bonds" differs operationally from usual rote memorization except in that it organizes the material to be memorized in what might (and possible should) appear to a child as a bizarre way.

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  • $\begingroup$ You don't have to use the jargon to use the concept. $\endgroup$ – Sue VanHattum Jul 4 at 16:10
  • $\begingroup$ @SueVanHattum "You don't have to use the jargon to use the concept." - certainly. But the mores of American education is inventing new jargon, not the usage of well-known concepts. $\endgroup$ – Rusty Core Jul 4 at 20:38
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I agree with the now-deleted answer from @Jedrek_Mansfield. It's not so much that one is better than the other. The approaches complement one another.

I have had plenty of experience with adults who still have trouble with basic addition and subtraction. Yesterday a student couldn't do 104 - 8 in her head. I showed her that I broke the 8 down into 4 and 4. Of course she knew 104 - 4, and it turned out she could do 100 - 4 also. I don't use the phrase 'number bonds' in a situation like that.

Once in a while I have a student who is willing to put some work into improving their basic arithmetic skills. And then I might mention the term to help them see the concept. I most often focus on the number pairs for 10. Many of my students don't see how to use those to their advantage: 16 + 7 = 16 + 4 (now I've 'made a ten') + 3 more = 23. Practicing breaking a problem down this way can be a tremendous help to students who didn't have strategies for this sort of thing before.

There is a lovely picture book, Quack and Count, by Keith Baker, that uses this concept well, showing all the combinations of ducks that add to 7. “Slipping, sliding, having fun, 7 ducklings, 6 plus 1.” I have long wanted to teach a math for elementary teachers course, so I could get students to make picture books like this for other sums. Years ago, my son was in a preschool and I watched the teacher play a game with them where the squirrel was hiding nuts. Here are 3, something is hidden, now there is 1. The kids loved telling her that 2 were hidden.

If you're interested in how life-changing something like this can be, you might want to read the chapter The Math Haters Come Around, by Tiffany Bearup, in my book, Playing with Math: Stories from Math Circles, Homeschoolers, and Passionate Teachers. (You can download a pdf of this CC-licensed book for free. But I believe it's worth the cover price, if you want to read more than that one chapter.)

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From the Wikipedia article Number Bond

In mathematics education at primary school level, a number bond (sometimes alternatively called an addition fact) is a simple addition sum which has become so familiar that a child can recognise it and complete it almost instantly, with recall as automatic as that of an entry from a multiplication table in multiplication.

When I try to paraphrase this, I get "A number bond is to simple addition what strong memorization is to the multiplication table." In effect, the phrase itself is not a distinction of two processes, rather something closer to a grade school teacher saying, "last year, M. Smith did so well with their students, memorizing their addition facts, that all but 2 have really internalized it, forming a Number Bond, and not grabbing the calculator as much as they used to." In other words, it appears to be used in context as a level of understanding not a process in and of itself.

Note: The article also states "the term number bond is sometimes derided as a piece of unnecessary new mathematical jargon." A worksheet asking students to find a number, which when added to the first, sums to 20, is linked as an example. These sheets have been around for 50+ years, and were just referred to as "addition drills".

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    $\begingroup$ I think that's the first time I've thought a wikipedia math answer was wrong. $\endgroup$ – Sue VanHattum Jul 4 at 16:12
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    $\begingroup$ @SueVanHattum - thanks, if the actual process is different than the article described (or my interpretation), I should delete. On the other hand, I'd love to understand it, and go fix that reference. $\endgroup$ – JoeTaxpayer Jul 4 at 16:15
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    $\begingroup$ The first thing that comes up when I google ("A number bond is a mental picture of the relationship between a number and the parts that combine to make it. The concept of number bonds is very basic, an important foundation for understanding how numbers work. A whole thing is made up of parts.") seems better to me. But it still doesn't get at the idea of working with families of sums, like all the pairs that add to 10, the most useful part of this concept. $\endgroup$ – Sue VanHattum Jul 4 at 16:18
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During this time I've been playing around with a Japanese abacus (soroban) which is a base-5 method, and I'm thinking it's more valuable for learning addition & subtraction quickly than rote memorisation and number bonds.

With a soroban, you have to mechanically operate the beads to get the correct answer. You start off moving them one by one and as you run out of beads on a rod you start a new rod. Sometimes you get stuck... eg. I need to add 4 beads in the units rod, but there's only 2 beads left. So I'll move 2 beads in the units rod, reset the rod to 0 and move 1 bead on the next adjactent 10's rod, then move one more bead in the units rod again. With some practice, it start to become second nature and you stop moving them one by one and just start forming the image of what you know the answer will look like.

It gets even trickier when you have to add 10, minus 5 and add 1 just to add a single bead. But once you get the hang of that you can do the same bead operation for any size number. Hundreds, thousands, billions, they all look exactly the same on a soroban so they're not at all intimidating because it's an identical operation.

So if I want to do 17 + 36, what I'd do on a soroban is 1+3 in the 10's rod, 7+6=3 in a unit rod and add 1 to the 10's rod. I get to see the answer 53 without having to use any intuition. With practice I start to figure out how and why it's that way.

I might do away with all rote memorisation of tables and just focus on the abacus. So long as my child does it often (haven't been able to so far) he'll manage to learn number combinations AND traditional number+number calculations with a single method.

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    $\begingroup$ "So I'll move 2 beads in the units rod, reset the rod to 0 and move 1 bead on the next adjacent 10's rod, then move one more bead in the units rod again." - did you mean two more beads, not one? I still think that a ten-bead abacus is simpler, but whatever works for you. $\endgroup$ – Rusty Core Jul 5 at 3:04
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    $\begingroup$ 10 bead abacus is simpler initially but far less efficient. As the soroban is Bashe-5, you see the relationships better. Eg. If you know 1+2=3, you’ll automatically learn that 6+7=13 because the beads look exactly the same (there’ll just never a single ‘5’ bead above the bar. It’s really helped me as an adult with poor arithmetic. $\endgroup$ – Humanities Guy Jul 5 at 3:42
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Another option is to just teach the addition tables, then tell him that subtraction uses the same numbers. So 3+2=5, therefore 5-2=3, and 5-3=2.

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I'm a big fan of number bonds.

They're a nice visual representation of the idea that you're breaking the number apart (especially if you actually show how some objects go into one bubble and the rest go into the other).

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