Experiencing True Inquiry
I don't think I was formally taught to decompose and re-compose numbers as part of learning math and it's a tactic I don't think I use as much as I might.
Decomposing to make adding or subtracting easier
This is really targeted at younger students learning to add but it's worth reviewing even with older students as it reminds them that equations can be a bit fluid.
8 + 6 is trivial for most adults, but it may be easier for a younger student to understand the benefit of thinking in terms of 10s. They first decompose the 6 into 2 and 4 and then add the 2 to the 8 to make 10 and then it's easy to add the remaining 4 to get 14.
Most older students will recognize that a number can be broken up into it's smaller components in different ways. For example, the number 6 can be broken up many ways into (2&4, 1&5, 3&3, 1&2&3, 2&2&2, etc.). Understanding this helps to understand fractions as well. A fraction like 7/8 can be broken into smaller components (1/8&6/8, 2/8&5/8, 3/8&5/8, etc.) and being able to further resolve these components into more simplified forms also helps to understand the size of the fraction. For example, 7/8 is the same as 3/8 + 4/8 which is the same as 3/8 + 1/2 or 1/2 + 1/4 + 1/8. This becomes more meaningful if the student is confronted with the issue of adding unlike fractions such as 7/8 + 1/2. With the knowledge that 7/8 = 3/8 + 1/2, it's easy to make the connection that 7/8 + 1/2 = 3/8 + 1/2 + 1/2 = 1 3/8. Rather than immediately converting to a common denominator and then adding, some decomposing and recomposing gets the solution more intuitively.
Decomposing to help multiply more easily and estimate quickly
For students who are learning their times tables, decomposing can be helpful in estimating and generally improving fluency with numbers. Decomposing numbers additively can help when learning about multiplying. Suppose you want to calculate 6 x 7. Working with 7s is hard but working with 5s and 2s is easy. If you visualize that a 7 is a group of 5 and 2, you can work with easier numbers.
8 x 7 is 8 groups of 5 and 8 groups of 2.
(8 x 5) + (8 x 2) = 40 + 16 = 56.
Unfortunately, if we use a more difficult example, it turns into more work potentially than following the traditional method:
256 x 23
= 250 x 23 + 6 x 23
= 250 x 20 + 3 x 250 + 2 x 23 + 2 x 23 + 2 x 23
= 5000 + 750 + 46 + 46 + 46
= 5750 + 50 + 42 +46
= 5800 + 88
= 5888
Is this useful? It's probably more work than its worth compared to just multiplying out the two numbers but it's useful thinking for estimating purposes if you don't go all the way through the process.
256 x 23 is going to be in the range of 250 x 20 (2 x 25 = 50 and there are two more zeros so 5000) which gives you an order of magnitude. And it's not hard to get a closer estimate by thinking that you could add in another 3 x 250 to get to 750. So, your estimate would be 5750 which isn't a bad estimate without doing any hard math in your head.
438 x 36 is a more difficult combination because they don't round easily to nice numbers but let's estimate:
400 x 30 = 12000 which is a good starting estimate.
6 x 400 = 2400 This helps refine the number a bit more.
Current estimate is 14400 and the actual number is 15768.
So it's not a method for arriving a the exact number but it's a quick method for making a reasonable estimate
Decomposing and Recomposing for Division
If the outcome of a division is likely to be a whole number, then decomposing can help. Consider the dividing 91 by 7. For most people, there is no obvious solution and they will start doing long division to solve. However, decomposing the number into two parts that are divisible by 7 will make a solution easier.
91 ÷ 7
= (70 + 21) ÷ 7
= (70 ÷ 7) + (21 ÷ 7)
= 10 + 3
= 13
Even with a larger more difficult number
192 ÷ 16
= (160 + 32) ÷ 16
= 12
How about using bigger numbers?
372 ÷ 31
= (310 + 62) ÷ 31
= 12
Bigger?
4557 ÷ 31
In this case, a suitable decomposition isn't obvious so long division is probably easier.
I haven't investigated whether this thinking is useful when the division is not going to be a whole number.
Summary
The value of decomposition, re-composition, and visualizing numbers for the purpose of developing fluency with numbers is clear to me. While it doesn't necessarily offer a solution to every problem, it expands the number of approaches a student has to tackling a problem.
Closing
This article was the result of reviewing a number of videos on the subject of number sense and composing and decomposing numbers including the following:
Funza academy is some fun and interesting problems and observations about manipulating numbers and solving problems. https://www.youtube.com/channel/UCnjlICG8FM8k1DpThYfCcPQ
Brillian.org Math and Science Done Right has some interesting stimulating math problems: https://brilliant.org/