I am teaching my daughter, who is currently about $46$ months old, additions. She is very curious and asks a lot of good questions. For example, when I told her that $2+6=8$ and $4+4=8$, she asked me the following question:
Why are they the same?
Surely I know the logical answer this question using Peano Axioms and the definition of natural numbers. But she has not learnt the Peano Axiom at this age.
So how to answer my three-year-old daughter's question that
Why $2+6$ is the same as $4+4$?
I'm nearly sure I did this with my child when she was young.
First, establish that she understands that a number, like three, is equal to $1+1+1$. Hold three fingers up and ask her "how many is this"?
Then spread them out and ask the same question. Are we adding $1+1+1$?
Try holding up eight fingers (keep your thumbs down, for example) and ask her to count them.
Then hold them together, but move your hands apart. What numbers do you see added? What is the total?
Then make a V with one hand, holding two fingers apart from the other six. What numbers are being added? What is the total?
Then hold three fingers apart from the other five. What numbers are being added? What is the total?
Finally, hold one finger apart from the other seven. What numbers are being added? What is the total?
Then you can talk about different ways to add numbers to make $10$. This time, have her act it out with her hands.
Blocks work well for thinking about addition. Have her count out 8 blocks, and then ask her about all the addition problems that have 8 blocks as the answer.
A lovely children's book which looks at all the sum pairs for 7 is Quack and Count, by Keith Baker. (You can buy it used here.) It has luscious pictures, a driving rhythm, and a lovely storyline. (“Slipping, sliding, having fun, 7 ducklings, 6 plus 1.”)
A good answer to this question is one that (is correct, and) she finds convincing.
As kids are growing up and making sense of the world around them, experimentation is often one of their key sources of truth. Perhaps, experimentation can also serve as a reliable and effective gateway to the abstract world of mathematical truths!
Here's one possible roadmap that leads convincingly to arithmetical truths.
Experiment with an Object: Take $2$ blocks, add $6$ blocks. Count the total number of blocks to discover that the result is $8$ blocks. Therefore, $(2 \mbox{ blocks}) + (6 \mbox{ blocks}) = (8 \mbox{ blocks})$.
Discover that these results apply to all Other Objects: Experiment to discover that this rule applies to other objects such as apples, almonds, chairs, tables, toys, candy and anything else you can think of. Perhaps she would be willing to take the leap with you to agree that this would apply to other things that she has not yet experimented with, such as ships, planets and whales. Therefore, $(2 \mbox{ things}) + (6 \mbox{ things}) = (8 \mbox{ things})$.
General result: Since these results are valid irrespective of the object or thing being added, it is therefore acceptable (and more efficient) to write our experimental result concisely as: $2+6 = 8$ without reference to any object in particular.
Soon we point out that, in particular, these rules also apply to fingers (just like any other object), which are always at hand(!) so we can use them no matter what it is that we are in fact trying to add.
The key feature of the approach just outlined is that the act of writing $2+6 = 8$ is a symbolic recording (by pencil on paper, or chalk on board) of an experimental truth that we have discovered by exploring the world around us. The question of logically or abstractly proving it, therefore, does not arise.
When we arrive at the question, "what is $4+4$?", we must now resort to experimenting with our fingers (or any other object) to discover the result of this addition. Having discovered the result, we can record it symbolically as $4+4 = 8$.
Of course, we can get creative with our experiments. For example, to count the number of dots below,
: : : :
we can count them as
: $+$ : : :
or as
: : $+$ : :
This provides an intuitive justification that $2+6$ and $4+4$ are really just two different ways of visualizing $8$.
Also, I can't help pointing out that the following is a staggering gap in her present education:
But she has not learnt the Peano Axioms at this age.
I hope you will remedy it at the earliest! ;)
None of the answers so far seem to give a name for the kind of concept in the inquiry. What you're asking is "how to make 8" or more generally "making numbers". See here and here.
"Making numbers" is usually taught late kindergarten or early grade 1. In this case, you might ask the school how they teach it and then teach it to your child the same way that that is taught (that the school will soon, but maybe not very soon, depending on curriculum, teach your child)? I think this is usually done using marbles or blocks to show that 6 marbles + 2 marbles = 4 marbles + 4 marbles. Thus, the kindergarten student (or early grade 1 student) learns to understand the uniqueness of the number 8 (the existence of the number 8 is of course known to the student): namely that the "8" as the combination of 6 and 2 is the same as the "8" as the combination of 4 and 4.
And of course this understanding is visual via the marbles and not of course an understanding that a grade 2 or 3 student would have already acquired.
(And of course you don't necessarily use the word "unique". Let me know if ever you think of a way to introduce the word "unique" in this context.)
Here is a written explanation to accompany Nick C's excellent 'hand-waving' answer. =)
$2+6 \quad = \quad 1+1 \quad + \quad 1+1+1+1+1+1$
$ = \quad 1+1+1+1+1+1+1+1$
$ = \quad 1+1+1+1 \quad + \quad 1+1+1+1 \quad = \quad 4+4$
Explain that adding the same numbers always gives the same result no matter which order you add them, so we can add the ones up in any way we like. Also explain that we define $2 = 1+1$ and $4 = 1+1+1+1$ and $6 = 1+1+1+1+1+1$, so if we choose the right order to add the ones up, we can get either $2+6$ or $4+4$.
Start with two equations that are the same in every way, like this:
2 + 2 = 4 and 2 + 2 = 4
Now, add "4" more to each of the two equations, one at a time, so you'll get 8 in each equation. But do it different ways in each of the two equations.
1.) Add all of the "4" more to the second "2" in the first equation: 2 + 6 = 8 (and 2 + 2 = 4)
2.) Now split up the "4" you're adding to the second equation into two "2's" and add one "2" to each "2" in the second equation: (2 + 6 = 8 and) 4 + 4 = 8
I think this is similar to the first answer someone gave above, but without using fingers. I explained it like this (a similar question - not this identical question) years ago to my five year old son and again to my four year old niece.
