Forget New Math, it does now work with young kids, and it is not needed for learning arithmetics in elementary school.
Instead, first teach the concepts of "same as", "more than", "less than" by lining up real objects like apples or backpacks or horses one against another, making pairs. Then you ask, how do we figure out "more" or "less" or "same" without bringing my horses and your horses to the river? We can abstract horses into counters and line up counters mine against yours and compare them. Then you ask, can we abstract other things like cows or women or children with counters? Do they have to be the same counters? Can we use different ones? Can we use sticks instead of stones? Can we compare without lining up counters? Well, we write counters on paper instead of actually carrying stones or sticks. Pictures of counters is an abstraction of physical counters. When we get too many things, it becomes unwieldy, can we do better? Well, we can "count" the counters - a whole new idea and a process, and come up with - ta-da! - numbers. Comparing 3 and 5 is the same as comparing ||| and |||||, just different representation. How is it better? Without introducing place value, not much. So you explain how place value works, and how with limited number of digits we can now represent any countable number, another level of abstraction.
But please, don't tell kids that numbers don't exist, that they represent the abstract notion blah-blah-blah. They tried it sixty years ago, and it did not work.
Let me quote Why Johnny Can't Add by Morris Kline:
"Is 7 a number," asks a teacher. The students, taken aback by the
simplicity of the question, hardly deem it necessary to answer; but
the sheer habit of obedience causes them to reply affirmatively. The
teacher is aghast. "If I asked you who you are, what would you say?"
The students are now wary of replying, but one more courageous
youngster does do so: "I am Robert Smith."
The teacher looks incredulous and says chidingly, "You mean that you
are the name Robert Smith? Of course not. You are a person and your
name is Robert Smith. Now let us get back to my original question: Is
7 a number? 0f course not! It is the name of a number. 5 + 2, 6 + 1,
and 8 - 1 are names for the same number. The symbol 7 is a numeral for
The teacher sees that the students do not appreciate the distinction
and so she tries another tack. "Is the number 3 half of the number 8?"
she asks. Then she answers her own question: "Of course not! But the
numeral 3 is half of the numeral 8, the right half."
The students are now bursting to ask, "What then is a number?"
However, they are so discouraged by the wrong answers they have given
that they no longer have the heart to voice the question. This is
extremely fortunate for the teacher, because to explain what a number
really is would be beyond her capacity and certainly beyond the
capacity of the students to understand it. And so thereafter the
students are careful to say that 7 is a numeral, not a number. Just
what a number is they never find out.