It may be thought the Peano Axioms and that successor notation 1=0′, 2=1′, etc is for the top of the curve. However, the bottom may profit from these as well.
2+0 = 2
2+3′ = (2+3)’
These equations are the definition of addition for these specific cases. They lead to a discussion of them.
The second equation defines addition for 2+4 using the already defined value for 2+3. We count on from the 3 in 2+3 as an addend. At the same time, we count on from the sum on the right hand side.
We count on the sum to count on the addend.
“Count on” is a phrase widely used to explain addition. What these equations say is that counting on actually defines addition when we structure our explanation properly. This is a magic insight at an early age.
Numbers are easier than geometry. For the bottom half of the curve, the place to learn axioms is whole numbers, not plane geometry.
Moreover, the above pair of equations can be written in grade K. Grade K already starts number line work, counting on, and simple addition equations. Grade 1 adds to it.
By adding the use of the prime notation for counting on just one number, we can express the recursive structure of addition. This is what we are teaching them with counting on and number lines. We just don’t say so. By saying so, we teach them structure. Structure sparks the imagination because structure gives you the feeling of understanding.
The bottom half of the curve needs the feeling of understanding in order to persevere. They are the ones who give up because they don’t understand. Even more than poor grades, the feeling of not understanding and that they can’t understand impacts them.
This is the core of where the self esteemers are coming from. The Peano Axioms build self esteem because they can be mastered by the lower half of the curve and they can master the definition of addition given above. This is very powerful precisely because this is real math and real understanding because it is the actual real structure of elementary math.
The reason the Peano Axioms excite old math profs is because they are the real deal. That is why they will excite the bottom half of the curve.
Whole numbers are more practical than plane geometry. So their axioms have more value. Those who turn their back on the Peano Axioms for tots are taking away real math that those tots can actually master.
Mastering the real definition of addition of whole numbers in terms of counting on is within the grasp of the bottom half. It is real math and real confidence to learn them.