Is the Wolf Child the paradigm for self-discovery? A wolf child learns neither language nor math.
Euler in his 1765 Algebra is searching for answers but presents his half baked answers as if complete. But they aren’t. We know that from Grassmann, Dedekind, and Peano.
The definition of addition by right additive identities is
x+0 = x
x+y’ = (x+y)’
where n’ means the next natural number. 1 = 0′, 2=1′, etc.
Not many people self discover these two equations. In effect, these are one off discoveries by Grassmann. Dedekind then developed them.
It simply makes no sense not to teach these two equations. It is irrational. Teachers have no leg to stand on in opposing the teaching of these two equations.
They don’t self discover them, and neither do their students. These two equations encapsulate the meaning of addition of natural numbers in terms of counting by one. Successor or n’ is counting by one.
Addition of natural numbers is derived from counting by one. These two equations show how that deriving happens.
Since teachers and the education establishment are so against teaching them, we have to ask why.
One reason is they don’t realize what they don’t know. They dismiss Grassmann, Dedekind and Peano without reading them. They may give a cursory look at some university webpages that have Peano Axioms and a few theorems. These tend to be formal and not appealing as easy to teach to lower grades.
The education establishment is overly familiar with the plus sign and equal sign. These signs are more abstract than they realize. Grassmann and Dedekind in effect defined the plus sign in terms of counting by one. But the education establishment doesn’t grasp that.
Keith Devlin doesn’t grasp that addition and multiplication are about keeping one’s place in the count.
There is a tendency to think that the plus sign is easy and not abstract and that a letter for a number is abstract. In fact, this is opposite. A letter for a number is simpler than the plus sign in abstraction.
You can use letters for numbers with prime notation before the plus sign is defined.
Letters can be used to make different words which have meanings independent of the letters used to spell them. This is more abstract than a letter can stand for different numbers.
It is idiotic to teach using letters to make up different words and sentences with different meanings and think that using a letter to represent different numbers is too difficult. The last sentence is an example. If you can teach it, you can teach using a letter to represent different numbers.
The car is black.
The car is not black.
These sentences are made with letters and have not only different meanings but opposite meanings. How can students grasp that letters can be used to mean opposite things?
What about repeated letters. How can students grasp the use of repeated letters such as the double t in “letter” and a single t in “repeated”? Shouldn’t this double the meaning? It has no impact on the word meaning. They learn that. Why can’t they learn to use t to mean a number?