Michael Paul Goldenberg was kind enough to point out the following link to me on the history of New Math in the 1950s and 1960s.
Perhaps the most obvious manifestation of this in the case of new math was the appearance of books in which the first chapter was devoted to sets, but then the rest of the book never used them.
In contrast, the Peano Axiom proof of associativity of addition and other laws uses sets.
Suppose we start with addition defined by
x+0 = x
x+y’ = (x+y)’
where ‘ is successor, so 1 = 0′, 2=1′, etc.
We want to prove that
x+0 = 0+x
We shall say that x is green if this is true for x.
0 is green. This follows because
0+0 = 0+0.
Assume that y is green. This is the induction step:
y+0 = 0+y
Now y’+0 = y’ by the first equation in addition which applies to x=y’.
Substitute for y in the right under the prime by 0+y, so
y’+0 = (0+y)’
By the second equation in the definition of addition, we have
(0+y)’ = 0+y’
y’+0 = 0+y’.
Thus y=0 is green, and y is green implies y’ is green.
Thus by the principle of green induction, all natural numbers are green. So the proposition is true for all natural numbers.
1950s and 1960s new math did not try to teach this type of proof. Or even concept.
They did not have a blueprint of what New Math should be. That blue print already existed in the books of Grassmann, Dedekind, and Peano. However, in the meantime, the Edmund Landau Foundations of Analysis book switched to the Kalmar definition of addition to avoid the Dedekind path of proving order and then the recursion theorem which allows definition by induction not just proof by induction. The Kalmar definition squeezed too much into the definition of addition and was not understandable. Thus the spread of the Peano Axiom approach was halted.
Grassmann was himself an elementary and then high school teacher in Germany. He was brother published in 1861, which is close to self published today. The links to these books are in my book and in some cases in other blog articles.
Place value notation is recursive. The place value representation and place value algorithms are all recursive. Their properties can all be proven working from the Peano Axioms.
One can introduce affine forms such as
y = q B + r
where B is the base, r the remainder less than B and q the quotient or coefficient of B. If q is larger than B this can be repeated. This gives us the place value representation.
This affine form is also a route into fractions.
New Math in the 1950s did not try this path.
The associative and other laws, place value notation, place value algorithms and fractions are the meat and potatoes of elementary education. The use of abstract methods should have been focused on those, defining what they mean and proving their properties. Instead, they didn’t know how to connect the main parts of arithmetic to the abstractions. The reason was that the work of Grassmann, Dedekind and Peano had been separated from the math ed people and even most mathematicians by the unfortunate Kalmar definition of addition in the Landau book. Moreover, Landau made the mistake of not doing the recursion theorem at all and not doing order before addition as Dedekind did.
When you do order first, you realize that natural numbers are about keeping your place in the count. More abstractly about indexing. Addition, place value notation, multiplication and affine forms can all be seen as a way of keeping ones place in the count more efficiently.
Multiplication is using arrays to keep ones place in the count. The affine form lets you keep track of a partial row. Seen this way, natural numbers are always about order and addition and multiplication of them are algorithms or methods for making order comparisons more efficient. So is place value notation.
This way of thinking is not the same as geometry. Geometric point sets are not the meaning of counting with whole numbers. That point of view is confusing because it is wrong. It is the wrong way to understand fractions. Instead fractions arise from affine forms and from functions on subsets of natural numbers that are multiples of the reduced form denominator of the fraction.
Grassmann published his book in 1861. Dedekind his in 1888 and Peano in 1889. This really was the New Math of the 19th century. If the 1950s movement had used this it would have made sense to parents and others. This is a roadmap through the core of arithmetic using abstraction for a purpose, to explain the meaning of what you are doing when you do elementary arithmetic with whole numbers.
The answer is you are always keeping your spot in the count, but with more elaborate methods to make it more efficient. These methods, such as addition, are defined and their properties for keeping the place in the count proven. So addition is repeated successor. Multiplication is repeated addition. Both are methods to keep your place in the count. This is where these ideas originally came from thousands of years ago.
The Grassmann, Dedekind, Peano program codifies that original counting purpose. This codification is an Occam razor application. They removed what was not the explanation of whole numbers, place value and the 4 operations and what is left was the explanation, since they had found out what to include to make sure it was. This was to include a codification of counting and keeping your place in the count.
This vision connects counting and the development of algorithms and symbols to the abstract codification of Grassmann, Dedekind and Peano. This makes sense and can be explained to parents and stakeholders.
By building on this foundation, students can go onto other abstractions in math such as fractions of whole numbers. That can lead to understanding the real numbers. Dedekind did both. A real number is an infinite decimal, but it can also be called a stopping number. It stops a set of fractions from going up or going down forever. That is, it is a least upper bound or greatest lower bound of a set of fractions.
The use of letters for numbers can be introduced in a way that is practical, to define, explain and prove the place value algorithms. These are recursive. I go through that in detail in my book for 1 and 2 digit addition including using separate letters for carries and other details.
Algebra today does not redo arithmetic using letters for each digit and proving how it works. This is a lost opportunity. It reinforces arithmetic and at the same time shows letters for numbers as practical. It is difficult to do the mechanics of place value notation that proves how it works in all cases without using letters for each digit, each carry and each digit in the result or intermediate stages.
This way of doing arithmetic into algebra links them together and makes sense to students and parents as practical and useful. The GDP approach is a practical and useful approach to the central part of arithmetic. Following it teaches the abstractions that are used in algebra.
Confidence with using letters is built up by working through the mechanics of arithmetic using letters for each digit and stating the algorithms in general form and proving their properties. For example for addition of 2 numbers, the carry is always 0 or 1 for each digit. This can be proven with letters standing for numbers.