Procedural v conceptual is highly ambiguous on the conceptual fork and even somewhat ambiguous on procedural. If procedural is the same as recursive math or constructive math or computable math, then procedural covers most of the subject matter of algebra and arithmetic covered in K-12. Conceptual is open ended on what it means.
Procedural v structural is a more fruitful direction to try to take to get a meaningful fork for math education. Procedural should be heavily algorithm oriented. Structural should be axioms, definitions, theorems and proofs.
Historically, we could say that procedural arithmetic and algebra is what existed in 1800. We can take Euler’s elements of algebra as an example of this type of teaching. Euler’s attempt to define 7 divided by 3 is an example of this type of thinking. He simply says we can divide a line segment of length 7 into 3 parts. The same argument is used today with pizza.
See paragraph 68 at following link (in English)
68. When a number, as 7, for instance, is said not to be divisible by another number, let us suppose by 3, this only means, that the quotient cannot be expressed by an integer number; but it must not by any means be thought that it is impossible to form an idea of that quotient. Only imagine a line of 7 feet in length; nobody can doubt the possibility of dividing this line into 3 equal parts, and of forming a notion of the length of one of those parts.
Structural math for arithmetic and algebra did not exist in 1800. This was created in the 19th century. This is what we call new math. However, it was actually created in the 19th century as a direct reaction to the pure procedural math in Euler and similar textbooks available by c. 1800.
Structural math is some variant of the Peano Axioms together with truth tables, some propositional calculus (i.e. symbolic logic), set theory and related subjects.
So we can talk about Euler Academy (or Lacroix, Lagrange and Laplace Academy to give it a Parisian flavor) v Grassmann, Dedekind and Peano Academy.
Euler Academy teaches procedures out of Euler’s Elements of Algebra. GDP Academy teaches structure out of their textbooks in the 19th century and other similar ones. Along the way they might add Russell and Whitehead, Turing and Church, and the Knights of the Lambda Calculus.
Euler has the mechanics of fractions using letters for the numerator and denominator of two fractions being added or otherwise operated on. What Euler does not have is 1) set theory 2) a theory of natural numbers.
Euler is moving in the direction of the natural numbers as the sequence starting from 0 obtained by adding one to the last element. This statement appears explicitly in his text, whose first draft was in 1765. We are told it sold over 100,000 copies. So it was the Khan Academy of its time.
19. In the same manner, therefore, as positive numbers are incontestably greater than nothing, negative numbers are less than nothing. Now, we obtain positive numbers by adding 1 to 0, that is to say, 1 to nothing; and by continuing always to increase thus from unity. This is the origin of the sequence of numbers called natural numbers; the following being the leading terms of this sequence:
and so on to infinity.
However, Grassmann, Dedekind and Peano grew up being taught at Euler Academy and rejected it as not being conceptual enough and not being structural enough. What was wrong with Euler Academy? Euler Academy was procedural. It lacked structure. It lacked foundations, i.e. the start of it was lacking.
Euler basically picked up with letters, took the natural numbers as self evident and then started to teach rules of signs and rules of fractions and solving simple linear equations using an unknown. So Euler does the same as much of K-12 education today does. We are still Euler Academy.
However, this is precisely what the foundations of math movement of the 19th century rejected as inadequate. They were not satisfies with the procedural explanations of Euler texts as students or as teachers and professors. Grassmann was an elementary and high school teacher and he was the one who took the next step from Euler to define the natural numbers by inductively adding one.
Grassmann didn’t just say that the natural numbers are obtained by adding one to the last. He started the process of proving the commutative and associative laws of addition using this definition of natural numbers. This is what goes from a vague concept which was present in Euler of adding one starting from zero to become structural math with axioms, a definition of addition of natural numbers and proofs of the properties of addition of natural numbers so defined. This is structural math.
This is what the 19th century replaced the purely procedural math in Euler Academy with. This is what we call New Math. Grassmann published his book in 1861. So New Math did start in the 60s, just the 1860s not the 1960s. New Math started with Grassmann Academy in a book published by Hermann Grassmann’s brother, i.e. practically self published especially when we consider the brother helped contribute to the book according to some sources.
The text most in direct contrast to Euler’s Elements of Algebra 1765 is Dedekind’s “The Nature and Meaning of Numbers” of 1888. This is the procedural v the structural. If you don’t want to teach at Euler Academy you must go to Dedekind Academy. You must go from Procedural Academy to Structural Academy. You must go from Old Math Academy to New Math Academy. You must go from Skills and Drills Academy to Definitions, Theorems and Proofs Academy.
If you don’t want to teach purely procedural math you must teach structural math, i.e. you must teach New Math. New Math is conceptual math. New Math is what Grassmann, Dedekind, Peano and others created in the 19th century when they rejected the procedural math in Euler’s Elements of Algebra. There is no other road out of the Euler Alps except over the three bridges of Grassmann, Dedekind and Peano.
Attractive and easily accessible html version of Euler’s Elements of Algebra.
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Pre-Algebra New Math Done Right Peano Axioms
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