As a project I will be publishing ebooks on the Common Core Standards from the point of view of the Peano Axioms.
Place value notation and counting with tens and hundreds start in grade K. This logically requires multiplication before place value notation.
Place value notation has the same recursion in it as multiplication does. If one starts from the Peano Axioms this is obvious. Part of this is covered in my main book on Peano Axioms already, which went through place value notation as recursive.
The Common Core Standards are not mathematically logical. This means when they claim to teach concepts, they don’t. They are teaching behavior when they are claiming to teach concepts.
Virginia grade K lesson involving counting to 100.
100 is 10 tens. This is already multiplication. It is already recursive. Counting by 5’s is already recursion.
m’*5 = m*5 + 5
Counting by 5’s is to increment the m value. This is recursion. This is multiplication.
To do grade K, one has to introduce, logically, whether one teaches it to students or not the following.
- Successor function.
- Addition including right additive identities. This is recursion.
- Multiplication as recursion.
- Place value notation as an expansion.
- A hidden symbol for ten like A in hex notation. Even if not taught, this is there logically in place value notation. Failure to use such a symbol makes place value notation harder the same way avoiding prime notation and the addition identities complicates learning addition.
If these are not taught to students, then we are not teaching concepts, we are teaching by rote. This is valuable to understand by itself.
By understanding the Peano Axiom math behind the grade K common core standards, we can see that we are not teaching the concepts. This by itself is valuable.
It exposes the claim that we are teaching concepts in grade K to be false.
If we want to teach the concepts of the common core standards in grade K, then we have to teach the above items from the Peano Axiom based Theoretical Arithmetic. Otherwise, we are not teaching the concepts.
The concepts are, it is true, half taught in some ways. Grouping is already recursion. We might as well call it that. Grouping is the same recursion as Peano Axiom multiplication. If we said that, we would make progress for ourselves, and eventually for students.
Keith Devlin’s opposition to teaching recursion and to teaching multiplication as repeated addition is a sort of anti math standard. At least as far as concepts goes it is. In real time, Devlin was spreading his anti recursion propaganda at the same time the Common Core Standards were proposed and adopted. Thus he was preaching against teaching the concepts in the Common Core Standards. This applied even to the people adopting them or writing out the standards.
The CCS were written by people who did not understand the math of what they were writing. But they thought they did.
Because of this, the CCS are not in logical order. Multiplication starts in 2nd grade, but place value which depends on multiplication starts in grade K. Both use the same recursion from Peano Arithmetic.