## Theorems for Tiny Tots

As much as possible the very simplest and most obvious statements in elementary math should be stated as theorems.  These should then be proven.

Let us consider ones place addition of two numbers.  So they are i and j.  The sum of i and j is less than or equal to 18.  This is a whole theorem on its own.

A corollary of this theorem is that the carry out of the ones place in adding two numbers is always less than or equal to one.

These need to be presented as theorems with formal proofs.  Students should learn the statement of the theorem and the proof.

We start with the simplest theorems.   These statements are told to students already but not presented as theorems with proofs to be learned.  This is missing an opportunity.

By proving little lemmas about place value notation and its algorithms, students learn proofs.  They also learn the structure of math. The structure of math is the meaning of math. The meaning is the inspiration.

By taking out the inspiration what is left are the drills.

Applications can’t make up for leaving out the meaning of elementary math.  And in practice, they don’t.

Applications can never inspire the same way as teaching the structure of math can.  Understanding math itself is the inspiration.  Take that away and you can never put it back in, no matter what applications or projects or games or videos you try to get it back with.

The teacher herself needs to feel that she understands the structure of elementary math.  She then conveys that feeling to the student. The student learns that he too can learn the structure of elementary math.  This is the inspiration transfer from teacher to student.  Without this, learning is drills and high stakes testing.