Al Baker of New York Times writes a stunning news item that almost half of eligible teachers were denied tenure in New York City in 2012.

Many New York City Teachers Denied Tenure in Policy Shift By AL BAKER

Published: August 17, 2012

Only 55 percent of eligible teachers, having worked for at least three years, earned tenure in 2012, compared with 97 percent in 2007.

Last month in New Jersey, Gov. Chris Christie signed legislation overhauling the nation’s oldest tenure law and making it easier for teachers to be fired for poor performance.

The last few blog posts are about how learning Peano Axioms and proofs of arithmetic laws like addition of natural numbers is associative can help engineer types transition to be math teachers. Math foundations is explaining elementary math so it

- Makes sense
- Is consistent in terms, notations, etc.
- Doesn’t leave out parts of definitions, theorems, proofs or algorithms.
- Explains why algorithms work within a consistent axiomatic framework.
- Links set theory all the way to the algorithms and arithmetic laws of place value notation.

The Jennifer Asmuth and Lance Rips papers on psychology of learning math induction and math concepts in elementary math assume you know Peano Axioms and mathematical induction at least a little. And more helps.

The Asmuth and Rips papers are a way for an engineer to learn how to teach elementary math. They are written from how a math person thinks about teaching elementary math. So that is the right math ed material for an engineer to start reading, because it starts where he does. But to get there, learning the Peano Axioms at a slow pace is a good idea.

The e-book Pre Algebra New Math Done Right is about 390 pages on getting from set theory to the proofs of the associative and commutative laws of addition. It also covers some math ed and history of math teaching.

In contrast, most webpages on Peano Axioms or math books go from the Peano Axioms to addition law proofs in less than 10 pages with few or no examples and exercises on the successor notation they use.

=Blurb on book

The book has 178 Examples, 463 Exercises, 98 definitions, 45 Lemmas with proofs, references to over 96 webpages, 58 chapters, and is 391 pages when formatted as pdf with Latex.

Typical treatments of the Peano Axioms cover the same material as in the book in about 20 pages or less. They usaully have few examples and the few exercises are as difficult as the theory.

In contrast, this text has many examples too trivial for the current texts on the Peano Axioms to cover. Building at a very slow pace with many numerical examples, the reader is taken through the Peano Axioms themselves, simple consequences, order of natural numbers, and simple identities used to prove the properties of addition. This build up includes many very simple proofs by mathematical induction.

There are no quadratic or higher algebraic formulas in the book. Complicated algebra formulas are the main stumbling block to learning mathematical induction. None are in the book, yet there are many worked out proofs of simple relationship using mathematical induction and simple problems for students to do.

Summer Sale 2.99 e-book.

Buy Pre-Algebra New Math Done Right Peano Axioms at this vendor

Buy Pre-Algebra New Math Done Right Peano Axioms at an alternative vendor

==

The math teachers who did not get tenure might have been weak on

- Understanding the concepts of elementary math.
- Understanding the problems students have in learning elementary math concepts.
- Lack enthusiasm or burned out.

The Peano Axioms build enthusiasm for teaching elementary math. Learning the Peano Axioms is the difference between being a maestro of elementary math and being a soda jerk of elementary math. If that sounds harsh, half the teachers did not get tenure. So it is reality.

Learning the Peano Axioms, the structure of math proofs of the associative laws and so forth of addition builds an understanding of elementary math concepts. It then opens up reading math ed literature that is better understood with such an understanding.

The history of math foundations of arithmetic is inspiring for teachers. Hermann Grassmann was an elementary and high school math teacher in Germany i the 19th century. Grassmann invented the definition of addition using induction and recursion and invented the first proofs of the associative and commutative and other laws using induction. The Grassmann definition was published in 1861 by his brother, practically self published.

When you learn the Peano Axioms and teach elementary math using them, then you are a missionary for Grassmann, Dedekind and Peano. You are not just a clock watcher in a class room wondering if the subway will be crowded when you go home.

From NYT article:

“If New York City hopes to have a great school system, it will need to come up with better methods of helping teachers develop, not only at the beginning but throughout their careers,” Mr. Mulgrew said.

Like actually learning what elementary math means so they can teach what it means? They could start by buying Pre-Algebra New Math Done Right Peano Axioms which is 391 pages to go just from the Peano Axioms to the definition of addition and proofs of its properties. This also includes material on basic set theory, order of natural numbers, the recursion theorem, some history of math ed, the Asmuth Rips type experimental psych studies on learning mathematical induction, and place value notation as a recursive algorithm.

The book has 178 Examples and 463 Exercises, many just elementary drills on successor notation. Math prof webpages on Peano Axioms don’t have hand holding to learn the very start of it. They go fast with few numerical examples.

Not only does the book hand hold the teacher, it teaches the teacher how to hand hold the student to learn this material if taught formally. It is also an example of how to teach math concepts even in this era before mass adoption of starting the Peano Axioms in first grade.

Peano Axioms for Number Line suitable for tweeting.

- Zero a tick.
- A tick then a tick.
- Zero no mom.
- Not zero, one mom.
- Zero green, green transfers, all green.

See http://newmathdoneright.com/peano-axioms/ for more ditties on the Peano Axioms.