Book description:

Pre-Algebra New Math Done Right is the book that 1960s era New Math should have done but didn’t. 1960s era New Math failed to focus on how we understand order and addition of natural numbers. Richard Dedekind in 1888 worked out all the elements for understanding order and addition of natural numbers. New Math in the 1960s provided some pieces of that such as set theory, ordered pairs, relations and functions.

However, New Math in the 1960s did not include how to define addition recursively. It did not show how to prove the properties of addition of natural numbers from a set of axioms, now known as the Peano Axioms or Dedekind Peano Axioms. Dedekind did this in his 1888 book, “What are and what should be the numbers?”.

New Math Done Right Peano Axioms covers the start of the Dedekind 1888 New Math. 1960s New Math only included bits of it. This is the reason that 1960s era New Math failed. It was not focused on getting to order and addition of natural numbers, their properties, place value notation and place value algorithms.

Currently, there are materials on the Internet on the Peano Axioms but these are at college level. Books for math below college level do not cover the Peano Axioms or how to define addition. Even for the number line, these books include little actual explanation of what it means.

New Math Done Right Peano Axioms has versions of the Peano Axioms for a pair of number lines as well as for a pair of bead strings. Using a pair of number lines is a superior method to a single number line for understanding addition. This comes from using the Peano Axioms to develop a way to use and teach addition using the number line.

New Math Done Right Peano Axioms shows how to define addition recursively and how this definition links to pairs of number lines, pairs of bead strings and to other such examples the teacher, parent or student may wish to create. This is a major step forward in making these aids to learning addition of natural numbers also teach the math concepts.

Current textbooks on arithmetic do not give the teacher, parent, or student much real explanation for the natural numbers or defining addition of natural numbers. Typically, they only give drills without explaining what is going on conceptually. Dedekind in 1888 worked out what is needed to explain addition of natural numbers. Dedekind created New Math in 1888. However, 1960s era New Math failed to include that part.

The reason this was left out in the 1960s was that the book Edmund Landau, Foundations of Analysis had put addition before order of natural numbers. This was done using a complicated definition of addition by Kalmar. This was to avoid a technical point in Dedekind’s work called the Recursion Theorem. The Recursion Theorem in Dedekind is a way to define addition up to a fixed maximum. One does this, however, for all maximum values. One then combines these definitions which are consistent on the values they overlap.

This is tedious work, and the Kalmar definition was intended to avoid it. However, the Kalmar definition was so complicated, that it ruined the explanatory power of the work. So when 1960s era New Math came along they did not understand Peano Axiom addition very well and ended up not including that part in New Math. Thus New Math was destined to fail since it did not include all the elements Dedekind showed were needed in 1888.

New Math Done Right Peano Axioms fixes these problems and introduces the natural numbers and addition in a way that is understandable but does not leave out parts. Along the way it also teaches mathematical induction at a slower pace and in an easier way than most treatments.

The book also surveys studies in teaching mathematical induction, proofs, and math reasoning. These are compared to inductive reasoning studies.

Place value addition is taught as a recursive algorithm. This is done using letters for each digit and the carries.

Author: Mark S. Tenney

Publisher: Mathematical Finance Company