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. The book is however not a pdf and is 100 percent html with no pdf, jpeg or pngs.
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.
The e-books in this series are 100 percent html. No pdf and no jpegs or pngs. They are completely in html and will resize on any device used to read them. They are fully searchable. You can take notes on any field or equation.
The books display similar to the html on this blog including the formulas typed in using html on some blog pages. The e-book uses an extensive system of numbered notes for each type of note, such as definition, example, exercise, solution, lemma or a comment on an external html link. The list of these is viewable in one e-vendors preview. The lists at the front for each of these types of notes are live html so you can click on them and go to that note field in the book. This makes the book highly organized which makes it easier to understand the structure of the e-book. Many e-books give you the feeling you are lost because you don’t have the physical book to thumb through the pages. The extensive numbered note system used in this book helps avoid that. Because it is 100 percent html with no jpegs or pdf it works to allow navigation and searching with the standard e-book utilities as well.
The note system makes it easy to keep track of where you are in each subsection, section, and chapter. The notes are all numbered using the chapter number and note of that type. So definition 4.5 is chapter 4, 5th definition type note in chapter 4. The definitions are listed in the front of the book with live html and the title of the definition not just the number in the list. These titles are hand crafted to try to either remind you of the definition or for short ones, they repeat the entire definition in the title. A similar method is done for titles of examples, exercises, lemmas and comments on external links.
This means the lists at the front of definitions, lemmas, examples and exercises are a type of study guide and overview. They are readable and give an understanding of the book and its structure. This avoids the lost in the e-book feeling that is particularly bad in math e-books compared to physical math books.
In one of the vendors you can see a pre-view that shows the full table of contents and lists of all definitions, examples, lemmas, exercises, and html links to external webpages. These are all full working links. There are links to math education history including in the 19th century and to math psych studies on induction such as Jennifer Asmuth and Lance Rips.
The Edmund Landau Foundations of Analysis book is actually hardest at the very start, the definition of addition using the Peano Axioms. The above book was written to focus on this transition and it covers from the Peano Axioms to proofs of addition laws for natural numbers only, not even going into multiplication. It also covers the set theory needed for the Peano Axioms. This is intended as a below college level introduction into the key part of math foundations for teaching elementary math from counting to transition to algebra in K-8.
Comments at the above thread on the Edmund Landau book show excitement of learning math foundations for elementary math. But as one comment makes clear, the Landau book is actually hard to read. The e-book promoted by this blog is to ease that transition and contains extensive examples and exercises.
This blog is focused on the topic of teaching the Peano Axioms, proofs of the laws of natural numbers with them, mathematical induction, and related topics in math education.
New Math Done Right is the name of a series of books I am writing on this topic. The first volume already contains close to 200 examples and over 400 exercises on the start of Peano Arithmetic. This is more than in any other book or university lecture notes.
The basic thesis is that 1960s era New Math failed to include the Peano Axioms and thus failed to have a sufficient focus on teaching arithmetic using set theory, axioms, proofs, etc.
R. James Milgram in his math education work has stressed the Peano Axioms and proving the associative, commutative and distributive laws. In a recent blog post, I link to some of his articles.
I have also linked to articles on teaching mathematical induction. Lance Rips at Northwestern Univ Psychology Department has studied the teaching of math induction and has emphasized its importance in proving the laws of natural number addition and their importance to education. This is also linked to in a recent blog post.
You do not, of course, have to agree with them or me, but it is an important area of research in math education.
Formulas I have published with David Beaglehole in mathematical finance on solving partial differential equations are widely cited in finance.
Our papers contain closed form solutions for Green’s functions for partial differential equations of parabolic type. We added many closed form Green’s functions to human knowledge. These formulas are in textbooks and widely cited.
I also provide assistance to committees on the Academy of Actuaries for regulation of risk of insurance companies. I am currently on the Model Efficiency Work Group that is working on applying techniques for reducing the number of scenarios required in stochastic scenario modeling of insurance company risk. This is part of a new set of insurance regulations for US insurance companies to use stochastic scenarios for reserves.
I will be developing my education series in math through calculus and then to probability, stochastic processes, and finance. The start of Peano Arithmetic includes the Recursion Theorem for defining recursive functions such as addition. Similar types of definition are made throughout probability and into finance.
Pre-service teachers in Germany are already taught the Peano Axioms at PH-Heidelberg.
“In der Arithmetikvorlesung aus dem Wintersemester 2010/11 an der PH Heidelberg spricht Prof. Dr. Christian Spannagel über die Folge der natürlichen Zahlen.”
About 6 minutes in he starts the Peano Axioms.
American pre-service teachers are not taught the Peano Axioms. This is a gap in their knowledge compared to pre-service teachers at PH-Heidelberg. Milgram also emphasizes the use of Peano Axioms as a basis of Russian elementary school education lessons.
So there is a need for the American teachers to be further informed on this area of research. This is supported by their importance, psychological learning studies by Lance Rips et al, and practice in other countries.
Pre-Algebra New Math Done Right Geometry of Addition is now on sale. This volume is sold in parts. These range from .99 to 3.99 currently.