Keith Devlin of Stanford University will have a MOOC course on transition to college math. This will not be a full course, but a partial course. He is hoping to get schools to use it as part of their courses. Devlin states in his MOOC blog post
Such courses typically comprise a mix of some elementary mathematical logic, proof techniques, some set theory through to an analysis of relations and functions, with a bit of elementary number theory and introductory real analysis thrown in to provide examples.
Given the problems students typically have when they meet this material for the first time, doing this at a distance is a challenge. Even if they did well at math in school, most beginning university students are knocked off course for a while by the shift in emphasis, from the K-12 focus on mastering procedures to the “mathematical thinking” characteristic of much university mathematics. Clearly, offering such a course as a MOOC is a huge experiment.
This is a good description of the new text Pre-Algebra New Math Done Right Peano Axioms. It is aimed at below college level. It starts with the Peano Axioms. It contains slow paced versions of the proofs of the first steps in the Peano Axioms. This includes naming and teaching the zero and successor or shift identities.
Right Zero Identity
x+0 = x
Right Successor Identity
x+y’ = (x+y)’
where x’ means successor, i.e. what most people think of as x+1.
From these right identities, we can prove the left identities
Left Zero Identity
0+y = y
Left Successor Identity
x’ + y = (x+y)’
We can also prove various other special cases. These are done slowly. In addition, there are many numerical examples and simple induction proofs for the student to do. These do not use any algebraic formulas above linear, i.e. no polynomial formulas as usually encountered in learning mathematical induction. Students complain this is the hardest part of learning mathematical induction, the difficult algebra of the formulas used as examples and tests.
In contrast, the text Pre-Algebra New Math Done Right Peano Axioms is the slowest pace lowest level introduction to proofs by mathematical induction. Thus it is an excellent part of such a course. High school students and others can follow the Devlin MOOC course and use PANMDRPA as a text to learn and practice proofs by mathematical induction of the laws of natural number addition.
The text lightly touches on the Recursion Theorem and does somewhat informally order before addition. Extensive references on these are contained in the text including to the important David Groisser notes on doing the initial segments of the natural numbers first before addition. That complements Dedekind’s original sequence of doing order first using tail sets of the natural numbers and then doing the recursion theorem and then doing addition, all in 1888. The text contains some basic elements of set theory needed for the material.
The text also contains the alternative definition of addition using constant sum lines, or pitch lines. These are introduced prior to addition. The text Geometry of Addition in preparation emphasizes the pitch line definition of addition and the geometry of pitch lines and induction along finite pitch lines and induction from pitch line to pitch line. This covers the plane of lattice points of pairs of natural numbers in a way different than the standard nested inductions.
Standard nested inductions are more complicated than pitch line induction. An example of nested induction is the Kalmar definition of addition in Edmund Landau Foundations of Analysis. Avoiding that complexity was the original motivation of the new definition of addition using pitch lines and also pitch line induction. However, the geometry of addition itself is a good way to learn proofs by mathematical induction avoiding algebraic formulas.