Lance and Rips state in the conclusion to their paper on page 394:
We’ve argued elsewhere (Rips, Bloomfield, & Asmuth, 2006) that math induction is central to knowledge of mathematics: It seems unlikely that people could have correct concepts of number and other key math objects without a grip on induction. Basic rules of arithmetic, such as the associative and commutative laws of addition and multiplication, are naturally proved via math induction. If this is right, then it’s a crucial issue how people come to terms with it.
Lance J. Rips
Note that Lance Rips and Jennifer Asmuth are in the psychology department not the math department. This is a psychology paper on learning mathematical induction in contrast to inductive reasoning (not a proof) and deductive reasoning. Inductive reasoning is what is taught in cognitive math education. It is not proofs.
The above paper will be difficult for most to read. This is a review paper not an independent study. It does however, discuss experiments. It is a mixture of college level Peano Axioms, psychology, references to other papers, and anecdotes about colleagues and students. Its informal and casual nature make it fun to read. It uses some logical symbols and assumes a certain familiarity with Peano Axioms, mathematical induction, and logical symbols.
IAx is defined on page 377. This is mathematical induction in logic symbols. IA stands for Induction Axiom. This is an axiom of the Peano Axioms. It says we have some set M. M only contains natural numbers. Suppose M contains 0. Suppose if M contains a natural number, then it contains the next natural number, then M contains all natural numbers. The next natural number after a given natural number is the successor of the given natural number. So 1 is the successor of 0.
Despite the use of logical symbols at some points, this paper is a valuable discussion and gives insight into how people learn mathematical induction and how it compares to deductive reasoning. This is in contrast to self discovery style inductive reasoning without a proof. Students in elementary school are taught inductive reasoning, which is not proofs. This is different than mathematical induction which is a method of proof.
As Lance and Asmuth point out, to learn and understand arithmetic and algebra, one has to learn mathematical induction and proofs by mathematical induction. The book Pre-Algebra New Math Done Right Peano Axioms (PANMDRPA) is a transition to proof book. It takes the most gentle slope possible to learn the Peano Axioms and proofs by mathematical induction.
One way that PANMDRPA is a gentle introduction to mathematical induction is that it contains no complicated algebra. There are not even quadratic formulas in it. All the mathematical induction proofs are about relations of numbers or pairs of numbers. Thus it is a pure play on conceptual understanding of mathematical induction. This is then applied to prove that addition of natural numbers is commutative and associative.
To be able to read papers like Asmuth and Rips one has to learn the Peano Axioms, proof by mathematical induction, basic set theory, and some logical symbols. The text PANMDRPA will help, although it does not cover logical symbols. Educators, standards setters, teachers, and parents will have to learn this basic material to read valuable papers on psychology of education like Asmuth and Rips. You can’t learn the math ed if you don’t learn the math.