The paper “From numerical concepts to concepts of number” by Rips, Bloomfield and Asmuth dispels the idea that just throwing number examples and drills actually teaches children number concepts well. The concept that the natural numbers 0,1,2, etc do not loop around is one that is not well taught in the current system. It is mostly implicitly taught, as opposed to formally taught.
Citation as it appears on Lance Rips homepage
Rips, L. J., Bloomfield, A., & Asmuth, J. (2008). From numerical concepts to concepts of number. Behavioral and Brain Sciences, 31,623-642.
Their affiliations in 2008 are from the paper:
Lance J. Rips
Psychology Department, Northwestern University, Evanston, IL 60208
Department of Psychology, DePaul University, Chicago, IL 60614
Psychology Department, Northwestern University, Evanston, IL 60208
This paper dispels the idea that the typical approach of teaching children to count will actually teach them number concepts. The picture books, lesson plans, etc. that are used do not actually teach the critical concepts of what makes numbers what they are. These are encapsulated by the Peano Axioms in the successor function as Rips et al spell out. They then discuss how much of that children learn and when based on a variety of studies cited in the paper.
The paper contradicts the idea that typical methods of teaching counting and of grouping objects actually teaches number concepts as in the Peano Axioms.
The paper is followed by criticisms from colleagues and then a response by the authors. This is all in the above linked pdf. The concepts in the Peano Axioms are used extensively by the authors and in the criticisms. Moreover, these are linked to learning numbers as a concept as opposed to mechanical learning that falls short of this.
Mechanical learning for example falls short in why there is no largest number, or in proving that addition of natural numbers is associative and commutative.
Teachers who do not know the Peano Axioms or the proofs of arithmetic laws will find this paper difficult to read. This may be why this body of research has had so little impact on teaching and on the education debate.
The debate in the paper is what we need to really change math ed. The Khan Academy video controversy, #mtt2k, was useful to get people interested in debating math, but it fell short of the debate in the above pdf. This is in part because many in the debate do not know the concepts of elementary math as well as Asmuth, Bloomfield and Rips.
The teachers in the Khan Academy debate were among the best, but even they don’t seem to manifest any awareness of the level of debate in the Asmuth, Bloomfield and Rips article and the responses of the critics. This covers not only math but psychology and neural science in learning numbers as a conceptual framework. That debate can only be joined by teachers when they themselves know the numbers as an articulated conceptual framework which currently means the Peano Axioms and proofs of arithmetic laws based on them.
Teachers don’t know what they don’t know when it comes to the Peano Axioms and to the Lance Rips et al literature on studying the natural number concept in children and adults too for that matter. This is a major problem in teaching math in K-8. When children get to algebra one in the 8th grade, many chances have already been missed to teach them to articulate number concepts like the natural numbers don’t loop around. These lead naturally into using letters to do proofs of properties of natural numbers. This teaches algebra concepts in a more meaningful and articulated way as to what the concepts are.
Natural Numbers are taught in a mishmash method. The logical structure of the Peano Axioms are not taught nor proofs of seemingly obvious properties like the natural numbers do not loop around. If they did, some number n would be the successor of two different numbers, and that is ruled out by the Peano Axioms. The formal proof requires careful consideration of what is proven in what order. That is discussed more in the book Pre-Algebra New Math Done Right Peano Axioms.
5.3. Math principles
What information must children include in their math
schema in order to possess the concept of natural
number? As we mentioned earlier, it is hard to escape
the conclusion that they need to understand that there is
a unique initial number (0 or 1); that each number has a
unique successor; that each number (but the first) has a
unique predecessor; and that nothing else (nothing other
than the initial number and its successors) can be a
natural number. These are the ideas that the Dedekind-
Peano axioms for the natural numbers codify (Dedekind
1888/1963), and our top-down approach suggests that
these principles (or logically equivalent ones) are acquired
as such – that is, as generalizations – rather than being
induced from facts about physical objects.
In other words, showing pictures of ducks or manipulating physical objects does not teach articulated math concepts such as there is no largest natural number and that this is linked to the properties of the successor function, in particular each number has a unique successor, and no number has two distinct predecessors, and no number fails to have a successor. Any candidate largest number would have a distinct successor, which would be larger than it. This depends on defining larger as having a chain of successors from the lessor to the larger. Such chains are discussed explicitly in the text Pre Algebra New Math Done Right Peano Axioms.
Most university web pages that discuss the Peano Axioms do addition before order and they never discuss that chains of succession don’t loop around in much detail or at all. This leaves a gap for reading the Lance Rips et al papers and in teaching this concept to children.
5.3.2. The successor function is one-to-one. It is the
one-to-one nature of the successor function that makes
the natural numbers unending. Children must learn that
each natural number has just one successor (so the successor
relation is a function) and that each natural number
except one has just a single predecessor (so the successor
function is one-to-one). Because of these constraints, the
sequence of natural numbers cannot stop or double
back. Evidence concerning children’s appreciation of
these facts suggests that they appear rather late (Hartnett
1991). Although children in kindergarten are often able to
affirm that you can keep on counting or adding 1 to
numbers, it takes them a while – perhaps as long as
another year or two – to work out the fact that this
implies that there cannot be a largest number. Counting
skill is not a good predictor of the ability to understand
the successor function, although knowledge of numbers
larger than 100 does seem predictive. It may be, as Hartnett
suggests, that children who can grapple with larger
numbers have learned enough about the generative rules
of the numeral sequence (i.e., advanced counting) to
understand their implications about the infinity of the
This paper shows that teachers need to learn the Peano Axioms and proofs on them. But university web pages give very short treatments heavy on proof and short on numerical examples and hand holding. This gap is filled by the text Pre Algebra New Math Done Right Peano Axioms. It only goes from basic set theory, the successor function, and the Peano Axioms to the definition of addition and proofs it is associative and commutative. This is done in roughly 390 pages. That also includes some other material on the above type of research as well as the history of math education and place value notation as a recursive system and that place value algorithms are recursive algorithms.
=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. The e-book is not a pdf and is 100 percent html with no pdf, jpg or png.
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.