The search

has interesting results. This blog is 3rd, which is good and bad.

http://www.cse.buffalo.edu/~rapaport/191/recursion.html

http://www.mathcircles.org/node/864

==

## Towers of Hanoi for Elementary students

Topic Classification: Towers of Hanoi, Recursion, Sierpinski triangle, Patterns |
Tags: |

Level: Primary education |
Difficulty: Moderate |

A lesson used at the East Lansing Elementary Math Circle for exploring the Towers of Hanoi with second-fourth graders. Includes an introduction to recursion, a connection to the Sierpinski triangle (via the graph of allowable moves), and a connection to the English ruler.

==

http://www.wyzant.com/Thornton_CO_elementary_math_tutors.aspx

http://www.amte.net/sites/all/themes/amte/resources/EMSStandards_Final_Mar2010.pdf

=Excerpt

I. Content Knowledge for Teaching Mathematics

EMS professionals must know and understand deeply the mathematics of elementary school as well

as how mathematics concepts and skills develop through middle school. This knowledge includes

specialized knowledge that teachers need in order to understand and support student learning of

elementary mathematics.

a. Deep understanding of mathematics in grades K–8.

Elementary mathematics specialists are expected to acquire the habits of mind of a mathematical

thinker and use mathematical practices such as precision in language, construction and

comparison of mathematical representations, conjecturing, problem solving, reasoning, and

proving. They need to able to use these practices in the following domains:

Number and Operations

• Pre-number concepts: Non-quantified comparisons (less than, more than, the same),

containment (e.g., 5 contains 3), 1-to-1 correspondence, cardinality, ordinality.

• A comprehensive repertoire of interpretations of the four operations of arithmetic and of

the common ways they can be applied.

• Place value: The structure of place-value notation in general and base-10 notation in

particular; how place-value notations efficiently represent even very large numbers, as

well as decimals; use of these notations to order numbers, estimate, and represent order of

magnitude (e.g., using scientific notation).

• Multi-digit calculations, including standard algorithms, mental math, and non-standard

ways commonly created by students; informal reasoning used in calculations.

• Basic number systems: Whole numbers (non-negative integers), integers, non-negative

rational numbers, rational numbers, and real numbers. Relationships among them, and

locations of numbers in each system on the number line. What is involved in extending

operations from each system (e.g., whole numbers) to larger systems (e.g., rational

numbers).

• Multiplicative arithmetic: Factors, multiples, primes, least common multiple, greatest

common factor. Proportional reasoning and rescaling.

Algebra and Functions

• Axioms: Recognize commutativity, associativity, and distributivity, and 0 and 1 as identity

elements in the basic number systems; understand how these may be used in computations

and to deduce the correctness of algorithms. The need for order-of-operations conventions.

• Algebraic notation and equations: Literal symbols, as shorthand names for mathematical

objects, or, in the case of numerical variables, as indicating an unspecified member of

some class of numbers (the “range of variation”). The process of substitution of particular

numbers into variable expressions. The solution set of an algebraic equation or relation.

Transformations of equations (or relations) that do not change the solution set.

• Modeling of problems, both mathematical and “real world,” using algebraic equations and

relations.

• The concept of a function as defining one variable uniquely in terms of another.

Familiarity with basic types of functions, including constant, linear, exponential, and

quadratic. Representations and partial representations of functions: Formula, graph, table;

or, when the variable is discrete, by recursion.

• Finding functions to model various kinds of growth, both numerical and geometric.

==

To really have a deep understanding of many of these concepts requires knowing the Peano Axioms and the structure of elementary math in terms of the Peano Axioms. The latter has not really been written yet. I am, in effect, writing it. Some of it is already in my books.

See e-vendor. They have a free ap to download to read their books on standard computers and many hand held devices.

What the group above says they are supposed to know is as difficult as knowing the Peano Axioms and recursion in elementary math. It is a matter of refocusing on the structure of elementary math. This provides a connecting thread. This makes it easier to learn.

The successor function as a set of ordered pairs of the form (n,n’) for n a natural number is just links from one whole number to the next. Moreover, by induction, we get to any ordered pair eventually.

Eventually, even the common core standards in math turn up in this search.

http://www.corestandards.org/assets/CCSSI_Math%20Standards.pdf

The seemingly unscalable wall of apathy towards the Peano Axioms may just turn out to be scalable with persistence.