has interesting results. This blog is 3rd, which is good and bad.
|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.
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
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
• 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
• 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.
The seemingly unscalable wall of apathy towards the Peano Axioms may just turn out to be scalable with persistence.