Understanding Fundamental Ideas in Mathematics at a Deep Level
A number of mathematicians and mathematics education researchers
have recognized the special nature of the mathematical knowledge needed for K-12
teaching and its implications for the mathematical preparation of teachers. In
particular, the interviews with Chinese elementary teachers in Liping Ma's 1999
book Knowing and Teaching Elementary Mathematics awakened many
mathematicians to this issue and its mathematical substance. The mathematics to
which U.S. schoolchildren are exposed from preschool through eighth grade has
many aspects. However, at the heart of preschool, elementary school, and middle
school mathematics is the set of concepts associated with the term number.
Children learn to count, and they learn to keep track of their counting by
writing numerals for the natural numbers. They learn to add, subtract, multiply,
and divide whole numbers, and later in elementary school they learn to perform
these same operations with common fractions and decimal fractions. They use
numbers in measuring a variety of quantities, including the lengths, areas, and
volumes of geometric
figures. From various sources, children collect data that they learn to
represent and analyze using numerical methods. The study
of algebra begins as they observe how numbers form systems and as they
generalize number patterns. Mathematics is often taught in elementary school as
a set of algorithms without developing the conceptual understanding needed to
move to higher levels. US
teachers often have very good procedural understanding of the arithmetic of
integers, fractions and decimals, yet a profound
conceptual understanding in teachers is essential, as they must provide their
students with this needed understanding for reaching
algebra and even higher levels of mathematical thinking.
This seminar aims to show participants that deep understanding of elementary
ideas like place value is attainable in elementary
classrooms, and that one way to cultivate this understanding is through
irresistible problems. The seminar takes the position that learning mathematics
can be motivated by interesting problems. The trick is to come up with problems
whose solutions either require or strongly motivate the development of the area
of mathematics to be learned. One could also take the narrow position that
mathematics is about problem solving.
Fortunately, there are plenty of arithmetic and geometric problems that motivate
the need for algebraic thinking. And on top of
that, solving interesting mathematical problems in an appropriate social setting
can really be fun. Have a look at the problems
below. You might not be able to solve any of them on the fly. But with two or
three partner teachers, you can solve them all. Some of the problems below can
be used to build entire lessons. For example, the first problem could motivate
the entire section on place-value.
Below is a set of possible topics for the seminar.
List of Topics
Place
Value An essay by Roger Howe, Yale University and Susanna Epp, DePaul
University. Arithmetic, first of whole numbers, then of decimal
and common fractions, and later of rational expressions and
functions, is a central theme in school mathematics. This essay
attempts to point out ways to make the study of arithmetic more
unified and more conceptual through systematic emphasis of place
value structure in the decimal number system.
Ma, Liping (1999). Taken from an online review: `Elementary school
teachers are expected to teach almost everything: math, reading,
science, social studies, and writing; along with nurturing,
soothing, and encouraging. It's not an easy job. It's also hard to
be an expert in any one piece of the job. But now, many are
hearing that we're losing the ``math race'' to other countries.
The drums of ``teacher competency" are booming... and any wise
teacher knows where the drum sticks will be landing next!''
Using parity, KenKen, and difference triangles.
Base phi and Fibonacci Representation, dynamic one-pile nim
Conway's Rational Tangles...Tom Davis' style.
Euler's Formula, using Zome Tools
Single Pile Nim Games, and Bouton's Nim, Puppies and Kittens
Solving Linear and Quadratic Equations in Z_7
Other References:
The Major Topics of School Algebra by Wilfried Schmid and H. Wu
http://math.berkeley.edu/~wu/NMPalgebra7.pdf An essay
listing the topics in high school algebra essential for advanced
mathematics in college.
Arithmetic for Parents: A Book for Grownups about Children's Mathematics
by Ron Aharoni. Online review: `Ron Aharoni writes clearly and
deeply about the crucial concepts of fundamental maths, how to
teach them and how not to teach them. He explains the layered and
subtle structure of elementary maths and how missing a layer can
lead to frustration and maths anxiety. "There's no royal road the
maths", an Euclidian quote he emphasizes which summarizes well the
message in this book. I'm not sure the book is for ``Parents'' as
its title suggests, but I highly recommend it for both lovers and
``haters'' of maths, regardless of their ``parental status.''
Looking forward to Ron's next book. '
Here's a collection of units written by New Haven
mathematics teachers as part of the Yale-New Haven Teachers
Institute:
http://www.yale.edu/ynhti/curriculum/units/2004/5/