Instructor: Dr. Sergei Abramovich

Office: Satterlee 210; Phone: (315) 267-2541 (office);

E-mail: abramovs@potsdam.edu

**GRED 531: Creative problem
solving - Mathematics**

**Course description**

A major goal of teaching school mathematics is to encourage students to think clearly and correctly through exploring, analyzing, and resolving problematic situations, or, in short, to solve problems. This course concentrates on solving a variety of problems relevant to the National and NY State K-12 mathematics core curriculum. The focus is on exploration of various mathematics contexts through solving problems and their extensions, posing new problems, and communicating mathematical demonstrations. As far as high school mathematics content is concerned, the primary ground rule in this course is that problem situations can be investigated with pre-calculus mathematics using a variety of tools &endash; pictures, diagrams, graphs, numerical arrays, and manipulative. Another focus of the course is a student-centered discussion of some publications on problem solving in mathematics.

Problems that will be considered in the course will come from many sources and contexts. Some problems will arise in counting contexts; their solutions will require inductive reasoning and the development of a system in counting (e.g., counting the number of handshakes among n people). Several problems will arise in the context of measuring; their solutions will require the development of deductive arguments related to geometric objects. Many problems will arise through an extension, alteration, variation, or modification of a problem for which a successful strategy/solution has been already developed. Some problems will arise in a geometric context and will require the development of algebraic reasoning (for example, finding the largest area of a rectangle with a given perimeter). Other problems will arise in non-geometric contexts, yet their solutions will be developed through geometrization of a problem situation (e.g., explorations with linear and quadratic equations). Quite a few problems will be pure mathematical without an immediate application to a real-world situation; however, methods developed through solving these problems will help connect different mathematical ideas and concepts. Yet, many problems will stem from a holistic context allowing for the appreciation of mathematics content, methods, and meaning. In many instances, modeling with manipulative and computing technology will be incorporated into problem solving. This would make in possible to address the NY State Learning Standards' emphasis on the use of technology in mathematics classroom.

**Problem solving topics**

(i) Patterns; (ii) Visualization; (iii) Recreational mathematics; (iv) Finding sums of numbers with special properties; (v) Exploring digits of a whole number; (vi) Concepts of divisibility and factoring as problem solving tools; (vii) Algebraic word problems; (viii) Pythagorean triples and related topics; (ix) Investigations in plane geometry; (x) Investigations in solid geometry; (xi) Combinatorics and probability; (xii) Inequalities; (xiii) Problems involving parameters.

**Recommended materials and useful web
sites**

Curriculum and Evaluation Standards for School Mathematics. National Council of Teachers of Mathematics. Reston, VA: The Council, 1989.

Polya, G. (1945) How to Solve It: A New Aspect of Mathematical Method. Princeton, NJ: Princeton University Press.

Schoenfeld A. H. Learning to Think Mathematically: Problem Solving, Metacognition, and Sense-Making in Mathematics.

(Available on-line: http://www-gse.berkeley.edu/Faculty/aschoenfeld/).

Schoenfeld A. H. What Do We Know about Mathematics Curricula.

(Available on-line: http://www-gse.berkeley.edu/Faculty/aschoenfeld/).

Lesson plans in mathematics

(http://ericir.syr.edu/Virtual/Lessons/Mathematics/index.html).

NCTM Curriculum and Evaluation Standards for School Mathematics (http://www.enc.org/reform/journals/ENC2280/280dtoc1.htm).

NCTM Principles and Standards 2000

New York State Education Department

Problems "with a point" website.

A list of 101 resources for mathematics teachers.

**Text book and local server**

There is no textbook for the course. Consequently,
course assignments and curricular materials will be available as
handouts. In addition, several materials for the course may be put on
a local server (**ClassFiles **volume of the **Zeus** Server,
folder **GRED 531** within the folder **abramovs**). Everything
placed in **ClassFiles** is automatically "published" on the Web
at **http://zeus.potsdam.edu**. This option is convenient for
those students who will be using non-campus (home) computers. To
access ClassFiles on the Internet, make your way, via your browser,
to the folder GRED 531 and see available documents. Holding down the
OPTION (Mac) or SHIFT (Windows) key while clicking on a document will
cause the document to be downloaded to your computer. You can then
double-click on the document and open it, assuming that you computer
has the same program that was used to create it.

**Evaluation criteria**

Students are expected to attend, be prepared for and participate in each class. This includes the ability to discuss regular homework assignments and readings, share emerging ideas during problem solving sessions, use technology, present creative solutions to the class, turn in 3 write-ups and a final project in a due time. The course work will include keeping notes of the class discussions, and assignments in special mathematics portfolio. At the end of the session the students will be expected to submit portfolio with records of class and home works for the course. A final project will consist of the development of a lesson plan based on solving a problem relevant to K-12 curriculum and allowing for the implementation of Polya's four-step process for problem-solving.

Following are the components of the grade for the course:

Write-ups (typed on a computer) 40%

Portfolio for the course including all notes (may be hand-written) 20%

Presentation of readings 20%

Final project 20%

*In addition, extra points may be given for
best, creative solutions*.

**Documented Disabilities**

Any student who feels she or he may need accommodations based on a documented disability should see me after class, during my office hours, or by appointment. Students needing an Accommodation Plan should see Disabled Student Services.

*It is expected that all work will be the
students own otherwise documented. Failure to credit others for
direct quotations and ideas will be considered plagiarism and will
result in the student receiving a grade of 0.0 for that
assignment.*

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You can **visit
a web site** created by
the instructor in support of the NYS Mathematics Core Curriculum
(Learning Standard 3).