Mastery Project

Mathematics and Science Teacher Education

in Yemen

May 16 - 20, 2009

Jim Wilson

Workshop Leader

This is the web site page is for use in workshop, in Sana'a Yemen, for the Mastery Project, Mathematics and Science Teacher Education in Yemen.

The material is taken from the EMAT 6600 Problem Solving in Mathematics course that is taught at the University of Georgia, as organized and led by Jim Wilson. Other instructors at the University of Georgia and elsewhere use some of the material in similar courses but no links to other courses is provided here. The material for the workshop draws heavily from EMAT 6600. Click here to go to the EMAT 6600 web site if that is desired.

EMAT 6600 Problem Solving in Mathematics is taught in the Department of Science and Mathematics Education. At other universities a similar course is often taught in a Department of Mathematics.

Special Web instruction: Use the browser back key to return to this workshop page. Links provided in the problem sets will return you to the EMAT 6600 page rather than the workshop page. I have placed a link on the EMAT 6600 page to return here if you inadvertantly go that route.

Last
modified on** April 13, 2009**

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e-mail to **jwilson@uga.edu**

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Wilson's Home Page**

-- To explore what a course on mathematics problem solving for secondary school teachers might be.

-- To explore the objectives and assessment of a mathematics problems solving course.

-- Doing mathematics requires doing problem solving.

-- Using technology when appropriate

-- Other descriptions of Problem Solving -- inquiry, exploration, reasoning, proving

References:

• Polya, G. (1945) How to Solve It: A New Aspect of Mathematical Method. Princeton, NJ: Princeton University Press. Second edition 1956.

• Polya, G. (1965) Mathematical Discovery, Vol I & II. New York: Wiley.

Chapters 1-4 of Volume 1 contain extensive problem sets that Polya developed for The Stanford-Sylvania Mathematics Institutes for Teachers.

• Polya, G. (1954) Mathematics and Plausible Reasoning. Vol.1 Induction and Analogy in Mathematics; Vol. II Patterns of Plausible Inference. Princeton, NJ: Princeton University Press.

•Polya, G., & Kilpatrick, J. (1974) The Stanford Mathematics Problem Book. New York: Teachers College Press.

•Brown, S. I., & Walter, M. I. (1990) The Art of Problem Posing. 2nd edition. Hillsdale, NJ: Erlbaum.

• Courant, R., & Robbins, H. (1941) What is Mathematics? An Elementary Approach to Ideas and Methods. New York: Oxford University Press. (Revised Edition prepared by Ian Stewart, 1996)

•Taylor, H. & Taylor, L. (1993) George Polya: Master of Discovery. Palo Alto, CA: Dale Seymour Publications.

•Steinhaus, H. (1950) Mathematical Snapshops. New York: Oxford University Press.

Other Resources

Polya, G. (1964) Let us Teach Guessing. Film/Video. Washington, DC: Mathematical Association of America

Math Forum(alias Geometry Forum). Formerly at Swarthmore. Now at Drexel University. (HIGHLY RECOMMENDED)

Mathematical Association of America

Mathematical Sciences Education Board

National Council of Teachers of Mathematics

A Visual Dictionary of Special Plane Curves

MacTutor History of Mathematics Site -- St. Andrew's University

SIAM

**EMAT
6600 Syllabus** - General
information about the objectives and operation of the course.

**PAPERS:**

**Synthesis of
Research on Problem Solving**. This paper was published as Chapter 4 in

Wilson, P. S. (Ed.)
(1993). *Research Ideas for the Classroom:
High* *School Mathematics*. New
York: MacMillan.

The book was part of
the National Council of Teacher of Mathematics Research Interpretation Project,
directed by Sigrid Wagner.

The bibliographic
reference for the published version is

Wilson, J. W.,
Fernandez, M. L., & Hadaway, N. (1993). Mathematical problem solving. In P.
S. Wilson (Ed.), *Research Ideas for the
Classroom: High* *School Mathematics*
(pp. 57-78). New York: MacMillan.

**Squares.** X What is the ratio of
areas of the two squares?
This is a discussion of some exploration and extensions of this problem.

**Roots 2
and 5**. This paper examines sets of equations that have graphs crossing
the x-axis only at 2 and 5. For a preview on one family of such graphs, click **here**.

**Problem Solving
with Heron's Formula**. X This is a paper on the development and demonstration of
Heron's formula for the area of a triangle given the lengths of its three
sides. Problems and explorations are included for using Heron's formula.

**An
Investigation with Parametric Equations**. This paper examines the movement
of triangles when one vertex is moved along the x-asis and another is moved
along the y-axis. We trace trace the movement of the third vertex.

**Extended
Concurrencies of the Triangle**

**PROBLEMS:**

**Project InterMath **

**Project InterMath has a web site with many investigations
deemed appropriate for Middle School mathematics teachers. I believe secondary
mathematics teachers may find some challenges here as well. Please access the
site: Project INTERMATH**

**Environment Problems**

**Volume of an
Irregular Solid ** X

**The Three
Point Problem from Geology**

**Volume of a
curved trench, Trapezoid Cross-section**

**Angle
Bisector of two line non-intersecting line segments** X

**Arc Length
equal to a given segment**

**Bisector of an
angle of a triangle** X

**Bisectors Problem in 120 degree Obtuse
Triangle** X

Bouncing Barney X

Ceva's Theorem X *new*

**Circles of
Apollonius for a Triangle ABC** X

**Color a Circle**

**Coloring Circles**

**Divide a Square into a Set of Acute Triangles X** *new*

**Equilateral
Triangle Altitude Theorem**

**Find
Inscribed Square for Given Triangle** X

**Find
Inscribed Rectangle with Maximum Area for Given Triangle** X

**Folding a sheet
of paper into equal areas** X

**Half the Area
of a Triangle: A Line Parallel to a Side**

**Half the
Area of a Triangle: A line Through a Point on the Side**

**Inscribed
Equilateral Triangle in a Square** – Construction X

**Inscribed Equilateral Triangle in a Square** – Problem X

**Inscribed
Triangle** in a given triangle X

**Isosceles
Right Triangles--Path of the Mid-Point**

**Isosceles Right
Triangles With a Common Vertex**

**Isosceles
Trapezoid -- Equal areas** X

**Kite** X

**Lines
of Symmetry in a Polygon**

**Locus
of intersection of two secants**

Minimum Path X

Networks of
Minimal Length

**Pairs of
Segments with the Same Sum of Lengths**

**Parallelogram with Integer Sides and
Integer Diagonals**

**Partition
Square into Acute Triangles ** X

**Perpendicular
Chords in a Circle**

**Points closer to the Centroid than the
Sides**

**Projections: Homothetic Similarity**

**Quadrilateral
Inscribed in a Semicircle**

**Quadrilateral
with Maximum area**

**Quadrilateral
with Squares on the Sides**

**Ratio on a line
segment** -- Something Golden

**Ratio: Segments
cut off by an angle bisector to the adjacent sides **

**Right
Triangle with perimeter 60, altitude to hypotenuse 12**

**Square
Inscribed along a base of any Triangle** X

**Squares
Inscribed in a Right Triangle** X

**Squares on the
Sides of a Parallelogram**

**Tangent
Lines Common to Two Given Circles **X

**Three Circles Problem**

**Three Intersecting Circles**
X

**Trapezoid
Inscribed in a Semicircle**

**Triangle
areas/Triangle Built on Outside of a Given Triangle**

**Triangle Area
and the Circumcircle**

**Triangles
with Integer Area and Integer Sides**

**Volume of Holes
Left by Tree Spade **

Algebra Problems

**Solve**
by iteration.

**
****Comparision of Two Radical Expressions** X

**Linear
functions tangent to their product function **

**Solve**
by iteration

Cryptarhythm Problems

"Mean" Problems

**Arithmetic Mean --
Geometric Mean Inequality ** X

**Areas
in square of sides a+b** X

**Area of a
Sector of a Circle ** X

**Comparing
Segments in two circles**

**Comparison
of altitude and median in a right triangle**

**Maximum
and Minimum of (x/(1+x^2))**

**Maximum of
f(x) = (1-x)(1+x)(1+x)**

**Minimum
Surface area of a can of fixed volume** X

**Sum of
Two Sines or Sum of Two Cosines**

Mixture of all

**Four
Number Challenge Problem ** X

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Wilson's Home Page**