 # What I will talk about

### The Case for Geometry

a. We have done a poor job in helping students learn geometry (except in my class . . .)

NCTM President Hank Kepner --- Don't Forget Geometry

Former NCTM President Glenda Lappan -- Geometry, the Forgotten Strand

b. Why learn geometry?

Geometry begins with the study of the space in which we live. It provides us concepts and relationship to help us describe and understand that space.

Geometry supports the learning of other areas of mathematics

c. What is geometry?

The study of shapes and configurations: mathematics concerned with size, shape, and relative position of figures; descriptions of our physical world.

A context for developing reasoning, justification, proof, and axiomatic systems.

d. Where does geometry fit into today's curriculum?

Concepts of geometry are essential to the development of mathematics throughout the curriculum.

e. How should we teach geometry concepts

Lappan notes: ". . .we [must] give students the time throughout their school mathematics to explore geometry to its fullest--to play, observe, analyze, conjecture, imagine, represent, transform, create arguments, and simply experience the beauty and joy of geometry." (NCTM News Bulletin, December 1999)

Geometry is not just about axiomatics and proof. Approaching geometry via appropriate problem solving and using technology appropriately can make geometry understandable to students and lead to better understanding of mathematics concepts and processes.

# Potential benefits of appropriate technology use

### Generative tools for constructing one's further mathematics study.

Why Problem Solving?

To explore problem solving in mathematics as

. . . a curricular goal

. . . an instructional strategy

. . . the essential core of mathematics

. . . a process for doing mathematics

To develop a "can do" approach to mathematics problems solving.

To understand and describe mathematics problem solving as more process than product.

To become a mathematics problem solver.

To use technology to solve mathematics problems.

To use problem contexts to create mathematics demonstrations.

To use Contextual Teaching and Learning concepts.

To use problem solving to construct new ideas of mathematics for yourself.

To engage in mathematical investigations.

To engage in some independent investigations of mathematics topics from the secondary school curriculum or appropriate for that level.

To communicate mathematics ideas that arise from mathematics investigations.

To consider ways to assess problem solving performance.

# InterMath Students

### Limited mathematics background.

Centers of a triangle

Centroid, Orthocenter, Circumcenter, Incenter, Euler line
Medial triangle, Orthic Triangle, Triangle from Midpoints of Orthocenter/vertex segments
Nine Point Circle

Locus Explorations

Travels in 1992 with Bill (Orthocenter), George (Centroid), and Ross (Circumcenter)

Orthotravels. What is the locus of the orthocenter when one side of a triangle is fixed and the
third vertex is moved along some path?

Extended Concurrencies of the Triangle

Fermat points, Napoleon points, and others.
Kiepert hyperbola.

Pedal Points and Pedal Triangles

Pedal.gsp. Open the file, hide the perpendiculars and move point P anywhere in the plane.
Trace the lines defining PQR as P is moved in a circle.
Trace the lines defining PQR as P is moved around the Circumcircle of ABC. (
Pedal1.gsp)
Trace the center points of PQ, QP, and QR as P is moved in a circular path.

Squares. What is the ratio of areas of the two squares? Conics

Parabola, Ellipse, Hyperbola: Directrix is a line or circle.

Second order conics: Directrix is a conic.

GSP Lessons

Inscribe a parabola in a triangle. GSP Sketch.

Problems. The Web Site for my Mathematical Problem Solving Course.

Folium of Descartes -- and what if?

Folium

Translation to (x - a, y - b)

Transform to (sin x, siny)

Solve: ABC = 4; 3A + 2B - C = 3

Investigate Research into Practice An article on a Synthesis of Research on Mathematics Problem Solving prepared by me, Nelda Hadaway (a classroom teacher) and Maria Fernandez (an experienced teacher and doctoral student).

Some Research Questions (Maybe)

-- How does emphasis on measuring inhibit reasoning and proof?

-- Particular issues of learning indirect proof.

-- Role of axiomatics; rigor vs rigor mortis

-- Geometry in a GPS curriculum . . .

-- Role of technology

--Impact of technology on learning

-- geometric concepts
-- proof

-- Problem solving in geometry

-- How to teach geometry in a GPS (Georgia Performance Standards) environment.

-- Teaching solid (i.e. 3-D) geometry

-- Comparison of Geometry learning in different countries

-- Does the Van Heile Framework for Levels of Geometry Learning provide guidance for us?

-- What is the role of constructivist epistemology or social constructivism in our understanding of Geometry, Problem Solving, and Technology?