Geometry: An Interactive Journey to Mastery [repost]

Inscribed over the entrance of Plato’s Academy were the words, “Let no one ignorant of geometry enter my doors.” To ancient scholars, geometry was the gateway to gaining a profound knowledge of the world.$1#$ Today, geometry’s core skills of logic and reasoning are essential to success in school, work, and many other aspects of life.

Like other math fields, geometry teaches us how to think. It leads students to uncover new truths based on already established ideas and facts. It requires us to test and examine the conclusions of others. It teaches us to lay out our thinking clearly, describing each step so that others can follow along and verify our results.

This systematic way of thinking is essential in many fields. Drawing conclusions from experimental data is the basis of scientific discovery. Our justice system depends on compelling evidence to render a verdict in a court of law. And we use logical reasoning in everyday conversations to win friends over to our point of view.

In fact, the great Greek scholar Euclid demonstrated just how much you can do with logic. He worked out that basically all of geometry stands on just 10 core principles. You can build the rest using straightforward, logical reasoning.

In short, geometry is among the great intellectual feats of humankind. However, geometry goes far beyond being just an intellectual exercise. Its real-world applications extend to navigation, architecture, engineering, physics, technology, and even art.

Botanists use the geometry of triangles to estimate the heights of trees.
Astronomers use an understanding of ellipses to describe the orbits of planets.
Quantum physicists use the mathematics of rotation to explain aspects of subatomic physics.
Architects use principles of symmetry to develop aesthetically pleasing buildings.
Engineers use the properties of parabolas to design headlights and satellite dishes.
With its powerful blend of intellectual accomplishment and practical application, it’s no wonder that most schools consider geometry a core subject. Yet as award-winning Professor James Tanton of The Mathematical Association of America shows in Geometry: An Interactive Journey to Mastery, geometry can be an exciting adventure at any age. Those who will benefit from his 36 clear and accessible lectures include

high school students currently enrolled in a geometry class;
their parents, who seek an outstanding private tutor for their students;
home-schooled students and others wishing to study high school geometry on their own;
collegestudents who are struggling with math requirements and who need to strengthen their grasp of this fundamental subject; and
anyone curious about the intellectual challenge of logic and reasoning that underlies mathematics, the sciences, and our technological world.

Professor Tanton’s excellent teaching style makes the course ideal for those students who have ever believed they’re “not good at math” or have had challenges understanding geometry in the past.

A Different Way to Learn Geometry

Even students who have done well in other math courses such as algebra can sometimes find geometry a challenge. More so than algebra and other equation-based math, geometry places particularly strong focus on making logical inferences from facts and building a story of reasoning. Plus, geometry involves a more visual approach—working with shapes and patterns from the real world.

Many geometry courses begin by teaching the results of geometric thinking—by listing a set of beginning rules first. But how can one build the foundations of a house without first having a sense of what the house should be? Professor Tanton encourages students to start by playing with ideas of the mind (and acts of the hand!) to develop a feel for geometric rules and a context for those rules.

In Geometry: An Interactive Journey to Mastery, Professor Tanton guides students as they build an understanding of geometry from the ground up. With this approach, the instruction focuses on the intellectual play of the subject and its beauty as much as its utility and function. Students begin with elementary building blocks like points, lines, and angles and observe how those basic units interact.

From a clear understanding of the fundamental principles, students use logical reasoning to expand their understanding of geometry. Like building a house brick by brick, each new discovery stands upon the others—without any sudden or confusing jumps.

Lectures:

1 Geometry—Ancient Ropes and Modern Phones
2 Beginnings—Jargon and Undefined Terms
3 Angles and Pencil-Turning Mysteries
4 Understanding Polygons
5 The Pythagorean Theorem
6Distance, Midpoints, and Folding Ties
7 The Nature of Parallelism
8 Proofs and Proof Writing
9 Similarity and Congruence
10 Practical Applications of Similarity
11 Making Use of Linear Equations
12 Equidistance—A Focus on Distance
13 A Return to Parallelism
14 Exploring Special Quadrilaterals
15 The Classification of Triangles
16 "Circle-ometry"—On Circular Motion
17 Trigonometry through Right Triangles
18 What Is the Sine of 1°?
19 The Geometry of a Circle
20 The Equation of a Circle
21 Understanding Area
22 Explorations with Pi
23 Three-Dimensional Geometry—Solids
24 Introduction to Scale
25 Playing with Geometric Probability
26 Exploring Geometric Constructions
27 The Reflection Principle
28 Tilings, Platonic Solids, and Theorems
29 Folding and Conics
30 The Mathematics of Symmetry
31 The Mathematics of Fractals
32 Dido's Problem
33 The Geometry of Braids—Curious Applications
34 The Geometry of Figurate Numbers
35 Complex Numbers in Geometry
36 Bending the Axioms—New Geometries