Teacher resources and professional development across the curriculum

Teacher professional development and classroom resources across the curriculum

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Mathematics Illuminated is a 13-part, integrated-media resource created for adult learners and high school teachers. The series covers the broad scope of human knowledge through the study of mathematics and its relevance in the world today. It reaches beyond formulas and computations to explore the math of patterns, symmetry, relationships, multiple dimensions, and more, all the while uncovering the secrets and hidden delights of the ever-evolving world of mathematics.

Mathematics Illuminated unites the strengths of traditional and new media learning through the coordinated 13 half-hour videos, online texts, web interactive activities, and group activities.

Developed with an Advisory Board of leading scholars and high school teachers from across the United States, the course examines mathematics from a holistic perspective, offering an enthusiastic approach that will help learners understand the broad spectrum of disciplines that make up the field of mathematics. The information and activities will help all learners solidify unfamiliar and/or complex concepts by delving into the "why" and not just the "how" of approaching mathematical questions. Mathematics Illuminated explores many of the great historical figures in mathematics and the problems they confronted; it also introduces you to some of the fascinating people who are embracing and studying today's emerging questions. Mathematics Illuminated provides an overview of how the content of the course connects and supports national and local standards in mathematics. Using these materials, teachers can simultaneously build content knowledge and explore and engage with new mathematical applications.

For information on graduate credit, click here.

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Several goals have guided all aspects of the development of the Mathematics Illuminated course materials.

The goals of the project are to help teachers and those with an interest in math to:

The course approaches mathematics as a means to greater understanding, rather than as a successive set of skills to be mastered. The series is designed to awaken one's senses to the power of math in one's everyday life, while enriching one's knowledge of the variety of inspiring subjects within the field of mathematics. The course integrates content throughout each medium (video, print, Web), in order to support different learning methods and styles.

The materials were specifically developed with two primary and one secondary audience in mind. In general, the materials were created to help interested, educated individuals, as well as high school teachers, develop a better understanding of the nature of mathematics. One primary target audience is educated adult learners who have either a mathematics undergraduate degree or an interest in mathematics (or both). The second primary target audience is in-service or pre-service high school mathematics teachers seeking a professional development experience focused on new, in-depth mathematical knowledge, as well as inspiration for viewing the world of math as an integrated whole.

The secondary audience for the series is non-teaching mathematicians who will use the materials informally for personal and professional development, or anyone interested in getting an overview of the mathematical landscape as it stands today.

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The selected topics and themes covered in the course were carefully chosen under the guidance of the full Advisory Board to align with the project goals. The Board, comprised of high school math teachers, university professors, and professionals in the fields of education and professional development, chose topics spanning the depth and breadth of the field of mathematics, from humankind's earliest study of prime numbers to the cutting-edge mathematics used today to reveal the shape of the universe (see details below under "Unit Descriptions"). The Board's goal was to help the audience appreciate math's wonder, breadth, relevance, and usefulness. Together they chose topics that could be appreciated by a diverse collection of people with varied interests and talents.

Some of the topics explored in the course, such as game theory or chaos theory, may be new to some, yet review for others. Other topics are rooted in traditional mathematical concepts, such as prime numbers and geometry. However, all of the topics covered introduce a new way to approach learning and understanding as they lead learners on a journey to new discoveries, or "Aha!" moments, within the themes covered.

A rigorous test of the course was conducted by an independent evaluation company (more information below). Adult learners and math teachers evaluated a prototype of the video, text, and Web site materials. Results of the evaluation were taken into consideration, and the final course components were adjusted based on the results and recommendations of the evaluators.

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In the creation of the materials for this course, it was assumed that the audience would be diverse in its exposure to, and level of understanding of, mathematics and its role in the world. For this course, we focused on individual adult learners, and high school teachers, acknowledging the variety of experience in how these groups would engage with, and learn and/or teach the mathematical content. Teachers will have familiarity with some of the concepts covered in the series, and may also have a selection of state and/or local mathematics standards to adhere to, while others may either have a very specific interest in one content area, such as prime numbers, or topology, or a broad interest in mathematics as a source for understanding the world around us. Regardless of the level of motivation or experience, we assume that all are pressed for time and/or resources and therefore expect our materials to clearly yet concisely illuminate mathematics: its theory, history, inherent beauty and role in past and present understandings of our world.

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Mathematics Illuminated is a multimedia course. Each of the 13 content units is comprised of a half-hour video program, an online textbook chapter, and group or individual hands-on learning activities. In addition, nine of the thirteen units have a web-based interactive. This web site provides access to all of the materials, as well as additional resources, including:

Any component can be used independently, or they can all be used together. If used together, we recommend progressing through the materials unit by unit, viewing the video first, then reading the online text, and finally completing the hands-on activities. The Web interactives, and all other Web content are meant to support the learning experience.

Video Series
The videos reach the broadest audience by providing poignant, thought-provoking entry points into the concepts explored in the series. There are three major components in each of the 13 half-hour videos. First, they provide the historical context of the mathematical concepts, bringing to life some of the great mathematicians of all time, along with their struggles and breakthroughs, through visually compelling historical re-enactments. Next, they provide a deeper understanding of the concepts through interviews with leading experts. These enlightening discussions are augmented with illustrative animations that reinforce and explain graphically the concepts that are being explored intellectually. Finally, they take you on a field trip to view real-world applications of the concepts and introduce you to people who are using the mathematical ideas and techniques every day in their own unique, relevant, and wonderful ways.

Online Textbook
The online textbook chapters explore in greater detail the ideas and themes examined in the videos and Web interactives. Each of the 13 chapters has an overview with clear unit and sub-section objectives that go deeper into the content, while modeling the arc provided in the videos. Most units begin by explaining the historical context of the unit content, and then go on to explore the field of study and its relevance. Completing the experience are numerous high-quality explanatory diagrams and illustrative pictures.

Web Site and Interactives
This web site serves as the entry point to all of the text and video materials, while providing links to additional resources that support the course themes. It includes Web-based learning interactives that enable the user to engage in deeper explorations of some of the more powerful, beautiful, and challenging concepts from the series.
The nine interactives are:

For example, the Metronome interactive simulates the correlation of a metronome's ticking to a sine wave. The Robot interactive allows participants to experience hands-on learning of an abstract concept.

This site also contains a glossary, organized by unit, and The Math Family Tree, which maps all of the content from the series on a dynamic timeline. This timeline illustrates how early mathematicians and their breakthroughs (as far back as 25,000 B.C.E.) influenced those that came later.

Users with a broadband connection can also view streaming video of the series and can download the textbook chapters.

Facilitator's and Participant Guides
Guides for each unit strengthen teachers' understanding of the content through hands-on activities for individuals or group workshops. Activities include concept application, problem-based learning activities, and opportunities for analyzing data—all in an inquiry-based model. A framework for applying concepts to state and local standards is included.  The activities seek to model best teaching practices by requiring the participant to be an active explorer rather than a passive recipient of information and are designed to be used between each of the video segments and to draw from content in the unit's video, and the text chapter.

The Facilitator Guide includes notes on how to conduct the session, the list of materials needed, tips, and answers.

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Unit 1. The Primes
The properties and patterns of prime numbers—whole numbers that are divisible only by themselves and one—have been a source of wonder across cultures for thousands of years, and the study of prime numbers is fundamental to mathematics. This unit explores our fascination with primes, culminating in the million-dollar puzzle of the Riemann Hypothesis, a possible description of the pattern behind the primes, and the use of the primes as the foundation of modern cryptography. Guests interviewed include Terence Tao, UCLA, and Ray Perlner, National Institute of Standards and Technology.

Unit 2. Combinatorics Counts
Counting is an act of organization, a listing of a collection of things in an orderly fashion. Sometimes it's easy; for instance, when counting people in a room. However, listing all the possible seating arrangements of those people around a circular table is more challenging. This unit looks at combinatorics, the mathematics of counting, or arranging, complicated configurations. In an age in which the organization of bits and bytes of data is of paramount importance—as with the human genome—combinatorics is essential. Guests interviewed include Jenny Quinn, University of Washington, Tacoma, and Terry Gaasterland from Scripps Genome Center, University of California, San Diego.

Unit 3. How Big Is Infinity?
Throughout the ages, the notion of infinity has been a source of mystery and paradox, a philosophical question to ponder. As a mathematical concept, infinity is at the heart of calculus, the notion of irrational numbers, and even measurement. This unit explores how mathematics attempts to understand infinity, including the creative and intriguing work of Georg Cantor, who initiated the study of infinity as a number, and the role of infinity in standardized measurement. Guests interviewed include James Tanton, St. Mark's Institute of Mathematics, and Ivars Peterson, Mathematical Association of America.

Unit 4. Topology's Twists and Turns
Topology, known as "rubber sheet math," is a field of mathematics that concerns those properties of an object that remain the same even when the object is stretched and squashed. In this unit we investigate topology's seminal relationship to network theory, the study of connectedness, and its critical function in understanding the shape of the universe in which we live. Guests interviewed include Greg Leibon, Dartmouth College, and Jeffrey Weeks, author of The Shape of Space: How To Visualize Surfaces and Three-Dimensional Manifolds.

Unit 5. Other Dimensions
The conventional notion of dimension consists of three degrees of freedom: length, width, and height, each of which is a quantity that can be measured independently of the others. Many mathematical objects, however, require more—potentially many more—than just three numbers to describe them. This unit explores different aspects of the concept of dimension, what it means to have higher dimensions, and how fractional or "fractal" dimensions may be better for measuring real-world objects such as ferns, mountains, and coastlines. Guests interviewed include Greg Leibon, Dartmouth College, and Karl Richter of LAIKA/house.

Unit 6. The Beauty of Symmetry
In mathematics, symmetry has more than just a visual or geometric quality. Mathematicians comprehend symmetries as motions—motions whose interactions and overall structure give rise to an important mathematical concept called a "group." This unit explores Group Theory, the mathematical quantification of symmetry, which is key to understanding how to remove structure from (i.e., shuffle) a deck of cards or to fathom structure in a crystal. Guests interviewed include Rosa Orellana, Dartmouth College, and Mary McGrath, Structural Chemist at Gilead Sciences, Inc.

Unit 7. Making Sense of Randomness
Probability is the mathematical study of randomness, or events in which the outcome is uncertain. This unit examines probability, tracing its evolution from a way to improve chances at the gaming table to modern applications of understanding traffic flow and financial markets. Guests interviewed include Raissa D'Sousa, University of California, Davis, and Anthony Baerlocker, International Game Technology.

Unit 8. Geometries Beyond Euclid
Our first exposure to geometry is usually the system proposed by Euclid, in which all triangles have 180 degrees. As it turns out, in curved space the sum of a triangle's angles can be more or less than 180 degrees. This unit explores these curved spaces, which are at once otherworldly and yet firmly of this world—and may hold the key to understanding the human brain. Guests interviewed include Daina Taimina, Cornell University, and Martin Styner and Brad Davis of the Computer Science, and Psychiatry Department's, University of North Carolina, Chapel Hill.

Unit 9. Game Theory
Competition and cooperation can be studied mathematically—an idea that first arose in the analysis of games like chess and checkers, but soon showed its relevance to economics and geopolitical strategy. This unit shows how conflict and strategies can be thought about mathematically, and how doing so can reveal important insights into human and animal behaviors. Guests interviewed include David Krakauer, Santa Fe Institute, New Mexico, and Craig Packer, Dept. of Ecology, Evolution, and Behavior, University of Minnesota.

Unit 10. Harmonious Math
All sound is the product of airwaves crashing against our eardrums. The mathematical technique for understanding this and other wave phenomena is called Fourier analysis, which involves the disentangling of a complex wave into basic waves called sinusoids, or sine waves. In this unit we discover how Fourier analysis is used in creating electronic music and how it actually underpins all digital technology. Guests interviewed include Elizabeth Stanhope, Mathematical Sciences, Lewis and Clark College, and David McCraaken, Jason Daniello, Aaron Grooms, and Cyril Lance of Moog Music, Inc.

Unit 11. Connecting with Networks
Connections can be physical, as with bridges, or immaterial, as with friendships. Both types of connections can be understood using the same mathematical framework called network theory, or graph theory, which is a way to abstract and quantify the notion of connectivity. This unit looks at how this branch of mathematics provides insights into extremely complicated networks such as ecosystems. Guests interviewed include Raissa D'Souza, University of California, Davis, and Neo Martinez, Rocky Mountain Biological Laboratory.

Unit 12. In Sync
Systems of synchronization occur throughout the animate and inanimate world. The regular beating of the human heart, the swaying and near collapse of the Millennium Bridge, and the simultaneous flashing of gangs of fireflies in Southeast Asia—these varied phenomena all share the property of spontaneous synchronization. This unit shows how synchronization can be analyzed, studied, and modeled via the mathematics of differential equations, an outgrowth of calculus, and the application of these ideas toward understanding the workings of the heart. Guests interviewed include Steve Strogatz, Cornell University, and Glenn Fishman and Charles Peskin of the New York University Medical Center.

Unit 13. The Concepts of Chaos
The flapping of a butterfly's wings over Bermuda causes a rainstorm in Texas. Two sticks start side by side on the surface of a brook, only to follow divergent paths downstream. Both are examples of the phenomenon of chaos, characterized by a widely sensitive dependence of the future on slight changes in a system's initial conditions. This unit explores the mathematics of chaos, which involves the discovery of structure in what initially appears to be randomness, and which imposes limits on predictability. Guests interviewed include Steve Strogatz, Cornell University, and Martin Lo, Jet Propulsion Laboratory, California Institute of Technology.

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