ISLAND SCHOOL MATHEMATICS 

Program Overview

The Island School Mathematics program seeks to fully utilize the resources inherent to our special place. We aim to produce capable, creative problem solvers who can understand the world through the lens of mathematics. Specifically, our mathematics program is guided by four enduring understandings we strive to instill in each of our students:

• Mathematics is the study of how to think;
• Mathematics is a skill that needs to be practiced;
• Everyone is capable of learning and mastering mathematics;
• Mathematics explains the universe in which we live.

We hope to show students that mathematics is a creative, thoughtful enterprise that leads directly to understanding our world. In accordance with these aims and The Island School’s commitment to fostering an experience that is truly place-based, our mathematics program teaches the theory and practice of celestial navigation. 

The Island School Mathematics course in celestial navigation seeks to develop an appreciation for the power of mathematics to analyze the world in which we live, as well as nurture a sense of wonder about the night sky and the ocean.  Our program focuses on challenging students to solve the classic problems from the history of science, mathematics and navigation: What is the circumference of the Earth? How do we find the longitude? What does the altitude of a celestial body at my meridian indicate about my latitude? What is the angular speed of the moon’s revolution around the Earth? What is the arc-measure between any two points on the surface of a sphere? Where am I? These questions, and others like them, are explored in detail during the 13-week Island School term.

Celestial navigation connects the most interesting problems in modern geometry and trigonometry to the practice of determining one’s location on the surface of the Earth. In addition, celestial navigation links the study of mathematics to the artistic, scientific and philosophical musings about the night sky that are ancient as human history.  In this way, the course becomes a multidisciplinary synergy of mathematics with the other partitions of the academy.  We are pleased to offer our students this exciting opportunity that is fully integrated with our special place.

Celestial Navigation—Course Description

Celestial navigation is an applied trigonometry course that teaches students the theory and practice of navigation by starlight. Until the advent of GPS, celestial navigation was the bedrock of all navigation science because it is failsafe, elegant and remarkably accurate.  A skilled navigator, in optimal conditions, working only with a sextant and a watch, can determine the location of his or her ship to within .2 minutes of arc—a position error of only 400 yards. The scientific theory at the foundation of celestial navigation teaches students about the motion of the Earth, the seasons, the wandering daylight problem, the stars, the planets and the celestial sphere.  Within this rich conceptual framework, students develop spherical geometry and spherical trigonometry in order to model the surface of the Earth and its luminous container, the night sky. In addition, students learn the practice of celestial navigation; our campus features uninterrupted views of the Northern, Southern, and Western horizons, allowing students the opportunity to develop skill using a sextant to find the altitude of a range of celestial bodies. They are taught to use a nautical almanac to find ephemeris data, to reduce sights with and without the use of H.O. tables and to plot the position of a vessel at sea on a chart.  In particular, the practical skills students acquire include:

• Charting a dead reckon track
• The Use, Maintenance and Correction of a Mariner’s Sextant
• Shooting the noon sun
• Shooting a dusk star (or planet)
• Shooting a twilight star (or planet)
• Taking a bearing with a compass
• Correcting a compass for magnetic deviation
• Plotting a position fix on a chart

The mathematical component of the course is driven by world problems that develop the geometry and trigonometry required for celestial navigation.  Problem sets are preceded by readings that equip students with the conceptual framework to create the mathematics necessary to answer a particular problem. Through this process students develop their mathematical modeling and problem solving skills.  The aim is to graduate students who are confident, mature problem solvers.

The course is an applied geometry and trigonometry course.  While many ideas from the plane geometry of Euclid are useful when modeling navigation problems, such as finding the distance to the horizon, students are also introduced to the geometry of curved surfaces.  The familiar properties of lines and shapes inherent to figures incident with a plane do not hold for figures lying on a sphere.  In particular, students study non-Euclidean geometry including great circles, loxodromes, spherical triangles and tangent planes.  The specific list of mathematics topics covered during the course includes:

I. Coordinate systems:

              a. Definition of a coordinate system;

              b. Coordinates on a plane;

              c. Coordinates on the surface of a sphere;

II. Spherical Geometry:

                 a. The elliptical parallel postulate;

                 b. Great circles, small circles and defining distance;

                 c. Arc measure, arc length, central angles, latitude angles;

III. Tangents and parallel lines:

              a.  Definitions of parallel lines;

              b.  Definitions of tangent lines;

              c.  Relations between tangents to a circle and parallel lines;

IV. Introduction to trigonometry:

              a. Special right triangles;

              b. Trigonometric ratios defined on a right triangle;

              c.  Trigonometric ratios defined on the unit circle;

              d. Solving triangles;

              e. Inverse trigonometry;

V.  Constructions:

              a. Introduction to the history, theory of compass and straightedge constructions.

              b. Constructing line segments and circles;

              c. Bisecting angles and segments;

              d. Constructing perpendiculars, parallels, similar triangles;

              e. Constructing the trigonometric ratios;

VI. Applied trigonometry:

              a. Law of sines;

              b. Law of cosines;

              c. Problems relating angular diameter, parallax angle and distance;

VI Spherical Trigonometry:

                 a. The angle-arc relations of spherical triangles;

                 b. The Navigational Triangle;

                 c. The spherical law of sines;

                 d. The spherical law of cosines;

Other mathematics topics may include vector algebra, systems of linear and non-linear equations, the geometry of map projections and a comparison of orthodromes to loxodromes.  The problem sets sample a broad spectrum of assumed mathematical knowledge; this wide range allows students to be challenged at their individual level of mathematics. 

              In addition to hours spent in the classroom and during study hall, the course includes frequent opportunities for students to take their mathematics education in their own hands, into the world.  To better understand the movements of the celestial sphere, students often appeal to the night sky to illustrate the spherical coordinate systems that describe the positions of objects in the sky.  Field trips include excursions to a nearby sandbar to investigate the distance to the horizon, afternoons with sextants on the beach and on boats to develop skill shooting stars and treks to chart the latitude and longitude of uninhabited cays. Through connecting mathematical theory to the practice of answering interesting, available questions, the course cultivates the students’ ability to solve problems that require creativity, patience and persistent effort. 

              A sample reading from The Island School’s Mathematical Introduction to Celestial Navigation is available here, complete with sample problems. A sample examination is available here. 

              Please contact Justin Dimmel, head of The Island School mathematics department, at: justindimmel@islandschool.org if you have any questions.

Student Comments:

"The difference is that in this class it's more applicable and easier to understand when we are actually doing real life math problems and not learning out of a book."

"The projects are a group effort, everyone has a role for that project and it is different every time"

"I can't think of any one of the classes that didn't enhance my mathematical reasoning ability. I especially like when we do activities outside and apply what we've learned with the resources around us."

"I think that everything we have done has greatly heightened not only my affinity for math, but the way in which I think about problems."

"By doing hands on work the concept formulizes in my head clearer and reinforces the importance of understanding concepts rather than memorizing formulas or numbers"

"Prior to attending The Island School, I had no tangible examples of where aI could find or implement the theorems and equation I was learning in class. Having no application for my knowledge, I immediately wrote off math and concentrated on other things"

"It helped me to get math off of the page and into my head. A person can have a different connection with an idea when they have experienced it and actually seen how it is used in life."