Mathematics
 | | "Euclid alone has looked at Beauty bare."--Edna St. Vincent Millay |
Our goal in teaching students mathematics is threefold:
Math and development of the adolescent
The development of abstract thinking is the overarching achievement of the teenage years. Its use in understanding the world can yield order, harmony, and even beauty; it is balm for the turbulent teenage soul. The logic and rigor involved in the study of mathematics supports the development of abstract thinking. Triangles are everywhere around us, for example, but to understand them we must separate them from the physical world. This is sense-free thinking; this is abstract thinking.
Mathematics is also fulfilling for the adolescent because there is almost always an exact answer at which we can--with rigor, logic, and thoroughness--arrive. As we develop this skill of "pure" abstract thinking through mathematics, we can apply it to other less exact domains, such as history and literature.
Math and the real world
No branch of mathematics, no matter how formal or abstract, is without significant applications in understanding the world around us. Knowledge of the physical and social worlds has been advanced and deepened through mathematical modeling and thinking. The connection between abstract math and real world utility is mysterious: Why should the seemingly chaotic real world be amenable to mathematical description in field after field? This profound connection, namely the power of using mathematics to understand the world, is of practical interest for students. No matter what field of study or work they pursue, they will find math useful.
The wonder of math
Many students arrive at Waldorf believing that they are not capable at math or that math is a dull and dry subject. Essential goals of the math program at Waldorf High School are to build confidence and to reveal the fun and beauty of mathematics.
 Permutations & Combinations, Ninth Grade Main Lesson, Mr. Claus
Math Course, Ninth Grade, Mr. Raizen
Trigonometry, Tenth Grade Main Lesson, Mr. Raizen
Math Course, Tenth Grade, Mr. Raizen and Mr. Claus
Projective Geometry, Eleventh Grade Main Lesson, Mr. Raizen
Math Course, Eleventh Grade, Ms. Ruggiero and Mr. Raizen
Math Course, Twelfth Grade, Mr. Raizen and Mr. Claus
Permutations & Combinations, Ninth Grade Main Lesson, Mr. Claus
Counting is the most basic of all mathematical skills--even toddlers can do it. Counting all the possibilities when a set of objects is grouped together in various ways is not so easy. If an automobile license plate consists of three letters followed by three numbers, how many possible license plates are there? How many different ways can a group of twenty students shake hands? How many possible five card poker hands are there and how many of them beat a pair of sevens? This course introduces the precise mathematical formulas that are used to count such groupings and continues by delving more deeply into the properties of whole numbers and of probabilities.
-Go to top-Math Course, Ninth Grade, Mr. Raizen
This course begins with a thorough review of basic math and algebra skills. We continue with such algebra topics as inequalities, polynomials, rational expressions, roots and radicals, complex numbers and linear and quadratic functions. Word problems and problem solving are introduced.
-Go to top-Trigonometry, Tenth Grade Main Lesson, Mr. Raizen
How can we accurately measure things that are so big that using a ruler is either impractical or impossible? The ancient Greeks invented trigonometry to accurately measure the path of stars in the night sky. Trigonometry was rediscovered during the Renaissance, and was used not just for astronomy, but in maritime navigation. Trigonometry is still widely used today in surveying, for example in mapping the route of a new highway. In this block, we will study the theoretical foundations of trigonometry , and then put it to practical use measuring building heights and other things.
-Go to top-Math Course, Tenth Grade, Mr. Raizen and Mr. Claus
This course is an in-depth study of geometry, focusing on the properties of triangles, quadrilaterals, polygons, circles and solids. We cover the Euclidean constructions and deductive logic, including reading and writing proofs. The course may also cover conic sections, symmetry and tessellations. Ninth grade algebra is reviewed. The extended path covers more advanced topics, while the general path has more time for review.
-Go to top-Projective Geometry, Eleventh Grade Main Lesson, Mr. Raizen
Projective geometry is essentially a geometry that agrees with our sense of sight. The sides of a square that is moving away from us seem to become smaller. If the square is rotated, the right angles deform, and the square becomes a trapezoid. Parallel lines appear to converge and meet. Projective geometry arose from the attempts by Renaissance artists to accurately capture perspective on a two-dimensional canvas. Projective geometry captures the fundamental principles of such perspectives in simple and elegant ways that can, at the same time, be difficult to imagine. Flexibility of thinking and new concepts of space are developed.
-Go to top-Math Course, Eleventh Grade, Ms. Ruggiero and Mr. Raizen
The general path continues with the study of algebra, covering such topics as exponents, roots and radicals, linear and quadratic equations, and functions. The extended path begins an in-depth study of precalculus with a focus on understanding functions in general, and then polynomial, logarithmic and exponential functions. Both paths include work with applications, word problems, and problem solving.
-Go to top-Math Course, Twelfth Grade, Mr. Raizen and Mr. Claus
The general path completes the study of algebra with a focus on quadratic, logarithmic and exponential functions. Additional topics will be taken up to round out the students' mathematics education by showing the beauty, fun and/or utility of math. The extended path completes an in-depth study of precalculus, including advanced trigonometric functions. We will end with either a brief introduction to calculus, or with a statistics unit, depending on the interest of the class. In both paths, applications, word problems and problem solving are integrated into the coverage of each topic.
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