Andrew Starzynski

Who am I? Born and raised in Chicago, IL, USA. Waldorf lifer; both parents Waldorf teachers. Bachelor’s in philosophy, mathematics and computer science. Master of Science in Applied Mathematics. Teacher training in Wilton, NH under Jamie York. Taught high school and middle school mathematics in Waldorf Schools for the past 11 years. Co-Wrote Making Math Meaningful high school math workbooks and curriculum with Jamie York.

Philosophy and mathematics? What is more philosophical than mathematics? “Let no one ignorant of geometry enter here”. There is beauty in mathematics!!

What is most important? That the students are enthusiastic about mathematics because they see the beauty in it. That the students understand not only how to work in the world but how the world works. That the students are prepared to move forward.

Mathematics in Waldorf Schools Waldorf Schools have main lessons. Students are allowed to focus on one subject and have it introduced to them properly. They are allowed to see how ideas developed. There is time for hands-on as well as head and heart activities. Concepts are introduced, put to sleep and then practiced over and over again. Students have to time to struggle and fail which ultimately leads to success and triumph!

Mathematics in History West East Geometry Form Drawing Beeswax, handwork, Designs in ML books Grade 5 Free-Hand geometry Grade 6 Geometry with Tools Grade 7 Geometric Theorems Grade 8 Solid Geometry Arithmetic Numbers The Four Operations Grade 4 Fractions Grade 5 Decimals Grade 6 Percents Grade 7 Intro to Algebra Grade 8 Furthering Algebra

Mathematics in History West East Geometry Grade 9 Descriptive Geo Grade 10 Geometric Proofs Arithmetic Grade 9 Algebra Completion Grade 10 Logarithms, Growth Grade 11 Analytic Geometry, Projective Geometry Grade 12 Calculus

Human Development Each human being’s development mirrors the evolution of human consciousness. In order to understand a concept fully, one must struggle with it similarly to the way in which the innovators of that idea struggled. Learning occurs when an idea is brought at its proper time and taught in an enthusiastic and complete way.

Bringing Math Concepts to Life Grades 1 – 4 Giving heart to the four operations Using movement to teach times tables Playing math games Chores (measuring) & Art (knitting)

The Four Operations Tumble Times Melancholy Minus Judge Duly Divide Placido Plus

Bringing Math Concepts to Life Grades 5 – 8 History and biography Discovering patterns Real world phenomena

Grade 7 Algebra Formulas Gauss Galileo Einstein Solving Equations using scales

Gauss’ Sum Carl Fredrich Gauss (1777 – 1855) The Prince of Mathematicians At age 7 or 8, he added up the the numbers from 1 to 100 in 30 seconds. How?

Gauss’ Sum Carl Fredrich Gauss (1777 – 1855) The Prince of Mathematicians At age 7 or 8, he added up the the numbers from 1 to 100 in 30 seconds. How? ……

Gauss’ Sum Carl Fredrich Gauss (1777 – 1855) The Prince of Mathematicians At age 7 or 8, he added up the the numbers from 1 to 100 in 30 seconds. How? …… = (1+100) + (2+99) + …. = 50(101) = 5050

Gauss’ Sum Add 1 + … What is a formula for adding consecutive whole numbers beginning at 1?

Gauss’ Sum Add 1 + … What is a formula for adding consecutive whole numbers beginning at 1? What is the process for adding these numbers?

Gauss’ Sum Divide the largest number in half and multiply by the sum of the largest number and one. (N/2)(N+1)

Grade 7 Geometry The Golden Ratio = φ π = C:D The Pythagorean Theorem is not a 2 + b 2 = c 2

Area What is Area? O th Dimension is a point 1 st Dimension is a line or distance 2 nd Dimension is area 3 rd Dimension is volume Measure, measure, measure. What is the area of your room, your desk, a piece of paper? How did the idea of area develop in history?

Area

A = base x height or length x width

Area A = base x height or length x width A = 18 cm 2 There are eighteen 1 cm x 1 cm squares that make up this rectangle.

Area A = 18 cm 2 A = 9 cm 2

Area A = 18 cm 2 A = 9 cm 2 A = (1/2)(base x height) OR (base x height)/2

Area A = (5x5) + (1/2)(3x5) + (1/2)(4x5) = 42.5 cm 2 A = (1/2)( )(5) Area of Trapezoid is (1/2)(b 1 + b 2 )h

Area How do we calculate the area of curvy shapes? Digital Technology

Area Finding the Area of a Circle C = πD = 2 π r A = π r 2

Grade 8 Number Bases and Computer Science Imagine if you only had 8 fingers… How would you count? 0, 1, 2, 3, 4, 5, 6, 7, 10, 11, 12, 13, 14, 15, 16, 17, 20, 21, 22,… Imagine if you only had 2 fingers? 0, 1, 10, 11, 100, 101, 110, 111, 1000, 1001, 1010, 1011, 1100, 1101,...

Grade 10 Geometry Archimedes’ Approximation for π 3 10/71 < π < 3 1/7

Grade 11 Projective Geometry Imagine if Parallel Lines met?

Grade 12 Calculus What is Calculus?

The Tip of the Iceberg Read about the History of Mathematics. Understand the development of human consciousness and the struggles it took to accomplish change. Try to replicate the activities of the greats of mathematics as much as possible.

Resources Jamie York’s Making Math Meaningful series Zero : The Biography of a Dangerous Idea Charles Seife Is God a Mathematician Mario Livio Waldorf Academy