High School

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“Projective geometry has the capacity to open minds and broaden thinking. I learned about things like perspective and duality, and all of this came together at infinity to create an understanding that I knew I didn’t have at the start of this block.”

Problem Solving and Perspective

The central point of mathematical activity in the Waldorf high school is problem solving.  The important thing is learning how to solve problems, not what the answer is.  With this as the focus, high school mathematics builds on both bases of mathematics:  inspiration (induction) as a beginning and logical conclusion (deduction) at a later stage in the mathematical activity.

The most important aim is to develop the students ability to think with a wide range of approaches until they get to the logical conclusion, and to give them confidence in themselves and in their thinking.  Another important goal is to prepare the students to apply calculations methods to everyday life and also to give them the foundation for further education.

Geometry is the mathematical discipline that deals with the interrelations of objects in the plane, in space, or even in higher dimensions. More than any other mathematical discipline, the field of geometry ranges from the very concrete and visual to the very abstract and fundamental. In one extreme, geometry deals with very concrete objects such as points, lines, circles, and planes and studies the interrelations between them. On the other side, geometry is a benchmark for logical rigor, the elegance of axiom systems, logical chains of proof, and the parallel world of algebraic structures.

In tenth grade, students study the projective properties of geometric figures

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Beyond Mathematics

In high school, children reach a new stage of development where an individual’s inner life confronts the outer world in a relationship that still has to find a form. In an integrative education, even geometry has its place in the deep work of young adults. The deeper concepts of mathematics around perspective, infinity, transformations, angles, boundaries, and duality lead to new insights and broader understanding of not just geometry, but of life.

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Introduction to Projective Geometry from a student’s main lesson book:

The Euclidian geometry we have worked with up until this point has dealt with the finite, the measureable. In the consciousness of the ancient Greeks, even the realm of the gods was considered in finite terms. Of course this finite or measureable nature implies ideals; for in actuality we can never be exact. As soon as we try to represent a point or line on paper, it is only an approximation, or rather a two-dimensional representation of the ideal. A point, as defined by Euclid is that which has no part, and a line is breathless and thus can never actually exit in the physical.

Projective geometry takes the elements of Euclid but stretches them in space toying with the idea of infinity. This geometry has seen application in the perspective drawings done already during the Renaissance by such artists as DaVinci and Durer. Projective geometry challenges Euclid’s elements asking us to see points as lines of infinity and whole planes becoming points. The mysteries of infinity order the random and obscure the ordered.

This block is an exploration of space, projecting lines and points to infinity with geometric nets and conic sections, observing the phenomena as they occur. We can wrestle with the ideas, but this course also gives us the opportunity to step back and relish the beauty and magic of these lines and points as we strive for exactness and perfection.

Students need to develop an intuitive understanding of geometric relationships and how to manipulate them. Learning how to do geometric proofs with compass and straightedge is an essential part of developing that knowledge. That knowledge will be used by an architect in many ways, from the creation of complex computer models to hand-sketching. In fact, one of the first things they teach in architectural perspective drawing class is how to use basic geometric principles we all learned in 10th grade geometry to quickly draw realistic and correctly-proportioned perspective images.

The relationship between mind and hand through pencil and paper is very direct (same with sculpting clay, for that matter). You lose that direct connection when a computer interface is involved. Once you know and have intuitively internalized the principles, the computer allows you to magnify that knowledge in practical applications.

I insist on seeing a demonstration of hand-drawing skills even for prospective employees who will only be doing computer drafting or modeling. What they can do with a pencil shows me in a very direct way how their brains work and whether or not they really understand what they’re doing when they try to graphically represent spatial concepts and systems,

So, yes, I think it’s important that students still learn how to do geometry the old fashioned way. Even though a computer will automate a lot of the calculation and construction for you, you still need to understand the geometric principles at work in order to use them. – Archinect

Every year our 12th grade students take a week-long trip to Hermit Island in Maine. They join about 100 seniors from other Waldorf schools for a week-long course on invertebrate zoology. They take daily trips to the tide pools and mud flats to investigate sea plants and animals. They discover creatures only visible in tide pools and under the microscope, as well as sea urchins, squid and sea stars. There are opportunities to experience the glowing of comb jellies and bioluminescence in the ocean. The students hone their observational skills by identifying various species of crabs and snails living in this vibrant ecosystem and come together as a group to discuss the week’s theme of earth as an organism.

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Our 9th grade was recently “away” on their Agricultural Practicum. We use the word away loosely. They were away and here at the same time. Students worked all week at Camphill Village Kimberton Hills, helping them harvest in the CSA, herb garden and orchard. While there, the students participate in the life of that community, work and share meals with villagers and co-workers. They also get a taste of work in the cow barn and in general farm maintenance. All of this plus staying overnight in the Garden Building!  While there, they work with Celia Martin in the evening preparing beef jerky for their backpacking trip in the spring.  Kimberton Waldorf students are introduced to a variety of complex issues around food and nourishment through the Agriculture Practicum, our gardening program and through exploring topics around food justice and food insecurity.

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The senior year at a Waldorf school is designed to be a synthesis of the students’ education and a preparation for their next step in life whether it be college studies or professional life. A highlight of the senior year at Kimberton Waldorf School is the Senior Project. The Senior Project is an opportunity for students to show personal initiative and independence in a study or work of their choice. Students must design a project that includes a research component, artistic or practical component and stretches their abilities mentally, physically and/or emotionally. Teachers step back while the students work under the guidance of a mentor. This creates a space for growth toward academic freedom.

Mirabelle Kunz is a senior at Kimberton Waldorf School. Mirabelle has been working on designing and creating a line of clothes that highlights her love of fashion and is full of color and texture. “I chose to make a clothing collection because I am interested in studying fashion design. My goal is to present my collection by holding a fashion show in the spring.” As part of her preparation for her senior project presentation, Mirabelle recently did a photo shoot in New York City to highlight her clothing line. “I planned a photo shoot to take pictures of my collection that I can now use in my college portfolio. I like fashion design because I enjoy the individuality that clothing can give a person!” Mirabelle hopes to study fashion design in New York City in the fall.

Past senior projects at Kimberton Waldorf School have included photography exhibits, pottery collections, mastering a foreign language, dance, building a 3-D printer, creating mobile phone apps, music and art. 2016 graduate, Hannah Wolfram flipped a house for her senior project. After purchasing a house in need of repair, Wolfram spent every weekend and holiday working on the house. Hannah worked to repair a leaky roof, replaced the kitchen and bathroom, updated electrical and plumbing, dry walled, sanded floors and painted. Learning the importance of budgets, timelines, inspections, collaboration and planning became a secondary level of education during her project.

Students can spend more than 100 hours working on their projects. They encounter real life problems as they work to overcome challenges, dead-ends and unexpected complications. They learn to persevere through difficulties, find new resources, and examine the subject from multiple perspectives. In the spring of their twelfth-grade year each senior presents an extensive written report and an oral/visual presentation of their work. The project is presented to a committee of faculty and outside community members, and before the entire school, families and friends. Students learn valuable public speaking skills as they explain and defend their work before a group. For many of our students, the senior project is one of the most challenging, memorable and ultimately valuable experiences they have in their senior year.

An Advisor’s Perspective by Celia Martin

I had been hearing about “the Tenth Grade Odyssey Trip” for years. That title denotes challenges, struggles, and obstacles to be overcome, but also triumph at the end. I had viewed photos of the trip, heard stories and seen the tired yet confident students limping or walking slowly down the halls the Monday after, and I wondered what the trip was really like. How difficult was it? Would I be able to do it? This year, as a tenth grade advisor, I found out.

After a seven hour drive on Sunday, October 7, we (28 students and three adults) arrived at our campsite along the Appalachian Trail in Virginia in a cold drizzle and set up our tents in a wet fog. We managed to get the charcoal lit and then worked to hack off slabs of frozen ground beef and cook the “hamburgers” in a steady rain. Thirty people were trying to hover over the sizzling meat, not only to keep the rain off the burgers and prevent it from putting out the fire, but also to feel a little bit of that wonderful dry warmth. There were no tables and no chairs so we stood around awkwardly, not wanting to sit on the cold, wet ground. We were chilly and damp and it was so foggy that the flashlights couldn’t cut through the mist. This was already an Odyssey! We were all very grateful for the warm, delicious food. Almost miraculously, the wet wood that we added to the charcoal after dinner started to burn and we crowded around a big campfire and laughed and talked and sang, the fire lifting our spirits. We looked forward to crawling into nice warm sleeping bags and we hoped that our tents wouldn’t leak.

The next day we were paddling down the James River and even though it was misting a bit, and a little cool, it was great to be paddling down that beautiful stretch of water. I couldn’t think of anyplace I’d rather be on that Monday morning, surrounded by the peace and tranquility of the river and that fantastic group of students. Everyone was full of energy and in high spirits.

For two more days we followed the river going through riffles and rapids and stretches of calm. We saw turtles and Great Blue Herons and the Kingfishers went chattering by. Each night we had a big campfire with songs, riddles and a story or two from Andy Dill. Everyone had their jobs to do so while some gathered wood or scooped water from the river, others cooked or washed dishes. Everyone was so willing to pitch in that it never seemed like work at all. We were tired from the long days of paddling and it felt good to crawl into our tents at night.

On our third night we camped just above Balcony Falls and we listened to the loud sound of the water pouring over the rocks all night. In the morning Andy charged the students with creating their own canoeing partners so that everyone felt confident about getting over the falls safely. After much discussion and rearrangement, we were ready to challenge the falls. Those on shore shouted encouragements to each pair as they prepared to go through, guided by Andy standing out on a high rock giving signals. Everyone was nervous but once we were all safe on the down river side, albeit a bit wet, we felt re-energized to keep going.

Later that day we traded in our canoes for backpacks and hiked three miles to our first campsite on the Appalachian Trail. The outhouse there was much appreciated after having nothing but the trees for three days. Some already had blisters and other foot problems. After just a few miles with those heavy packs, we all decided to eat the dinner that weighed the most so we wouldn’t have to carry it the next day. After a delicious and filling meal of lentil stew with vegetables, we had another wonderful campfire filled with fun and laughter and went to bed early in preparation for the nine mile uphill hike the next day.

Our first full day on the trail was very challenging. Our packs were full and heavy and the trail was very steep and rocky. It wasn’t easy for anybody but we all kept going and we elevated our spirits by singing, joking, playing word games and by believing that soon, very soon, we really would be at the top. At one point we were treated to a beautiful view of the landscape below and there, far, far below us, was the James River winding around the base of the mountains where we had just been the day before. That was the first time we had a sense of how high we had climbed, and it felt very gratifying.

At our campsite that night we found that the spring was very shallow which made it difficult to scoop out the water. A group of dedicated students worked for hours into the darkness scooping and filtering water to painstakingly refill everyone’s bottles. I was amazed by how well everyone worked together and how irrepressible this group of kids was, despite the difficulties. They smiled and sang through the struggle and helped each other always.

The next day was technically not as difficult as the previous but because we were so tired from the day before, it was a challenge. Andy spent time each morning caring for foot problems and blisters and now we also had some wrapped knees and ankles and sore hips where the backpacks sat. Almost unbelievably the trail still continued to go up, but not as steeply as the day before. The views were awe inspiring and gave us a reason to pause to catch our breath. When the first group arrived at the campsite that afternoon, a few of the boys left their packs and ran back to help those at the back of the group who were still about a mile out. We were really tired that night but there were tents to erect, water to be purified, meals to cook and dishes to clean. Remarkably, the students were still singing and laughing and helping each other through their exhaustion. We talked about how much we missed the conveniences of home but no one was complaining.

At the campfire that night we calculated that if we wanted to be home at 8 pm the next evening, we needed to get up extra early at 5:30 am. We all readily agreed and the next morning we quickly took down our tents, ate breakfast, loaded up our backpacks and were silently hiking out of our campsite in the dimness of the morning at 7:10 am. After two miles we reached one of the vans along a road and were able to leave our heavy backpacks to continue the last five miles carrying only water and food. This was our steepest elevation rise yet – a 2000 foot gain in only a few miles. It was an arduous climb even without heavy packs; steep, rocky and seeming to go up forever. Whenever we thought we were at the top, the trail kept going higher. We could feel the air getting colder and colder. Finally we arrived at a beautiful grassy meadow stretching out along the top of the ridge. It was sunny and peaceful and many of us just wanted to lie down and take a nap. But we were so close to the end of the journey that a whole new energy overtook us and we hiked the last mile to the vans full of energy and triumph. We had done it!

On Monday morning, there we were – limping or walking slowly down the hall, tired yet very satisfied, each of us feeling an unspoken connection to everyone else. We had struggled together, overcome obstacles together, supported and helped each other and kept each other going. We had met the challenge and met it well.  Now we knew, really knew, what it means to go on an Odyssey.