upper grades

Waldorf Geometry :: Math in the middle grades

Ge•om•e•try | noun


Middle English- via Old French from Latin “geometria”, from Greek, gē ‘earth’+ metria ‘measurement’.

Earth Measurement. This sounds like something entirely different from most of our own experiences with Geometry in school, yes?

Over thousands of years, geometry has become a standard part of math class and yet it sits in the modern math curriculum isolated from its true origin.

Ancient scholars, the first geometers, understood geometry to be the act of measuring the complete human experience of living on Earth. They set out on this study in order to understand the design behind everything they were experiencing in the physical world. In ancient Greece, the latin word “Mathematikos” meant “desire to learn” and the latin word “mathema” meant “knowledge/study.” In other words, the measurement and study of the physical world in order to understand the human experience on earth and how everything around us is created.

Inspiration is needed in Geometry, just as much as in poetry.
— Alexander Pushkin

Rudolf Steiner and Waldorf education place the utmost importance on the Geometry curriculum and have created a different path for students than the typical math curriculum offers. In truth, in Waldorf schools children begin their study of “earth measurement” with their first lesson on the first day of first grade - by drawing straight lines and curved lines. Straight lines and curved lines are nature’s design tools and they will become the essential building blocks of writing and drawing.

Printing, cursive, numbers, and music notation are constructed with lines and arcs. As the students progress through the early grades, patterns of lines and arcs are part of almost every lesson. Where there are patterns, there is geometry. Clapping games establish rhythm and order. Eurythmy and dance involve expression by way of patterns in movement. Students sit in rows, circle up, and line up.

Music and geometry go hand in hand as well. Rhythm, intervals, and patterns are the geometric design of music and poetry. As the students learn more about music, they can observe how different geometric patterns in music have unique qualities that induce a particular mood or feeling. Major chords sound “happy” and minor chords sound “sad.” The interval between a root note and the 5th sounds and feels much different than the interval between the root note and the seventh. Music is a great example of geometry as the tool behind the expression through sound. The idea of math as a universal language actually goes much deeper than numbers and calculations on paper being the same everywhere in the world. Ideally, a child’s education in the early grades is full of geometry.

Early experiences with Geometry

In grades one through four, there is no formal geometry class. Geometry comes by way of everyday school life (music, writing, movement, etc) and Form Drawing class. Forms are patterns of straight and curved lines that become more complex as the children advance through the grades. Simple repeating forms in grades one and two help with the development of coordination for printing and cursive writing. More complex forms in grades three and four help to develop a sense of spatial awareness and symmetry. In the fourth grade, students practice forms that are also symbols from different cultures (i.e. Nordic symbols) and learn how to tie knots - a three dimensional version of the form drawings.

Moving into the middle grades

Grade Five - Freehand Geometry

Grade five is a special, transitional year, symbolic of the peek of childhood. The students study great civilizations of the past, including the “Golden Age” of Greece. The students themselves are in the golden age of childhood. They are at the peak of their development in their child bodies and have gained as much mastery of their physical bodies as they are going to before heading into puberty. Fifth grade also marks the transition from form drawing to geometry. The goal of grade five “freehand geometry” is to lock the archetypal geometric forms into the body by drawing them without the use of tools. Classes also set out to find geometric forms in the world outside of their classrooms.

Freehand Geometry.JPG

Grade Six - Geometric drawing tools

With the arrival of adolescence, grade six becomes a year of re-birth both physically and in the curriculum. In geometry, we need to go all the way back to the beginning with straight lines and curved lines. This time, however, the children learn to construct lines and arcs with geometric drawing tools. A straightedge (ruler) constructs a line and a compass constructs an arc. The students will now begin a second journey through geometry that mirrors grades 1-5. The three-fold approach to learning (doing, feeling, thinking) is beautifully displayed in the geometry curriculum. In grade 6, the focus is on the physical doing aspect of learning. Learning to use tools to construct and measure is a physical task and it challenges the students as they begin to develop new physical bodies. The focus on “doing” in grade six mirrors the 1st and 2nd grade curriculum where an understanding of lines, arcs, and patterns comes from drawing forms with both the hands and feet, walking patterns, and rhythmic clapping games, etc.

Sixth grade Waldorf geometry.JPG

Grade Seven - Inward and outward exploration

Grade Seven in Waldorf schools is generally thought of as “the year of exploration.” A seventh grader, now fully immersed in adolescence, is ready to outwardly explore the physical world and inwardly begin an exploration of the question “who am I?” The feeling life of the seventh grader takes center stage in the curriculum as we use geometry to understand the human body and the natural world we live in. Da Vinci’s Vitruvian Man and the work of Fibonacci give us clues about the mathematical formula behind all that is living on our planet. It’s an incredible moment when students discover that the same ratios and patterns found in pentagrams, roses, apples, insects, hurricanes, and the milky way galaxy, are also found in the structure of our own bodies. The observational studies of nature performed by the likes of Plato, Pythagoras, Eratosthenes, Fibonacci, and Leonardo Da Vinci are mirrored by the observational powers of the seventh grader (they see everything, yes?).

Seventh grade Waldorf Geometry.JPG

Grade Eight - Digging deeper

In grade eight, the geometry curriculum enters the thinking realm. Armed with knowledge about how to use the tools of the geometer and the experience of searching for one’s self both in nature and the human body, the students are ready to become mathematicians, in the ancient Greek spirit of the mathematician being one who studies all things. We can now venture into the world of abstract thought and theory. The students will grapple with how to measure and study the three dimensional world on the two dimensional surface of the paper. By the 8th grade, it is the hope that students can see that there is always more than meets the eye, and they now have the physical and intellectual tools to dig deeper through observation, making precise measurements and calculations, and articulately describing what they see. They are mathematicians, scientists, and philosophers training to see things on multiple levels. On one level, numbers define quantities and help us with measurements and calculations. On another level, each number has its own unique set of qualities. Like the circle, the number one represents wholeness and the beginning. Plato called the circle “the mother of all shapes.” When the whole is cut into two, polarities are created (up, down, left, right, life, death, hot cold, positive, negative, male, female, etc.). The number three has a very balanced quality (tripods, tricycles, triangles). This way of understanding the quality of numbers is similar to the phenomenological way of teaching science. When a botany student imagines a plant, the goal of the teacher is to get the child to have a fluid mental picture that includes the entire life cycle of the plant, as opposed to just a fixed image of the plant in a particular moment in time.

With geometry, the goal for the 8th grader is to help them think of numbers not just as quantities but as parts of patterns in nature with their own qualities that shape the world around us and our experience of the world.

Grade eight Waldorf Geometry.JPG
“Let no one ignorant of geometry come under this roof.”
— Latin inscription above the door of Plato’s Academy

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•We offer a comprehensive Geometry curriculum, covering the grades 5-8.

Perspective - Art in the middle grades

Teaching perspective drawing to seventh graders has always been a highlight for me. This is one of those magical moments where the curriculum meets the students everywhere they need to be met.

 Thirteen year olds are always right.

 Just ask them ;)

 A seventh grader is desperately trying to form his or her own point of view and beginning to understand that we all see the world through our own, unique lens. The most important concept of perspective drawing is the establishment of point of view. In perspective drawing, the artist must constantly ask "how would this look from my perspective?"

If you'd prefer to listen to the audio version of this post, you can do that right here:

 The Horizon

The horizon represents the limit to how far the eye can see, assuming we can look beyond the buildings, trees, and mountains that might be in the way. In perspective drawing, the horizon is a straight line that establishes the "eye line" or point of view of the artist. In reality, we know that the horizon is not straight because the earth is round. We draw it as a straight line because that's how we perceive it. (More on perception versus reality later)

It's fitting that a seventh grader should grapple with the idea of learning to understand the world between him/and the flat horizon which, in turn, begs the questions: "What's beyond the horizon? and... Isn't the world round?"


Vanishing Points

 In the sixth grade, students are often satisfied with isometric three dimensional drawing. All of the lines of an isometric box are parallel. It looks real! Life gets a little more complicated in 7th grade, however,  as the students move farther away from the simplicity of childhood. The typical seventh grader begins to question everything (especially the teacher!).

 When it comes to teaching perspective drawing, I like to teach by asking questions. "If the sides of this cardboard box are equal in length and parallel in real life, why do they look like they're getting closer together as they go off into the distance?" Soon they discover that straight lines going into the distance appear to line up with vanishing points on the horizon. Now they are ready to construct rules for drawing the world as it appears to us.

 On more subconscious level, there is another phenomenon at play. As human beings, we can venture beyond the horizon in the physical world, and we can explore the depths of our inner selves. The vanishing point on the horizon mirrors the vanishing point inside each human being. Interestingly, both the horizon and the vanishing points are not fixed.

They are simply boundaries placed by the artist based on his or her unique perspective at a single moment in time.

Art as metaphor for life.


Perception vs. reality

 I love taking seventh graders through this journey of building a set of rules to create reality and then realizing that we need to keep bending the rules when a new piece of information is introduced.

 "Why does it seem like there's more than one vanishing point?"

 "Why does it seem like all the vanishing points change in reality when I move my eyes?"

 "Is there really a point out there?"

 Some students are happy to live inside the set of rules for perspective drawing and some edge closer to the idea that this set of rules is a convention that humans created. It's a method of taking our visual perception of the three dimensional world and putting it onto a two dimensional surface in a way that accurately represents the artists point of view at that particular moment in time.

The process is akin to learning a language in order to express your point of view. It's all the more valuable if our students can be guided towards developing this set of rules on their own. As they work with the drawing exercises, I encourage you to try not to give in to the temptation of TELLING them what they are experiencing. The questions that lead to the rules will naturally flow out of them if they are given many opportunities to EXPERIENCE the drawings!

~Brian (& Robyn!) Wolfe

*Want more? Read another piece from our Art in the Middle Grades series - Black & White drawing, G6.

Black and White - Art in the middle grades

The Waldorf art curriculum is smartly designed to meet children where they’re at in each phase of their development. If we provide children with the right tools at the right time, the whole experience nourishes them on a deep level.

In the early grades (1-4) students live in a playful artistic realm. The experience of playing with colors and finding basic form is more nourishing than technique and detail.

Students use art as an extension of their imaginations and the stories they hear. Around fifth grade, students are learning about ancient civilizations and how art was used as a method of communication and story telling. From early cave art to the hieroglyphics of Egypt and the mandalas of ancient India, the painting and drawing styles were full of imagery and color. Early art tends to also be fairly two dimensional. It's primary purpose was to tell a story, honor the gods, and represent culture. This is perfect for a child (such as the average 5th grader) who is on the verge of puberty but still lives in the imaginative world of childhood. 

As puberty sets in, the child moves away from the colorful imagery of these ancient peoples into a world of realism and perspective. From the beginnings of civilization and the golden age of Greece the curriculum moves toward the fall of Rome as children enter the 6th grade. Their arrival in Rome begins the next phase of artistic expression.

Black and White - Art in the middle grades

Teaching art (or any subject!) in the middle school is challenging yet meaningful work. Students generally are starting to compare their own work with that of others. They are also realizing that they are very different from each other. Each student is on his/her own journey both internally and externally. Most of us remember the battles we faced each day in middle school. The inner struggle to understand how and why the body is changing is contrasted by the outer struggle to find one’s place socially. The child is thrust into a world of polarity. These polarities are often expressed verbally by middle school students. “I HATE my drawing!” “This is SO TOTALLY fun!” “My teacher has NEVER liked me!” “This is the BEST class!”

There may be no better way to meet this moment in their development than to introduce the polarities of black and white drawing along with the multitude of greys that come with studying the nuances of light and dark.

Students begin to study optics in Physics and using a phenomenological approach are encouraged to look at the world as infinite expressions of light and dark. By learning to see light and shadow and positive/negative space, elements of realism and depth begin to emerge. 6th graders are also generally hungry to learn technique. “It doesn’t look right” is the most common complaint I hear from middle school art students. Their ultimate goal is to make their drawing match what they see (either in reality or in their mind’s eye). Learning how to effectively use lines and shading helps the students become better able to draw what they see, as everything we see is based on relationships between light and dark.  Black and white drawing, with it's emphasis on using lights and darks to shade & add depth, is the best way to help them begin to achieve more realism in their art work. This new skill meets the children perfectly as they are beginning to see their social environment through their own internal polarity lenses. Things are fair or not fair, someone is telling the truth or they are lying. That experience was either the BEST, or the WORST. Sound familiar?

Black and White - Art in the middle grades

Emotions tend to run hot and cold at this age. Socially, students tend to feel “in” or “out.” As teachers, we strive to have the experience of the grey scale between light and dark be therapeutic for them, on a deeper unspoken level. While black and white is the main theme at the start of adolescence, the shades of gray in between hint at the next stages of complexity in the development of the human being. As students move beyond the 6th grade towards the end of middle school, vanishing points will expand artistic horizons and bring an entirely new perspective in the grades to come.

~Brian Wolfe

**Looking for some black & white drawing inspiration? We made a Pinterest board for you loaded with resources and ideas!