There is a deep human need to understand, explain and predict the natural world. Math is the system that allows us to do that. Few children today really don’t understand math – why it works and how it is connected to the real world. Since they have only memorized ways to solve specific problems, they struggle when the look of a problem changes slightly or they’ve forgotten one step in the process. In contrast, math in a Waldorf school is characterized by an individual experience of the truth of numbers, and an exploration of mathematical concepts that will yield confident and competent problem solvers.

Waldorf education lays the foundation for each individual to experience the internalization of mathematical thinking. Math lessons are brought through many subjects and modalities, while mindfully educating and experiencing math through the hands, heart and head. Waldorf math education involves movement, music, rhythm, art, form drawing, language, creativity, curiosity and wonder, creating a truly multi-sensory approach to mathematics.

### How do we make math (a subject that can be so abstract) lively and imaginative?

Waldorf starts off the introduction to math by asking a seemingly simple question, “What is the largest number in the universe?” 1 (one) is the biggest because I am one.” Other responses discussed in class are “One is the biggest because without it there isn’t any 2, or 3, or even a million.” “One is the biggest because there is one Universe.” “One is the biggest because it can be any number it wants.” All sorts of philosophical and mathematical truths become evident through just this “one” discussion. This gets them thinking in a whole new way about numbers, and how they relate to us and the world. Eventually the children arrive at “I am one!” they see how their bodies are shaped like the number one, they relate themselves to the vastness of the Universe, and realize at that point that they are co-creators. All of this happens in 1st grade. So, from the start, children are made aware of the significance of numbers and enter into a multi-dimensional relationship with them. Once they’ve experienced numbers with their imaginations, they can use their will to execute stars and polygons. They move their bodies through the math facts of all four processes (+ – / x) each day, and create personalities for each math function. There is Tina Times, Mina Minus, Pam Plus and David Divide. Each character is known by how they appear and how act. For example, David Divide has a sword and always chops things up, sometimes in half or more.

We start math in the Early Childhood program with an exploration and observation of the qualities of numbers in nature and in stories, developing an inner sense of many and few, large and small. In 1st grade, students explore the concept of one and wholeness, laying the groundwork for later work with fractions. They learn all 4 processes (add, subtract, multiply, and divide) by bringing tangible objects together in groups or trading them with friends to really understand what it means to add together or divide into equal groups. Only once they have had many opportunities to see quantities shift in these physical transactions do they learn to use numbers and equations to represent those interactions. In this way, math is grounded in the real world rather than a set of abstract concepts and rules to be memorized.

In 2nd grade, these processes are revisited with the goal to perform all of them as mental math and then in 3rd grade they are learned again with shortcuts (borrowing and carrying, long division, multiplication tables). By approaching the 4 processes together and returning to them each year with a different way to understand them, we give the children the chance to understand on a deeper level how numbers interact, setting them up for success when more complex mathematical concepts are broached in later grades.

Third graders measure the length of the hallway by laying elbow to elbow and eighth graders use the Pythagorean Theorem to predict distances of a large field. As the students age, we increasingly draw out into broader and more abstract applications of mathematics, but because of the strong foundation built in the early years, our students are more able to understand those abstract applications and how they work.

In the elementary grades, creating an imaginative picture around a mathematical concept feels forced. We do it through stories because we recognize that for the elementary age child, the path to interest is through the feeling life, and imagination is a huge part of the child’s feeling life.

But the truth is, a story is just one way to capture your students’ imaginations. We do it by connecting the math work with art. If imagination is king for the elementary school child, art is queen. Those beautiful geometry drawings, paper folding activities and free-rendering activities, allow our students to explore mathematical concepts in a way that comes alive for them. Like in 4th grade fractions block, divide up a cake/dosa and then eat your fraction.

Exploring math socially, by getting students working on a project together, talking it through, and explaining to each other. Middle schoolers especially learn really well in a social environment. As students’ progress into middle school, they no longer enjoy stories. But rather discuss inspiring biographies of scientists and mathematicians of the world. Along with math games, puzzles and riddles to ignite thinking.

Throughout the grades, we keep this focus on feeling and experiencing math to build connections and meet a variety of learning styles. Although math is typically thought of as a left-brain activity, by bringing movement and artistic representation into the experience of mathematics, we are able to engage the right hemisphere and enhance cross-brain connections. We build on a child’s natural connection to rhythm by pairing activities such as clapping and jumping rope with counting and times tables. Until grade 5 we have free hand geometry, while 6th graders use real compasses and rulers to make intricate designs of overlapping shapes, turning their math explorations into beautiful works of art and precision.

### How do we make sure our students get enough practice without it becoming “kill and drill”?

The amount of practice students need depends somewhat on their age. Here’s how we see it.

Grades 1 & 2 — We take up number concepts, the feeling for numbers, place value and how the processes and our number system works. Practice comes in movement, rhythm and circle activities. They do need some practice writing the numbers and solving problems.

Grades 3-6 — These are the prime skill-building years, and this applies to every subject. Skill-building requires practice. They’re learning and practicing how to use the processes and they need daily practice using them — over and over and over.

Grades 7-8 — In these years we go back to the big concepts. Here they have solid computation skills and they are being presented with lots of different opportunities for using them. They’re figuring out how and when to use those skills. Regular practice is required and ideally their skills are solid enough that they can explore the world mathematically.

### How do we keep our students feeling confident and capable about math, especially when they struggle?

There are different kinds of math students. There are the natural thinkers and the rule followers. Both can be wildly successful though. What if a student does not have a natural connection with Math. Some prefer to read or write. When it came the time to do math, he/she may just be a Rule Follower. They just need someone to lay things out and explain them very clearly — step-by-step. Teach them the steps (the Rules to follow) and they will implement them flawlessly, eventually developing an understanding for the concept and how and when to use the rules.

Better still to help them approach math with a spirit of exploration and we encourage them to make connections and discoveries on their own. And truly, this is a fantastic way for them to learn. Math students LOVE being presented with a real-world math situation and then thinking it through to come up with the answers. With math tools at their disposal, they are ready to conquer the world, make connections and find solutions.

Mathematics is not taught in isolation from other subjects at Waldorf schools. It is part of a holistic learning experience that connects with the child’s inner self and body through muscle-memory and other exercises. For the Class one Waldorf child math is really some kind of kinesthetics or whole body learning experience. Math is also closely related to and taught through music – furthering the important connecting between the child’s body and their understanding of numbers. As adults we all know that the most lasting memories are always those that involves more than one sense, for e.g.; going past a biscuit factory on the way to and from school and breathing in the aromas.

By using bodily movement to aid the understanding of numbers in the early grades – even before learning to read and write – the child develops a proficiency much like a musician memorizing scales. It is a slow and unhurried approach that does not push the child to count or read too early – an approach that does not taint a child’s passion to learn, but one that rather ignites that passion. For e.g.; clapping and stamping exercises to learn multiplication tables or playing with bean bags or glass beads – learning to add and subtract without even knowing that they are doing so. They get to participate in imaginative math fairy tales, requiring them to solve the same problems the main characters have to solve. They use manipulatives (small stones and shells) to work through exciting math tales. This experiential learning allows a real ‘living’ understanding of math to develop within the children.

When children begin writing, they begin with roman numerals and integrate this lesson within their form drawing block. Roman numerals have much easier forms and more straight lines than our common curvy numerals. Each number, 1-12, invites a discussion that requires a deep intensity of imagination. Waldorf math begins with Roman numerals.

This multi-faceted learning approach continues into Class 2. Column algorithms, vertical addition 1, 10,100. Children practise many sums and wrote some in our books. Work the times tables in many different ways, always with rhythm: sticks, walking, clapping, bean bag throwing, etc. of 2, 4, 8, and 11 times table. A genuine love of math can only be enhanced by a practical approach in the mid to later grades.

Children take part in music classes involving flute, voice and other instruments, allowing them to ‘feel’ the beauty and rhythm of the numbers. Math is the key to participating in Waldorf music lessons. Math is everywhere. Waldorf education seeks to help students develop and integrate math, music, building, movement, storytelling, art and more – all at once.

### Math through the Grades:

The rhythm of the day, of nursery rhymes and poems, and the social considerations of how many friends need a place setting or a swing are all integral parts of the youngest child’s day in a Waldorf early childhood classroom.

Waldorf First Grade Math Story in Grade 1 – The story of King Plus, Queen Minus, Magician Multiply and Doctor Divide

In **first grade**, students learn that numbers exist everywhere in the world, especially in nature. Through this holistic approach to learning math, the special significance of the number one is discovered (as in one universe, one human being) and students explore the numbers that are found within each person who has two eyes and two ears, four limbs, and so on. In this way, the mystery of numbers is introduced and is further explored through the grades. In first grade, the four math processes are taught simultaneously because they reinforce each other (multiplication is fast addition, division is fast subtraction) and while learning math facts we begin to develop a general number sense which is so important for subsequent work in mathematics.

In **second grade** – Multiplication tables are taught through patterns. The four processes are further enhanced through the practice of place values and long division and multiplication.

In **third grade**, practical math activities such as measuring, understanding the calendar, and furthering comfort with the four mathematical operations (addition, subtraction, multiplication and division) are the bulk of the math program.

In **fourth grade**, working with fractions is a perfect topic because the children are experiencing an “existential fragmentation” of their world as they begin to separate from their parents and the journey toward puberty begins. Fractions exist in our everyday life. We share by giving half of something to a friend, we use fractions in measuring while cooking, and find the fractions of time when we clap to the beat of music. By the fourth grade, children have developed an initial sense for fractions, but now we can begin to understand their meaning.

Beginning with the whole, we use manipulatives that we can divide and make into fractions. Food is a concrete example in which to create fractions and provides a great time to share. The students use their hands to fold and cut paper, labelling the parts of the whole as we go along. We build an understanding of 1⁄2 , 1⁄4, as well as mixed numbers such as 2 1⁄2. Furthermore, we find the fractions of time through musical notation and clapping on the whole beats, half beats, quarter beats. Along the way, we begin to add and subtract fractions, building confidence as we move towards more abstract constructs.

Fractions are arguably one of the most complex mathematical concepts that are brought to elementary students. This is why we work with many different kinds of manipulatives with lots of variation as well as repetition. Fractions can cause some confusion.

As they begin to understand the world around them, they note the differences and likeness of their surroundings, community, and even themselves. Fourth graders begin to notice and comment how they are different or unique from their peers, while still revelling in how they are the same. And as they are part of the whole—their whole family, class, school community and glorious world—each child brings their own individualism. And that is where the math doesn’t quite add up, for we certainly are greater than the sum of our parts.

In **fifth grade**, comfort with decimals as additional expressions of fractions is a central math theme. Concept of fractions have already been introduced in 4th grade, decimals are a natural progression for them in the 5th grade. In this next step they take in learning number concepts along with the place value system using decimal fractions. An understanding of place value in numbers less than one is cultivated. Practice with decimals in all operations is needed to create familiarity with the relationship between fractions and decimals.

**Sixth grade** is a time to deepen the math learned thus far, and be introduced to the concepts of business math and more formal geometry lessons. Business mathematics brings the students in touch with daily life and practical knowledge of finance at a time when the students are really asking, “When will I be applying this?” – Statistical graphs; formulae; currency exchange rates; sales tax.

Geometric drawing from art (compass & straight edge); basic Euclidean geometric constructions; measurement & area.

In **seventh grade**, learning about ratios (relationships of one number to another) complements the child’s experience of working through relationships between themselves and the world. During the seventh grade year, we continue with geometry studies and add formal algebra into the curriculum (although algebraic thinking has been part of the math work through all of the grades).

In 7th grade, students are growing more comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. The discipline of mathematics requires a positive disposition and self-direction. We cultivate a sense of monitoring and assessing one’s mathematical thinking. This approach includes perseverance in searching for patterns, relationships, and meaningful solutions.

Number Sense, Properties, and Operations; Exponents & square roots; percentage; ratios; ratio in a square; ratio in a circle (pi);Patterns; Functions; and Algebraic Structures : simplifying expressions and solving equations; Data Analysis, Statistics, and Probability; Rates ; distance, time, speed problems; puzzle problems, Shapes, Dimensions, and Geometric Relationship: Area of rectangles, parallelograms, non-right triangles, the shear and stretch; similar figures; geometric drawing & construction; Pythagorean Theorem; angle theorems.

The culminating year, eighth grade, is dedicated to deepening the geometry of solids, they get to experience this by working with the pentagon & the golden ratio; Fibonacci sequence and Euclidean to Platonic geometries. This year might also include work with number bases and loci, among other math topics. Along the way, math work is beautifully complemented by many handwork activities, music and eurhythmy designed to bring mathematical understanding into the will.

In **8th grade**, students check their answers to problems using different methods, and are able to ask themselves, “Does this make sense?” They can understand the approaches of others to solving complex problems and identify correspondences between different approaches. Mathematically proficient students are able to consider the meaning of a problem and look for ways to begin. They analyse givens, constraints, relationships, and goals. Students can consider the form and meaning of the solution and plan a solution pathway rather than merely jumping into a solution attempt.

Students notice patterns that exist in nature and society; mathematics provides the grammar and structure that make it possible to describe these patterns. Mathematics is a discipline grounded in critical thinking and reasoning. In mathematics, students set up problems, devise and carry out strategies. Students then also evaluate the solutions, and justify these methods, strategies, and solutions.

Other topics for 8th grade are Number Sense, Properties, and Operations; Base systems of numbers; square root algorithm; calculations using Pythagorean theorem; percentage changes; Patterns, Functions, and Algebraic Structures with rational numbers; Proportions and ratio; order of operations; distributive property; Data Analysis, Statistics, and Probability; Dimensional analysis (converting metric U.S.); polynomial curves; puzzle problems; Shape, Dimension, and Geometric Relationships, Area and volume; Platonic & Archimedean solids.

The overarching theme of mathematics in Waldorf Education is to develop adults who can solve problems rather than simply compute numbers. Teachers present the students with a question and challenge the students to think of as many ways as possible to answer it. Freed from the constraints of memorized rules, students are able to bring a true understanding of math to develop their own approach to solving the problems. They leave here with the belief that problems are solvable, phenomena are explainable, and they just need to think through it to come up with an answer.

Like all Waldorf curricula, math lessons are carefully planned to meet the needs of the developing child. Waldorf math education involves movement, handwork, music, rhythm, art, form drawing. By teaching in this way, the math curriculum is designed to bring mathematical understanding into the will. As a result, Waldorf students acquire a deep mathematical understanding that they carry throughout their lives.