The Waldorf Approach to Teaching the Four Basic Operations of Arithmetic in Grades 1-3
by Vivian Jones-Schmidt, Waldorf Teacher at Lotus & Ivy
The Waldorf approach to arithmetic is different from that of any other educational system in a few distinct areas.
In first grade, we start with whole numbers and make sure children are fully grounded in operations with whole numbers before proceeding to working with partial numbers [decimal and/or common fractions]. The entire process takes three years, beginning with the four basic processes (addition, subtraction, multiplication, and division) in First Grade; more practice with the four processes with larger numbers and introducing regrouping in Second Grade; and long multiplication and division in Third Grade. Hence, Fourth Grade is when fractions are introduced and when students are ready to really grasp the concept of partial numbers. In Fourth Grade, we work with all four processes with fractions, equivalent fractions, common denominators, and GCF. This is the first major difference between the Waldorf approach and that of other systems.
The second major difference has to do with how each operation is introduced. Waldorf teachers create lessons to teach the child’s intellectual thought, imaginative thought, and moral life. Moral and ethical values permeate everything we teach, but we rarely think about this aspect of teaching arithmetic. As I explain how we teach each operation later in this article, I’ll also point out the underlying values that are being taught.
In Waldorf arithmetic, we work within a three-year cycle: a concept is introduced in the first year, practiced in the second, and mastered in the third. Second Grade arithmetic thus continues the practice of the four operations, using larger numbers and including regrouping. Third Grade arithmetic, in addition to a focusing on long multiplication and long division, also explores various themes regarding measurement, time, money, and weight — all accompanied by intriguing and practical projects.
Unlike other methods of teaching arithmetic, Waldorf students are never given “blind procedures”, but are led by their teacher to the discovery of mathematical concepts. For instance, in later years, we do not tell them a-squared + b-squared = c-squared when teaching the Pythagorean Theorem, but we help them discover it by exploring patterns with triangles. In the early grades, this means we help the child experience, through stories, that 12 = 4 + 8 and not simply memorize the fact.
And in addition to these differences, as a teacher I add another layer. When teaching arithmetic to children, I talk about number stories, number journeys, or number exercises; but I never refer to arithmetic “problems.” I firmly believe that the language we use can be powerful: it can, subconsciously, create a mood or an expectation that lasts a lifetime. I want nothing to do with children equating “arithmetic” with “problems,” so I don’t use that word.
Quality of Number
After kindergarten years rich with manipulative materials, construction projects, measuring sand and water, and imaginative play, formal instruction in arithmetic starts in First Grade with an exploration of the quality of numbers. We don’t often think about this as adults, but what is “one”? And what does “one” mean? Where do we find oneness in our daily lives? Often, a class will soon begin to talk about the Sun, because most children have noticed the Sun and its significance as the largest body in the sky as well as its work in providing light and warmth.
We discuss and illustrate each number, 1-12, in this manner. I always end this introductory period by engaging the children in talking about the twelve months of the year.
Addition
The four operations in arithmetic are often taught in Waldorf first grades with stories of gnomes. Some teachers give the gnomes names like “Patty Plus,” but I just call them by their signs: Little Plus, Little Minus, Little Times, and Little Divide. Little Plus is associated with the color green. When we teach addition, we start with the sum. So, 12 = 4 + 8. Let’s think about the ramifications of that. Traditionally, addition is taught with the addends: 4 + 8 = 12. Not only does this approach limit the possible combinations of numbers for 12, but the subliminal focus of addition in this teaching method is on acquiring more and more. Acquisitiveness and greed are not moral qualities that we want to teach.
On the other hand, if we start with the sum, one already has the total and does not necessarily want more. Not only that, but think of the flexibility possible when one starts with the sum. How can you make 12?
12 = 4 + 8 12 = 3 + 9 12 = 6 + 6 and so on.
There are many ways to make 12, many correct answers to the question, “What makes 12?”
Subtraction
Little Minus is associated with the color blue. Poor Little Minus keeps dropping things. When we teach subtraction, we start with the number of “gems” Little Minus finds in a pocket when the castle is reached. The second number is the number Little Minus remembers counting along the way as each gem was picked up. So, 7 = 12 - ?. We want to find out how many gems were dropped by Little Minus. Here we emphasize a certain compassion toward Little Minus. After all, we all lose things.
We can also teach subtraction by talking about sharing: “Little Minus has 5 apples. After sharing some with a friend, Little Minus has 2. How many did Little Minus give his friend?” 2 = 5 - ?
Multiplication
Little Times is associated with the color yellow and is always in a hurry to give “gems” away. But if she has 12 and gives 3 to each friend, how many friends will she be able to share her gems with? 12 = ? x 3. In other words, how many threes are contained in twelve?
Multiplication is also taught as “fast addition,” for 4 x 3 is the same as 4 + 4 + 4, but it’s a much faster calculation.
Division
Little Divide is associated with the color red and always wants to make sure that each person has the same number of “gems.” So, if Little Divide has 12 gems and 6 friends, how many gems can he give to each friend?
12 divided by 6 = 2. Both division and multiplication stress generosity.
Division, of course, is fast subtraction. Dividing 12 by 6 is the same as subtracting two from twelve, six times; but it’s much faster.
After all four operations are introduced, we bring out 24 counting stones and work with all four operations on that number. This again reinforces flexibility of thinking, as we begin to see “24” from so many different directions. Of course, by the end of first grade, we do transition to the more traditional algorithms of each operation, as in
6 + 6 = 12 12 - 4 = 8 3 x 6 = 18
But along the way, we’ve avoided acquisitiveness and greed, as well as the idea that there’s only one way to work with any particular number. We’ve taught ethics and flexibility of thinking through story, discovery, and practice. The virtues of generosity, sharing, and compassion continue to be taught, discussed, and considered throughout the elementary years—even in the Business Math block in Sixth Grade.
Vivian Jones-Schmidt, Class 1 Main Lesson & Math Teacher, 9am ET
BRINGING MAIN LESSONS TO LOTUS & IVY CLASS 1 STUDENTS FROM VIRGINIA, UNITED STATES
Vivian has been teaching for over thirty years, and this will be her fourth First Grade class. She never planned to be a teacher, but her college mentor suggested that she explore early childhood education, and she did. After graduating from the College of William and Mary with a degree in Government, she travelled up the road to the University of Virginia, where she received an M.Ed. Having discovered that she loved teaching, she was one of the first kindergarten teachers in Virginia’s new public kindergarten program.
As music and especially singing had always been part of her life, Vivian sang in the College Choir and in Gilbert & Sullivan productions throughout her undergraduate years. G&S called to her again in Charlottesville, and that’s where she met her husband, a college history teacher. She then worked as an artist for several years before their daughter was born.
One day, Vivian walked into a Waldorf kindergarten and felt she had found Heaven. When their daughter, Rebecca, reached kindergarten age, Vivian started studying Waldorf Education. She knew she was drawn to the school, but what was this unknown education about, anyway? Finally satisfied that, in fact, Waldorf Education resonated with everything she’d always thought an education should be, she began teaching Handwork at the same time Rebecca entered kindergarten.
After a few years as a handwork teacher, class teaching called her, and she started with her first class of sparkling and challenging students, most of whom she is still in contact with, and some of whom now have children of their own. Vivian has spent most of her teaching years at the Charlottesville Waldorf School but was also privileged to teach for two years at the Richmond Waldorf School. In addition to teaching Handwork and Chorus, she has also served in many leadership positions at CWS. She has mentored several teachers as they’ve started their Waldorf journeys, and she was one of the earliest members of the Editorial Board of Renewal: A Journal for Waldorf Education. She has written many poems and songs for her classes, and a few of her middle school plays have been gathered into the book Three Plays for Small Classes.
Vivian is blessed with two adorable granddaughters.