How Does Waldorf Teach Mental Arithmetic?

Waldorf teaches mental arithmetic through a few minutes of spoken, moving practice every day and through number stories. Your child counts, claps and steps through number facts in the morning circle. Then you tell a small story that ends in a question, and your child works it out in their head. Each new sum is done with counters first, then said aloud, and only then written. Written methods mostly wait for grade 3, when the numbers already live in your child's head.

This article is about that daily practice. For how the four processes are first introduced, see the four processes in grade 1.

What mental arithmetic looks like in a Waldorf morning

In a Waldorf class, the main lesson usually opens with a rhythmic part: a verse, a song, some movement. That's where the mental arithmetic lives.

Peter van Alphen wrote a training manual for Waldorf teachers in East Africa, and it sets out what goes into that time. In grade 1 math blocks, children practise "a few forms of counting (e.g. starting at 20 and ending at 40)." They count backwards and in 2s, 3s, 5s and 10s, and work on one times table. All of it comes "with lots of varied movement."

At your kitchen table, a morning might go like this:

1. Stand up and count from 20 to 40, stepping on each number.
2. Count back from 40 to 30, whispering.
3. Clap the 2s up to 24, stamping on each one.
4. Toss a beanbag back and forth counting in 10s.
5. Sit down for one number story.

That's about ten minutes. It works best at the same time every day, so it becomes part of your daily rhythm and not a lesson your child has to be persuaded into.

Why the movement matters

Prerana Waldorf School describes children who "move their bodies through the math facts of all four processes" each day. They clap and stamp the times tables and add and subtract with beanbags and glass beads. The school's reasoning is that memories last longest when more than one sense is involved.

A child who has stepped the 3s round the garden often finds the pattern is there when they need it. Over time the counting grows, from 20 to 50 and then to 100. Times tables in grade 2 has more ways to move them.

What is a Waldorf number story?

A number story is a small spoken tale with a sum inside it. Waldorf teachers also call them story sums. Here's one from van Alphen's manual:

Grandmother went to the market with 12 loaves of bread. She sold 2 loaves to the farmer's wife and another 3 loaves to the carpenter's wife.

Before anyone works out the answer, the teacher asks which helper the story needs. In many Waldorf classes the four processes are characters, often the math gnomes, and your child names the one who takes things away. Only then do they work out that 7 loaves are left.

Van Alphen gives the reason: the story sum "develops the ability to think clearly about a situation, to understand what needs to be done, and select the correct operation." The story also gives the sum a reason to matter, because your child wants to know what happened to grandmother's bread.

Some teachers are careful with the word they use. Lotus & Ivy, a Waldorf-inspired homeschool writer, talks about "number stories, number journeys, or number exercises" but never arithmetic "problems." A sum is something to find out, not something wrong.

How to tell a good number story

Keep it short and keep it close to your child's life. The hens laid 8 eggs today and we used 3 for pancakes. The bus had 6 people on it, and 4 more got on at the bakery.

A few things help:

• Use your child's world: the garden, the shop, the grandparents, the dog.
• Start small, with numbers up to about 10 in early grade 1, then up to 20.
• Don't give the answer. Van Alphen suggests letting a child who knows it whisper it in your ear, so a younger sibling can still work it out.
• Add a "what if." If 24 tomatoes fill packets of 4, how many packets? And what if the packets held 8?

In grade 2, van Alphen's manual often puts two or more steps in one story. The farmer picks 20 apples, gives 5 to the neighbor, then shares the rest between 3 baskets. Your child has to hold the first answer in their head while working the second.

Why does Waldorf teach mental math before written math?

Most Waldorf teachers follow the same order for a new sum: do it, say it, write it. Van Alphen calls it the DO-SAY-WRITE procedure. Each sum "is first done with counters, then spoken, and finally written down."

It fits Steiner's advice to begin arithmetic with things a child can see and hold. In a 1923 lecture he said the teacher should "start from the sum, which is always given," meaning the whole heap of things in front of the child, and let the child find its parts. The counters come first because the number is real to the child there. Saying the sum puts it into words. Writing it is the last and smallest step.

Van Alphen treats the move from counters to the head as one of the big steps of these years. The children "move from working with the concrete" to "being able to do the bonds in their heads, without needing the counters." He's also clear that children who still need their counters "should be allowed to use them, until they can let go of them." Fingers are the same. They're a stage, and most children leave them behind in their own time.

How mental math grows from grade 1 to grade 3

Grade 1. Counting forwards and backwards, a little further each month. Skip counting in 2s, 5s and 10s. Number bonds: all the ways to make 10, then 6, 7, 8 and 9. Short number stories with all four processes.

Grade 2. Prerana Waldorf School says the four processes are "revisited with the goal to perform all of them as mental math." The numbers get bigger, and bundling matchsticks or sticks into tens helps with place value. Stories grow to two steps. The times tables are said, clapped and walked until they're known by heart.

Grade 3. Prerana says the processes "are learned again with shortcuts." This is when the written column methods usually arrive, carrying and borrowing on paper. A child who can already do 36 + 27 in their head usually finds the written method makes sense, because they know what answer to expect.

Mental arithmetic doesn't stop in grade 3. It keeps its place in the morning, now with bigger numbers, money and measurement.

Practice in the car and on walks

Mental math travels well. Melisa Nielsen of Waldorf Essentials suggests practising "in the car while running errands" and tossing beanbags. She also turns the question round: rather than "what is 6 x 2?", ask "what is 12?" Your child might say 6 and 6, 2 groups of 6, 10 and 2, or 15 take away 3. Every answer is right, and each one shows how well they know the number.

A few more that need nothing at all:

• Count the steps to the gate, then count back on the way in.
• Take turns adding 2 to the last number, then 3, round the table.
• On a walk, guess how many trees are left to the corner, then count.

If you're worried that all this talk and no worksheets will leave your child behind, Is Waldorf math rigorous enough? sets the Waldorf sequence against conventional school, grade by grade.

A week to begin

Pick one time each morning and keep it. On Monday, count to 30 and back, stepping. On Tuesday, add the 2s with claps. On Wednesday, tell your first number story, using your child's own name and something from yesterday. By Friday, let your child tell you one. Most children love setting a sum for a grown-up.

Starpath's free guides for word problems in grade 1 and problem solving in grade 2 set out how number stories grow across a block, with example stories to tell. For whole curricula, see the best Waldorf homeschool curriculum for grade 1.