Why Do We Put the Whole Before the Parts (Math)?

In Waldorf education there are variations on how the times tables are taught. However, one constant remains – that we always work from the whole to the parts. But what does that mean and why do we do that?

In Waldorf classrooms different teachers recite the times tables with their classes in different ways. In “The Waldorf Way” David Ruenzel describes his math lesson in this way:

“….The movement, then, is highly purposeful and characterized by a sort of choreographed fastidiousness. Auer’s 1st graders stood rhythmically clapping their hands and stomping their feet as they chanted their multiplication tables: 9 is 3 x 3, 12 is 4 x 3, 15 is 5 x 3. “

However, in a Waldorf school in Maine, the lesson sounds a little different. Their lesson is described like this;

“There are recorders being played, verses being recited in unison, feet stomping, hands clapping. “Two times four is eight.” (Clap.) “Three times four is 12.” (Clap.) “Four times four is 16.” (Clap.)” The 26 pupils in Sarah Van Fleet’s fourth-grade class are standing in a circle reciting their multiplication tables, a timeless exercise in mathematical memorization, but one with a difference. While reciting, they clap out a rhythm and pass around orange beanbags.”

In one case the whole is recited first, in the second case the parts are recited first. So if the parts can come before the whole in recitation what did Rudolph Steiner mean when he spoke about the parts coming before the whole in Waldorf education?

When one speaks of putting the “whole before the part”, the order in which the numbers are found in recitation is certainly one way to accomplish that. However, the concept is rooted in a much deeper methodology. The concept of putting the whole before the parts is based on HOW the child learns the basic concepts of addition, subtraction, multiplication and division, not in how they may recite these facts later after they already know them or are practicing them.

In teaching math it is important for the child to be able to view the whole of the concept before they break it into computational parts. Steiner believed that,

“All teaching matter must be intimately connected with life. In counting, each different number should be connected with the child or what the child sees in the environment. Counting and stepping in rhythm. The body counts. The head looks on. Counting with fingers and toes is good (also writing with the feet). The ONE is the whole. Other numbers proceed from it. Building with bricks is against the child’s nature, whose impulse is to proceed from whole to parts, as in medieval thinking. Contrast atomic theory. In real life we have first a basket of apples, a purse of coins. In teaching addition, proceed from the whole. In subtraction, start with minuend and remainder; in multiplication, with product and one factor.”

He continues on to say,

“Instead of offering, say, three apples, then four more, and finally another two, and asking the child to add them all together, we begin by offering a whole pile of apples, or whatever is convenient. This would begin the whole operation. Then one calls on two more children and says to the first, “Here you have a pile of apples. Give some to the other two children and keep some for yourself, but each of you must end up with the same number of apples.” In this way you help children comprehend the idea of sharing by three. We begin with the total amount and lead to the principle of division. Following this method, children will respond and comprehend this process naturally.

According to our picture of the human being, and in order to attune ourselves to the children’s nature, we do not begin by adding but by dividing and subtracting. Then, retracing our steps and reversing the first two processes, we are led to multiplication and addition. Moving from the whole to the part, we follow the original experience of number, which was one of analyzing, or division, and not the contemporary method of synthesizing, or putting things together by adding.”

It is interesting that using this method division is then seen to Waldorf students as simpler than multiplication, The way most people were taught division it was more complicated!

A good illustration of how looking at the whole, instead of the parts would work is an example of talking about “Rays of Light”. When I was in science class we had a unit on “Rays of Light” and we learned about them as individual rays. It was a hard and abstract concept to grasp as a child.

However, in Steiner’s world you could replace that lesson with a photo of mountains reflected in a lake – this lesson would be much easier to grasp. You can break the reflection down into rays at a later time.

However, these story poems are not about learning math, they are about practicing the times tables. You will first be teaching your child the four mathematical processes in addition to reciting “math facts” (see E-book “Sixth Sense Math”). Once they have grasped the concept of the process they are able to move on to recitation and even more advanced – the recitation or “acting out” of story poems.

Because different teachers use different methods in their recitation of the math facts Earthschooling includes two versions of the math fact story poems in second and third grade. In one version the product is first and in the other version the factor is first. You should use whichever set of poems synchronizes well with the other lessons your child is doing.

 

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