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Beads Before Numbers

The abacus as the oldest hands-eyes-symbols machine: what each bead means, how a full rod hands on a ten, and how to make one from beans and string so the grandparent can teach.

Spark & Whetstone · sparkwhetstone.org/learn/beads-before-numbers · CC BY-NC-SA 4.0. Answers are at the end, so you can try each question first.

Lesson 1. The oldest counting machine

Long before written sums, people counted with pebbles on a board and beads on rods. Counting boards were used in ancient Greece and Rome; bead frames have been used in China, Japan and Russia for centuries.

What they share is the idea children now meet in school as hands, then pictures, then symbols: you move a thing, you see a pattern, and only then you write a number.

Nothing in this course is about exercising your mind. It is about how a beautiful machine works, and about the pleasure of teaching it to someone.

Question. What do counting boards and bead frames have in common with how children learn number now?

  1. You move real things and see the pattern before you write a symbol.
  2. They avoid numbers altogether.
  3. They only work for adding.

Say it back. In your own words: why might moving beads come before writing numbers?

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Lesson 2. A bead is a place

On a Japanese soroban each rod has one bead above a bar and four below. A lower bead pushed up to the bar counts one; the upper bead pushed down to the bar counts five.

So seven on one rod is the upper bead down and two lower beads up: five and two. Each rod to the left is worth ten times the one to its right, exactly like the places in a written number.

Question. On a soroban rod, how do you show seven?

  1. The upper bead down and two lower beads up
  2. Seven lower beads up
  3. The upper bead down and seven lower beads up

Say it back. Say it back: what is the upper bead worth, and why do you need it?

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Lesson 3. Handing on a ten

When a rod has no room for what you want to add, you hand a ten to the rod on the left and take the difference back from this one.

To add 9 to a rod showing 6: you cannot fit nine more. Add one bead on the tens rod (that is ten), then take one away here, because ten is one more than nine. The rod now shows 5, and the tens rod shows 1: fifteen.

This is the "carry" from school, done with your fingers: the pairs that make ten are the whole secret.

Question. A rod shows 6 and you want to add 9, but there is no room. What do you do?

  1. Add one on the next rod to the left and take one away from this rod.
  2. Push up nine more beads somewhere on this rod.
  3. Clear the rod and start again from zero.

Say it back. In your own words: how does a bead frame carry a ten?

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At the table: Build a bean abacus

Players: a grandparent or grown-up and a child, 6 or older. You'll need: string or pipe cleaners, dried beans or beads with holes, a cardboard frame or a shoebox lid.

  1. Thread three strings across the box lid: ones on the right, tens in the middle, hundreds on the left. Put nine beads on each.
  2. The grown-up shows a number with beads and the child says it aloud; then swap.
  3. Add two numbers together on the frame, saying each hand-on of a ten out loud.

Answers

  1. You move real things and see the pattern before you write a symbol. — Both start with something you hold and move. The written number comes last, as a record of what your hands did.
  2. The upper bead down and two lower beads up — The upper bead is five and each lower bead is one, so five and two make seven.
  3. Add one on the next rod to the left and take one away from this rod. — Ten goes on the tens rod; because you only wanted nine, one comes off this rod. 6 + 9 becomes 1 ten and 5 ones.