Why Do Children Find Abstract Columns Difficult?
Written algorithms (column addition and subtraction) can be challenging. Children encounter rules such as: "one carries over to the next column", "borrow a ten and cross out the digit".
These ideas may be hard to picture. Explaining why each step works can help connect the notation with the operation.
The CPA Method (Concrete - Pictorial - Abstract)
The CPA (Concrete – Pictorial – Abstract) approach moves from physical objects, through pictures, to symbolic notation:
- Concrete: The child handles physical objects (blocks, candies, sticks).
- Pictorial: Objects are replaced by visual models, graphics, and animations.
- Abstract: Connect the familiar models with digits and operation symbols.
A physical model may help a child understand carrying and regrouping. Return to it whenever the written notation becomes difficult.
Column Addition: What Does "Carrying One" Mean?
When introducing written addition, explain why a small digit 1 is written above the tens column.
How to Explain It?
Imagine ones are little friends living in the right room. When exactly 10 friends gather in the room, it gets too crowded! They merge into one big "Super-Ten" and move over to the room on the left.
**In Number Land:** In the minigame *"Trampoline Party"*, monsters bounce on a trampoline. Once 10 assemble, they physically merge into a pack of tens and fly to the left column. The child can observe where "carrying one" comes from.
Column Subtraction: Regrouping Tens Step by Step
Subtraction across tens (e.g., $42 - 15$) can be challenging. Where do we get more ones when the top digit is smaller than the bottom one?
Answer: Unpacking a Pack of Tens
Instead of the confusing term "borrowing", let's use the metaphor of unpacking a box:
- On the left shelf stand sealed boxes (each containing 10 candies).
- On the right tray lie single candies.
- When a hungry monster wants 5 candies, but there are only 2 on the right tray - the child takes one whole box from the left shelf and opens it!
- The box breaks down into 10 loose candies on the right tray. Now we have 12 candies, and can take away 5.
For place value and four worked examples, see starting column addition and subtraction.
