CPA at Home: Concrete, Pictorial, Abstract
The three-stage learning sequence at the heart of Singapore Math — and how to apply it in your home sessions.
What You’ll Learn
The Three Stages
Jerome Bruner’s research established that learners — at any age — move through three stages when encountering new mathematical concepts. Singapore Math is built on this sequence. Skipping a stage does not save time; it creates gaps that reappear later.
Concrete — The student handles physical objects. Counting beans, grouping blocks, building fractions with paper strips. The concept lives in their hands before it lives in their mind.
Pictorial — The student represents the concept visually. Drawing bar models, sketching groups, creating diagrams. The concept moves from hands to eyes.
Abstract — The student works with symbols, equations, and numbers without physical or visual support. This is where most curricula start — and why students struggle.
Why the Sequence Is Non-Negotiable
Moving a student to abstract work before they’ve built concrete and pictorial understanding is the single most common source of math anxiety. The symbols become arbitrary marks — memorized procedures with no meaning underneath them.
The Diagnostic Question
If a student can solve an abstract problem but cannot show you what it means with objects or a drawing, they are working abstractly without understanding. This is a red flag, not a success. Return to concrete before advancing.
The Bar Model: Your Primary Pictorial Tool
The bar model is Singapore Math’s signature visual representation. It is not a supplemental strategy — it is the core pictorial stage tool for virtually every concept from addition through algebra.
Part-Whole Model
Used when you know the total and need to find a part, or when you know the parts and need to find the total. A single bar divided into sections.
Comparison Model
Used when comparing two quantities — how much more, how much less, how many times. Two bars drawn side by side with the difference labeled.
When to Move Between Stages
The three-question diagnostic tells you when a student is ready to advance:
- Can they explain the concept in their own words?
- Can they apply it to a new, unfamiliar problem?
- Can they sustain it — retrieve and use it a week later without re-teaching?
All three must be yes before advancing to the next stage. This is not perfectionism — it is the minimum condition for the next stage to work.
Reflection
Take a moment before moving on. These questions are where the lesson becomes personal.
Pick a concept your child is currently working on. Which stage are they actually in — and how do you know?
Have you ever moved your child (or been moved yourself) to abstract too quickly? What did that look like?
Try drawing a bar model for the next math problem your child works on. What does the drawing reveal that the numbers alone don’t?