In the 1960s, developmental psychologist Jerome Bruner proposed that learning moves through three modes: enactive, in which knowledge is represented through action and physical experience; iconic, in which knowledge is represented through images and pictures; and symbolic, in which knowledge is represented through abstract symbols.
Bruner's framework was not developed specifically for mathematics, but its application to mathematics education has proved to be one of the most practically powerful ideas in the field. In mathematics education, this sequence is most commonly called the concrete pictorial abstract progression, or CPA, and it describes the order in which mathematical concepts should be introduced to produce genuine understanding rather than hollow procedural compliance.
The sequence is: physical, hands on experience with concrete materials first; then visual, pictorial representations of those physical experiences; then abstract, symbolic notation.
That order matters. It is not arbitrary. And deviating from it, which most traditional mathematics instruction does by beginning with the abstract and adding concrete illustrations afterward, produces a specific and predictable kind of mathematical fragility that shows up most visibly when the mathematics becomes more complex.
Why the Order Matters
The reason the CPA sequence produces more durable understanding than starting with abstraction is grounded in how the brain builds mathematical concepts.
Abstract mathematical symbols, the numeral 7, the operator +, the fraction notation one half, are conventions. They carry meaning only for someone who has connected them to something more fundamental. The numeral 7 means something because it has been connected to the experience of seven objects, seven steps, seven counts. The symbol + means something because it has been connected to the physical experience of combining groups.
When abstract notation is introduced before the concrete experience it represents, children can often learn to manipulate the notation correctly without understanding what it represents. They can add two digit numbers without understanding what addition means or what the digits represent. They can simplify fractions without understanding what a fraction is. The notation is there. The meaning is not.
This is the specific failure mode that the CPA sequence prevents. By building the physical experience first, then connecting it to a visual representation, then connecting the visual representation to the abstract symbol, the sequence ensures that by the time the notation arrives, there is something for it to mean.
What Each Stage Looks Like
Concrete. The concrete stage involves physical objects that a child can handle, move, arrange, and observe. The objects can be purpose made mathematical manipulatives: base ten blocks, fraction tiles, connecting cubes, counters, algebra tiles. They can also be everyday objects: coins, dried beans, small toys, pieces of food.
What matters is that the child is physically manipulating a real world representation of the mathematical concept. They are not looking at a picture of blocks. They are holding blocks. They are not reading about combining groups. They are physically moving objects from one group to another and counting the result.
The concrete stage should last long enough for the concept to feel physically familiar. A child who has spent adequate time in the concrete stage understands what is happening before any notation appears. The physical experience is the understanding, and the notation that comes later is a record of that understanding rather than a replacement for it.
Pictorial. The pictorial stage introduces visual representations of the physical experience without the physical objects. The child who was combining actual groups of objects now draws or works with pictures of groups. The child who was placing counters in a ten frame now draws dots in a ten frame diagram. The child who was manipulating fraction tiles now colors in fraction bar diagrams.
The pictorial stage is a bridge between the physical and the abstract. It retains the concreteness of the physical experience in a form that can be reproduced on paper, communicated to others, and referred to without physical materials. It is also the form that many mathematical representations take in curriculum materials, including bar models, array diagrams, number line representations, and area models.
Children who have not passed through the concrete stage frequently struggle with the pictorial stage, because the pictorial representations assume a physical understanding that has not been built. A ten frame diagram is meaningful to a child who has placed counters in a physical ten frame. It is an abstract diagram to a child who has not.
Abstract. The abstract stage introduces the conventional symbolic notation of mathematics: numerals, operators, fraction bars, variables, and all the other symbols that constitute formal mathematical language.
For a child who has passed through the concrete and pictorial stages, the abstract notation arrives into a space already occupied by genuine understanding. The notation is a compact and efficient way of recording something the child already knows through experience and image. The equation 3 + 4 = 7 is not new information. It is a succinct representation of something the child has physically experienced and visually represented.
For a child who has not passed through these stages, the abstract notation arrives into empty space. It can be learned as a set of rules for manipulating symbols, but the rules have no anchor in experience, which makes them fragile and difficult to reconstruct when memory fails.
How CPA Applies Across Different Mathematical Topics
The CPA progression is not a technique for early childhood mathematics only. It applies every time a genuinely new mathematical concept is introduced, regardless of the student's age or the sophistication of the concept.
Place value: counters organized into groups of ten and groups of one before base ten notation; diagrams of tens and ones before the abstract digit in a position representation.
Multiplication: equal groups of physical objects before array diagrams before the abstract equation; arrays of objects before the area model before the multiplication algorithm.
Fractions: physical fraction pieces before fraction diagrams before fraction notation; folded paper before shaded rectangles before the symbolic fraction.
Negative numbers: physical representations such as two color counters before number line diagrams before abstract integer operations.
Algebra: balance scale models before algebraic diagrams before symbolic equations; physical experience with maintaining balance before the abstract rule of doing the same to both sides.
Geometry: physical shapes before geometric diagrams before symbolic formulas; measuring real objects before calculating with abstract measurements.
In each case, the sequence is the same. Experience builds image. Image builds symbol. The symbol means something because the experience and image are already there.
The Most Common Mistake
The most common deviation from the CPA sequence is introducing the pictorial before the concrete is genuinely established. This produces a specific problem: children learn to read and produce pictorial representations without understanding what they represent.
A child who learns to interpret a bar model diagram without having physically acted out the situation the bar model represents can use the bar model as a procedure for solving problems without understanding why the procedure works. When the problem is slightly different from what the bar model diagram was designed for, the child cannot adapt.
The concrete stage is the one most often skipped in the pressure of formal instruction, because it takes more time, requires more materials, and produces more mess than symbolic instruction. But it is also the stage that does the most foundational work, and skipping it produces exactly the kind of procedural without understanding mathematics that shows up as struggle when the material becomes more complex.
Implementing CPA at Home
For homeschooling families, implementing the CPA sequence is more straightforward than in a classroom, because you can take as much time as needed at each stage and can use a wider range of materials.
Invest in a few key manipulatives. You do not need everything. The most versatile are: a set of base ten blocks for place value and whole number operations; a set of fraction tiles or fraction circles for fraction concepts; a double sided set of counters for integer and operation concepts; some connecting cubes for measurement and pattern. These four sets of materials cover the vast majority of elementary mathematics concepts that benefit from concrete representation.
Before introducing any new concept symbolically, spend time with the physical representation. Ask the child to show you the mathematical situation with the materials before you write anything down. Build the physical experience first.
Move to pictures only after the physical experience is solid. Draw diagrams of what you have been doing with the materials. Connect the drawing to the physical arrangement explicitly: "This bar in the diagram represents these eight blocks."
Introduce the symbolic notation last, and connect it explicitly to the pictures: "This equation, 8 minus 5 equals 3, is a compact way of writing what our diagram shows."
Keep the materials accessible throughout the abstract stage, so that when understanding fails, the child can return to the physical representation rather than being stranded with a notation they cannot reconstruct.
Bruner's foundational framework Bruner, J. S. (1966). Toward a Theory of Instruction. Harvard University Press. Bruner's original articulation of the enactive iconic symbolic progression in learning provided the theoretical foundation for the CPA approach in mathematics education, arguing that genuine understanding develops through physical action before visual representation before abstract symbolism.
CPA in Singapore mathematics and its international influence Kho, T. H. (1987). Mathematical models for solving arithmetic problems. In Proceedings of the Fourth Southeast Asian Conference on Mathematical Education (pp. 345 to 351). Institute of Education. The systematic application of Bruner's progression to Singapore mathematics education, documented in this and subsequent work, produced one of the highest performing mathematics education systems in international comparisons.
Research on concrete manipulatives and mathematical learning Sowell, E. J. (1989). Effects of manipulative materials in mathematics instruction. Journal for Research in Mathematics Education, 20(5), 498 to 505. This meta analysis of sixty studies found that long term use of concrete manipulatives produced significantly better mathematical achievement and more positive attitudes toward mathematics than instruction without them, with effects strongest when manipulatives were used systematically across the CPA progression.
The importance of bridging between concrete and abstract Goldstone, R. L., and Son, J. Y. (2005). The transfer of scientific principles using concrete and idealized simulations. Journal of the Learning Sciences, 14(1), 69 to 110. This research documented the critical importance of explicitly connecting concrete and abstract representations, finding that merely providing both without bridging between them did not produce the transfer of understanding that systematic progression produces.
CPA and students with mathematics learning disabilities Witzel, B. S., Mercer, C. D., and Miller, M. D. (2003). Teaching algebra to students with learning difficulties: An investigation of an explicit instruction model. Learning Disabilities Research and Practice, 18(2), 121 to 131. This study demonstrated that the CPA sequence produced significantly better outcomes for students with mathematics learning disabilities than abstract only instruction, establishing the progression as an evidence based practice for diverse learners.
The National Research Council on representations in mathematics learning National Research Council. (2001). Adding It Up: Helping Children Learn Mathematics. National Academy Press. The NRC's comprehensive synthesis of mathematics learning research emphasizes the role of multiple representations, including concrete and pictorial representations, in building the conceptual understanding that abstract mathematical competence requires.
