The Science of Singapore Math

Many of us were taught mathematics through a system of rote memorization and procedural drills. We learned that to solve a problem, we simply needed to follow a specific set of rules—carry the one, borrow from the tens place, or invert and multiply. While this approach might have helped us pass tests, it often left us without a true understanding of what those numbers actually meant. When we transition to homeschooling our own children, it is incredibly common to default to these familiar methods. We sit down with flashcards and worksheets, expecting our children to absorb the rules just as we did. Yet, this often leads to frustration, tears, and the early onset of math anxiety when a child inevitably asks, "But why does this work?" and we realize we do not actually know the answer.

The problem with teaching math purely as a set of abstract rules is that it skips the critical foundation of conceptual understanding. When children memorize a procedure without understanding the underlying logic, they are building a house on sand. If they forget the rule, they have no logical framework to fall back on to solve the problem. The value proposition of modern, evidence-based math instruction is direct: we must build deep, structural understanding before we ever introduce abstract symbols. This is not about making math "easier" by avoiding hard work; it is about making math logical, coherent, and deeply internalized.

The thesis is clear: the remarkable effectiveness of Singapore Math is largely driven by its reliance on the Concrete-Pictorial-Abstract (CPA) framework. This evidence-based instructional sequence transforms mathematics from a confusing set of rules to be memorized into a logical system that children can physically interact with, visually represent, and ultimately understand at a profound level. By understanding and applying the CPA framework, homeschool parents can move beyond rote memorization and equip their children with a robust mathematical foundation.

The empirical foundation for the effectiveness of this approach is robust. A large-scale cluster randomized trial by Jaciw et al. (2016) assessed the impacts of Math in Focus, a widely used "Singapore Math" program built explicitly on the CPA framework. The researchers found that this program positively and significantly impacted the mathematics achievement of elementary students. This study is crucial because it provides rigorous, gold-standard evidence that the Singapore Math approach is not just a passing trend, but a systematically effective method for improving student outcomes at scale. It validates the core premise that structuring math instruction differently leads to measurable gains in understanding [1].

To understand why this approach works so well, we must look at the theoretical engine driving it. Chang, Lee, and Koay (2017) provide the definitive theoretical grounding for the Concrete-Pictorial-Abstract sequence. They synthesize the foundational learning theories of Jerome Bruner, Zoltan Dienes, and Jean Piaget to propose a clear model for teaching with representations. Their work clarifies that learning mathematics is not a leap directly into numbers, but a progression. It begins with the concrete (physical manipulatives), moves to the pictorial (visual representations like bar models), and only then arrives at the abstract (numbers and symbols). This sequence mirrors the cognitive development of children, ensuring that abstract concepts are always anchored in physical reality and visual logic [2].

The benefits of the CPA framework extend beyond mere test scores; they fundamentally change how children feel about math. A recent, large-scale mixed-methods study by Iyamuremye and Burns (2025) involving over 700 eighth-grade students demonstrated that CPA instruction significantly enhances both learning motivation and mathematical achievement. While this specific study was conducted in rural Rwanda—highlighting that CPA is effective across vastly different cultural and geographic contexts—the underlying mechanism remains the same. When students use the CPA method, they develop a deeper understanding of the material, which in turn increases their confidence and intrinsic motivation to tackle challenging problems. They stop seeing math as a mysterious code and start seeing it as a puzzle they have the tools to solve [3].

However, implementing the CPA framework is not without its nuances. A recent qualitative case study by Leong, Toh, and Cheng (2026) revisits the CPA heuristic, exploring the practical complexities teachers face when applying it to more advanced topics like quadratic expressions. This research acknowledges that moving fluidly between the concrete, pictorial, and abstract stages requires intentionality and practice. For homeschool parents, this means recognizing that the CPA framework is not a rigid script, but a dynamic tool that requires observing the child and knowing when to introduce manipulatives or when to encourage abstract thinking. Furthermore, practical applications, such as the action research by Salimi et al. (2020), show that the CPA approach can be successfully targeted to specific areas, like improving elementary students' understanding of geometry concepts. Together, these studies confirm that the CPA framework is a versatile, evidence-based strategy that parents can confidently use to build deep mathematical fluency [4] [5].

Translating the CPA framework from academic research into your daily homeschool routine requires a structured, intentional approach. It means resisting the urge to jump straight to the worksheet and instead building a multi-sensory progression for every new concept. Here is how you execute the three stages at home.

Step 1: The Concrete Stage (Do It)
This is the "doing" stage, and it is entirely physical. Before you ever write a number on a page, you introduce the mathematical concept using physical manipulatives. These can be specialized tools like base-ten blocks, Cuisenaire rods, or an AL Abacus, but they can also be everyday objects like dried beans, buttons, or LEGO bricks. The goal here is for the child to physically interact with the math. If you are teaching addition, the child is physically combining two piles of blocks. If you are teaching division, the child is physically distributing items into equal groups. This stage grounds abstract ideas in tangible reality, allowing the child to feel and see the mechanics of the operation.

Step 2: The Pictorial Stage (See It)
Once the child is completely comfortable solving the problem with physical objects, you transition to the "seeing" stage. Here, the physical manipulatives are replaced by visual representations. The child draws the problem. This is where Singapore Math's famous "bar models" come into play, but it can also involve drawing dots, tally marks, or simple diagrams. The pictorial stage is the critical bridge. It requires the child to hold the concrete concept in their mind and represent it visually on paper, moving them one step closer to abstract thought without losing the underlying logic.

Step 3: The Abstract Stage (Write It)
Only after the child has mastered both the concrete and pictorial stages do you introduce the "writing" stage. This is the realm of traditional mathematics: numbers, symbols (like +, -, =), and standard algorithms. Because the child has already built a deep conceptual understanding through blocks and drawings, the abstract symbols are no longer arbitrary rules to be memorized. They are simply a faster, more efficient shorthand for the physical and visual processes the child already thoroughly understands.

What to Avoid: Rushing the Abstract
The most common mistake parents make when using the CPA framework is rushing straight to the abstract stage. It is incredibly tempting to skip the blocks and the drawings, especially if your child seems to "get it" quickly or if you are feeling pressed for time. You might think, "They know what three plus four is, why do we need to get the blocks out?" This is a trap. Rushing to the abstract stage builds a fragile understanding based on memorization rather than deep comprehension. Even if a child can memorize a math fact quickly, skipping the concrete and pictorial stages deprives them of the structural logic they will desperately need when math becomes more complex later on. Always build the foundation, even when it feels slow.

If you want to implement the CPA framework without having to invent the lessons yourself, there are several excellent, research-aligned curricula designed specifically around this progression.

Singapore Math (Primary Mathematics or Dimensions Math)
This is the original source material. These programs are built entirely around the CPA framework, heavily emphasizing bar modeling in the pictorial stage to solve complex word problems. They are rigorous, logically sequenced, and excellent for building deep conceptual understanding.

Math in Focus
This is the Americanized version of the Singapore Math curriculum, and it is the specific program studied in the Jaciw (2016) randomized trial mentioned in the research section. It follows the exact same CPA methodology but often provides a slightly more guided, structured approach for the teacher, making it highly accessible for homeschool parents.

RightStart Mathematics
While not explicitly branded as "Singapore Math," RightStart is deeply aligned with the concrete stage of the CPA philosophy. It relies heavily on a specialized manipulative called the AL Abacus to build an intense, physical understanding of quantity and place value before ever moving to written algorithms.

Did you learn math through rote memorization or conceptual understanding? What curriculum are you currently using with your kids? Hit reply and let me know.

[1] Jaciw, A. P., Hegseth, W. M., Lin, L., Toby, M., Newman, D., Ma, B., & Zacamy, J. (2016). Assessing Impacts of Math in Focus, a "Singapore Math" Program. Journal of Research on Educational Effectiveness, 9(4), 473–502. https://www.tandfonline.com/doi/abs/10.1080/19345747.2016.1164777

[2] Chang, S. H., Lee, N. H., & Koay, P. L. (2017). Teaching and Learning with Concrete-Pictorial-Abstract Sequence – A Proposed Model. The Mathematics Educator, 17(1), 1–28. https://repository.nie.edu.sg/bitstreams/1bc521cc-ef36-4ad8-b994-4cab3ee7d37b/download

[3] Iyamuremye, E., & Burns, D. (2025). Concrete-Pictorial-Abstract instruction: enhancing students' learning motivation and achievement in mathematics. Cogent Education, 12(1), 2558303. https://www.tandfonline.com/doi/full/10.1080/2331186X.2025.2558303

[4] Leong, Y. H., Toh, T. L., & Cheng, L. P. (2026). A revisitation of Concrete-Pictorial-Abstract. Educational Studies in Mathematics, 122, 1–22. https://link.springer.com/article/10.1007/s10649-025-10455-4

[5] Salimi, M., Suhartono, S., Hidayah, R., & Fajari, L. E. W. (2020). Improving mathematics learning of geometry through the concrete-pictorial-abstract (CPA) approach: collaborative action research. Journal of Physics: Conference Series, 1663, 012046. https://iopscience.iop.org/article/10.1088/1742-6596/1663/1/012046/meta