Very often, we see students struggling with multiplication facts, and this extends to multi-digit multiplication, long division, fractions and later Algebra. How do we support these students? Here are some suggestions.

- Concrete – Pictorial – Abstract (Bruner, 1960)

Jerome Bruner stressed that learning is an active process and for students to acquire full conceptual understanding, students move through three stages – enactive, iconic and symbolic. In Singapore Math, it is re-labelled as Concrete- Pictorial-Abstract (CPA approach). In multiplication, students in Singapore move through the three stages of C-P-A as well. Instead of rote memorization, students connect concrete experiences to noticing relationship between facts. By building 4 groups of 3s, they discover that 4 x 3 means 4 threes. By separating 12 into 3 groups, they discover that 12 divided into 3 groups give 4 in a group. Through concrete manipulatives, they also discover that 12 divided by 3 also means separating 12 into groups of 3 each. This concrete experiences are translated into pictorial arrays before moving to abstract representations.

2. Phases of basic fact mastery (Baroody, 2006)

In a recent article by Kling (2015), “Students who learn multiplication facts through traditional approaches generally do not retain the facts because the method attempts to move students from phase 1 directly to phase 3 of Baroody’s (2006) three development phases.”

So what is Phase 2? How do we use derived facts? This can be illustrated with a simple example below. To derive 6 x 3, students think of 5 groups of 3 = 15 and adding one more group of 3. So, the answer is 18.

In another simple example here, to derive 9 x 3, students think of 10 groups of 3 = 30 and subtracting one group of 3. So, the answer is 30 – 3 = 27.

In Singapore Math, the distributive property is the building block for multiplication and students are expected to apply the strategy fluently, accurately and efficiently.

## Related Resources

For more related resources, please refer to our Multiplication and Division page.