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