By: Amy Bilek, K-8 Math Instructional Coach at The Frances Xavier Warde School in Chicago
Throughout the last five years, many schools and districts across the United States have adopted or consider adopting “Singapore math.” It is important to know that “Singapore math” is in fact the same math as the math we know and the math generally taught in schools. The math has not changed. What “Singapore math” refers to is the methodologies and approach to instruction.The reason it has gained such global popularity is that it has proved highly effective.
If your child, class, or school is transitioning to a Singapore math-based program, here are three great tips which will aid in the transition and lead to more success in the new program:
1) Embrace Multiple Methods
When you walk into a math classroom aligned with Singapore methodologies, one thing you are sure to notice is that solving problems in multiple ways is valued. Teachers often elicit multiple methods from students and present them side by side on the board.
So, a new student transitioning to Singapore math should start exploring how problems can be solved in multiple ways. Being flexible with numbers and operations is part of the process as well as learning how to evaluate different perspectives.
A typical Singapore math lesson begins with posing a task. Students are expected to explore solving the task before direct instruction is given. Thus, students must be willing to take risks and attempt various strategies. These mindsets might be new for some students. As such, it is important to encourage them to be open in accepting other’s ideas and challenge other perspectives if necessary. Students should also work to practice recording ideas with both visual and numerical models.
2) Practice Decomposing Numbers
An important aspect with Singapore methodologies is the value placed on developing students’ number sense. Math instruction is given to help students develop deep conceptual understanding, which requires a solid foundation of number and number relationships. Across the primary grades, students need to be proficient with strategically decomposing numbers.
In kindergarten, students begin this work by breaking apart numbers in various ways. This work (examples shown below), helps students develop a deeper concept of number and value. Often students record this work with a number bond. The total is the number at the top and then the two parts are shown coming out and below from the total.
In image 1, you can see here how a student decomposed 8 into 5 and 3, 6 and 2, 7 and 1, and 4 and 4. This ability to break down numbers is helpful when moving to basic fact addition and subtraction. When given 8 – 3, students may imagine a number bond with 8 as the total, 3 as the known part, and then a missing part. Being fluent in decomposing 8 this way, helps them see that 5 is the missing part, and thus the answer to this subtraction fact.
In second grade, students engage in more work to break apart numbers being mindful of place value.
Above we see a second grader that used number bonds to decompose a 2-digit plus 2-digit subtraction problem (showing 42 as 40 and 2 and 11 as 10 and 1). This allowed the student to focus on subtracting the ones from ones (2-1=1) and tens from tens (40-10=30). Then, finally adding these together to arrive at a final answer (31).
This comfort with breaking down numbers is also very important when introducing regrouping for subtraction. Below you see how when subtracting 16 from 32, it is more effective to think of 32 as 20 and 12 (rather than 30 and 2), because that regrouping allows you to subtract away the 6 ones. A student that has developed their ability to mentally decompose numbers can more easily understand the regrouping that happens in the algorithm.
The importance of flexibility, when breaking down numbers continues in third through fifth grade with multiplication and division. Here we see that when dividing by 3, it is most efficient to think of 87 as 60 and 27, because both of those numbers lend themselves to being divided by 3.
Number bonds have proven an effective way for students to record how they are breaking down numbers. As students move through their primary years, continually improving skills and flexibility when decomposing numbers, it is incredible to see how they grow in their number sense and their ability to calculate mentally. You can view this progression throughout the think!Mathematics series by watching the video entitled “Decomposing Numbers-Operations,” on the following page. Once you’ve opened the page scroll down to the border which reads “Math Concepts.” There will be several titles, look for “Decomposing Numbers-Operations.”
3) Explore Bar Modeling
A final common attribute of Singapore math programs is the emphasis placed on visual representation. Bar models are rectangles used to represent numbers and quantities. They are used to translate verbal information in word problems into pictorial representations. Bar models are a bridge between the story and the abstract equation and help students see what operations are most suitable for that situation. Listen to think!Mathematics author, Dr Yeap Ban Har explains solving word problems when using bar models here. This will take you to a heading, “About the Development of think!Mathematics. Browse through the titles listed to locate “Solving Word Problems Using Bar Models.”
Encouraging students to start exploring how to represent story problems with diagrams, is a key step in transitioning Singapore math. You may wish to begin with actual cut strips of paper to represent the bars that the children can physically move around and manipulate, and simple problems such as the one below. Notice how you can move from using connecting cubes, into the bar model.
In third, fourth, and fifth grade students begin solving more complex story problems by first representing the story with a visual bar model. Once proficient at creating bar models, students are able to use this tool to access and solve problems that would otherwise be much beyond their years and require algebra to solve.
In transitioning to Singapore math, it’s not the math that changes, it’s the mindset of how to approach the math; and that’s where the students will need ongoing support and encouragement.
Students must be willing to take risks and approach things in multiple ways. They must value solving tasks with multiple strategies and practice taking on the perspective of others. Students must work to decompose numbers in various ways and consider which way is most suitable for the particular problem. Further, they must connect the abstract mathematical symbols and stories with visual representations. And perhaps most of all, students must be ready to actively engage in their math learning.