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Thinking Math: A Conceptual Approach to Math Instruction

By Vicky Placeres

01/03/2020

Thinking Math: A Conceptual Approach to Math Instruction
Mathematics instruction has evolved over the years. Some parents may associate math class with workbooks and problem sets from their own school days. These exercises emphasized learning the procedures for calculations and problem solving. While drill and practice still play an important role in mathematics learning, nowadays there is more focus on conceptual learning. In other words, the goal is that students be able to understand and explain why they are doing what they are doing. With a conceptual understanding, students are more likely to be capable of transferring their math skills into new, unfamiliar, and “real life” contexts.

This year, the elementary school at Uruguayan American School adopted the Math In Focus program, based on the Singapore Math Approach for Grades 1–5. This program emphasizes more collaboration, discussion, the use of models, and multiple problem-solving strategies than does our former mathematics approach.

So what does a typical elementary school mathematics class look like? A class will usually begin with an anchor task, which is a problem or question posed to help students construct the meaning of a concept. For example, if the lesson objective is “using objects to find number bonds; find different number bonds for numbers to 10,” then the anchor task might start with the teacher posing a question. “Marcos bought four toys. How many ways can he put his four toys in two chests to put them away?”

The teacher will then model an approach to the problem, which contains several steps. First, the teacher will ask students to represent the problem concretely. Students might draw or use blocks to show the number of toys Marcos has. Second, the teacher will record students’ answers on the board and draw the answers to help students visualize the solution. Third, the teacher will then present a variation on the anchor problem to see if students can solve it without teacher assistance.

“What if Marcos bought five toys? How many ways can he place his five toys in the two chests? Draw all the ways you find on your paper.” Throughout the demonstration, or exploration stage, the teacher is constantly describing his or her thoughts, or “thinking aloud,” so that students can follow the problem-solving process.

Students then choose the strategy that works best for them and work independently or in pairs to solve the problem. Afterward, the class discusses not only the solution to the problem but also how they arrived at the solution, and why they chose a specific problem-solving strategy.

As the class continues, students work on other exercises while the teacher monitors their progress. Those students that demonstrate that they understood the strategy will work independently, while the teacher assists individuals or small groups that need more support. The lesson closes with a class reflection about their thinking process and a discussion of the most effective problem-solving strategies.

In effect, the teacher aims to make students aware of their thinking, i.e. build their metacognition, so that they understand what they are doing and why they are doing it.
With the Singapore Math Method, classes tend to be more active and collaborative. When you walk into the room, students might be discussing a problem, sharing different problem-solving techniques, drawing a pictorial representation, or debating the answer.

Providing a model and time for students to practice with concrete pictorial representations of the problems has been key to helping them internalize new mathematical concepts and skills and to understand that math is about thinking as well as getting the right answer. As a Grade 1 student aptly put it, “I do not just guess answers. I think about the problem.”

Vicky Placeres is Elementary School Principal at Uruguayan American School.




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