Many teachers are unsure of what metacognition – thinking about thinking – involves and how it can be taught in the math classroom. Dr Lee Ngan Hoe shares some strategies that you can use, especially with students who are weak in math.

Why is Metacognition Important?

When asked why he chose to focus on how to teach metacognition to students weak in math for his doctoral research, Assistant Professor Lee Ngan Hoe shared how his encounters shaped his decision.

“The role of metacognition is explicitly stated in the Singapore Math Curriculum, and it dates as far back as 1990 when the curriculum framework was conceptualized,” says Ngan Hoe.

“But as late as 1998, after 6 years of implementation, I attended a conference where there were a few hundreds of math teachers, but when asked ‘what’s metacognition’, no one appeared to be confident enough to explain nor cite examples.”

So, What is Metacognition?

Metacognition is basically “thinking about thinking” (Ministry of Education, 2006, p. 7). We can view metacognition as “being aware of and regulating of one’s thinking”. “It is just a big word, but it is actually a very common practice when we solve problems,” says Ngan Hoe.

“During the problem-solving process, we will normally try several approaches before we arrive at the solution. When we know we are not getting anywhere with an approach even before we reach a dead end, we will stop and try another approach.”

From his 6 years of classroom teaching experience, Ngan Hoe noted that students weak in math tend to lack this skill of monitoring their own thinking process. “When they are faced with non-routine problems, a lot of them believe that there is only one way of doing it,” he says.

They use what he calls the “bulldoze way”; such students simply “bulldoze their way until they hit a brickwall” and then they will give up. These students lose confidence after repeated failed attempts at solving problems and begin to believe that they are not good in math.

Using a Classroom-based Approach

Ngan Hoe wanted teachers to view research on metacognition not just as theories but as something that can be effectively used in the classroom.

For his intervention programme, he offered his services to a neighbourhood school where he taught the “coaching classes” – an after-school programme that the school developed to provide extra help to their students.

Four groups of Secondary 1 Normal (Academic) students – one experimental group and three comparison groups – were used in the study. These students attended coaching classed each week over a period of 10 weeks.

For the first few sessions, Ngan Hoe went through each of the components of the Problem Wheel (see Figure 1). At this point, the students did not need to solve the problems.

For example, for the first lesson, students were provided with a list of problems and asked to focus on the “given” component. They were instructed to write down all the information that was given in each problem. The subsequent lessons focused on the other components.

Ngan Hoe later got the students to practise solving problems on their own, with the Problem Wheel to guide them.

The Problem Wheel

In an earlier study with gifted students, Ngan Hoe successfully applied Richard Paul’s reasoning wheel (see Lee, 2008, p. 65). For this intervention, Paul’s reasoning wheel was simplified to become the Problem Wheel (Figure 1).

Figure 1. Problem Wheel.

The Problem Wheel comprises 5 components:

• Given: What is the problem about? What information is given in the problem?
• Find: What you are supposed to find?
• Picture: Using the information you have found, draw pictures to represent the problem. This helps students to visualize the relationships among the information given.
• Topic: Think of the topics you have studied, e.g., Algebra, Geometry, Fractions. Which topics are relevant to this problem?
• Formula(e): Focus on the formulae you have learned in these topics. Choose the most appropriate formula(e) to solve this problem.

The Four Metacognitive Strategies

Using the Problem Wheel as a basis for questioning, Ngan Hoe also used four strategies to help the students to be more aware of their thinking process: mathematical log writing, effective questioning techniques, identification of structural properties of problem and pair and group problem solving.

• Mathematical log writing: Students write about their mathematical learning, which helps to make their thinking visible to themselves.
• Effective questioning techniques: Teachers and students ask questions about the problem, which helps them develop productive habits of problem solving.
• Identification of structural properties of problem: Encourage students to compare different problems, which makes them aware of how they can apply prior knowledge to solving similar kinds of problems.
• Pair and group problem solving: Get students to question each other, which makes their thinking process visible to all in the group, including themselves.

In class, students were allowed to talk out loud while solving the problems, whether individually or in pairs/groups.

He would give them fairly similar problems and prompt them to compare the problems and see if there were similarities and differences. This helped the students appreciate that they could tap on past experience to solve new problems.

At the end of each session, the students wrote a reflection of what they had learnt in their mathematical log.

Becoming More Confident Students

When Ngan Hoe compared the math exam results immediately after the intervention and a year later, he found that the difference between the coaching class and the other three comparison classes had closed up. From being the worst class, tthe students in the intervention group were now on par with the top class in terms of math achievement and problem solving.

Before the intervention, they would “stare and say I don’t know how to do” to the problems given to them. By the end of the intervention, he could see them trying to make sense of problems given by asking themselves questions and drawing pictures.

The Problem Wheel helped the students to kick-start the problem-solving process. Going through the components of the Problem Wheel enabled them to see that “they actually know something about the problem before starting to solve the problem”. This helped to build up their confidence, and in turn, their attitude towards math also improved.

Ngan Hoe’s experience demonstrates how important it is to equip our students with metacognitive skills. By teaching them how to systematically think through the problem, to think about what they were doing and thinking, even the weaker students were able to independently solve math problems.

References
Lee, N. H. (2008). Enhancing mathematical learning and achievement of Secondary One Normal (Academic) studies using metacognitive strategies. Unpublished doctoral dissertation, National Institute of Education, Nanyang Technological University, Singapore.

Ministry of Education. (2006). Secondary mathematics syllabuses. Retrieved October 1, 2009, from https://www.moe.edu.sg/education/syllabuses/sciences/files/maths-secondary.pdf

Further reading
Visit https://math.nie.edu.sg/kywong for more information about metacognition.