Building students’ flexible mental strategies enables them to solve maths problems efficiently in their head. This article will discuss:

• What are flexible mental strategies and automaticity?
• What can we do, as teachers, to help build up these skills in our students?

What are flexible mental strategies?

Flexible mental strategies are the different methods that students use in their heads to solve maths problems. These strategies are essential for supporting mathematical development both in school and in everyday life.

Students need to learn how to select the best mathematical strategy for any given problem, based on the numbers in the maths problem.

What do flexible mental strategies look like?

Watch this video that illustrates the different mental strategies students might use to solve a problem. Before watching the video, consider the problem: 150 + 160 + 39 = ?

• How might you solve this problem in your head?
• What other strategies can you think of that might be used to solve this problem?

 

In this video, we see a demonstration of various strategies that students might use to solve the problem. These include:

• applying an algorithm
• split and bridge to 100
• partitioning
• doubling
• counting on
• using benchmark numbers
• compensation.

For a deeper dive into the ‘what’ and ‘how’ of different mental strategies, explore the online learning module, Explicitly connecting known strategies to flexible mental strategies.

Strategy selection

Knowing which strategy to use when solving a problem is key to developing flexible mental strategies. It is important for students to have a toolbox of different strategies to draw on, but they also need to know what strategy might be the ‘best fit’ for each problem.

We want students to be flexible and efficient in their approach to problem-solving.

• Flexibility comes from fluency in multiple mathematical strategies.
• Efficiency comes from selecting strategies that make sense for both the problem and the student's level of understanding.

For example, a primary school student may use strategies such as repeated addition or drawing marks, but you would not expect these strategies to be employed by older students. The goal is for students to choose strategies that are both appropriate for the problem and efficient for their stage of learning.

Automaticity

A key component that underpins a student’s ability to apply flexible mental strategies is automaticity. In mathematics, automaticity refers to the ability to solve problems or recall facts quickly and effortlessly. With automaticity, students can apply number knowledge fluently, reducing cognitive load and allowing them to tackle more advanced mathematical reasoning. For example, a student with automaticity immediately recognises that doubling a number means multiplying it by 2 or adding it to itself. Similarly, they instinctively know that 75% is equivalent to three-quarters. Building students’ automaticity strengthens their ability to make connections between different mathematical concepts and solve more complex problems.

So, what can teachers do in the classroom to help students develop flexible mental strategies?

Planning and problem selection

Teaching a lesson that focuses on flexible strategies requires intentional lesson planning and selecting examples with numbers that will help develop the strategies students need. Anticipate the strategies students might use and prepare prompts or questions that encourage them to refine or extend their thinking. Plan prompts or questions that will encourage students to use more efficient strategies or extend their thinking to apply these strategies in new contexts.

It’s also important to consider what not to include in examples. For example, if you want students to move beyond doubling and halving to solve a problem, include odd numbers in the problem.

Developing students’ flexible mental strategies enhances both their adaptability and efficiency in mathematical thinking. To dive in deeper, or to look at some more examples of flexible learning strategies, listen to this podcast episode or head over to the online learning module.