Thank you for reading this far! By daring to lead with the words ‘Active’ and ‘Learning’, I expect many potential readers are already a dozen items further down their feeds. Those of you who have remained have likely done so with a slight tightening of the lips and perhaps a narrowing of the eyes.

Hang in there a little longer!

The term ‘Active Learning’ has been used in education for decades yet despite a formal definition from Bonwell and Eison in 1991, teachers’ interpretation and implementation of active learning varies considerably. Yet another needless polarising educational debate, to some, active learning represents a radical departure from the safe haven of ‘traditional’ teaching, aimlessly venturing into the unchartered waters of discovery, problem-based, and collaborative learning. To others, active learning is yet another in a long list of educational ‘fads’: star jumps to the five times table, treasure hunts and relay races.

This article offers another definition of active learning – one more aligned with that of Bonwell and Eison – and presents some evidence to support the incorporation of genuine active learning opportunities in mathematics lessons. The second part of the article introduces (or re-introduces) the reader to a range of teaching approaches designed to capitalise on the true value of active learning.

## What is active learning?

Active learning is a teaching approach that seeks to involve students in the learning process rather than passively listening to the teacher or following instructions. It is less concerned with physical activity and more with cognitive activity. Active learning has its roots in the constructivist approach which promotes experiences that allow students to construct or build their understanding. However, it is important to recognise that this definition of active learning doesn’t expect students to learn by themselves or in groups with the teacher acting purely as a facilitator. On the contrary, active learning requires highly skilled teaching that uses a wide range of instructional techniques. Of course, it could be argued that all learning is active as (most) students are listening, writing and doing something in most lessons. However, to be truly active in their learning, students must also compare, organise, explain, justify, make decisions and solve problems.

## Why use active learning?

Active learning has many benefits for both students and teachers. Research shows active learning can increase student performance, engagement, motivation, and retention in mathematics and other STEM subjects. For example, a meta-analysis of 225 studies found that active learning increased average examination scores by about 6% and reduced the failure rate by 1.5 times compared to traditional lecturing. Taking a more active role in learning can help students develop a deeper understanding of mathematical concepts, as well as skills such as communication, collaboration, and resilience.

Active learning can also make teaching more enjoyable and rewarding, as it allows teachers to use a variety of instructional methods to support and challenge their students. Teaching strategies designed to encourage active engagement in learning can foster a positive classroom culture where students feel more confident, curious, and respectful of other ideas.

We should acknowledge the excellent teaching and learning that happens every day in mathematics classrooms. But we should also strive to improve our practice and address the issues that arise in classrooms where teaching and learning are not up to the mark. One of these issues is that students often get low-level tasks that require them to mimic a routine or procedure without the need for deep thought. Students mostly get information from the teacher and have little chance to participate actively in the lesson or explore alternative methods. Students do not get enough time to explore and understand the mathematical concepts they are learning. Students also do not get enough time to explain their reasoning or compare and contrast the merits of different approaches. Dr. Malcom Swan studied 779 GCSE pupils and found the most common learning strategies in mathematics classrooms. Here they are, from the most common to the least:

- “I listen while the teacher explains”
- “I copy down the method from the board or textbook”
- “I only do questions I am told to do”
- “I work on my own”
- “I try to follow all the steps of a lesson”
- “I do easy problems first to increase my confidence”
- “I copy out questions before doing them”
- “I practise the same method repeatedly on many questions”

These strategies may work if the goal is narrow, i.e. to pass exams, but if we want students to develop a coherent understanding of mathematical ideas, to think creatively and critically, and to apply skills in non-routine ways to problems in unfamiliar contexts, then we should seek to expand the range of learning strategies they experience.

## Some Active Learning Strategies

There are many active learning strategies we can use in our classes and which you choose to incorporate will depend upon your expertise, your goals, and your students. Here are a few examples of some common and effective strategies. Some you may already be familiar with.

- Example-Problem pairs or ‘I do, we do, you do’ where students are expected to respond to a teacher’s modelled example by completing another example collaboratively and/or by themselves. When expertise is low, this strategy can help students learn new concepts or procedures by observing the teacher’s steps and then applying them to similar problems. For example, the teacher can model how to solve a linear equation, and then ask the students to solve one on their own. As student expertise increases, this strategy can also be used to introduce variations or extensions to the problems, such as changing the coefficients, signs, or terms.

- Worked examples where students study a worked example before attempting a similar problem themselves can help regulate the cognitive load and encourage students to focus on the essential features and steps of the solution, rather than any superficial details. For example, students can be given a fully worked example of how to find the area of a trapezium, the students then study this example before attempting to find the area of another trapezium with different dimensions.

- Worked examples with self-explanation prompts require students to not only complete an example but verbalise their thinking too. This strategy can help students deepen their understanding and improve retention by making them articulate their reasoning and justify their choices. For example, the teacher can provide a worked example of how to simplify a fraction, and then prompt the students to explain each step and why it is valid. By carefully choosing examples, teachers can use this strategy to draw out students’ misconceptions or identify gaps in knowledge.

- Multiple-choice questions where students not only select the correct answer but must explain the errors that could lead to some of the wrong answers. This strategy can help students develop their metacognitive and analytical skills by making them reflect on their thinking whilst they identify the sources of errors. For example, the teacher can present a multiple-choice question on how to find the median of a data set, and then ask the students to explain why some of the other given options are incorrect. This strategy can also be used to address common misconceptions or errors by designing the distractors based on them.

- Think-Pair-Share activities where students are required to explain their thinking to a peer. This strategy can help students enhance their communication and collaboration skills by having them share their ideas, strategies, and solutions with a partner. For example, the teacher can pose a problem on how to find the volume of a composite solid, and then ask the students to think about it individually, before pairing up with a partner and sharing their answers and explanations. This strategy is useful as students are often more willing to share their thinking in pairs rather than in a whole-class setting. Think-pair-share involves all students and can be used to encourage peer feedback, support, and learning, by asking the students to compare, critique, or even improve their partner’s work.

- Inclusive Questioning can be used to encourage all students to participate in class discussions. If you’re concerned about passive students, it’s vital to avoid allowing them to opt out of classroom activities. Instead, aim to use techniques like cold calling, think pair share, and show-me boards every lesson. This way, every student knows they are expected to generate and share answers one way or another. Cold calling can take time to embed so perhaps start with pair share and show-me boards, and then ask students to explain their answers. If you notice that Student A is less engaged than their partner, try asking Student A what their partner was saying. This way, you can ensure that every student is included in the conversation and feels valued.

- Testing transfer rather than retention requires students to apply recently presented knowledge to solve new problems. Most practice questions given to students are testing retention or remembering, and few test understanding or transfer. Yet, transfer is our goal if students are to have success with unfamiliar problems. For example, the teacher can ask the class how knowing 5 + 7 = 12 helps us to state the value of 5 + 17. Or, how knowing that 2x + 1 = 7 allows us to state the value of 4x + 3 without solving for x.

- Activities which require students to give multiple examples that satisfy a given criteria can help students broaden their creativity and flexibility by making them generate and explore different possibilities and solutions. For example, the teacher can ask the students to write down as many examples as they can of subtractions that result in 314. With a little planning, it is possible to increase the challenge and complexity by adding more constraints or conditions to the criteria. For example, the teacher can ask the students to write down a 3-digit by 3-digit subtraction that uses 6 different (non-zero) digits and results in 314.

- A non-example illustrates what a concept is not. They are used to highlight required characteristics and help students learn the boundaries of a concept. For example, 8 : 9, 7 : 3 and 11 : 2 : 5 are examples of simplified ratios whilst 5 : 10, 8 : 2 and 12 : 4 : 6 are non-examples. To be effective, teachers should provide lots of examples and non-examples which need to be chosen carefully to highlight essential features of the concept.

- What is the same and what are different activities (or which one doesn’t belong) can help students develop their comparison and classification skills by making them identify and explain the similarities and differences between two or more objects, concepts, or problems. For example, the teacher can show the students three quadrilaterals, such as a square, a rectangle, and a rhombus, and ask them to list the properties that are shared and the properties that are not. This strategy can also foster critical thinking by asking the students to justify their choices and opinions.

- Questions with no answer can help students improve their problem-solving and reasoning skills by making them test and evaluate their hypotheses and methods. For example, the teacher can ask the students to draw a triangle with sides 3, 4 and 10, and then ask them to explain why it is impossible. This strategy can also be used to introduce new concepts or topics by asking the students to explore the conditions that make the question possible/impossible to answer.

- Always, Sometimes, Never True. This strategy sees students being given statements and being asked to decide whether the statement is always, sometimes or never true. Importantly, students should be asked to justify their answer and where possible, provide examples (or counterexamples). It is also possible to ask students to rewrite the ‘sometimes’ statements so that they are always (or never) true. For example, the statement “The sum of three numbers is odd” is sometimes true. To make the statement always true, the statement can be rewritten as “The sum of three odd numbers is odd” or, to make it never true, it can be rewritten as “The sum of three even numbers is odd”.

I hope this discussion of ‘Active Learning’ has taken us beyond educational buzzwords and deeper into the nuances of effective teaching strategies. Whilst some will continue to debate the diverse interpretations of the term, I hope you have seen that the Active Learning described here is not an alternative to ‘traditional’ teaching but a powerful set of techniques that deserve thoughtful consideration. If you are seeking ways to increase student engagement and improve understanding, I would encourage you to incorporate some of these strategies into your teaching. Try one out with your classes, and evaluate and develop the execution before deciding whether it is something you will add to your repertoire of strategies.

Let me know how you get on.

Until next time,

Thanks for reading.

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