## What Makes for a ‘Good’ Maths Lesson?

**The question of how to plan and deliver an effective mathematics lesson holds no easy answer. A ‘good’ maths lesson can be as varied in scope, structure, content and delivery as the degree of difference in teachers and classrooms – potentially infinite! There is certainly no such thing as a ‘one size fits all’ approach and what works well for you at your school may not work well for me at mine – and vice versa.**

It’s also important to note up front that no maths lesson exists in isolation. Each lesson you teach will be one component of a broader topic, which will be one component of a broader continuum of mathematical understanding, and so on. A ‘good maths lesson’ will always necessarily be a part of a sequence of lessons or learning experiences which will ideally build mathematical understanding, improve fluency, build problem solving capacity and then develop mathematical reasoning skills.

What follows is a set of observations and suggestions from a classroom teacher who has yet to have delivered that ever-elusive ‘perfect maths lesson’. Some of my own lessons have been spectacular failures – even some of those in which I have been meant to be modelling ‘good practice’ to colleagues!

Nevertheless, it is my intention that the following might at least provide food for thought and conversation on what constitutes effective mathematics teaching and learning, at least within the structure of classrooms and schools that we currently inhabit.

*You* are encouraged to contribute to an ongoing conversation in your school about what works – and doesn’t work – as you continue on the maths learning journey with your students.

**1. ****Good Structure**

**(a) Using ‘Warm-up’ or Ignition Activities**

The purpose of these is simply to start children ‘thinking mathematically’ – to establish a mathematical mindset. Games, loop activities, short open ended problems (for which solutions can be shared or discussed) – anything which will have students ‘lace up their maths boots’.

Keep this brief – about 5 to 10 minutes – and accessible. Choose something all or most students can readily participate in. Inclusiveness is important. Warm-ups also don’t necessarily need to be directly related to that lesson’s content – although it’s handy if your warm-up activity can be used to ‘bridge’ into your explicit teaching session.

If students have access to devices in the classroom, using maths apps to warm up is fine – but *be selective *and “ask three” at the conclusion of the warm-up session about what they did, what they found difficult and what they learned.

**(b) The Role of Explicit Teaching**

There is *absolutely* a place in your maths class for teacher directed instruction (note that by this we are not referring specifically to the teaching methodology called ‘explicit direct instruction’ (EDI), but rather, the general pedagogy of ‘teacher explanation’: “This is how we do it / solve it / plan it…”).

Spend some time in your lesson either introducing new concepts or revising existing concepts. Remember to involve students as you do this. Ask questions unsolicited (try a ‘no hands up’ policy in your classroom!). Ask students to assist you with working out.

Time taken for explicit teaching will vary depending upon the complexity of the concept you are teaching, but usually fifteen minutes is ideal; even adults will have trouble actively concentrating on someone explaining detail for longer than this!

Use stories / analogies / role plays to get your message across. Only use digital instruction clips in active teaching sessions if (a) they are short, and (b) you can pause them easily to check for understanding. Remember – *you* are the most effective resource / piece of technology your students have in their classroom!

**(c) Improving Fluency with Practice**

Generally speaking, we often tend to ‘overdo’ practice in mathematics lessons. This is because most maths textbooks and resources emphasise this through the provision of lots and lots of closed questions and problems for ‘drill and practice’. Whilst there is no doubt that it is important that students reinforce their understanding of mathematical concepts with practice on set exercises, the temptation can be for this to become ‘busy work’ by requiring to students to work through long sets of repetitive, similarly formatted maths exercises.

Sound ‘practice’ (or development of fluency) in mathematics will ensure that student exercises are *scaffolded* appropriately (e.g. similar examples; prompts; one-to-one or small group coaching where available; provision of tactile and/or visual learning resources such as blocks, diagrams, counters…) to make exercises accessible for students still not confident with concepts.

As students become increasingly confident, remove or ‘fade’ these scaffolds and encourage them to tackle exercises using ‘pen and paper’ working and their own mental strategies. Remember that the development of fluency in mathematics can be defined as a continuum along which students are encouraged to move from concrete to increasingly abstract conceptualisations.

**(d) Practising Problem Solving and Using Group Work**

Once a level of mathematical fluency has been established, there is great value in having students work either individually or together on a problem (related to the focus for the lesson). Doing so will develop higher order thinking skills such as problem solving and reasoning, especially when they are pushed just outside their comfort zone or ‘zone of proximal development’. Building in group work and/or individual problem solving sessions allows students to experiment, work through, persist and learn from mistakes.

Productive discussion should be encouraged during, and at the conclusion of, these sessions. This could be structured (e.g. asking groups or pairs to share their working / thinking with the class) or less structured (e.g. small groups working through an open ended problem together, where there will inevitably a degree of discussion).

As a teacher, listen carefully for ‘student voice’ by prompting students with questions which will promote mathematical discourse, e.g.:

*‘Tell me about how you arrived at that answer’**‘Can you convince me that your answer makes sense?’**‘What was easy in this problem? What did you find difficult?’**‘Who arrived at a different answer? Why might there be differences’*…

Better still, choose problems and tasks that are *open ended* and can be easily differentiated (e.g. multiple solutions; students able to set different parameters to make the problem easier or harder).

Provide resources and working materials – blank paper and/or grid paper, counters, MAB blocks, plasticine… whatever the task lends itself to. This will assist students with ‘tinkering’ with possible strategies and solutions.

Choose mixed ability groups if your intention to benefit from peer-on-peer learning / coaching; choose ability differentiated groups if you want to vary the challenge levels in tasks (but do so discretely – perhaps even allowing students the choice regarding the level at which they would like to challenge themselves.) You can also use this session to focus more individualised attention on students needing support.

Remember – the objective here is to move your students beyond understanding and fluency towards ‘problem solving, communicating and reasoning’. Plan problem solving and group work activities carefully and be prepared for a variety of student responses. Developing a culture of openness to experimentation and making mistakes will make these sessions less threatening and more fun as students give themselves and one another permission to explore and play with mathematics.

** (e) ****Remembering Reflection**

Use whatever your learning intentions for the lesson were to recap and reflect at the end of a lesson or learning sequence. These focus questions in turn should be guided by the curriculum focus – e.g. Year 4 Fractions:

**‘Recognise that the place value system can be extended to tenths and hundredths, and make connections between fractions and decimal notation’**

**–> “Who can tell me about some fractions that are easy to convert into decimals? Why is that?”**

It’s easy to run out of time, especially when students are engaged in their work or when packing up takes longer than expected. When this occurs, perhaps take time after the break to quickly recap and refresh on some of the key questions in the lesson.

Again, using questions to generate mathematical discussion is especially useful here.

Try to reflect at least once on context for the maths covered – *how could this mathematics be useful in the world beyond the classroom?*

Some teachers use the idea of a ‘maths journal’ – where students take 5 minutes at the end of each lesson to write one thing down they have discovered, or a mathematical question to be considered later in discussion (e.g. *“What happens with trading when I have to subtract decimal numbers?”* or *“What is the actual use of doing 2D Shapes?”*).

**2. Covering Content**

Covering the appropriate content, at the appropriate level of depth, in a reasonable amount of time will *always *present a challenge for teachers. Some tips to make doing so as painless as possible and to avoid getting ‘bogged down’ in unnecessary detail or content are as follows:

**Start with the syllabus document as your guide.**This tells you clearly not only the generalised outcomes for students at each Stage and then Grade level, but also the expected content to be covered (as well and ‘Background Information’ and useful ‘Language’).**Use a good scope and sequence**that has been worked out with your grade buddies and/or curriculum leaders. Following a textbook scope and sequence is not always a good idea as you may find yourself getting bogged down in too much detail in some areas and neglecting others through lack of time. Let your reference text follow your scope and sequence – and not the other way around. Ensure you have some room for ‘give’, to allow for re-covering some content students find difficult and moving more quickly through areas students seem confident with.**Be confident in your own understanding of the content,**whilst also allowing students to see that even you make mistakes and struggle from time to time. Never give the message “Maths is not my thing” – but rather “I may find this a bit challenging but I will keep working at it.”**Always try to find some ways in which the maths you are teaching relates to the real world**– and include this as a feature in your lessons.*For example, if teaching division (sharing between equal groups) to Year 1 or 2, explain how this could be used for sharing out batches of cakes in a bakery to ensure the same number of cakes in each bag.*

**3. Mindful Delivery**

Delivery in teaching is what we might otherwise refer to as the ‘art’ of our profession. It is the way through which we engage our audience, maintain their attention, make our teaching and learning experience ‘un-forgettable’ – and importantly, build trusting relationships with our learners.

‘Mindful delivery’ simply means approaching our lessons remembering that getting into the minds of our students usually requires conscious effort. Try to answer for yourself the following question: “If I were (one of my students), what might it take to attract my attention, maintain my enthusiasm and develop my understanding?”

Some ideas that might help you develop mindfulness of your maths lesson delivery:

**Allow mathematics to be playful.**Remember, we remember much more effectively that which we associate with positive emotions (humour, camaraderie, team spirit, playfulness). Where you can, use games, role plays, outdoor activities and creative tasks to reinforce concepts. Remember the old saying: “Teaching is 90% theatre” (I don’t know who first said this, but I will credit it to John Kouimanos, my first Head Teacher and the current Chairman of the Teachers Mutual Bank).**Tell stories to exemplify concepts.**Especially stories about yourself – students love getting to know you! If you’re not comfortable with them knowing too much about your life outside of school… make it up! Using narrative and oral tradition to help our learners to recall important concepts and details is a trick that is as old as human civilisation itself – and something that the First Australians did instinctively and effectively! (See also our recent ‘Maths in the Media’ story on the work of Dr Chris Matthews.).**Use a range of resources.**Don’t just stick to one textbook, one lesson format, one digital maths resource. Provide a variety of materials to assist students that require visual and concrete aids to their learning.**Remember the importance of ‘student voice’.**Having kids verbalise what they’re thinking about is critically important, not just to have them share their ideas but also to have them become consciously aware of their own ideas and learning processes. During ‘teacher talk’ time students are often either passive ‘sponges’ or simply tuning out altogether. Productive student talk encourages your class to engage more actively in their learning, and gives you highly valuable assessment insights.

For further information and insight into developing mathematical discourse, see the Calculate ‘Munch’ articles ‘Learning Conversations in Mathematics’ and ‘Hints for Turning Teacher Talk into Student Talk’.

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**4. Creating Context.**

Context in maths for children and young adults is especially critical. As educators we are all too familiar with the catchcry from our students, “Why are we even learning this stuff, Miss / Sir?” Very often, objections of this nature indicate some underlying difficulty with content. The real question underlying their grumbling: “If I’m going to be bothered persevering with this, is it really going to be worthwhile at the end?”

Giving examples of where maths is used in the world outside of the classroom shows our students that maths is, at its heart, a tool. With that tool we can describe, explore, discover and predict things in the world around about us in ways and at levels of precision that words or pictures alone could never allow.

It also shows us that without a *context* based in the real world, mathematics often appears nonsensical – in the same way as would a foreign language or a word for which we do not yet know the meaning. However, by persevering with mathematics (as with learning a new language), we are unlocking wonderful new ways of understanding our world and solving problems.

Ways to incorporate context into your lessons include:

**Commencing new topics in mathematics**with an anecdote from your own experience, a Youtube clip or film excerpt or a short talk from a friend or community member;**Pick a maths topic focus and have students take a stroll**with you around the school or the local neighbourhood / community, noting down anything they see that might relate back to their understanding of the topic;**‘Make the maths visible’ in other subjects**, by drawing attention to, or providing a verbal explanation of, how mathematics is used across in other academic disciplines as and when you are teaching them;**Consider maths excursions**, class talks or activity days held in partnership with local business people or industry partners; or**Using ‘authentic’ and ‘rich’ tasks**in the maths classroom, drawn from problems or situations in which students might need to apply or use mathematics in the real world.

For further detail on the importance of context in maths teaching and learning, see the Calculate ‘Munch’ article ‘Why Context is Critical…’.

Remember also, however, that as Bertrand Russell said, “Mathematics, when rightly viewed, possesses not only truth, but supreme beauty.” For a proportion of your students, mathematical inquiry and complexity will be appealing in and of itself, without requiring you to make a real-world application visible (indeed, many mathematical phenomena and relationships do not necessarily have a ‘real world’ application – at least not yet!). Such students should be encouraged to continue to enjoy the beauty of maths for its own sake, bearing in mind that such students may well grow into the advanced mathematical thinkers and problem solvers of the future!

**– Marcus Garrett**