A Practical Guide to Teaching Mathematics in the Secondary School

Clare Lee, Robert Ward-Penny, Clare Lee, Robert Ward-Penny

• Publisher: Routledge
• Year: 2019
• eBook ISBN: 9781351060691
• Edition: 2
• Topic: Education
• Subtopic: Education General

A Practical Guide to Teaching Mathematics in the Secondary School offers straightforward advice, inspiration and a wide range of tried and tested approaches to help you find success in the secondary mathematics classroom. Illustrated throughout, this fully updated second edition includes new chapters on using ICT in the classroom and promoting a positive learning environment, as well as fresh and easy to use ideas that can help you engage your pupils and inspire mathematical thinking. Covering all key aspects of mathematics teaching, it is an essential companion for all training and newly qualified mathematics teachers.

Combining ideas and practical insights from experienced teachers with important lessons from educational research, this book covers key aspects of mathematics teaching, including:

• planning effective lessons
• using assessment to support learning
• encouraging mathematical activity
• integrating ICT into your teaching
• making lessons engaging
• building resilient learners.

A Practical Guide to Teaching Mathematics in the Secondary School is an essential companion to the core textbook Learning to Teach Mathematics in the Secondary School. Written by expert practitioners, it will support you in developing imaginative and effective mathematics lessons for your pupils.

Chapter 1

Planning mathematics lessons

Robert Ward-Penny and Clare Lee

For an emerging mathematics teacher, planning is fundamental. A good lesson plan can provide a foundation for both effective teaching and successful learning. Planning can also help to tackle many of the concerns and fears that teachers have as they start their careers; in the words of the author Alan Lakein, ‘planning is bringing the future into the present so that you can do something about it now’.

As people think and organise their thoughts in different ways, planning is a personal process. You must develop techniques and habits that support your own teaching, and this chapter is intended to set this process in motion. It contains a number of practical activities that you can use to develop your planning skills while you begin your teaching. As you read through this chapter, you might find it useful to have to hand a copy of any pro forma documents provided by your institution or mentor and, if possible, some examples of lesson plans that you have already written.

Beginning to plan

There is no one correct way of planning a lesson. Some teachers find it useful to start with the learning objective, or aim of the lesson, and then sketch out some rough ideas before starting to construct a formal lesson plan. For instance, you might start with a large piece of paper and write down everything that comes to mind about the mathematical concept or procedure that you are teaching. You can then pick out the most relevant aspects and appropriate resources for your particular class and draw a path through the ideas (Figure 1.1); this path forms the basis for a more formal lesson plan.

Figure 1.1Informal planning diagram for ‘adding fractions’

Task

Sketch an informal planning diagram for teaching coordinates, and draw a path through it in a similar manner to Figure 1.1. Do you think you would find this approach useful when starting to plan a lesson?

One of the most common ideas currently used in planning mathematics lessons is the three-part lesson: starter, main and plenary. This is a useful starting point, as it reminds you that pupils’ attention spans are limited, and that moving between tasks can maintain pupils’ focus and bolster their learning. However, it is intended as a structure, not a straitjacket; for instance you might find that a longer lesson requires a mini-plenary in the middle, or two iterations of each phase. On the other hand, extended work such as investigations may call for a more holistic approach; examples of such tasks are offered in Chapter 8.

Task

Observe a range of experienced teachers and focus on the ways in which they structure their lessons. How do they use or adapt the idea of the three-part lesson? How do they alter the structure of their lessons to suit different classes?

Planning effective starters

An opening or introductory activity can serve a number of different roles in a lesson:

• Linking back – start a lesson by reflecting together on what your pupils already know. If a class is in the middle of a series of lessons, you might choose to briefly recap the previous lesson. For instance, you might ask pupils to work in groups to create a spider diagram of everything they know now about ‘area’. This information can then be added to throughout this and subsequent lessons.
• Looking forward – begin with a problem that the pupils cannot yet solve efficiently. For a lesson on the nth terms of sequences, you might start with some simple linear sequences. Can the pupils find the tenth term of each sequence? What about the millionth term? Returning to the same problem at the end of the lesson can help pupils explicitly recognise their own progress and the purpose of the learning.
• Mental/oral starters – a good opportunity to develop each pupil’s facility with mental mathematics. This might take the form of a generic game such as bingo or ‘Countdown’, or be tailored to support the learning objectives. For instance, if you were teaching a lesson about angles on a straight line you might present a 3 × 3 (or 5 × 5) grid of numbers, where four (or 12) pairs of numbers add up to 180 – can the pupils find which number is left over? Similarly, a mental starter on multiplying fractions can support a lesson on tree diagrams and pre-empt difficulties.
• Real-world starters – a starter involving manipulating numbers drawn from the real world. For example, pupils may work in groups to answer ‘how many toilet rolls do you think the UK uses each year?’ or ‘how many letters do you think fit in a post-box?’ and then defend their responses. Questions can also arise from recent headlines or from the calendar, for instance: ‘how many gifts were given in total during the song “The Twelve Days of Christmas”?’
• Focusing starters – starter activities can also be used to help manage behaviour. Having an activity such as a puzzle ready at the start of a lesson can direct pupils’ attention as they enter the room. Pupils could use five minutes at the start of the lesson to work in pairs and see how many playing cards they can make into a stable house, or play a reaction testing game on the interactive whiteboard, generating data to be used later in a lesson about decimal numbers or summary statistics. In each case, pupils who arrive and settle promptly benefit from doing so.

Task

Choose a mathematical topic that you might soon have to teach; for example, calculating the mean from a set of discrete data. Which of these types of starter do you think would be most appropriate for this topic? Sketch out two or three ideas for starters you might like to try for yourself.

Now try this

Drawing on the ideas offered above, your own ideas and your observations, write down a list of at least five different starter activities. When you are in school, try these out in the classroom and make notes. What worked well? What was less successful, and why? Did any types of activity particularly suit different groups?

Planning for learning

The learning objective or learning intention is central to planning mathematics lessons. You will usually draw this from a medium- or long-term planning document. However, you should also think about learning outcomes or success criteria in your planning. These will help you to structure and differentiate pupils’ learning in more detail. As an example, consider this learning objective: be able to visualise and use plans and elevations of 3-D objects. You might choose to deconstruct this objective into three learning outcomes. Can you:

• identify 3-D shapes when given plans and elevations?
• draw plans and elevations of basic 3-D shapes?
• work with plans and elevations that include hidden (dotted) edges?

This list offers a logical structure for progression within the lesson: you might start with a whole-class exercise where the teacher projects plans and elevations of real-world objects onto the board for the pupils to identify; move onto a worksheet with simple 3-D shapes from which the pupils have to draw plans and elevations; and then conclude with a practical exercise where pupils make a structure with blocks and then sketch the corresponding plan and elevations. Breaking down the pupils’ learning in this way can also help you identify exactly what new mathematical ideas and concepts you will need to explain. By focusing on the learning you will be in a better position to plan the content and timing of your explanations, and to select appropriate questions and worked examples that will move your pupils’ understanding on.

Plan what you say. Clear explanations are key so if you have a particularly complex topic to cover, write out what you are going to say first. This helps to avoid babble and stress because you have considered probable queries and areas of misunderstanding pupils may have.

Madeleine Adams

Teacher of mathematics, Oxford

Task

Starting with the learning objective ‘understand Pythagoras’ theorem and be able to find missing side lengths in right-angled triangles’, write down a set of outcomes for your pupils that would indicate a clear progression towards a good understanding of, and ability to use, Pythagoras’ theorem. How might you develop these outcomes into activities for the main part of a lesson?

Planning for variety

If you ask your pupils what they want from a ‘good’ lesson they are likely to mention variety, group work and choice. It can be challenging for a new teacher to plan for variety, since transitions can be difficult stages to manage. One rule of thumb is to change the activity roughly every 15 to 25 minutes, unless what the pupils are doing demands more time. Signal the change before it happens, for example by saying ‘in five minutes I will want you to put those blocks away and get some spotty paper’ or using something like a countdown clock on your interactive whiteboard to keep both you and the pupils to time.

Variety in lessons involves a balancing act between several aims. For example, you will need to balance:

• giving your pupils time to build their understanding of mathematical concepts with setting aside time for consolidation and practice;
• encouraging the pupils to think and reflect individually and allowing them to talk through ideas with others;
• providing ways for your pupils to see, feel and touch with requiring them to read diagrams and develop their ability to visualise.

Task

Think back over several lessons that you have observed or taught and think about the ways that pupils worked in those lessons. How did the teacher plan for variety?

It is likely that many of your lessons so far have included individual work using textbooks or worksheets, partly because that is the way mathematics has been taught in the past, but also because that might feel safe while you learn to keep control in your classes. The chapters in this book are full of practical suggestions that can help you introduce variety into your teaching. However, you can also vary the way in which you use the textbook itself.

Making the most of textbooks

Textbooks are a common feature of mathematics classrooms, and they can be a valuable source of practice material. Over time, however, pupils will tire of simply working through lengthy exercises, so it is worth considering different ways in which you might plan to use a textbook.

• Not every question – which questions do learners need to attempt to work towards the learning outcome? Would it suffice to only do the odd questions, or the prime numbered questions? Perhaps the pupils could decide how confident they are and choose themselves, for instance by selecting five questions from a set of ten.
• Reverse engineering – start by considering with the pupils how the questions are graduated. What makes question 2f (simplify a²b × ab²) more challenging than question 1d (simplify a²b³ × a⁴b²)? Where do they think most people will make mistakes? Which question is the hardest, and why?
• Do it yourself – get the pupils to write a textbook page for themselves. How will they introduce the topic? Will they include examples? What questions will they include, and why?

Many modern textbook series come with teachers’ guides and linked multimedia resources. These might also provide you with ideas to support your planning.

Task

Find a mathematics textbook and choose a page at random. What do you think are the strengths and weaknesses of this page? If...

Table of contents

• Cover
• Half Title
• Series Information
• Title Page
• Copyright Page
• Contents
• List of illustrations
• Notes on contributors
• Series editors’ introduction
• Acknowledgements
• List of abbreviations
• Introduction
• Chapter 1 Planning mathematics lessons
• Chapter 2 Practical Assessment for Learning
• Chapter 3 Using ICT safely and effectively
• Chapter 4 ICT from the front of the class
• Chapter 5 Pupil-led ICT
• Chapter 6 Working collaboratively
• Chapter 7 Discussion and communication
• Chapter 8 Enquiry as a vehicle for teaching and learning mathematics
• Chapter 9 Going the extra mile
• Chapter 10 Promoting a positive learning environment
• Chapter 11 Developing subject knowledge
• Chapter 12 Action research – systematic reflective action to improve practice
• Index