Understanding and Teaching Primary Mathematics

Observing Sections

Chapter 3 - Observing ‘Problem Solving using Mathematics’

The activity uses the idea of a party and starts off with a table that seats 6 children, 2 at each side and 1 at each end. As more people arrive, tables are added to the end of the previous table. Watch the piece of video and work out for yourself what the pattern will be. Once you have a pattern that will allow you predict the number of people seated at any number of tables try to generalise the formula using an algebraic expression. Now explore other possible arrangements of tables. What would happen if you formed ‘L’ shapes with the tables? What would happen with tables which are different shapes?

Think carefully about how you would introduce this to a class you are working with. What questions would you ask to scaffold the activity? How would you leave the activity to be open enough so that you children move beyond simply filling in tables of results and how would you encourage them to generalise?

Chapter 4 - Observing 'Counting and Understanding Numbers'

On the companion website you can watch a session called 'Decimals Forever' taught by Jonny Heeley a teacher who is a previous winner of a Teacher of the Year Award. Although the session is put together especially for TV and described as a 'Master Class' there are lots of ideas that are directly transferable to most classrooms. The opening activity for example is a practical version of the activity suggested earlier in the chapter and helps pupils match decimals to their equivalent fractions and percentages. When you have watched this piece of the lesson think carefully about how you might organise this in your classroom. The key things to think about are which fractions you would choose and how you could differentiate the activity so that all your pupils could take part.

Focus on the opening 5 minutes of the session rather than watching all the way through and think how you could expand this for a full session. The first thing that I would do is work with the 'number families' so that they arrange themselves in ascending order. It would also be useful to ask your pupils to place themselves on a number line which you could place on the classroom floor or in the playground.

I was surprised at the decision to ask the pupils to explore recurring fractions by carrying out long division rather than using a calculator. If the purpose of a calculation is to explore the patterns which the results make it is much more sensible to use a calculator to carry out the calculation.

Chapter 5 - Observing 'Knowing and Using Number Facts’

Watch the video which shows a counting stick being used to support the learning of multiplication tables. A counting stick providers a powerful mental image which can support children in both learning multiplication tables and retaining these facts. They are easy to make either by buying a metre long piece of dowelling rod, or using a metre stick. Use coloured tape to define 10cm long sections on the stick .

Whilst you watch the video try to learn the multiplication table yourself by putting yourself in the place of a learner. Reflect on the following questions: How does the counting stick support learners in memorising new facts by drawing on facts they already know? How might you use the counting stick to model other aspects of the number line? (For example how could you use it to model decimals between 0 and 1, or multiples of 100,000 between 0 and 1,000,000)?

Chapter 6 - Observing 'Calculating'

Find the clip called 'How Many Peas Fill the Classroom'. This is a fascinating activity in which the teacher works with her class to try to calculate how many peas it would take to fill the classroom. As well as developing and practicing children’s calculation skills it also develops the idea of very large numbers in a real context. Children often find it very difficult to visualise very large numbers. The task also requires the pupils to break down a complex problem into small steps.

Although this clip is from a Year 7 group I have carried out very similar activities with learners in primary schools. Focus particularly on the segment from 4 minutes onwards which focuses on the teacher taking feedback from the groups. What might you do differently as a result of observing this section of the lesson? One concern might be that the large numbers they were working with did not carry very much meaning to the children as they didn't break the task down into chunks which they could visualise. Perhaps dividing the classroom up into more manageable 'chunks' to start with would have allowed the learners to make more sense of the answers they were coming up with.

Imagine you are feeding back to the teacher. What are 3 things that you feel were strengths in the lesson, and what is one key thing that you would suggest the teacher does differently? How would you adapt this activity for the group that you are currently working with?

Chapter 7 - Observing 'Understanding Shape'

Find the video which shows a teacher working with her class of 7 and 8 year olds in describing and constructing 3 dimensional shapes from 2 dimensional drawings. I would like you to focus on the section from 4 minutes 30 seconds into the video. This is the point at which the children are hiding 3D shapes behind a board and describing them to a friend who then has to guess which shape is hidden. Listen to the extract and jot down all the technical language that the pupils are using to describe the shapes.

If you were planning this activity which shapes would you use in order to maximise the vocabulary that could be developed? What do you see as the teacher's role in this sort of an activity?

A useful way of adapting this activity is to ask the children what the shape could be after each question so that begin to notice the sorts of properties that shapes have in common. So, for example, if a shape has 8 vertices it may be a cube, or another kind of cuboid. The children have to ask another question to be certain which shape it is.

Chapter 8 - Observing ‘Teaching Measuring’

Find the video which shows teacher introducing an activity which involves her pupils in measuring 500 millilitres of fluid. She sets an imaginary context for this by dressing up as a 'witch' who requires the potion. Watch the clip from the beginning and think through other contexts you could use to motivate the pupils. In this role play the teacher does not fully bring the role to life, she is really just the teacher dressing up. Could you enter into the role more convincingly?

Observe the different ways in which the children record their 5 measures which will total 500 millilitres. How could you encourage them to think of as many different ways as possible? How might you adapt this lesson for use with the pupils you are currently teaching?

Chapter 9 - Observing ‘Handling Data’

Here are 2 examples of how to explore the data handling cycle holistically. In the first lesson in the clip, the teacher takes a holistic view of the data handling cycle, collecting, recording and analysing data about school meals. He sets this as a genuine question suggesting that the new school cook wants some feedback. He chooses to use human charts as one way of representing the data. Watch the clip and then think about the following questions.

What are the advantages of using the pupils to create human bar charts? Can you think of other ways of involving learners so directly in representing data? Watch the second lesson in the clip which links the data handling cycle to athletic achievements. Here the children actively engage in ‘athletic’ type activities and record the results. After watching the clip, think about these questions. How would you safeguard children's self esteem if using this activity? How would you make sure that pupils who are less confident in their athletic ability were involved in the activities? Focus on the use of cameras and video to help in assessing these activities and the range of activities which require the use of different units in measurement. This activity shows how data handling can be used as a part of a cross curricular project.

Chapter 10 - Observing ‘Teaching in the Early Years’

The video clip describes how a school in Birmingham introduced personalised learning programmes for their learners in the Early Years in order to transform the view of mathematics for all learners in the Primary School. As well as offering lots of starting points for teaching this extended clip gives you a clear sense of what effective mathematics teaching in the Early Years might look like. The reflective discussions between the practitioners will help you understand the reasons for the developments they are making.

Watch the clip and then make notes in your portfolio about the following questions. What are the implications for your teaching in the later years? What are the key things that primary classrooms should learn from Early Years practice?

Chapter 11 - Observing ‘Inclusive Mathematics Teaching’

The video clip shows how a piece of IT called a visualiser is used to support deaf children in a mathematics lesson. This allows a teacher to ‘film’ their demonstration and record, or add text to the piece of video. This particular clip shows the equipment being used in a secondary school to demonstrate the construction of bisectors of angles. Watch the clip and then reflect on the following questions in your portfolio.

What could you use the ‘Visualiser’ for in your classroom? Can you think of a lesson you have taught recently which would have been enhanced by the use of a visualiser?