# Enhancing teachers’ subject knowledge and developing reasoning

## Written by Lynda Maple

It’s almost twenty years since I first started to lead Let’s Think Mathematics courses. Many things have changed that influence the decisions tutors make about the way different aspects are covered, not least the changes to the mathematics curriculum. While Let’s Think activities are not dependent on national initiatives, being aware of the climate in which teachers are working helps to forge positive relationships with the participants.

A focus on teachers’ subject knowledge comes and goes but it can’t be denied that this is an important aspect in a teacher’s repertoire of skills and competences. For me, one of the most interesting roles for a Let’s Think tutor is to spot opportunities within the lessons to enhance teachers’ knowledge of mathematics and to help them develop a deeper understanding of the relationships between concepts within the subject.

Many teachers in primary schools express their lack of confidence in the subject and the Let’s think programmes often provide a safe place to discuss what they see as deficiencies in their own learning. It’s a great moment when someone who feels nervous about their understanding about a particular topic, ‘sees the light’ for the first time in what might be many years. It doesn’t come about because the tutor has set out to teach something explicitly, but rather, through breaking the concept down to its basic level and building understanding from scratch. This is a fundamental approach when tutors simulate lessons as part of the training.

Some Let’s Think lessons seem to exemplify this theme time and time again. Take the lesson ‘Sharing an Apple’. Early on in the first episode, students are asked to explain the written fraction ½. In the simulation, teachers, like their students, offer a number of suggestions –

The 2 means two parts, the 1 is how many of those parts, the / is ‘out of’. So ½ means 1 out of 2.

Or, 1 divided by 2. Other fractions are explored including ¼, 2/3 and ¾.

It is usual for everyone to agree with the ‘out of’ description, but much discussion takes place about the ‘divided by’ aspect. Can ¾ be 3 divided by 4? How could you show this visually? It’s not unusual for disagreement in people’s views. An excellent example of cognitive conflict! This is further challenged when the tutor writes a/b and asks what that means. Someone will answer a divided by b and refer to their own time at secondary school. The small-group discussions that follow are fascinating and its common to hear it stated ‘I never knew this – now it makes sense’!

A lesson written for younger pupils ‘Giant’s Palace, explores place value using the relationship of smaller parts to a whole, comparing smaller and smaller footsteps, combining them and evolving a way of writing the total. The first time I used this lesson was a real eye-opener for me and the teachers. I took for granted that they all understood place-value and was not expecting the ensuing discussion. Several teachers were intrigued to see how the context of the story enabled the concept of place-value to be exemplified so visually. For some it demonstrated how the position of digits in a number related to their value. ‘Now I see’, said one teacher, to be followed by other similar comments and smiles from some quite unsure members of the group.

It seems that teachers of mathematics in secondary schools, can also develop clearer understanding and enhance their subject knowledge through the Let’s Think approach. One of my favourite lessons, Pencils and Rulers, starts by using a grid to find the total costs for different combinations, eg 4 pencils and 6 rulers, when the unit cost is known. The lesson develops where grids show a selection of total costs and the unit cost has to be deduced in order to complete the grid.

Shorthand methods are encouraged to show results

Eg 3p + 4r = 36, and 2p + 5r = 38.

I then write these underneath one other and ask the group where they have seen such notation.

3p + 4r = 36

2p + 2r = 20

Simultaneous equations!

They are then given pairs of equations and have to solve them using a grid. The groups are then challenged to explain how the grids and the formal notion are linked. It’s a really good example of developing reasoning – a key principle within the Let’s Think approach.

I’ve experienced all sorts of emotional responses to the activity. Many non-specialists start by feeling very insecure and even anxious when the topic of simultaneous equations is introduced, but by the end of the session there is a shared feeling of success and a sense of real achievement. Specialist teachers of mathematics can be quite surprised at how a very simple idea develops into something quite complex and difficult to teach. On one course, a head of department stated that she was going to teach the lesson to her colleagues and re-write the school’s scheme of work so the lesson could be included.

The richness of the Let’s Think lessons caters for very different levels of experience and competence, for students and teachers alike. For some, it is about building up knowledge and understanding from simple ideas within concrete contexts. For others, it’s the other way round. Start with the formal abstract idea or algorithm and unpick it to find the basic principles. In mixed KS2/3 courses, this happens all the time and one of the great benefits is that teachers’ subject knowledge is enhanced by working together. An idea that underpins Let’s Think work regardless of age or aptitude.