A major strand of the work of the CfEM programme was a randomised control trial, in which teachers in the intervention arms of the trial taught seven lessons specifically designed to exemplify a Teaching for Mastery in FE approach.

The Teaching for Mastery in FE approach is based on five Key Principles:

- KP 1: Develop an understanding of mathematical structure
- KP 2: Value and build on students’ prior learning
- KP 3: Prioritise curriculum coherence and connections
- KP 4: Develop both understanding and fluency in mathematics
- KP 5: Develop a collaborative culture in which everyone believes everyone can succeed

These Key Principles are incorporated into each of the example lessons, with each lesson emphasising one of them.

The design of the lessons takes into account both the mathematics to be taught and ways of working in the classroom.

In terms of the mathematics; lessons:

- address fundamental mathematical ideas/concepts
- highlight mathematical structure (e.g. by using context, representations, variation)
- foreground students’ common misconceptions through activity that provokes cognitive conflict
- provide for a range of likely approaches
- connect different areas of mathematics where possible.

In terms of ways of working in the classroom; lessons include:

- introduction through a context which is to be considered using mathematics
- drawing on, and valuing, students’ prior learning
- collaborative work and a culture in which everyone believes everyone can succeed
- active and collaborative student engagement on substantial tasks
- whole class discussions which emphasise approaches to the mathematics rather than answer-getting and answer-checking
- opportunities to develop both understanding and fluency.

You can access guidance for use of these lessons with Functional Skills students, with additional exam questions material under the lesson drop-downs below.

The resources below include brief descriptions of each of the lessons, all resources needed to teach the lesson and a links to online professional development activities which are designed to support teachers in preparing to teach the lessons.

When solving proportional problems, students often have just one strategy, most commonly an additive approach. This lesson uses the context of hotel stays and a double number line to explore different ways of solving these problems using both additive and multiplicative approaches. Students have the opportunity to think about the approaches in terms of efficiency for solving proportion problems.

Using representations to provide access to the mathematical structure of a problem is a key principle of teaching for mastery (Key Principle 1). In this lesson, the double number line is used to illustrate and compare the thinking which relies on addition with that which relies on the use of multiplication.

Proportion problems can crop up in many places in the curriculum, e.g. exchange rates, distance–time, rates of pay, conversion between units and similar triangles. This lesson could be used effectively in any of these different places in the curriculum, emphasising that, while the context may change, the approach stays the same. This helps to support students in making connections between mathematical topics, which is an important aspect of teaching for mastery (Key Principle 3).

In the trials, many students started the lesson using additive methods and by the end of the lesson were using multiplicative approaches.

Some students used double number lines effectively but many preferred to use other approaches or methods.

Students often get confused with the different ways in which part–part and part–whole relationships can be represented. In this lesson, ratios and fractions are presented together, with a single diagram that represents both the ratio and the fraction.

The diagrams provide insight into mathematical structure (Key Principle 1) and their use encourages students to see links between mathematical concepts, rather than viewing them as separate content. This is important in supporting a coherent and connected curriculum (Key Principle 3) and is essential in the FE sector, where there is limited curriculum time.

In the trials, most students started incorrectly, but self-corrected with the help of the diagrams.

The lesson worked best when teachers allowed students to work through their own cognitive conflict.

Students often view a letter in algebra as representing a specific unknown that needs to be found and find it difficult to answer questions in which the unknown is a variable. This lesson starts by introducing the context of buying a carpet with width 4m and variable length *n* to establish that *n* represents a variable. This provides students with a ‘concrete’ model that supports the concrete, pictorial, abstract approach to teaching for mastery.

Later in this lesson, area models are used to highlight the relationship between algebraic expressions in factorised and expanded form, and to support students in understanding multiplicative algebraic structure. Developing an understanding of mathematical structure through mathematical representations is a key part of the teaching for mastery approach (Key Principle 1).

This lesson’s focus on what it means to fully factorise an expression helps to establish links with common factors and highest common factors. This encourages students to see the links between mathematical concepts (Key Principle 3).

In the trials, most students completed the tasks correctly, but many of them took some wrong turnings before realising what the correct answers were.

Almost all students were able to answer related exam questions at the end of the lesson.

Students usually know how to solve linear equations. However, they may have had little practice at modelling and representing relationships involving unknowns, and they often find questions involving unknowns inaccessible. The aim of this lesson is to develop students’ algebraic thinking.

The lesson starts by introducing two contexts that are both horizontal in nature: building walls out of blocks and building train tracks. These contexts support students in developing an understanding of how bar models and other diagrams can be used represent mathematical structure (Key Principle 1) and provide a ‘way in’ for students, so that all students can make progress and have some success (Key Principle 5).

In the main task for this lesson, students match geometry questions and word questions with the same underlying mathematical structure, and use bar models to represent the mathematical structure. The focus is on students being able to solve problems using algebraic thinking, rather than algebra notation.

In the trials, students benefitted from using concrete resources, such as unit cubes, to model relationships.

Some students used bar models fluently, while for others using bar models was difficult.

Students often view percentage change as an additive process, calculating the percentage change amount and adding or subtracting to give the required percentage increase/decrease.

This lesson focuses on the opportunity that percentage change problems provide to explore the underlying multiplicative relationship (Key Principle 1) between the original and new values.

Considering additive approaches that students are already familiar with (Key Principle 2), alongside strategies that involve multiplicative reasoning supports students in developing both their fluency and understanding (Key Principle 4) as they learn to recognise when and how to apply additive and multiplicative approaches.

In the trials, students engaged well with different approaches to solving the same problem.

By the end of the lesson, most students were able to answer the exam questions correctly.

When working with frequency charts, students often struggle to see the relationship between data presented as a list and the same data represented on a chart. The use of sticky notes in this lesson, to both record and display data values provided by students, helps students to identify their own data values within a frequency chart and develop their understanding of the relationships between different representations of data.

Using multiple representations provides insight into the mathematical structure of data sets and exposes the way in which the three averages provide a summary (Key Principle 1). Developing both fluency and understanding is an important part of the mastery approach (Key Principle 4) and in this lesson, time is spent interpreting and comparing various data sets represented using frequency charts and summary statistics. By exploring these different representations, students are supported in developing a deeper understanding of the way in which the mode, median and mean represent the average of a set of data and the distinction between measures of average and range as a measure of spread.

This lesson can be used with Functional Skills students (see guidance below).

The key principle that is being focussed on in this lesson is KP4 ‘Develop both fluency and understanding’

Students often struggle to model a real-life scenario mathematically. In particular, in the case of linear equations, they tend to not understand the role of a constant (such as a fixed cost) and the rate of change (such as price per unit). In this lesson they develop their understanding of how to represent a situation algebraically, as a linear equation, and then to plot the straight line graph it represents. They use the graph to answer questions related to the scenario.

The linear relationships are presented as written descriptions, algebraic equations and graphical representations simultaneously to help students to see the **links between mathematical concepts**, which is important in supporting a **coherent and connected curriculum** (Key Principle 3). The use of multiple representations provides insight into the **mathematical structure** (Key Principle 1) and helps students understand how linear relationships are represented by straight line graphs.

This lesson is not suitable for use with Functional Skills students.

The key principle that is being focussed on in this lesson is KP3 ‘Prioritise curriculum coherence and connections’

Students often struggle to model a real-life scenario mathematically. In particular, in the case of linear equations, they tend to not understand the role of a constant (such as a fixed cost) and the rate of change (such as price per unit). In this lesson they develop their understanding of how to represent a situation algebraically, as a linear equation, and then to plot the straight line graph it represents. They use the graph to answer questions related to the scenario.

The linear relationships are presented as written descriptions, algebraic equations and graphical representations simultaneously to help students to see the **links between mathematical concepts**, which is important in supporting a **coherent and connected curriculum** (Key Principle 3). The use of multiple representations provides insight into the **mathematical structure** (Key Principle 1) and helps students understand how linear relationships are represented by straight line graphs.

This lesson is not suitable for use with Functional Skills students.

The key principle that is being focussed on in this lesson is KP1 ‘Develop an understanding of mathematical structure’.

Students at this level are likely to be familiar with the probability scale and to know how to calculate theoretical probabilities. They usually understand that the relative frequency of an event can differ from its theoretical probability, but can sometimes focus on procedures rather than developing their understanding of the ‘why’. In this lesson frequencies are explored and represented on a frequency tree diagram, and from this example, a probability model is developed. The shift from the actual frequency to the theoretical probability develops students’ understanding of mathematical structure (Key Principle 1). Students work with decimals and fractions within these models, encouraging them to make connections (Key Principle 3) with other areas of mathematics.

Carefully designed solutions are presented to students within the lesson to expose common misconceptions and provide opportunities to establish what students already know (Key Principle 2), as well as promoting a **collaborative community** (Key Principle 5), where students are encouraged to contribute and share their own ways of working.

This lesson is not appropriate for Level 1 functional skills students but could be used with Level 2 students. See below for guidance.

Students at this level are able to recall a number of angle facts and this lesson provides an opportunity to **establish what students already know** about properties of angles and their understanding of parallel lines (Key Principle 2). Students are encouraged to **make connections** (Key Principle 3) between different properties in order to develop an understanding of how they can be used to identify the necessary information to determine geometric features. The in-depth focus on reasoning in this lesson supports students in **developing both their fluency and understanding** (Key Principle 4) as they learn to recognise when and how to apply angle properties to a range of problems.

This lesson is not suitable for Functional Skills students.

For many students, factors and multiples are abstract concepts that are difficult to relate to. As a result, they often confuse factors and multiples, and highest common factors (HCF) and lowest common multiples (LCM). They rely on remembering a method or technique for finding these, rather than understanding the principles underpinning the techniques. This lesson uses the context of chocolate bars and packing trays to support the development of thinking about factors as the dimensions of a rectangular array and multiplication as the area of the array. Developing an **understanding of mathematical structure **through mathematical representations such as arrays is a key part of the teaching for mastery approach (Key Principle 1). The use of arrays aims to highlight the **links between mathematical concepts **(Key Principle 3).

Students at this level usually know how to find the area and volume of 2- and 3-dimensional shapes and can recall and apply the relevant formulae correctly. **Valuing and building on students’ prior learning **is an important part of the mastery approach (Key Principle 2). In this lesson, time is spent discussing *why *their area and volume calculations work; this establishes what students already know and supports them in developing a deeper understanding.

This lesson’s focus on the effects on area and volume of scaling the dimensions of rectangles and cuboids provides an opportunity for students to see the **links between mathematical concepts **such as area and volume, proportionality, enlargement and similarity. Helping students to make **connections across the curriculum **is an important aspect of teaching for mastery (Key Principle 3).

This lesson can be used with Functional Skills students (see guidance below).