Mastering Maths research lessons

On this page you will find sections on:

If you are looking for the Desmos self-study materials which introduce the lessons, here is the link. 

The design of the lessons


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

 

 

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





Overviews of each of the lessons are here (one PDF).

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The Key Principles and Research Questions are here (one PDF).

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All_CfEM_lessons_Overview.pdf

   


CfEM Lessons 1 to 12 LOs KPs and RQs.pdf

The rationale for the lessons (reminder: all the lesson resources can be found here).


Lesson 1: Multiplicative reasoning

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).

Lesson 2: Ratios and fractions

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.

Lesson 3:  Basic algebra (factorising and multiplying in algebra)

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).

Lesson 4: Algebraic thinking

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.

Lesson 5: Percentage chnage and best buys

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.

Lesson 6: Frequency charts and averages

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.

Lesson 7: Understanding straight line graphs

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.

Lesson 8: Understanding equations

Students are usually familiar with using letters to represent variables but can struggle to understand the mathematical structure of an equation. In this lesson algebraic equations are used to model real-life situations and students explore the effects on the meaning of equations and expressions when the definitions of the variables change.

Developing an understanding of mathematical structure (Key Principle 1) through mathematical representations is a key part of the teaching for mastery approach and in this lesson the use of representations is encouraged when exploring relationships.

Possible incorrect equations are discussed, to expose common misconceptions and provide insight into students’ existing knowledge (Key Principle 2). Providing different possible solutions can give students a starting point, building confidence and encouraging them to share their understanding (Key Principle 5).

Lesson 9: Using frequencies and probabilities

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. 

Lesson 10: Geometric reasoning

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. 

Lesson 11: Factors and multiples (previously Lesson A)

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).

Lesson 12: Areas and volumes  (previously Lesson B)

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).

Where can I find the lessons?

The lessons are hosted on the ETF website.

Closing the lessons

In many lessons, it is noticeable that teachers may leave things unresolved from the point of view of the students. In Mastering Maths lessons this may be because it has been something of an achievement to get students collaborating and working on mathematics. Why would you want to stop them?

Discussion of how to solve the problem(s) is very important, and needs time.In early trials of lessons it seemed that the teacher needed something like a quarter of an hour to close the lesson.

However, it is important to plan for this part of the lesson. It is difficult because teachers need to know where students have got to, and work with that, at the end of the lesson. They will want to work with student thinking and summarise collaborative thinking. In the close of the lesson the teacher can leave nothing to chance by pre-planning for lesson closure "in the moment". This does require the teacher to have really understood the way in which the classroom task and activity engage the students in working with important mathematical concepts.

Questions teachers can ask themselves to plan for the closing of the lesson might include:

It is more important to discuss approaches to solving the types of problems that students have been working on – such as the representations that are most useful – rather than the solution(s) to the problem(s). Above all, they should try to orchestrate a whole-class discussion that draws on everyone's thinking.

Although it is perhaps difficult to plan the end of the lesson in detail, teachers can prepare for it by thinking about where they think the students in each class might have got to with each particular lesson after the time they have. AND, importantly, they should think about where in the lesson they will have access to information and students' thinking that they might be able to draw on to inform a whole-class discussion. 

In the closing part of the lesson they should know what everyone has been thinking and how they might work with their contributions to sum up the learning (mastery) of the group.

The Mastering Maths lesson plans all provide guidance about running the part of the lesson that follows the main pair-work task. Teachers should read this guidance carefully.