Resources to support teaching and professional development

  1. Teaching for Mastery
  2. The Connective Model
  3. Assessment
  4. Inclusion
  5. Resources for home learning
  6. Professional Development resources

1. Teaching for mastery...teaching for understanding

Progress in mathematics learning each year should be assessed according to the extent to which pupils are gaining a deep understanding of the content taught for that year, resulting in sustainable knowledge and skills. Key measures of this are the abilities to reason mathematically and to solve increasingly complex problems, doing so with fluency, as described in the aims of the National Curriculum.

Teaching for Mastery Introduction (NCETM) 2015

Project exploring teaching for mastery in mixed-age classes

This small-scale action research project explored teaching for mastery in mixed-age classes, focussing on how to teach so that children move through the programmes of study for their year group at 'broadly the same pace', a crucial issue when focussing on 'closing the gap'.

The report includes a number of case studies and links to video clips of teachers talking about these case studies can be found below. The clips focus both on adapting the lesson structure to allow a focus on age-related expectations in a mixed-age class and elements of mastery (or quality first teaching) which proved to be significant during the project.

Structuring the lesson:

Keeping the class together where possible: a flexible approach

Providing separate teaching for individual year groups

Quality first teaching

Elicitation tasks

Pre-teaching

Rapid support and intervention

Questioning

Models and images

Rich tasks

Feedback and marking

2. The Connective Model

Learning mathematics and demonstrating understanding of mathematics involves connecting real experiences, contexts, mathematical images/pictures, language and symbols.

A pupil really understands a mathematical concept, idea or technique if he or she can:

NCETM adapted from John Holt 'How Children Fail' 1964

Below are resources which explain The Connective Model and others which provide support with the use of mathematical images to aid and demonstrate understanding. It is important to understand that children need to play with any image that is going to be used for teaching understanding of mathematical concepts, so that they can become familiar with the resource and how it is structured, otherwise they may not be able to attend to features of the resource as the teacher intends. As a result of play, mathematical ideas and questions will arise.

3. Assessment

Maths assessment: key principles

Key Understandings in Mathematics Single Age

Key Understandings in Mathematics Multi Age R/KS1

Key Understandings in Mathematics Multi Age KS2

Children as evidence: assessing understanding in mathematics by talking to children

4. Inclusion is about meeting the needs of all children

The National Curriculum states that:

The expectation is that the majority of pupils will move through the programmes of study at broadly the same pace. In order to support teachers to meet the needs of all children we offer a range of support including training in flexible and intensive interventions.

Resources

5. Home learning

Home learning support with maths for schools, parents and pupils

We offer support to all adults working with children at home, with the aim that we engage children in positive mathematical experiences where they are encouraged to think, notice and wonder. This is an opportunity for learners to experience the beauty and creativity of mathematics.

We have provided packages of ideas for years 1 or 2, years 3 or 4 and years 5/6 all starting from the same simple resource/experience that can either be found at home, created at home (with paper and pen) or accessed online; those working with children in schools may have access to alternative equipment for the same activities.

Where children are able to work with other children, we would encourage working on maths in a collaborative way, exploiting all opportunities for sharing ideas and building on each other's thinking. You will find there will be more than one way to engage with the activities and this is to be encouraged and embraced.

1. 'Nim'

This focuses on variations of a simple game, for all year groups, using objects from around the home (such as leaves, pasta pieces or Lego bricks) as counters in the games.

The games involve making decisions about choosing numbers to either subtract to reach zero or add to reach a target number; problem-solving and reasoning are key to identifying and explaining winning strategies in the games.

2. 'Nice and Nasty'

This focuses on the same simple game, for all year groups. The game involves applying understanding of Place Value; decision-making and explanation, both elements of reasoning, are central to the activities.

3. 'Pattern'

All the activities are focussed on exploring patterns. In Y1/2 the activities involve thinking about and creating repeating patterns, using things from around the house, whilst in Y3/4 and Y5/6 the activities all involve growing shapes resulting in number patterns (sequences).

The activities for all year groups involve articulating and explaining patterns underpinned by reasoning.

4. 'Balancing Numbers'

All the activities are focussed on exploring groups of numbers that have equivalent totals. In Y1/2 the activities involve thinking about balance scales whilst in Y3/4 and Y5/6 the activities all involve Magic Vs.

These are variations on the activity found on the Nrich website. The focus for all year groups is on Additive Reasoning, all underpinned by problem-solving and reasoning.

5. 'Numbers'

All the activities use pieces of paper numbered 1 to 9. These are shuffled and three or four cards dealt. By combining the cards dealt, the aim is to make numbers to meet given criteria (for example, even, odd, bigger than 10 or multiples of 3).

The focus for all year groups is on Calculation linked to Properties of Numbers, all underpinned by reasoning.

6. 'Measures'

All the activities use string, wool, cotton or something similar to measure body parts. In Y1/2 the focus is on Comparison, Counting and Estimating. In Y3/4 and Y5/6 the focus is on Multiplicative Reasoning, looking at the multiplicative relationship between the lengths of body parts. This includes fractions in Y5/6. The activities are all underpinned by reasoning.

7. 'Number Pyramids' (revisited)

We explore odd and even numbers combined with additive reasoning in all year groups, with Y5/6 also featuring multiplicative reasoning.

8. 'Calculators'

We explore number patterns generated on a calculator. The focus for all year groups is Number; understanding sequences leading to understanding properties of numbers, all underpinned by reasoning and representation.

In all year groups there are arithmetic sequences (produced by adding or subtracting the same number) generating odd and even numbers, multiples and non-multiples and (in Y5/6) negative numbers.

Geometric sequences (produced by multiplying by the same number) are introduced in Y5/6.

9. 'Playing Cards'

We explore the earlier resource; either a homemade (from paper) set of ten cards or ten cards from a playing card deck.

The focus for all year groups is calculation underpinned by reasoning and problem-solving: Y1/2 and Y3/4 focus on Additive Reasoning whilst Y5/6 focus on both Additive and Multiplicative Reasoning.

10. 'Money'

For these activities coins are used. The focus is on Additive Reasoning in Y1/2, Y3/4 and Y5/6; addition and subtraction underpinned by reasoning and problem-solving with a particular focus on working systematically to find all possibilities.

11. 'Dice'

For these activities dice or spinners are used to generate numbers. The focus is on Additive Reasoning in Y1/2, Y3/4 and Y5/6; addition and subtraction underpinned by reasoning and problem-solving. There is also a focus on Multiplicative Reasoning in Y3/4 and Y5/6; multiplication and division underpinned by reasoning and problem-solving.

12. 'Number Pyramids'

For these activities number pyramids are created by the pupils. The focus for all year groups is on Additive Reasoning; addition and subtraction underpinned by reasoning and problem-solving with a heavy focus on addition.

13. 'Calculator'

For these activities, you use a calculator to explore counting in different steps. The focus for all year groups is on Number; understanding number patterns underpinned by reasoning and representation.

14. 'Shape'

For these activities, you use sheets of paper to create different shapes which are then explored in a variety of ways. The focus for all year groups is on Geometric Reasoning; understanding 2D shape underpinned by reasoning and problem-solving.

15. 'Playing Card'

For the activities, you can either create a small set of cards from a piece of paper or use cards from a playing card deck. The focus for all year groups is on Additive Reasoning; addition and subtraction underpinned by reasoning and problem-solving.

6. Professional Development Resources

Pre-Teaching and Assigning Competence

Supporting children to be active and influential participants in mathematics lessons through effective use of assigning competence and pre-teaching

Following a year-long action research project, funded by Devon County Council, Jurassic Maths Hub and Cornwall and West Devon Maths Hub, the report (available to download below) examines the impact of class teachers pre-teaching and assigning competence to vulnerable children, with the aim of supporting them to be active and influential participants in mathematics lessons. Pre-teaching here is identified as additional individual or small group time with the teacher who will teach the lesson, in advance of the lesson. The focus of any pre-teach is on addressing the barriers the learners might have in accessing the lesson.

The impact has been extraordinary. We have had the privilege of witnessing teachers change children's lives through this project. Children who had no belief in themselves as learners in mathematics now believe in themselves, and are actively involved in their own learning and in the learning of others

Learning from the ICME international survey on teachers working and learning through collaboration

Many professional development projects, including ones set up by the maths hubs, involve teachers working and learning collaboratively.

In order to both support and challenge professionals who lead mathematics developments across the South West, Alison Clark-Wilson was invited to join the local PD network meeting in the summer term of 2017 and engage participants in exploring and furthering their understanding of collaborative working, through a collaborative process. The resulting report can be found below. We hope this is the start of many conversations about collaborative professional development, both locally and across the country.

Developing Spatial Reasoning Pilot Project report 2024

Three Year 3 and 4 teachers each chose six under-resourced children for a small pilot project which aimed to improve the children's attitude and self-belief in maths and their attainment. Spatial reasoning was chosen as the mathematical content as research has identified that it can impact not just on geometric understanding but also on number.

The children engaged in a workshop using pattern blocks and then they co-led a workshop for their peers back in class. Our findings revealed that despite the brevity of the project five children were more actively engaged, achieving more within their daily maths lessons and that one child had already improved their attainment. All the children commented favourably about their involvement and wanted to repeat the experience.

The children in each school created a presentation about the project:

Research Talks

Mathematics Research Talks #1 - Cardinality Counts: The Value of Two Revisited – Dr Ruth Trundley

Watch at YouTube

Dr Ruth Trundley explores the importance of cardinality, drawing on the case study used for her doctorate (which followed two children from the age of 17 months to 5 years 7 months) and more recent research undertaken with Dr Helen Williams.

Mathematics Research Talks #2 Subtraction - Using Variation to Support Children to Stop Counting and Start Using What They Know

Watch at YouTube

Stefanie Burke explores recent research undertaken with Carolyn Wreghitt to examine how children think and how they can be supported to use what they know when subtracting ones from two-digit and three-digit numbers, with a focus on the bead string as an aid to explanation.

Mathematics Research Talks #3 - Understanding Structured Number Lines

Watch at YouTube

Dr Ruth Trundley shares recent research, undertaken with Stefanie Burke and Helen Edginton, that ran in two parts from September 2018 to July 2020, with pupils from EYFS to Y6 focused on understanding structured number lines as a representation of the structure of the number system. Central to this is understanding the number line as a measurement model.

Mathematics Research Talks #4 - Supporting Active Participation in Maths Lessons Through Pre-teaching

Watch at YouTube

Dr Ruth Trundley explores findings from the project which focussed on the research question: How can we support all children to access age-appropriate mathematics and be active and influential participants in maths lessons through effective use of pre-teaching and assigning competence?

Mathematics Research Talks #6 - The Use of Colour to Support Understanding in Mathematics

Watch at YouTube

When maths is represented visually, colour can have a significant impact. It can both support and undermine understanding depending on how it is used. Dr Ruth Trundley shares research, undertaken with a group of teachers working with pupils from EYFS to post-16, that considered how colour can be used to enhance learning in three different contexts: Resources (manipulatives), Board work & Pupil recordings.

Mathematics Research Talks #7 - Supporting Reading Comprehension for Maths Word Problems

Watch at YouTube

Kate Palmer, speech and language therapist, explores the reading comprehension skills required for maths word problems, the development of a programme to support these skills through vocabulary and visualisation work, and findings from the pilot run of this programme.

Mathematics Research Talks #8 - Teaching for Mastery and the Voice of the Mathematics Textbook

Watch at YouTube

Textbooks matter. They play a part in shaping curriculum and pedagogy. In this way they can be seen as playing a role in the transformation of the broader subject discipline of ‘mathematics’ into ‘school maths’. This leads to questions about the nature of the influence exerted by a mathematics textbook during this transformation, and how this is enacted within the school maths curriculum. Andy Ash reports on a research study that was a collaboration between a university-based professor and seven teacher-researchers, along with the schools’ alliance lead as a co-researcher. Its aim was to investigate the interplay between the teachers and the textbook whilst they developed maths curriculum reforms within their schools. The research identified important changes to both classroom practice as well as teacher beliefs.

Mathematics Research Talks #9 - Solving Subtraction Problems by Means of Indirect Addition: findings from a Collaborative Lesson Research Cycle

Watch at YouTube

Subtraction problems of the type a – b = ? can be flexibly solved by various strategies,. This includes the indirect addition strategy (“how much do I have to add to b to get at a?”) which is particularly useful when a and b are close. However, often children do not choose this strategy. Dr Ruth Trundley will explore findings from a collaborative lesson research cycle which expose some of the reasons why this might happen and the challenges associated with addressing this issue.

Mathematics Research Talks #10 - Micro-teaching lesson study – an alternative model for maths teacher professional learning

Watch at YouTube

Micro-teaching lesson study is an adapted iterative lesson study where teachers collaboratively design and teach a learning episode. This is in contrast to traditional lesson study that involves designing a whole research lesson. Laurie will share how this model works for professional learning which was also used as research setting for her recent doctoral work. Eight primary teachers came together for 3 lesson studies, each with a different focus, to collaborate on the design of one-problem-multiple-changes procedural variation tasks. Three iterations of the same task were designed, revised, retaught in a single morning session. Using the case of one micro-teaching lesson study, Laurie will discuss how the model generated different types of data and how this data contributed to a window on the professional learning of the participating teachers. Laurie Jacques is an independent consultant (lauriejacques@smartpd.co.uk) and part-time research fellow at UCL Institute of Education (l.jacques.16@ucl.ac.uk)

External resources