In their webinar with The Education Hub, Associate Professor Fiona Ell and Dr Lisa Darragh from The University of Auckland discussed principles that teachers can use to develop mathematics and statistics programmes at primary school. They explored research-based ways to organise and design for learning that include all students and stimulate engagement and progress in mathematics and statistics. Here are some of the key insights from their webinar:

• It is important to have an overarching structure to mathematics programmes, because structure and organisation:
• frame what is possible to do and learn in relation to maths, by creating an organising structure in which individual lessons sit.
• communicate messages about the nature of maths learning and what is important (for example, some maths activities emphasise the performance of learning for the purposes of evaluation, while others emphasise learning and exploration).
• simplify planning and preparation, because a considered, co-constructed, and commonly understood overarching structure allows for straightforward planning of the components that sit within it.

• A well-rounded mathematics programme contains four key aspects. Not all lesson activities will involve all four aspects, but it is helpful to ensure all four feature when planning a sequence or unit of work. Theyinclude:
• Creating maths, which relies on key mathematical processes such as logical thinking, and  involves activities like statistical investigations and exploring the properties of numbers.
• Using maths to solve relevant, real-world problems.
• New learning of maths, which should be explicitly framed as such because it is important for students to be aware that they are learning something new.
• Practising the maths that has been learned, as repeated exposure to an idea serves to consolidate it (this might be an appropriate point to use online prgrammes or apps).

• Problem-solving is a valuable component of mathematics programmes, but it is important to approach it carefully. Problem-solving should focus on problems that do not have an obvious pathway to a solution, and which allow students to mimic the work of mathematicians by using what they already know to work logically towards what they do not yet know. Problem-solving can also be used to teach skills and knowledge, provided that the teacher has a strong understanding of the maths being used and the students have a positive attitude to using maths to problem-solve. It is particularly powerful to explicitly teach students about the relationship between the problem to be solved and the skills and knowledge needed to solve it.

• Maths games have a role to play in mathematics programmes but, again, they should be used thoughtfully and carefully. Like all the elements in a mathematics programme, games convey particular messages about the nature of maths and about learning and doing maths. For example, games that rely on being able to produce answers quickly and have a single ‘winner’ may communicate to students that maths is about speed, that there is only ever one correct answer, and that only a few will be successful at it. When choosing maths games to play, look for games that involve collaboration, multiple pathways to solutions, and time for thinking. Also consider whether or not the game involves everyone doing maths, or just one or two players. Fun is also important!

• Practice is an important part of mathematics programmes, particularly when it is framed as having immediate relevance for and impact on current work. Some important ideas to consider in relation to practice are:

1. Automaticity, which helps to reduce cognitive load. It applies to basic facts learning as well as to relationships, patterns, and subitising sets.
2. Mastery, which builds confidence. While it is beneficial for students to work at the limits of their understanding, it is also exhausting, so they should not do it all the time. Practice activities that are pitched at the right level give students a feeling of competence.
3. Understanding and insight, which are supported by practice activities that are well-constructed to highlight patterns and relationships between concepts.

• There are several ways to approach differentiation and grouping that reduce the potentially negative impact of fixed ability grouping. One way to do this is to use low-floor, high-ceiling activities, which allow students to approach the maths in ways that are appropriate to their level of knowledge and understanding. For example, a starter activity such as ‘The answer is 12’ allows students to offer a number of ways of making twelve, from 10 + 2 to 11 ¼ + ¾ . When it comes to learning new material, teachers can offer a range of ways to approach the new learning, whether through direct instruction from the teacher, practice activities, or application activities, and students can choose the approach that interests them. Universal Design for Learning (UDL) principles are also useful, and teachers can design inclusive programmes by building both supports and extensions into maths activities.

• Technology should be used judiciously in mathematics programmes. Online programmes such as TinkerPlots can be useful for exploring statistics and probability, and there are a number of free online programmes such as GeoGebra that teachers might consider using. However, it is worth exercising caution in the use of programmes that are expensive for schools and that collect a lot of data about student users.