By Rehana Akhtar
It has always been a recurring thought to me over my years of teaching as to ‘Why do pupils need to be re-taught and why do they not retain key skills and concepts?’
The Program for International Student Assessment (PISA) is a worldwide study by OECD (The Organisation for Economic Co-operation and Development) in 70 nations of 15-year-old students’ scholastic performance on mathematics, science and reading. PISA collect educational achievement data and provide information about trends in performance over time. The position for the United Kingdom in the 2015 PISA rankings was 27th. The top five places are consistently taken by East Asian jurisdictions such as Shanghai and Singapore. These countries use highly supportive and structured research-informed textbooks. The use of these resources is of central importance.
How can we try to increase our ranking and avoid the misery of re-teaching? Having a passion for the purposeful development of teachers, I have found that careful curriculum design and the use of high-quality resources are essential in creating engaging learning experiences for students. I have found that mastery teaching approaches and resources are hitting the nail on the head. Whilst I am in my early phases of implementing mastery in my department, I appreciate the essential components to lesson design. These include:
- Opportunities for pupils to think mathematically: this is achieved through class activities which encourage discussion, with questions to ensure that the steps have been understood.
- Motivating students’ learning by delivering an effective pedagogy
- Requiring students to support their answers with reasoning
- Concrete Pictorial Abstract: using CPA activities help to build a truly rich, deep understanding of maths.
- Encouraging the whole class to work and progress together.
- Intelligent practice: setting exercises, which employ ‘minimal variation’ in the numbers to focus attention onto the key learning point – the surest way to lead to ultimate success with problem-solving questions.
- Mathematical language: ensuring precise and consistent use of mathematical language at all times and in all answers.
- Formative assessments: using test results to inform purposeful next steps and ensure gaps in learning are addressed promptly.
To allow the above to flourish in lessons we have chosen our resources carefully. We primarily use resources from the Oxford series, Discovering Mathematics. This series includes student books, workbooks, teacher guides, Kerboodle online resources, manipulatives (algebra discs), and self-assessment booklets. These resources have helped move students forwards in the following ways:
- Guided Discovery: students discover new mathematical concepts for themselves with teacher guidance – through class activities, intelligent practice, and clear examples. This has helped to engage students and prevent them from becoming withdrawn. I ask students to work in groups and take a role where everyone has to contribute. Through careful guidance of tasks, all students are forced to engage with the problem. I also encourage students to explain their solutions; once all answers have been listened to, we discuss the most efficient method. Students are not afraid to ‘give a wrong’ answer: they have come to realise that this contributes to valuable learning. The key idea here is for students to engage fully with the problem and learn to solve it for themselves. My outlook is to get the students to do all the working with purposeful guidance, helping to channel their thoughts in the right direction.
- Concrete-Pictorial-Abstract (CPA) approach: Drawing on the research of the renowned educational researcher, Jerome Bruner, Singaporean schools make extensive use of the CPA approach.
- Experiencing concrete examples (e.g. manipulating algebra discs)
- Moving to pictorial representations (e.g. looking at diagrams on the page showing algebra discs)
- Progressing to abstract statements using words and symbols (e.g. working through actual algebraic problems, but without discs or diagrams)
I have found that students have been able to think deeper and give clearer reasoning whenever manipulatives have been used. Whilst problem-solving, students explore structures and make connections, engage in mathematical conversations and are able to reason and communicate concepts and ideas with conceptual understanding. As a teacher, this allows me to gain a greater understanding of where misconceptions lie and how deeply students’ understanding actually goes. It also helps pupils develop their ability to communicate mathematically and to reason. In fact, it is almost impossible for reasoning not to happen in a classroom where manipulatives are used regularly. The use of manipulatives has challenged even those students, who appear to have already grasped the learning. The pictorial stage acts as a bridge from the concrete towards the abstract; this stage allows students to make the cognitive link between the concrete and abstract notation. The ‘CPA’ approach helps bring excellence to the teaching and learning of mathematics.
- Precise use of Mathematical Language: discussion opportunities and open questions allow students to explain or justify their answers. This helps develop language, communication, and reasoning skills. I have asked students to search for definitions by directing them towards the relevant resources. Use of journals allows students to record their vocabulary; I find that students are more likely to recall words when they have processed the deeper meaning behind them. Drawing visuals allows them to make a stronger link to the word. This is particularly helpful for those who don’t have English as a native language (although many native speakers also struggle with subject-specific language).
- Formative and Summative assessments (such as can be found on Kerboodle or MyMaths): Dylan Wiliam, whose seminal work with Paul Black (Inside the Black Box, 1998) made formative assessment a priority for policy makers and schools alike, draws a distinction between AfL and formative assessment. He says that assessment for learning (AfL) – where the intention is to help you meet students’ needs better – only becomes formative assessment when the evidence you gather is used to adapt teaching to address students’ needs. Formative assessment is when you actively use evidence of student learning to make professional judgements about what to do next in your teaching. Within a mastery maths lesson you are constantly striving for this. The regular purposeful dialogue, questioning and execution of intelligent questions informs me of students’ understanding. It lets me know clearly where and when a student has a misconception or insecure understanding. This informs my next steps and has a positive impact on teaching and learning.
I have tried to give a snapshot of how mastery maths and high quality resources have upskilled me and my team to teach more effectively. We are on a journey that will only improve as time goes on. We hope this will in turn instil a love of learning in our students. As Whitney Houston once sang: ‘Children are our future, teach them well and let them show the way.’
Rehana Akhtar has been teaching for 13 years paying particular attention to maths pedagogy. Rehana has drawn upon a wide range of research into pedagogical approaches that aim to engage learners and lead to desirable outcomes in mathematics, aiming to deepen the understanding of educational practices that optimise opportunities for mathematics learners. Rehana is a Lead Maths Practitioner at St. Michael’s Catholic School and a NCETM Accredited PD Lead.
Rehana also featured in our Discovering Mathematics Video where she describes how the course has helped improve her classes fluency and understanding of mathematics. To find out more about Discovering Mathematics, click here.