: Building Mathematical Antibodies: Teaching Pupils to Tackle the Unknown How we can equip pupils with the strategies to solve unfamiliar mathematical problems.
—
Blog
Building Mathematical Antibodies: Teaching Pupils to Tackle the Unknown
How we can equip pupils with the strategies to solve unfamiliar mathematical problems.
Share on:
by Exchange Research School at Don Valley Academy
on the
Danny Peaker
Danny is a secondary Director of Maths for Castleford Academy Trust, working across multiple secondary schools to improve maths achievement and teaching quality. He supports the work of Exchange Research School as an Evidence Lead in Education (ELE).
Can we truly teach pupils to conquer the ever-shifting landscape of mathematical problem-solving? A significant portion of GCSE maths hinges on tackling unfamiliar scenarios, often presented in new contexts. This begs the question: is problem-solving an innate talent, or a skill that can be cultivated? Like a constantly mutating virus, new challenges emerge. Our mission becomes equipping students with the ‘antibodies’ – the critical thinking and problem-solving strategies – to adapt and overcome any mathematical obstacle.
Understanding why pupils struggle with problem solving is key to improving classroom practice. Challenges may stem from inconsistent planning and delivery, as highlighted by Ofsted (2023), which found weak integration of problem-solving across curricula. However, wider factors may also be influential. In the 2023/24 KS2 SATs, disadvantaged pupils scored 20% lower in mathematics, compared with a 38% gap in reading, suggesting that reading difficulties may hinder mathematical problem solving.
To identify where pupils struggle, it is important to examine the problem-solving process itself. Polya’s (1945) four-stage framework, still widely used, highlights common difficulties and aligns with the EEF Improving Mathematics in Key Stages 2 and 3 guidance (recommendation 3). Understanding the problem Many weaker students have low literacy, which increases cognitive load and limits their ability to understand and interpret mathematical problems. A common experience is students reading a problem, but not actually understanding what is being asked.
- Develop a problem-solving toolkit: Teach pupils transferable strategies and key combinations of facts and methods to tackle a range of problem types, since not all problems can be explicitly taught (Ofsted). - Use genuine problem-solving tasks: Regularly set non-routine problems without rehearsed methods so pupils practise applying and adapting strategies, as recommended by the EEF, thus fostering adaptability and critical thinking.
Devise a Plan If pupils have encountered a similar problem, they can often link conditional knowledge (knowing when to use something) and procedural knowledge (knowing how to use it). However, non-routine problems present a significant challenge. Students may struggle to connect the problem to previously learned mathematical concepts and resort to “brute force” attempts, often leading to frustration.
- Teach and Compare Different Approaches: Encourage pupils to explore and compare various problem-solving strategies. This not only enhances their flexibility but also deepens their understanding of the underlying mathematical concepts. - Master declarative and procedural knowledge: Pupils solve problems more successfully when key facts and methods are secure and automatic, reducing working memory load (Ofsted).
Carry out the Plan Once they understand the method, they can usually execute it, especially with well-rehearsed processes. However, if the plan is flawed due to a misunderstanding of the proposed problem, the execution, however perfect, will be incorrect.
- Structure Problem Presentation Strategically: The EEF suggests organising teaching to present problems with similar structures but different contexts together, and conversely, problems with the same context but different structures. This helps pupils discern essential mathematical relationships and transfer knowledge across diverse scenarios. - Promote Visual Representations: The EEF highlights the value of visual representations in aiding problem comprehension and solution development. Encourage pupils to use diagrams, graphs, and other visual tools to externalise their thinking.
Look back and check This crucial step is often neglected. Students may be so relieved to have an answer that they skip the vital process of checking and reflecting. Did the strategy work? Could there have been a more efficient approach? What have they learned?
- Foster Metacognitive Reflection: The EEF stresses the importance of requiring pupils to monitor, reflect on, and communicate their reasoning and strategy choices. This metacognitive approach empowers students to become self-regulated learners, capable of adapting and improving their problem-solving skills.
Just as a virus adapts and evolves, so too do the challenges our students face in mathematical problem-solving. As educators, our response must be equally dynamic and robust. This requires a multi-faceted approach, beginning with strategic curriculum planning that weaves problem-solving throughout the learning journey. In the classroom, we must arm our students with a diverse toolkit of strategies, encouraging them to explore, reflect, and learn from every encounter. And finally, we must foster a culture of resilience, reminding them that every ‘mutation’ of a problem is an opportunity to strengthen their ‘antibodies’ – their critical thinking and adaptability. By embracing this holistic perspective, we can empower our students to conquer mathematical obstacles, and become lifelong, adaptable problem-solvers in an ever-changing world.
References
George Polya, (1945), How to Solve It: A New Aspect of Mathematical Method, Princeton University Press
