Ofsted wants to reduce gaps in achievement by appealing to research. To that end, it has conducted seven curriculum research reviews. Teachers and school leaders will be reading these to find out how they should teach to satisfy inspectors. Unfortunately, its maths review – published to immediate calls for its withdrawal – ignores swathes of evidence and could lead to poor practice.
The impression given in the review is that memorising facts and procedures, followed by application exercises, is the watchdog’s evidence-based preferred way of teaching. But that is what mathematics teaching has been like in most parts of the world for decades – and it does not work.
The National Curriculum states that pupils should become competent in problem solving, reasoning and fluency, but Ofsted does not acknowledge the extent of research about these aspects of mathematics. They ignore findings stretching back over the past 60 years that focus on educating learners to do the thinking required to solve unfamiliar problems.
Adaptable and innovative thinking is required in digital, engineering, scientific and commercial endeavours. Ofsted refers to “solving types of problems” but its review fails to address research that shows deeper understanding can develop through tackling non-routine problems that expose structure, and hence contribute to learning. Learners who only practise solving known types of problems tend to develop only superficial knowledge.
Ofsted’s review is also ill-informed about the meaning of “concept” in mathematics. It portrays a concept as static knowledge. Yet in mathematics education, conceptual learning is known to be dynamic.
Consider “square”. The first time a child hears this they are probably looking at a plastic shape and their conceptual understanding coheres around such objects. Fast forward several years, and 9 is taught as the square of 3. Later, “negative 1” is the abstract square of an imaginary number. The concept is frequently revisited and adapted, and each use of the word builds on what has gone before. The concept is not static, but restructured and reconnected many times on the learning journey.
Ofsted sees concepts as second-order knowledge arising from learning factual, procedural and “conditional” knowledge. They identify conditional knowledge with the stem “I know when…”. They support this limited view with reference to a 2007 article by Miller and Hudson, but this article focuses on “I know why”, advocating exploration with concrete contexts, materials, apparatus, representations. In it, formalisation of facts and procedures is a final step, not a precursor.
So while Ofsted recognises that bridging between meanings is important, its emphasis on rehearsal of facts, procedures and strategies inexplicably ignores the importance of this range of multi-sensory exploration.
For example, for Ofsted the addition concept arises from learning number bonds (facts) and column addition (procedure). Rather, the mathematical concept of addition depends initially on pre-numerical concepts of adding to and taking away from quantities of stuff. Formalisation, including number bonds and column addition, comes later through the use of language and conventional representations, but the fundamental concepts are already there, ready to be expressed by counting, measuring and then symbols.
There are still more ways that this Ofsted review over-simplifies the complexity of mathematics teaching and learning. But these three – excluding dynamic conceptual understanding, excluding exploration and non-routine situations and excluding research that challenges their preferred model – are particularly egregious.
The “example, exercise, practice, revise” model is tried and tested, and the fact is that wherever and whenever it has been tried and tested, it has failed to generate strong mathematical capability for all. So we urge teachers and leaders to maintain the statutory aims of the national curriculum. These include knowing facts and procedures, but focus on sense-making and “knowing why”.
In doing so, the research will be on their side. To empower students as mathematical problem-solvers, teachers need to build on and adapt the conceptual knowledge they already have. And to achieve that, facts and procedures are not the main event but merely supporting tools.