Knowledge and skills in Maths
In this post I explore how the knowledge/skills debate might influence our thinking about curriculum in maths.
"Knowledge" is comprised of facts and methods:
For example: times tables, names of shapes, keywords
These are standalone items which need to be committed to long-term memory. It is not enough for them to be computable (e.g. working out the sum of angles in a quadrilateral by counting triangles, or working out 6 x 7 by doing 3 x 7 and doubling it) because this adds unnecessary demands on working memory. Rather, they need to be memorised to the point that they are automatic ("The sum of angles in a quadrilateral is 360 degrees", "6 x 7 is 42").
Facts can be memorised by frequent recall practice, for example using flashcards.
For example: dividing by 10, writing a number as the product of its prime factors
Anything which can be drilled is a method, and hence is a type of knowledge. This includes complex methods with multiple steps to success, such as finding a missing side using trigonometry. In every case, the principles of overlearning, spacing and low-stakes interrupt testing are essential to achieve fluency.
Methods are hierarchical, which means that the sequencing of the curriculum is important. For any given method, there are prerequisite facts and methods which need to be taught first. This principle of building new knowledge on existing foundations is crucial to forging strong storage strength in the way it is remembered.
Whereas patterns, relationships and cognitive shifts are best developed through procedural variation, well-designed questioning and rich tasks, methods by definition need to be learnt and followed in a prescribed way, and so direct instruction plays a central role.
Alongside knowledge, students also need to develop reasoning. Let's not call this "skills" because that's much too ambiguous - it's actually a bit of a red herring. Equally, let's not call it "problem solving" because this is also open to misinterpretation (the challenge is that, as with football/futsal, you don't just develop problem solving by solving problems).
Reasoning is what enables students to make sense of unseen questions, and it aligns with AO2 and AO3 at GCSE.
Strong underlying knowledge is crucial to this. We can only hold a certain amount of information in working memory at a time, and this is the first barrier to success when faced with a complex question for which we don't have rehearsed steps to success. The solution to this is developing fluency in the necessary facts and methods. Once knowledge becomes fluent, its cognitive load demands dramatically decrease and so students can start applying it to challenging problems.
However, reasoning is not a distinct layer that sits above knowledge. Rather, the two concepts are deeply interwoven through the notion of understanding. If we design a curriculum where the right knowledge is developed in a connected, relational way, then students will be well-equipped to make their own sense of the mathematics when it appears in new contexts.
As an example, say that we teach students to split an amount in a ratio by following the instructions "add, divide, multiply". If we then ask them a question which is slightly different, they will have no idea how to answer it because the method was too specific and never really made sense. On the other hand, if we teach students the notion of "parts" of a ratio and use this language when learning and drilling several different methods, multiplicative reasoning will become embedded in their cognitive schema and they will be much more versatile when faced with unfamiliar questions.
Exposure to many contexts
The other key element in developing reasoning is to expose students to a diverse range of questions, problems and mathematical experiences.
Even after students have learnt separately about expressions and angles in shapes, they are still likely to freeze up the first time they see an algebraic polygon question. It will seem overwhelming and the total lack of familiarity will lead to cognitive strain. However, after some live modelling and corrected practice this whole class of problems will become much more approachable. Moreover, students will be able to make further small cognitive jumps on their own, from the foundation of this now-familiar context. For example, they will cope well with algebraic angles in a pie chart even without that being explicitly taught.
As mathematicians, if we are asked how many ways 20p can be made with coins, we will start by applying the mathematical process of considering an easier problem: how many ways can we make 5p? We'll then build on this result by working systematically to make sure we don't miss combinations for the 20p question. There are several approaches such as these (also including trial and error, drawing a table and identifying key information) which can be applied to a lot of problems in maths. By naming and practising these techniques across a broad range of topics, students will start to identify patterns and use them on their own. Then as they experience more and more success, their resilience will grow.
In summary, the learning of mathematics must be underpinned by fluency in a large amount of knowledge. This is achievable by teaching for deep, relational understanding and by drilling and frequently revisiting methods. On top of this, we should build opportunities for students to experience success in mathematics in as many different contexts as we can conceive.