I’d like to share my thoughts on a handful of pedagogical statements and their implications for mathematics teaching.

### 1. “Students must know what they’re learning, not just what they’re doing.”

This really isn’t a big deal in Maths, because a purposeful approach to mathematical tasks rarely relies on metacognitive engagement with an objective. In some subjects, context can be very distracting – students might be writing about a poem but really learning about effective paragraphing. In Maths, if they’re reflecting shapes in diagonal mirror lines then that’s almost certainly their objective too.

As a further illustration, say that two classes are doing the same activity to provoke their thinking about surds. In one group, students can precisely explain that “We are learning to manipulate and compare surds”; in the other, they describe the task: “We’re matching cards with equivalent expressions”. Isn’t the difference chiefly one of articulation, rather than a contrasting quality of learning? By completing a well-designed task, students constantly test and correct their understanding – which is the basis of learning. Even if they can’t identify the title of the topic they’re studying, that doesn’t diminish their powers of pattern recognition which will help them tackle similar problems in the future.

### 2. “Students should always see the point in what they’re learning.”

This statement is also misplaced because it falsely encapsulates students’ motivations. In my experience, we study Maths for four reasons:

- Enjoyment
- To develop our abstract problem-solving skills
- To achieve a qualification
- Because in some cases, there will be applications to future study or work

I strongly feel that it’s more helpful to engage students with points 1-3 than to get too caught up in point 4. The reality is that beyond basic numeracy, a lot of what students learn in Maths lessons isn’t going to be of direct use in their later lives. And while pseudo-realistic contexts can be great for sparking interest in otherwise abstract topics, so too can mathematics be a context in itself. I had much more fun today with Year 13 discovering ‘cyclic integration by parts’, than with Year 12 calculating the cross-sectional area of a fictional river.

At this point, I risk coming across as dismissive of the role of context in Mathematics – which is actually pivotal – so I’d like to reframe the above statements into three others which I think are more constructive.

### 3. “Generalising and specialising are at the heart of mathematics.”

A lesson without the opportunity for learners to generalise is not a mathematics lesson.

– Professor John Mason

It is a centrepiece of subject pedagogy that mathematical thinking consists of conjecturing and testing those conjectures. “Is this always/sometimes/never true? Are there special cases or exceptions to the rule? Does this result apply to a broader set of problems?” In a sense, this eclipses the role of contextualising and de-contextualising learning in other subjects. It also presents a real challenge: do our questions and prompts test the limits of students’ understanding and provoke them to sharpen their internal conjectures? If not, we risk surface learning and uncovered misconceptions. Students need to experience mathematical ideas in a range of contexts to make important connections and strengthen their powers of generalisation.

### 4. “Students enjoy being on a journey towards mastery.”

Positive outcomes are most readily achieved when students are self-driven to learn – both through intrinsic curiosity and a thirst for progress. We talk about “students taking ownership of their learning”. One of the key ways we can make this happen is by engaging them in the narrative of their progress over time. An experienced colleague on my team uses a GCSE Route Map where students shade in topics on a tree of learning as they explore the subject. I am currently leading a disaffected Key Stage 3 group through a perpetual cycle of diagnostic feedback and practice. The common theme is that by sharing how each learning activity fits into students’ personal journeys towards mastery, we can impact positively on their emotional engagement. The ‘context’ for learning is their own growing understanding.

### 5. “Feedback focuses students’ attention.”

I have set out my view that tasks and objectives are inherently aligned in Maths, but perhaps this underplays the importance of targeted effort. Feedback is essential because it forces us to challenge specific aspects of our understanding. The objective itself is merely a context for the personal learning experiences of every student. A class may be outwardly ‘learning to draw pie charts’, while its constituent individuals variously attend to their fragmented conceptions of angles, proportion, equivalence and scale.

**What are your views on the role of context in mathematics? Share your thoughts in the comments.**

Really thoughtful piece James. I think your third paragraph hits the nail because there is nowhere to hide in the thinking and gives rise to speculation etc. Your description of the mastery map is very powerful – a brilliant example of seeing their progress over time. Satisfying for the student and teacher alike.

A few thoughts I have had are:

1. I am not sure I agree that what students are doing is always what they are learning. For example, statement 3, do students know they are generalising/specialising? Should they? I have, however, found it a difficult distinction because they can be the same/similar and it is a real issue in other subjects.

2. I don’t think context has to mean “real-life application”. Why is this topic/skill/concept important in mathematics? I have been teaching the trapezium rule to 10Ma1 (areas of shapes) but linking to sequences, integration and surface areas/volumes (proof by integration). Not necessarily brilliant but just trying to find the area under y=x squared and how quickly the limit becomes apparent is interesting as well as an opportunity to sport patterns and generalise. I do think students should understand the bigger picture and although the Routemap is useful it is not the whole thing.

3. The first page of the Routemap includes an introduction which explains knowing the stuff is only 50%. Not just for the exam but mathematics is about more than a long list of skills to be mastered. It has been helpful to focus students on what they need to be able to do for an exam but it is only half the answer… This is supported by the third statement and the Routemap risks focusing only on surface and coverage, not depth and I am mindful of this. I hope other teachers who use it are too.

The real challenge is to make this theory impact on Key Stage 3 students so they are prepared for the challenges they face…