New maths teachers often hear two questions. From students: “When are we ever going to need this?” From colleagues: “What’s your narrative?” These are, in fact, the same question asked from different perspectives. The aim of this project is to help new teachers answer, and ideally pre-empt, that question – to motivate the study of mathematics through what I call narrative questioning. On this page, I’ve begun compiling some of those narratives.
[Note: This project currently lives as a self-contained page on my main website. Eventually, it will have a dedicated site of its own.]
1. Background & Theory
This project has its roots in my final PGCE assignment (2021-22), where I presented an academic poster on the problem of creating an intellectual need for mathematics. Since then, my thinking has often returned to this question – especially as I learnt more about motivation during my MA (2022-25). (You can view the original poster here.)
I first came across the idea of intellectual need in an episode of the Mr Barton Maths Podcast featuring Dan Meyer (#009). But the concept can be traced back to Piaget’s theory of equilibration. According to Piaget, humans – especially children – are naturally driven to make sense of their environment. We experience equilibrium when the world aligns with our understanding. When we encounter something that doesn’t fit – an anomaly – we experience disequilibrium. This discomfort motivates us to adapt our understanding to restore balance. This cycle of equilibrium, disequilibrium, and adaptation is what drives cognitive development.
Teachers can harness this natural process by guiding students to the limits of their understanding, then nudging them over the edge into disequilibrium. This allows students to experience first-hand the gaps in their knowledge, creating an intellectual need to fill those gaps.
Guershon Harel describes several types of intellectual need in mathematics. One especially useful type is the need for computational efficiency. In short, we like to be able to work things out quickly! Consider this example from Paul Smith (@drpaas1001):
Without a calculator, work out:
Fully simplify your answer.
Students could use column method to multiple the numerators and then the denominators, before attempting to simplify the resultant fraction by searching for and cancelling common factors. What a pain! This creates an intellectual need for a more efficient method – namely, cancelling common factors before multiplying. Crucially, this need emerges from the gaps within the student’s own mathematical knowledge. It’s not imposed from outside via appeals to test scores or future careers they may not care about. This is what makes the method so powerful and sustainable.
This brings us to the topic of motivation. According to Self-Determination Theory, people flourish when three basic psychological needs are met:
- Autonomy – feeling in control of one’s behaviour
- Competence – feeling effective and capable
- Relatedness – feeling connected to others
When these needs are satisfied, people experience higher-quality motivation, resilience, and well-being.
Focusing for a moment on autonomy, a proven way to satisfy this need in the classroom is by sharing rationales – helping students understand not just what they are learning, but why. This is precisely what creating an intellectual need achieves. Better yet, it allows students to experience the rationale first-hand – a “show, don’t tell” approach that aligns with good storytelling.
Of course, autonomy alone isn’t enough. Students must also feel competent. Teachers should allow them to safely encounter the gaps in their knowledge, not throw them head-first into the void. The task should stretch but not overwhelm them, falling within what Vygotsky called the Zone of Proximal Development.
Carefully creating an intellectual need can not only enhance autonomy and motivation, but also make lessons more meaningful. According to one theory, a series of events feels meaningful when it can be interpreted as a narrative. The core elements of a narrative are setup, conflict, and resolution. Conveniently, these elements map directly onto Piaget’s cycle of equilibrium (setup), disequilibrium (conflict), and adaptation (resolution). Thus, the best teachers are in fact the best storytellers, crafting narratives in which students are the protagonists of their own intellectual journeys.
In summary: A good set of narrative questions (1) reminds students what they already know (equilibrium/setup), (2) exposes a gap in their understanding (disequilibrium/conflict), and (3) reveals a new concept or method to fill that gap (adaptation/resolution). That’s the pattern I aim to follow in the examples below.
I should clarify that this isn’t a wholly original project, but rather a modest extension of work already begun by others – especially Dan Meyer’s Directory of Mathematical Headaches.
2. Further Reading/Listening
- Barton (2016). Dan Meyer.
- Barton (2018). How I wish I’d taught maths.
- Boom (2009). Piaget on equilibration.
- Brown (2022). Learning and stories.
- Brown (2022). The Flammarion engraving.
- Chaiklin (2003). The zone of proximal development in Vygotsky’s analysis of learning and instruction
- De Bres (2018). Narrative and meaning in life.
- Deci & Ryan (2000). The “what” and “why” of goal pursuits: Human needs and the self-determination of behavior.
- Fuller et al. (2011). Intellectual need and problem-free activity in the mathematics classroom.
- Harel (2013). Intellectual need.
- Meyer (2013). The three acts of a mathematical story.
- Meyer (2015). If math is the aspirin, then how do you create the headache?
- Meyer (2023). If math is the aspirin, then how do you create the headache?
- Mooderino (2014). Story structure: Pity, fear, catharsis.
- Movshovits-Hadar (1988). School mathematics theorems – an endless source of surprise.
- Ryan & Deci (2000). Self-determination theory and the facilitation of intrinsic motivation, social development, and well-being.
- Willingham (2004). The priviledged status of story.
3. Narrative Questions
These questions/narratives were added as they occurred to me – in no particular order. One day I may organise them more systematically. In the meantime, I recommend using Ctrl+F (or Command+F) to search for topics.
Topics generally follow the Edexcel GCSE specification.
If you’re struggling with how to introduce a topic that doesn’t appear below, feel free to get in touch. We can figure it out together!
Mean from a frequency table
See my original poster, section 3. In short, ask students the calculate the mean for increasingly long lists of data, before revealing a more convenient way to organise the data (in a table).
Cross-cancellation before multiplying fractions
See above.
Converting recurring decimals to fractions
Operations with fractions are much easier than operations with recurring decimals. So, before teaching students how to convert recurring decimals into fractions, first ask them to perform – or try to perform – some operations with recurring decimals.
To set the stage, start with non-recurring decimals. For example, what is 0.3 × 0.83? Easy enough! But what about 0.333… × 0.8333…? Let them struggle for a while. If some think they’ve finished, ask them for ever more accurate answers (more decimal places).
Then, when the tedium starts to set it, reveal that the exact answer is simply 
Well, there is… Cue the lesson.
Later bonus/extension work could (should) return to operations with recurring decimals.
Trigonometry – sides
To start, ask students to find:
- The circumference of a circle with diameter 2cm [use diagrams];
- A missing angle in in a triangle, given the other two angles;
- The hypotenuse of a right-angled triangle, given the two shorter sides;
- A missing side in a right-angled triangle, given an opposite angle of 20° and a hypotenuse of 2cm.
Students should find tasks 1-3 easy enough, but struggle with task 4. As you go through answers, you should also struggle with task 4 (theatrically).
But now recall task 1: we found the circumference by multiplying the diameter by π, which is just a number (3.141…). Returning to task 4: if only there were a convenient number (like π) that we could simply multiply the known side by to find the missing side… Well, there is! It’s approximately 0.342 [= sin(20), but don’t tell them this just yet!]. So, the answer to task 4 is 2 × 0.342 = 0.684cm.
Just as π represents the ratio between the circumference and diameter of a circle, this number represents the ratio between side lengths in a right-angled triangle [Thank you for this useful addition, Miss Reid!]. The only problem is that, unlike π, this number is not fixed – it changes as the angle changes. But fear not! You don’t have to remember all the different numbers. Your calculator does it for you. All you need to know is the angle… Cue the lesson.
Start by using sine to find opposite sides, then the hypotenuse. You could even introduce the sine curve, which essentially tracks the number as the angle changes. After that, you can follow a similar narrative for cosine and tangent, before moving on to finding missing angles.
Rationalising denominators
Ask students to convert the following fractions into decimals using bus stop method:
to 3 decimal places
to 3 decimal places [give them this hint:
= 1.41421356237…]
to 3 decimal places
Students should find questions 1-3 easy enough, but struggle with question 4. As you go through answers, you should also struggle with question 4 (theatrically).
Then, reveal that