How to present mathematical content clearly, build intuition, and create effective maths exercises
Writing mathematics for students is different from writing it for publication. A paper assumes the reader is a peer who will verify your reasoning. A lesson assumes the reader is building their understanding for the first time. Every notation choice you make should serve that reader.
Start with Language, Then Formalise
The most common mistake in mathematical writing is introducing notation before the idea it represents is understood. Lead with plain language. Once the concept is clear, the notation gives it precision.
scene{
## What Changes When a Function Changes?
Imagine driving along a road. At every moment, your position is changing —
but sometimes you're accelerating, sometimes slowing down.
The *derivative* captures the rate at which your position changes at any
given instant. Not the average over a journey, but the exact rate right now.
definition{
The instantaneous rate of change of a function at a point.
}(term: Derivative)
}[
diagram{
curve(expression: x^2; colour: blue)
segment(start: (0, 0); end: (2, 4); colour: pink; dashed: true)
point(x: 1; y: 1; colour: orange)
}(
x domain: [-3, 3]
y domain: [-1, 8]
)
]
Build Intuition Before Rigour
Show a specific numerical example before stating the general rule. Students learn to identify a pattern before they learn to express it.
scene{
The derivative of $x^2$ is $2x$. Let's see why this makes sense:
- At $x = 1$, the slope of $y = x^2$ is $2$.
- At $x = 3$, the slope is $6$.
- At $x = -2$, the slope is $-4$.
The pattern: the slope is always *twice* the x-value.
equation{
\frac{d}{dx}(x^n) = nx^{n-1}
}
This is the **power rule**, which generalises the pattern.
}
Use Parameters to Reveal Structure
When a student can manipulate a variable and watch the result change in real time, they build genuine intuition — not just a memorised formula. Use the parameter element for this.
scene{
The graph below shows $y = ax^2$. Drag the slider to change $a$.
Watch what happens to the shape of the parabola.
}[
parameter(name: Coefficient $a$; var: a; range: [-3, 3]; step: 0.5; default: 1)
diagram{
curve(expression: a * x^2; colour: blue)
}(x domain: [-4, 4]; y domain: [-10, 10])
]
The parametrised exploration should come before you state the conclusion. Let the student discover it first.
Sequence Your Ideas
A worked example should follow the pattern: setup → process → result. Don't skip steps that feel obvious to you — they're not obvious to someone learning.
example{
**Differentiate** $f(x) = 3x^4 - 2x^2 + 7$.
Apply the power rule to each term:
- $\frac{d}{dx}(3x^4) = 12x^3$
- $\frac{d}{dx}(-2x^2) = -4x$
- $\frac{d}{dx}(7) = 0$
Therefore: $f'(x) = 12x^3 - 4x$.
}(title: Power Rule)
Choosing Between Inline and Block Equations
Use inline math ($...$) when the expression is part of a flowing sentence. Use a block equation when the expression is the main point of the moment and deserves emphasis.
A sentence like "the area is $\pi r^2$" uses inline math. A major result like the quadratic formula deserves a block equation on its own line.
Check Understanding with mathresponse
After a worked example, a mathresponse question gives students an immediate chance to practise. Keep the first question close to the example — nearly identical in structure.
mathresponse{
Using the power rule, find $\frac{d}{dx}(5x^3)$.
}(
answer: 15x^2
mode: algebraic
hint: Multiply the exponent by the coefficient, then reduce the power by one.
)