Not every maths activity needs to be a full lesson or an elaborate project.
Sometimes it’s the tiny demonstrations — the quick, memorable moments — that make a huge difference for students.
Good teachers build a repertoire over time: small, powerful ways to represent ideas. They know half a dozen different ways to show that fractions are equivalent, that negative numbers make sense, or that a straight line has a constant gradient.
These aren’t flashy or time-consuming, but they’re the glue that helps ideas stick.
Take, for instance, one of the simplest geometry demonstrations there is: showing that the angles in a triangle add to 180 degrees.
Rather than simply telling students, “there are 180° in every triangle,” try this quick demonstration:
-
Have them draw any triangle.
-
Tear their triangle into three pieces, leaving the corners intact.
- Line up the three pieces so that the points meet.
Every time, those three angles form a straight line.
That’s it — a 60-second activity, but one that turns a rule to remember into a relationship students can see.
It’s simple, quick, and memorable — but you might wonder: why does it work so effectively?
Why It Works
This tiny demonstration does more than confirm a fact — it builds understanding.
When students physically manipulate the pieces of their triangle, they’re not just seeing a property of angles; they’re connecting ideas:
-
A straight line is 180°
-
All triangles share that total
-
Angles adding to 180° aren’t arbitrary — they form a visible, geometric relationship
It’s a perfect example of moving from rote knowledge (“I can say it”) to relational knowledge (“I know why it’s true”).
These small, hands-on demonstrations are powerful because they engage what researchers call multiple representations — visual, physical, and verbal. Students see it, handle it, and talk about it. That combination strengthens understanding far more than hearing an explanation alone. It only takes a minute, but gives students a mental model to anchor later, more abstract ideas.
Professional Knowledge in Action
Knowing when and how to use these demonstrations is part of what’s often called mathematical pedagogical content knowledge (PCK) — the teacher’s understanding of how particular concepts are best represented, explained, and connected for learners.
It’s what allows a teacher to move beyond “telling” to showing, and to anticipate where students might need something concrete before the abstraction can make sense.
Building this repertoire doesn’t require special equipment — just curiosity, creativity, and a willingness to look for those small, powerful ways to make the maths visible.
