I love lightbulb moments. They’re the real reward for teachers. When thinking about a definition, I’d say that a lightbulb moment is when a pupil jumps from Remembering to Understanding. These moments are incredibly valuable. They increase students’ ownership over their learning, and are mammoth motivators. They make students feel successful in a qualitatively different way from just being successful on a test. They feel like their brain ‘gets it’. The associated dopamine release helps addict them to learning, and raise students who want to study and learn well after they leave my classroom.

Yet there’s a real problem with relying on these moments for your students’ learning. A lightbulb moment is analogous to the slipping of tectonic plates. It’s the unpredictable culmination of a process. Changes in conditions – heat, pressure, other movements – increase the probability of the slip occurring, but the actual moment of slipping is random and unpredictable. The same is true of lightbulb moments. The conditions you create can alter the probability of a given student having a lightbulb moment at a given time, but nothing you do can make it it happen, or predict exactly when it will.

Lightbulb moments are the joy of teaching.

This is a huge problem. I am not willing to rely on the good favour of probability for my students’ success. I cannot run my classroom solely on a model of improving conditions to maximise the probability of lightbulb moments, because there will be some students who the probability does not favour. I need to teach in a way that guarantees each student’s mastery of our key concepts. Teaching solely for lightbulb moments clearly doesn’t do this.

Yet neither am I willing to write off lightbulb moments completely. The promise of the motivation they create, the positive attitude they cultivate, and the increased feeling of self-efficacy for students is just too much.

This term I’ve been consciously working on a trade-off. I need an uncompromising approach to making sure every student succeeds, that still has adequate conditions to keep the probability of lightbulb moments high.

I can safely say that this term I’ve seen more lightbulb moments in one class than I have in the rest of the year. Why?

Sequencing Concepts

Think about what students say when they’re explaining a lightbulb moment. Most of the time it goes like this:

“Oh I get it! It’s like [insert other concept they’ve learned] but just [insert key difference]”

Understanding is all about links. Part of the reason students are slow to understand is that we start from scratch. We begin a “new” topic like it’s new. But it’s not. The curriculum just happens to be split up that way. Lightbulb moments come when students understand something new in the context of something already understood.

Maths is about links between a vast array of connected concepts.

My way around the trade-off was not to change my teaching, but to increase the proximity of related, already understood concepts. It’s all about sequencing.

My Year 7 class began this term with sequences – something they’ve all seen before. Everyone was comfortable with the idea of a sequence, and describing a term-to-term rule. We spent a lot of time working on constructing sequences, first physically (with Numicon, multilink or matchsticks), then pictorially, and then abstractly with numbers. They understood that a term in a linear sequence was composed of n differences and a constant, because that’s how they’d built each sequence. Cue dropping in something about writing this down in algebra.

The first, second and third terms of a Numicon sequence.

Throughout this time we’d also been playing games about functions and substitution that we’d learned earlier in the year. I needed to increase the proximity of these previously understood concepts.

The next concept was linear graphs. It’s always irritated me that students (and curricula) don’t see the connection between linear sequences and linear graphs. Part of the problem is that we insist on writing them in different forms, and treating them completely differently. Lightbulb moments demand that students make this often hidden connection.

We spent our time plotting sequences on coordinate axes, filling tables of values, and investigating gradients and intercepts. I taught linear graphs in the same way I normally do – just in close proximity to linear sequences.

It didn’t take long. One lesson in particular I remember – I was bombarded with students sharing their Eureka moments, one of whom did leap up like Archimedes from a bath!

“The gradient is just like the difference, it’s how much it changes”
“The y-intercept is where it starts from. Like the constant in a sequence”
“Steeper lines have bigger differences”

and even

“Sir this is the same as sequences. Why did you make it seem so difficult?”

The next thing we did was scatter graphs, and the connections didn’t take long. We finished the term with a lesson of Barbie Bungee. One of my top all time teaching moments came when a student, NC Level 4c, had estimated the equation of his line of best fit and used it to predict different bungee heights. I know students in top set Year 11 who couldn’t have done that.

Dropping a thrill-seeking Barbie makes for some excellent mathematics.

Proximity to previously understood concepts had fundamentally changed the conditions of the classroom in favour of lightbulb moments. Not everyone had them. Some students didn’t really figure out all the connections, and don’t have as strong an understanding. But those students weren’t sacrificed – I hadn’t given up on traditional teaching in the hope they’d find a lightbulb moment.

My key takeaway here is that although there often seems like a harsh dichotomy between discovery learning and direct instruction, this doesn’t have to exist in practice. You can encourage discovery and deep understanding by tweaking direct instruction. Knowing that discovery relies on making links with sound prior knowledge re-prioritises a classroom that values both.