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.

David, I *loved* reading this! Never have I ever read anything by anyone yet who gets it quite like this, and I am absolutely delighted to hear that it was as successful as I always believed possible.

Sequencing is everything, and we shall not have a mathematics education system we can be proud of until that mantra echoes up and down the halls of teacher training institutions up and down the country.

That line:

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

what do we make of that? Is the ‘difficulty’ they perceived an unavoidable starting place, until the connections are made and the clouds clear, or can we eliminate even that?

I think the difficulty is the difficulty of encountering something new. I did everything possible to minimize the amount of new stuff – we did y=mx+c as a formula when we were doing sequences, and I made them plot their sequences as graphs – but the new vocabulary and new types of tasks is enough to disorientate.

The problem is that students always think they’re building from the ground up, because that’s how it usually feels. They don’t expect it to be a familiar concept applied in a different way, because it’s never been like that before…

I have one difficulty with all this. Sometimes kids “getting it” is dramatic for all the wrong reasons. It’s because they haven’t listened up until then, or it has been badly explained, or they just didn’t try to understand.

My worry is we are in danger of celebrating the light bulb moment like we would celebrate someone pulling a child out of the way of an oncoming truck. It’s a really good thing, but better if they weren’t in front of the truck to begin with.

I always suspect the moments of epiphany my students experience come about solely because of my hitherto poor explanations.

Really enjoyed read this though. Bravo!

The case you point to definitely exists – when a student gets something simple that just should have been explained better. But there’s also the case where a student meaningfully finds and absorbs a connection that deepens their understanding, where concepts click in a way that can’t happen from just being told about a relationship. That relationship needs to be explored.

What’s important to recall here is that I’m not arguing about getting a shallow understanding of the concept being immediately taught, but making deeper connections between concepts that strengthen mathematical understanding as a whole. My treasured lightbulb moments are when students ‘get’ something bigger than the process I’m asking them to master that day.

Enjoyed this very much. I call them ‘pennydrop’ moments. When the penny drops and you hear that create sound: ahhhhhhhh…

Thanks for this. Thankks to DD for pointing me towards it. In reply to OA and DD I think there is something here that is in addition to just teaching stuff properly in the first place. It seems that understanding comes when a learner links two different ‘facts’ by recognising the similarities that are fundamental to the previously disparate bits of knowledge. I do think that is more than ‘pulling the child out of harms way’ and the ‘rescuing’ effect teachers will get when a child eventually ‘gets it’.

So the learning needs to be planned in a sequence something like:

teach [stuff]

Emphasise the particular features of the [stuff] that will link to later [stuff]

Close in time, teach the new [stuff] and emphasise the elements of the new stuff that the child will then see as relating to, linking to, the previously learned stuff.

Understanding results and the world is calm again.

Great stuff. I have a feeling that there are maths teachers who have never even noticed that the nth term of a linear sequence is almost exactly the same maths as y=mx+c, so they stand no chance of getting students to construct this understanding. Possibly it’s a good plan to learn the two things separately and then be shown that they are basically the same thing, rather than learn the two things simultaneously and stay confused. You can plan for to make the magic moment happen.

Er, the last maths I studied was o level in 1979 but am now teaching level 6 maths in primary. We didn’t do sequences in 1979, let alone link them to y=mx+c so I need my very own light bulb moment here. Sorry to horrify all you maths specialists out there.

If I’m getting it right, x=n, the ordinal place in the sequence (1st place, 2nd place etc) and y is the answer. ( I get the c bit- that’s obvious and m is the bit that changes by x ( or n) each time). Have I got that right? Us primary teachers could do with a refresher… Might take gcse next year if I can find a suitable place to do it- don’t really want an evening class full of 16 yr olds who failed it first time around. Never ever thought about using numicon to teach line graphs so thanks for that.

Talking about light bulb moments, it was during teaching y=mx+c last year that I finally really understood why c is the y intercept- 40 years after having been taught it!

You’ve got it right there! Understanding what graphs represent is so important, but is something often skipped over. They’re easy once you see the relationships involved.

Yes I like the way you linked it to what was basically a science experiment in the barbie lesson- makes it real. I’ve thought about doing this when teaching algebra; eg pressure equals force over area for example. Do you think this might be useful?

Definitely, makes functions more meaningful when you get the idea of what variables are. It’s good to use the same experiment structure and sheets as they use for science as well – genuinely useful cross curricular work!

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