I had a great teaching moment today so I’m going to tell you about it because… this is a blog.

I had a “high ability” (ugh…that phrase) Year 7 class and decided to teach a little off-centre. As a class we carried out an investigation into consecutive numbers (Borrowed mostly from nrich). I asked the students if they could create every number from 1-30 by using the sum of any amount of consecutive numbers. It took a while to get their heads around what I was asking, but they obliged and really got into it. After about twenty minutes or so, we discussed as a class which numbers you **couldn’t** get. There was some debate, and a really positive vibe when students disagreed over some of them. ‘Wrong’ answers merely provoked others to offer (nicely!) a way of getting that particular number. 28 for example, was initially dismissed as ‘not possible’, but picked up by someone else with a solution (1 + 2 + 3 + 4 + 5 + 6 + 7).

I loved the collaboration between the entire class. I was expecting it to take longer than it did, and we were left with around 15 minutes to go. Usually I’d pull out one of the many ‘short’ resources on my data stick, but this time I decided to be a bit more care free. I set the entire class a very vague task: “Ask yourself some questions about the numbers that you can’t make by adding consecutive numbers, and tell me something amazing about anything you find out”. I was convinced this would result in 33 blank faces. What happened next has really inspired me to really reflect about how I teach. Students were completely invigorated by the challenge I set them. I still got “We can’t impress you sir, you know EVERYTHING about maths” but I quickly insisted that their belief in me was reassuring, yet flawed!

Listening around the room, I started to hear really interesting conversations. Not ‘interesting…for year 7s trying to find something out that I already know and am hoping they figure out’, but GENUINELY interesting… I actually learned from them!

Remember these students are 11 years old, and that they only had around ten minutes…

Here’s what I learned, because they found it out, on their own, with literally ZERO input from me.

1.) We ALL discovered together that the numbers less than 30 that you can’t make from the sum of consecutive integers were 2, 4, 8, 16.

2.) They figured out and correctly predicted that the next number would be 32 (I knew that bit)

3.) One student managed to find out that 2 = 2 x 1, 4 = 2 x 2, 8 = 2 x 4, 16 = 2 x 8, and therefore the sequence is actually happening again. Cool, but obvious when you think it through (as an adult!)

4.) A small group of students figured out that using that sequence: 2, 4, 8, 16, you can multiply any consecutive sequence numbers together, and get a new number in that sequence. e.g. 2 x 4 = 8 (which is in the sequence), 4 x 8 = 32, which is in the sequence. etc

5.) One AMAZING student, who spent the first 8 minutes of the 10 saying they couldn’t find anything, out of NOWHERE discovered that even though 2 cannot be made by adding consecutive integers, it CAN be made by adding consecutive decimals: 0.2 + 0.3 + 0.4 + 0.5 + 0.6 … WOW!

I was completely gobsmacked.

There were other discoveries, but those were the coolest.

So I guess my point is this: kids can work out the most amazing things that will surprise you. You just have to enable them.

What did they actually learn in this lesson?

Hi Andrew. I was wondering when you’d chip in.

At the most basic level, they learnt that you can make every number from 1-30 by using consecutive integers apart from 2, 8, 16. They also learnt what ‘consecutive integers’ means.

They also picked up that there are patterns in numbers that go beyond simple counting, and therefore began to see the number system in a different light (although maybe I’m being optimistic about how much of that last part they noticed).

They made links between finding patterns and making predictions, and the limitations of this with their current skill set (we discussed how we could ‘test’ whether 1000 could be made from consecutive numbers, and students assumed there must be a ‘better way’ than to start with 8, double it, double it, double it etc).

Beyond that the lesson is about discovery, and a shifting of passively accepting mathematical rules and concepts to developing a sense of wonder and excitement about numbers and their hidden depths.

If I get rid of the waffle (things that sound great but actually you can possibly assess and know they learnt) in your answer, they learnt:

a) what “consecutive integers” means.

b) a single obscure number fact that is not in the curriculum.

And this took how long?

I don’t mean to be a grouch here, but you obviously thought this was such a wonderful lesson that it needed to be written up. Yet the actual identifiable amount of useful learning is probably what, equivalent to what can be taught in 5 minutes to students of that level? Does that not say something about what we are trained to think is “good” maths teaching, and how disconnected it is from learning?

I agree that the ‘waffle’ is less quantifiable, and seemingly therefore unacceptable to you. Aside from your issue with ‘useful learning’, which we could go round in circles of disagreement about (we won’t) forever; does it bother you that it wasn’t in the curriculum?

The reason I posted was not because I wanted everyone to teach that lesson. It was because I gave students a chance to go in their own direction and find things out, with no guidance, and they discovered things, things that required them to think mathematically, and that surprised me. I have never been a huge fan of investigative lessons. I enjoy them, but I accept there are many, MANY pitfalls with them – hence I was cynical about where the lesson would go.

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Found this entry on my ‘looking for inspiration’ web searches and enjoyed the write up. By taking a squinty view at the task and the fact you mentioned integers (without direction) I wonder how far they could have taken it if they had twigged an integer could be negative or positive. So -1 + 0 + 1 + 2 = 2 and -3 + -2 + -1 + 0 + 1 + 2 + 3 + 4 = 4 and so on…