What it Looks Like When I Atomise

A concrete example of my attempt at Routine Atomisation for Nth Term Rule

Podcast is AI generated, and will make mistakes. Interactive transcript available in the podcast post.

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I recently worked with a teacher who is on our Transformation Programme, and the topic under discussion was Nth Term Rule of a Linear Sequence.

The challenge ahead of us was a Year 9 bottom set, and how we could guarantee their success for a descending sequence; something where negatives were involved.

I had three gos at this before I was happy with the end result. I also realised that I was making certain choices along the way, and there’s a little something further I would now add, on reflection.

Naveen Rizvi

has an alternative approach. Her alternative has some interesting pros and cons. Despite being quite different, it is still follows all the principles of Unstoppable Learning.

So, in this post we’ll first look at the process I went through, thinking from scratch.

In the next post we’ll look at the atomisation of Naveen’s approach, by contrast.

What I used to do

First, how I’ve done this in the past. Say this is the sequence:

\(40, 35, 30, 25, 20, …\)

In the past I will have asked them to find the first order difference (maybe just asked them ‘how is it changing each time?’)

In this case, that’s -5.

Next, I have done one of two things:

Ask them to write out the sequence for -5n:

We can see that -5n hasn’t ‘worked,’ so now we can ask how we need to change the rule to get to our sequence.

One issue with this is that it is a lot of work. A lot of unnecessary work.

So another method I’ve used is just to focus on the first term.

If at the moment our first term is -5, what would get us to 40?

This is now ‘less work', but it still has two problems.

1. What do I mean by ‘what will get us to 40’? This is fine if kids can intuit and interpret my meaning, but it’s not exactly clear, not exactly logically faultless

2. Even for much, much simpler sequences than this, with all the correct working, kids still write down the wrong thing

In reality, the working for a very large proportion of them, even in the higher sets, seems to end up looking something like this:

I could argue that this is their all their fault, really. Too lazy to follow my working exactly as I set it out. Too quick to right down plausible numbers without thinking about what they were doing. No checking whether what they wrote generates the correct sequence.

I’m not going to, though. I know enough cognitive science to know that this is very natural, normal, and predictable human behaviour.

It does not matter that ‘well some kids got it right;’ that does not absolve me and my teaching of my responsibilities, it does not mean the teaching here is good enough. I know I can do better than this.

Atomisation Attempt 1

Here’s my first attempt. This is the full chain view.

The key to my thinking here was Atom 2, potentially a plutonium atom.

You find the difference as in the past.

But now before doing anything else, you write that down correctly.

If the first order difference is +3, you write +3n.

If it’s -17, you write -17n.

Whatever it is, you get it written down right away; no chance to jot down a note and then later on misinterpret your notes and put that -5 in the wrong place.

This also offers a very simple transformation sequence.

“Write the Nth term rule for this sequence:”

Prompt

Response

\(+3n+C\)

Prompt

Response

\(+3n+C\)

Prompt

Response

\(+4n+C\)

And so on.

The + C bit is necessary so that the response they give can be correct before we’ve figured out the value of C.

The problem with this atomisation, I later decided, was Atoms 3-5. Two problems with that:

1. I was told this group would largely refuse to answer if asked something like ‘what’s 15 - 6?’ If they were ultimately coaxed to respond, they would start counting on fingers. Would they really be comfortable with this sudden introduction of algebraic terms?

2. The simplest cognitive routines are single chains of thought, top to bottom. But here, Atom 3 breaks away from Atom 2. We’re noting down ‘-5n + C’ and then working on a new chain of thought to figure out the value of C… before later returning to -5n + C so we can substitute. It can work, sometimes these kinds of breaks are unavoidable, even. But they’re never as simple as a single chain of thought. There’s a risk here of split attention effect

So, can we get to a single chain of thought that doesn’t require any algebra?

Atomisation Attempt 2

Here is attempt 2:

This new version removes the need for ‘C’ and so on by asking them to take everything they know and write out the complete Nth term rule in full, as an unsimplified expression.

You could stop at Atom 2 and, technically, they wold have the correct Nth term rule!

The new version also dramatically reduces the number of steps, from five atoms down to just three, while maintaining a single flow of thought, no breaks away.

It does ask them to do more in Atom 2 - an argument can even be made that it’s more of a cognitive routine than a transformation, ‘write this, and then write this, and then write this…’ - but it’s still so simple, mostly just transcribing numbers from one location to another, that I’m placing the bet that all kids will be able to follow this via a transformation sequence.

What they’re learning is to ‘write the first order difference, then write n, then write the first term minus, then the first order difference again.’

The group in question won’t be able to process the negative arithmetic; no way around it, we’re going to have to let them use a calculator for that bit, at this stage.

It’s a lot simpler, I’m a lot more confident they would be able to learn each atom here via atomic instruction (the instructional sequences for teaching a transformation,) and the chain they have to learn is also much shorter now.

There’s just one final problem that’s niggling at me: split attention.

They first attend to the difference, write it down, then attend to the first term, write it down, then have to attend to the difference again.

I wanted to remove that split attention, so finally settled on this: