Small Steps and Atomisation: The Difference
One teaches processes. The other teaches mathematics.
Podcast is AI generated, and will make mistakes. Interactive transcript available in the podcast post.
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I recently had the pleasure of sitting down to dinner with Anna Stokke, and during our conversation, I think I finally struck upon the simplest way to explain the difference between ‘breaking a process down into small steps’, and atomisation.
We’ll look at this both through the higher level topic of solving simultaneous equations, and then through the much more basic topic of column addition.
Small Steps: Adding Equations
Say you’re teaching students to solve simultaneous equations by elimination. At some point they will need to add (or subtract) the two equations to eliminate a variable.
So, the small steps approach says ‘here’s a small part of the whole end-to-end process that we need to practise until it’s second nature’. The first task set might look like this, with students being asked to add each pair of equations:
This is an excellent idea, a fantastic approach, and the above is a brilliant task set. You should use it if you have this topic coming up soon.
However, it is treating ‘adding equations’ as a step in a process, and so, it is missing, overlooking, neglecting, a huge amount of conceptual richness.
Atomisation: Adding Equations
By contrast, when we take an atomic view, the first task set to practise adding equations might instead look like this:
As with the small steps task set we are repeatedly practising the act of adding equations. They share that in common.
The difference is that, this time, the concept of adding equations is the focus, not the step of eliminating a variable.
As a result, whether a variable is eliminated is incidental. In the task set above it only happens once, in task 4.
Instead, each task introduces a new idea to the concept, ‘expands’ the concept.
The small steps task set is still an option for us here, especially when we want to home in on Case 4 above, eliminating variables, to serve some greater purpose. But it’s no longer the starting point, and no longer the focus.
Instead, this sequence introduces a much broader range of applications of concept ‘add equations’. The task set can explore such a broach range, in part, because in many cases just one thing changes from one task to the next. When just one thing changes, students are able to focus their attention on how that small change produces a mathematically meaningful difference in outcome, not just on repeating the same action without variation. Each time, they have to think about what has changed, and what impact they might have. They have infer or deduce the correct response each time. What we have here, I believe, is a desirable difficulty. Having been through the sequence above with a mixed ability Year 9 class and seen every student at the bottom of that class be successful, it’s not ‘productive struggle’, they don’t try and fail, they try and succeed. The tiny variation, the small leaps in logic from one task to the next is what assures this.
If you’re a regular reader of this Substack, then the sequence above will be familiar to you. It’s an expansion sequence, and we’ve looked at a few of these in the past.
The Comparison
The small steps approach treats adding equations as a step in a process. It can help assure automaticity of that step. It will help students to master the process.
The atomisation approach treats adding equations as a mathematically meaningful concept in its own right. It not only assures automaticity of that step in the process - when we come to it - it also helps students to master the mathematical concept, seeing it applied in a large range of contexts, and rapidly learning to generalise. It develops a flexibility in student thought that means they can more easily handle non-standard tasks when they appear.
Column Addition
This was a topic that Anna brought up. I don’t get to think at this level as often as I would like, it tends to be more Naveen Rizvi’s specialism. But I wanted to give it a go. Specifically, Anna brought up the small step of carrying the one.
Again the small steps worksheet might look like this.
With the expectation that students respond something like this to teach task:
Again, the small steps approach will help to guarantee mastery of the standard algorithm. It’s a good approach; I happily recommend it if the alternative is to only work through comprehensive I do / We do examples from start to finish.
What about the atomic approach?
This time I had to come at it from an entirely different angle:
