Craig Barton interviewed me recently, during which I discussed a series of lessons I planned and taught on solving simultaneous equations.
I could be wrong, but I think this was the best planning and teaching I ever did.
Several people have asked if I would share examples of what I described during the interview, so I’m adding that here. It’s a bit lengthy, but hopefully provides the detail many people were asking for, as well as some insight into how Siegfried Engelmann’s Theory of Instruction can be applied to the classroom.
I’m splitting the post into four parts:
- Specification of content
- Sequencing of content
- Pedagogy / Instructional Approach
- Limitations of Atomisation
This is Part 2 – Sequencing of Content.
How should we sequence this content over the time we have available?
In some ways, this is the wrong question; rather than trying to cram content into the time available, we should allocate time based on what we believe is needed for 100% comprehension.
However, time is always a constraint for us teachers – in fact, it is the ultimate constraint. At the least, we should perhaps be prepared to argue ‘There isn’t enough time to cover all of what we might like to, so we will cut the content short… cut it off at X’ rather than trying to rush through content, and risk leaving pupils feeling that ‘they’re stupid,’ because they do not understand (when they couldn’t possibly, at the pace we were going.)
In this instance, we demonstrated this restraint when actively choosing not to teach solving by substitution. In recognising explicitly that we have not taught that, however, we can plant a flag that says we need to come back to it at a later date (e.g. in Year 10.) Using simultaneous equations for modelling (e.g. converting word problems into simultaneous equations, to solve the word problem) was also grudgingly left out, for now, as our time was cut shorter than initially expected.
The table below shows how the thirteen components settled upon were sequenced across the five lessons.
I – Initial instruction
R – Revisit (which can include interleaving concepts together)
It’s probably worth noting a couple of points,
1) We spent three lessons adding and subtracting equations, before we switched to having to decide whether to add or subtract, in order to eliminate one variable.
2) We spent four lessons (7 hours) before finally ‘putting it all together,’ i.e. before actually solving any simultaneous equations. This is in stark contrast with my own prior planning, when I would begin a series of lessons on this topic by showing pupils how to solve what I considered to be a simple pair of equations, from start to finish, right in the very first lesson.
3) In this instance, no more than two new things were introduced each time, yet anything from five to eight ideas could be tested, expanded, retested and integrated in any given lesson.
4) Finally, it might seem strange that deciding whether to add or subtract was covered in only one lesson. In the following lesson we moved on to multiplying to find a common coefficient; one of the great challenges for weaker pupils is in deciding between many options available to them i.e. ‘knowing what to do.’ To minimise the number of decisions they needed to make at this stage of the learning process, we taught them to always cross multiply the coefficients of x, then subtract – there was no longer any ‘decision’ to be made as to whether or not to add or subtract.
This certainly isn’t optimal. By approaching the task of solving this way, pupils will sometimes have to deal with larger numbers or more complicated mental arithmetic, when they could have used various short cuts, e.g. sometimes one coefficient is a multiple of another, so both equations don’t need to be altered, or sometimes multiplying by the coefficients of y would yield smaller numbers to work with, or sometimes adding would result in simpler arithmetic.
We made the choice fully aware of these limitations, which I often argue is what is most important. For now, we judged that ‘always do it this way’ was more likely to result in consistent success for the weakest members of our team. Also, recall that we have taught all of the components required to take advantage of those more efficient processes (addition as well as subtraction, and deciding whether to add or subtract.) When placed at the heart of the full length process, however, we knew that having to make those decisions would result in cognitive overload for the weaker pupils, whereas our stronger pupils would likely (and indeed did) leverage what we taught them to spot that there were more efficient processes they could make use of. For those who wouldn’t spot this on their own (most pupils) this is something we can come back to in Year 10, once we have more time available – we can say ‘You know that thing you can do really well? Turns out, we can make that simpler and easier for you – let me show you how,’ and we would then be focusing only on decision making, and efficiences of different processes that all result in the same solutions.
As we saw it, the choice was between ‘Try to teach them all the clever short cuts, that seem really easy to us as experts, or don’t. If we do, most will fail completely, only our strongest pupils will succeed. If we don’t, everyone succeeds, our strongest pupils will probably spot the short cuts independently, and we can still come back and fill in those gaps for every other pupil at a later date, once they’re 100% secure in the foundation process.’
So we opted for 100% success.
More on this in Part 3.