When is a mathematical process not a process? This changes everything.

When it’s a concept*.

The best example I have so far to explain this distinction, and its importance, is expanding single brackets compared with double brackets.

Transformations

The type of concept at work here, in Engelmann’s taxonomy, is a transformation.

There is an input, the transformation concept is applied to it, resulting in some output.  For this to be conceptual rather than procedural it should be possible to directly infer the transformation that is taking place by exposure to carefully chosen examples of input followed by output.

The transformation is the hidden concept we want pupils to infer from observation the changes from inputs to outputs

Single

Concept: Distribution

I used to think of expanding a bracket as a process.  There are many ways of communicating it, but one might be:

However the lines are drawn, however the working is set out, and whether you go for this or a grid method, or a variation of that which links it to rectangle areas, it will still largely be communicated as a step by step process.

Here’s why I don’t think it is a process.  Consider the following as an alternative method of instruction, the one I would probably opt for now:

Ideally these would be written on a whiteboard, with pupils able to witness which numbers you rub out and change, and which stay constant

By chance, I actually did something very close to this with my Year 7 bottom set class in my second year of teaching.  The sequence, language and examples were less carefully chosen, and I didn’t use the identity symbol, but the principle was the same: rather than treating this as a process, I used a few examples from which they were able infer what was happening.  This was a group of poorly behaved children who barely knew their times tables, but still they could see that I was multiplying the first number by the other two, 100% success.

Depending on the group, either the initial instruction sequence (the examples) or the later sequence of teacher-led pupil activity can be expanded to include all manner of important variations on that concept, including:

• Have three or more terms in the bracket
• Swap the order of the unknown and the constant
• Include negative terms
• Include more than one unknown
• Interleave with index laws
• Include a term outside the bracket which isn’t multiplied by its contents
• etc.

So much to cover, so little teacher exposition needed.  The concept is ‘multiplication is distributive over addition,’ and it can be inferred through carefully chosen examples (this declarative specification of the concept can be provided at some later date.)

As a caveat, I think verbally pointing out that ‘Ten times three is thirty, and ten times five is fifty,’ *is* an option, or ‘Ten times three ex,’ if you prefer, but I would suggest waiting until the second expression has been written, and pupils have had a few seconds to read it; explain verbally what happened *after* the fact, rather than while you’re writing it up on the board.  We tend to like doing this because it feels like it helps us manage behaviour.  Having tried both, I felt that silence on my part actually helped draw attention to my writing, while speaking at the same time possibly added to the cognitive load of the explanation; after all, it required reading and listening at the same time, which we all know aren’t possible! (split-attention effect)  So this part of the advice isn’t about silencing the teacher – as so much ‘Ofsted Outstanding’ training is wont to do – so much as it is about judicious use of language – Economy of Language, as Doug Lemov might call it.

Compare this now with expanding a pair of brackets.

Double

Process

First, my above suggestion will not work for a pair of brackets.  The reason is obvious:

It’s barely possible to infer directly how the input – the expression on the left – resulted in the output, the expression on the right.  At a stretch, a small minority might spot the relationship between the terms in the bracket and the first and final term in the right hand expression, but few, if any, will realise where the middle term came from.  If they did, how long would it take them to form that realisation?  It wouldn’t be instant at all.  Indeed, forgetting about the middle term is the classic mistake we see.

So my earlier method won’t work here.  Expanding a pair of brackets is not a concept.

I’m now going to set this out a little differently to how I most often did this, and how I’ve seen most teachers teach this, using neither FOIL, any of the many crossed line set ups, nor grid multiplication.  Instead I’m going to set out clearly the three lines of processing that are actually taking place.

Each of these three steps represents the application of a transformation.  The distributive property is applied twice, albeit in very different contexts, and simplification by collection of like terms is applied once.

This changes everything.

Here’s why this changes everything.

Traditionally I would have simply taught the process for expanding a pair of brackets, step by step.  I would have asked that pupils replicate my steps.  Now I would do something very different.

Each of those three lines I would treat as a separate concept, to be communicated separately.  Each one might take a lesson, more than one lesson, or perhaps two of them would appear in the same lesson.  They might be separated by only one day, or perhaps a hundred days; however this is structured, they are treated as separate ideas and fully explored independently of one another.

One lesson might consist of applying the distributive property to the pair of brackets only once (getting to Step 1, though it wouldn’t be called that yet.)  That first line becomes ‘the answer,’ no more.

Notice that, since each of these lines *does* express a transformation concept, they *can* each be communicated in the same way I outlined for single brackets.  It takes a little more time to see how you get from a pair of brackets to step 1, but not that much more, and importantly it *can* be inferred from carefully chosen examples.  It could be done with three and four term expressions in the brackets, rather than binomials, or with three or four pairs of brackets rather than just two.

The same is true in going from step 1 to step 2, and 2 to 3.

Eventually, once all three concepts are fully explored and embedded they would be ‘chained’ together in one lesson, or if needed, more than one.  Going from a bracket pair through to step 2, or from a given Step 1 to Step 3, or all the way from start to end.

By this point, pupils should be fully able to answer the traditional, slightly boring and pointless ‘expand this pair of brackets’ question on an exam paper, but they will also have a much deeper understanding of the concepts masked by that step by step process, and hopefully be better able to deal with anything an exam paper could throw at them.

 

What do you think?  What are the limits of this?  What won’t it work for?

 

 

* We could get into a discussion about whether or not a better categorisation of this is actually ‘procept,’ discussed here by Gray and Tall.

For the sake of simplicity, I’m not going to do that for now.