Order of operations
One of our induction tasks prior to the start of the PGCE course is to reflect on the podcast “BIDMAS or BODMAS? Order of operations in depth” with Craig Barton and Jo Morgan. They forensically analyse some of the confusions that arise from a decidedly suboptimal mnemonic for order of operations (BIDMAS).
Two takeaways for me were: (1) So much time is being spent on something that isn’t really a mathematical concept per se! (2) What does one really want to convey to students about these operations and ordering?
For more discussion on (1) see Order of operations — necessary or arbitrary where Foster et al argue for this topic being a matter of convention. It makes sense to me (and take this with a pinch of salt as I am speaking as an aspiring trainee teacher who has not even yet begun the training course!) that this topic should be framed and taught accordingly. For example:
- Paul has £50 and buys 4 books at £9 each, leading to a cost of: 50 — (9 x 4)
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- Mary makes 50 journeys across a bridge: the first 9 journeys are free, and the remainder are charged £4 each, leading to a cost of: (50–9) x 4
With no notion of order of operations, then Paul and Mary might both represent this (or two groups of students in a classroom activity might both represent this so that they self-discover the issue…) as
Such a pair of example scenario contexts illustrates the ambiguity — and thus should help students understand the need for either brackets …
… or some order of operations convention. Order of operations isn’t a deep mathematical concept. It is about notation and convenience! (Check out the podcast for confirmation of which order is the convention when no brackets are present…).
However, perhaps the more interesting aspect is (2): What do we want to convey about operations? Seems like there are plenty of interesting aspects we could cover: an operation acting on two ‘things’ of the same type, associativity, commutativity, inverses … all feel like relevant (and important!) concepts. More on that next time!
