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Language is a wonderfully malleable thing, providing endless opportunities to torture sentences into new and pleasing shapes that extend the form, pierce our thoughts, or tickle us with a double meaning (four candles anyone?). This lexical flexibility has led to great works of literary art, a rich array of slang and local dialect, and ever-evolving ways for students to baffle teachers by talking in an almost incomprehensible language of their own, some of which is bare inventive, mint and sway.
This flexibility is not without drawbacks however; the myriad ways in which the same basic sentiment can be expressed using English can also result in a certain casual attitude to language use, allowing misunderstandings and misconceptions to flourish. Rarely is this more starkly highlighted than in examination questions, where a carelessly constructed question can result in unexpected answers from students, such as the possibly apocryphal “find x”.
In fact, much of the time spent by awarding organisations on quality control is centred around anticipating the ways in which questions can be misinterpreted by students and tightening up the language to ensure that they answer the question that the examiners intended. While this is an essential exercise, it is a constraint that does result in some questions being predictable, allowing teachers to train students to spot keywords that “always” mean a certain mathematical technique or idea is required.
More generally, how we use language as teachers can have a significant impact on student understanding. Consider the words “volume” and “capacity”: what do these mean to you? Many of us use these words interchangeably, comfortable and secure in our understanding that volume is the space an object inhabits and capacity is what it holds. Now imagine that you are a student asked about the volume of a thermal travel mug: what does this really mean? Is it the amount of coffee it holds? Is it the quantity of water it would displace if submerged in a bowl with the lid on? Could it be the amount of plastic used in the construction of the mug itself? Worse still, we often present students with cuboid shaped “tank filling” problems in which the external dimensions are given but the “correct” answer uses these values to calculate the volume of liquid. As confident mathematicians we are happy with the assumption that the walls of the tank are of negligible thickness, but for a student is this clear? And what misconceptions are being established by failing to address these subtle complexities?
In the statistics sphere this was brought home to me after speaking to several teachers of maths and other subjects about when to use “Bessel’s correction” (using n - 1 rather than n) during standard deviation calculations. Even for confident mathematicians, it seemed that it was not obvious that the distinction is entirely dependent on the phrasing of the question, and often, because textbook questions are delivered in a specific context, no reference is made to whether the sample standard deviation or the population standard deviation is desirable (even in this case I should really be careful to say “an estimate of the population standard deviation”). Confused? Let me explain…
Let us consider the questions:
A semantic interpretation could be that Bessel’s correction should not be used if we are asked to “calculate” rather than “estimate”, but this is not clear-cut; in some ways it depends on whether or not a context has been provided with the data. Were we simply given a list of 10 numbers? Were we told that the numbers were a sample from some larger population? In the first instance, there is an implication that we do not need to use n - 1 as we are calculating the standard deviation of these specific data (though why anyone would want to do so remains to be justified). However, in the second – if we have the information that the data is from a sample – it remains uncertain. Can we be certain that the absence of the term “estimate” (as compared to “calculate”) is evidence that the sample standard deviation is required, or is it simply an oversight in the language?
As educators, it is essential that we take great care over our language use and consider whether the sentences we construct are actually creating opportunities for misunderstandings to develop and become entrenched. How far does this extend into precision in mathematical language, and which of those hills are you prepared professionally to take your stand on? You can tweet us @CambridgeMaths or comment below.