Since I started teaching, one of my favourite topics to teach has been ‘finding the nth term of linear sequences’. There’s a few reasons for this, including that it doesn’t take too long to come up with examples, and there’s loads of resources on this already available. Primarily, though, I love the fact that every single student in my nth term lessons has been successful. Regardless of their age, prior attainment, or their perceptions of maths, they’ve all managed to find the nth term of simple increasing sequences. After all, the procedure is straightforward and can be quickly replicated.
The way I’ve introduced this topic has varied over time – at points, I’ve spent significantly more time explaining what the nth term is and why it is useful, and at others, I’ve gone through a couple of silent examples before providing students with the opportunity to practice. Regardless of the teacher exposition section of the lesson, the practice questions would have looked something like this:
Find the nth term of the following sequences:
And typically, all my students would have been successful with such questions. Obviously, some would require more teacher support, and some would have been quickly moved on to decreasing sequences and fractional sequences, but all would have been well. However, although I mentioned that my students were initially successful, it remained that the vast majority (including high-attaining, well-motivated students) could not remember how to find the nth term when tested on it at any point in the future.
I have lost count of the number of students, who, when asked to find the nth term of:
3, 8, 13, 18
in a test will write down ‘n+5’ or ‘n+3’. I have also lost count of the number of students who will simple continue the sequence, writing down ’23, 28, 33′ in a desperate attempt to gain marks. Evidently, something was going wrong. Now, in part this is due to the distinction between performance and learning. As stated by John Mason, ‘teaching takes place in time, but learning takes place over time’ (Griffin, 1989). Arguably, the primary reason for my students failing to remember was that they were not given adequate opportunities to regularly revisit this skill, and making full use of retrieval practice to ensure that finding the nth term was something that had been learnt, rather than just being performed. Without doubt, this was part of the problem, but with reflection, I think something else was going on.
As stated earlier, the procedure for finding the nth term is relatively straightforward – once students have performed two or three similar questions, they are likely to be able to perform the method, in the moment, with little thought as to what they are doing. To quote Willingham (2010), ‘memory is the residue of thought’ – that is, we remember what we think about. The danger, therefore, with students practising the same style of question over and over and over is that they stop thinking; the moment they stop thinking, the chance of committing the procedure to memory is greatly reduced.
I was now unsure how to deal with this with relation to my own practice – to have students move on to a new procedure/concept after only three questions on finding the nth term seemed deeply counter-intutitive, but to have students mindlessly answering procedural questions was not yielding positive long-term results. In recent months, as a result of reading Craig Barton’s (2018) ‘How I Wish I’d Taught Maths’ and in an attempt to address this, I have started writing sequences of ‘minimally different’ questions, applying what I have learnt about variation theory to my own practice.
For example, if I now wanted students to find the nth term of linear sequences (in particular, when they are learning this procedure for the first time), I would present them with the following:
At first glance, nothing much appears to have changed from the set of questions earlier, and yet on closer examination, each has been chosen for a specific reason. Question (a) is straightforward, and then (b) is related to (a) in that each term in the sequence is 1 lower. (c) is related to (b) for the same reason, and yet this time, students are forced to think about the ‘+/-0′ – does it need to be there? Question (d) is again very similar, in that the common difference is constant and yet the starting point is 3 lower. With question (e), the starting point remains the same, and yet the common difference changes – how will this affect the nth term? With questions (g) and (f), sequence (g) has been doubled – will that double the entire nth term or just part of it? With question (j), perhaps not all students will reach this point, but some will – either way, this provides an excellent introduction to decreasing linear sequences.
I have only used this set of questions once so far, with my Year 8s, just before Easter. As such, I cannot possibly yet reveal the effects on their long-term learning. Indeed, if I do not give them the chance to practise and retrieve this skill in future, then it is still highly likely it will be forgotten. However, in the lesson itself, something felt very different to how my nth term lessons have felt in the past. Pupils, on the whole, were significantly more interested in trying to predict what the nth term would be, and were hugely delighted when they predicted correctly. With question (f), several students initially predicted that ‘the second part of the nth term would be the same, because they’ve got the same starting point’, and were astonished when this turned out not to be the case. The fact that their predictions were incorrect caused them to refocus and check their work independently, and then they began to reason as to why. When students are given the opportunity to predict and reflect upon their answers, then they are likely to ask deeper questions about the procedure or concept in question, which will hopefully lead to greater learning over time.
I have used ‘minimally different’ questions with all of my classes now, across a variety of topics and I have been astonished each time by the initial reaction of my students, and so I would really recommend trialling this. However, this needs to be done with caution. If you are creating a set of questions yourself, you need to constantly be thinking ‘what do I want them to notice here? What do I want my students to be thinking about?’. As you create, work out the answers simultaneously – I keep (despite being reasonably proficient at all GCSE Maths!) being surprised by what I’m noticing about how one change in a question can completely change an answer – make a note of that to discuss with your students. If you have found a set of questions from elsewhere, work them out yourself beforehand, regardless of whether the answers are included. Most importantly of all, you will need to model with your students exactly what is expected of them, as this will almost certainly not come naturally at first. If you give your students some carefully chosen practice questions, but don’t encourage them to predict and reflect, then the effort that has gone into the creation of these questions has been wasted. If you don’t stop the class at appropriate intervals and discuss their predictions and reflections, then the questions will have little impact above any other set of randomly chosen questions, as students will begin to mindlessly work through the exercise.
In short, carefully chosen, intelligently varied exercises will only have a positive impact on initial student learning if they are combined with discussion and forcing students to predict and reflect upon their answers. They will only have an impact on long-term student learning if students are given the opportunity to regularly revisit the skill or concept in question.