# Scenario: Alice teaching nth term of linear sequences

## Context –curricular, professional, other

Alice was an experienced mathematics teacher, working at a private girls’ school. The year 10 group she was teaching had just sat their end of school-year exams. Alice noticed that the majority of the group had incorrectly answered a standard question on the nth term of linear sequences. This lesson was intended as a revision lesson of the topic to correct their mistake in preparation for the terminal GCSE[1] exam in year 11.

## Scenario

Alice, an experienced mathematics teacher, was reviewing how to determine the nth term of a linear sequence with a Year 10 (pupil age 14-15) class. In their end of school-year exam, Alice noticed that many of her pupils had the misconception that if a linear sequence has a common difference of a between one term and the next, then it has an nth term of . This lesson was intended to correct her pupils’ misconception. Alice demonstrated the differencing method for finding the nth term of a linear sequence, before giving her pupils a paper-and-pencil worksheet as an introductory exercise. The first question on the worksheet depicted a sequence of F-shapes with an nth term of .

Some pupils experienced more difficulty with finding the nth term for this sequence than Alice had anticipated. To Alice, the question appeared to be a routine application of the method she had just demonstrated on the interactive whiteboard, replicating the style of question her pupils were likely to face in their terminal GCSE exam. Recognising her pupils’ difficulties, Alice stopped the class and went through the question briefly on the interactive whiteboard, before letting them continue with the worksheet.

Subsequently the pupils worked on a spreadsheet exercise intended to give them repetitive practice on finding the nth term. The spreadsheet exercise mimicked a textbook exercise, consisting of a series of linear sequences for which students had to find the nth term. When the students entered a potential nth term for a sequence, the spreadsheet provided immediate feedback: ‘well done’ for a correct answer and ‘try again’ for an incorrect one. For Alice the spreadsheet exercise had an important advantage over non-digital resources such as a textbook or paper worksheet: it incorporated a button that when clicked would randomly re-generate all the questions to be different. She used this button as a classroom management tool, to prevent pupils from copying their neighbour without the disruption of asking them to move seats.

# Knowledge Quartet Coding Commentary

## Scenario: Alice teaching the nth term of linear sequences

The examples Alice used were not intentionally selected by her to serve the stated purpose of the lesson, namely to counter her pupils’ misconception that if a linear sequence has a common difference of a between one term and the next, then it has an nth term of . Indeed, the first question on Alice’s paper-and-pencil worksheet, a sequence of F-shapes with an nth term of 3n + 3, might serve to reinforce her pupils’ misconception, due to the unfortunate repetition of 3. This is an example that obscures the role of variables (Rowland, et al., 2009) and would therefore be better avoided by intentionally selecting examples, such as 4n + 3 or 3n + 7, where repetition of coefficients is not an issue. In a post-observation interview, Alice did recognise that 3n + 3 might cause her pupils difficulties because of the repetition of 3. Her inclusion of this example therefore suggests an explicit consideration of the choice of examples she gave to students was not part of her original planning process for this lesson, representing an uncritical adherence to the resources she downloaded.

Alice’s use of the capability of the spreadsheet to randomly generate a set of questions to enhance her classroom management also suggests that she did not consider the pedagogic advantages and disadvantages of choosing specific examples over others when planning the lesson. Random generation of examples is likely to be inappropriate here since it may give rise to examples like  which obscure the role of variables and thus unintentionally serve to reinforce her pupils’ misconception. Rowland et al (2009) suggest that random generation of examples might be reasonable as a means of demonstrating the efficacy and general application of an established method. However in this lesson, as stated above, Alice’s aim was to counter a particular misconception she had noticed in the pupils’ recent exam. In this sense, her choice and use of examples, not only did not serve the purpose of the lesson, but may have contributed to reinforcing the very misconception Alice wished to counter.

## References

Rowland, T., Turner, F., Thwaites, A., & Huckstep, P. (2009). Developing Primary Mathematics Teaching: reflecting on practice with the Knowledge Quartet. London: Sage.

[1] General Certificate of Secondary Education, school-leaving exam in the UK