Why do some pupils find algebra so mystifying and difficult, but some just seem to “get it” and quickly develop both procedural fluency and conceptual understanding? Dietmar Küchemann carried out research in the 1970s examining how 14-year-olds interpreted the use of letters in algebraic contexts. His findings were reported and discussed in K. Hart, 1981, Children's understanding of mathematics: 11-16 and Children’s understanding of numerical variables in Mathematics in School, September 1978.
Küchemann identified six different modes of understanding. Recognising the way we use these different interpretations might help us illuminate where our pupils’ difficulties lie, and then to support them to overcome these. In the first three interpretations the letter is considered as a concrete entity (see The ‘algebra as object’ analogy: a view from school by Kate Colloff and Geoff Tennant, discussed in Issue 116) while the next three levels, on which we focus here, involve aspects of abstraction.
- A letter may be used as a specific unknown: “if e + f = 8, give an expression for e + f + g.” This requires the letter to be understood to represent a specific but unknown value. Common incorrect answers to this question were 8g, 12 and 9.
- The letter may be used as a generalised number where the letter could take a number of values: “what can you say about c if c + d = 10 and c is less than d?”
- Letters may be used as variables where there is a (second-order) relationship between the expressions: “which is larger, 2n or n + 2?”. In his research, Küchemann was looking for answers of the form “it depends” with some evidence of substitution of, say, n = 1 and n = 10. A commonly seen answer was “2n is bigger because multiplying makes things bigger”. This is a very prevalent and hard-to-eradicate misconception (and is coupled with “division makes things smaller”).
As mathematicians, we move adeptly between different interpretations of notation, suiting our reading of the variables to the context. But are our pupils as familiar, implicitly, with these interpretations? Are they accomplished in recognising which is applicable in different contexts? Küchemann used a number of carefully crafted questions to probe and clarify what pupils were thinking; these are available on the CSMS Tests area of the Increasing Competence and Confidence in Algebra and Multiplicative Structures (ICCAMS) website. You could use these items as an effective diagnostic tool for identifying the interpretations with which your pupils are comfortable and confident, and hence plan precisely the questions and activities which will first consolidate and then deepen their algebraic fluency, understanding and reasoning. ICCAMS, a major research project, was undertaken in 2008 to 2012 building on Küchemann’s research. Information and trial teaching materials can be found on the ICCAMS website: well worth reading before planning your teaching of algebra next year.