Pupils are intrigued by algebra. At first they find it hard to believe they must learn something that seems ridiculous. That it rapidly becomes almost incomprehensible is much less of a surprise and many spend the ensuing years fluctuating between bemusement and frustration.
My research in Kent and in Austria suggests that Austrian and English pupils aged 14 experience similar problems with algebra and that their teachers are uneasy about these lessons. In both countries the algebra content of the national curricula is seen as problematic.
The following list does not reflect a particular model of learning or a teaching approach. The points arose through discussion with teachers in both countries about the errors which secondary pupils make. If there is a common theme it is the need to maintain the complexity of algebra - for in this lies its power - while making it understandable and coherent. These are, then, tentative suggestions.
* Be explicit about when the understanding of variable is important (for example, to answer the question: Which is bigger, 2n or n+2?). The assumption that a letter has to stand for a particular unknown is widespread; as is the view that two different letters must stand for two different numbers.
* Dolots of simple formula work based on diagrams (for example, perimeters).
* Encourage checking by substituting fractions, decimals or negative numbers as well as positive integers.
* Be aware that textbook questions may encourage pupils to avoid or to ignore the algebra (for example, if a+b =23, a+b+2= ?).
* Use nth term expressions from sequences to discuss the values n can take.
* Actively teach and revise the vocabulary of algebra.
* Discuss the nature of the equals sign, particularly its dual role as a (procedural) prompt for a calculation or an answer and its (structural) use when stating equivalence.
* Analyse arithmetic structure as well as algebraic structure (for example, the tendency to detach sign from the signified: 50 - 10 + 10 + = 60 not 20!)
* Be aware that pupils revert back to earlier, possibly simplistic notions when algebra gets hard (for example, over-reliance on trial and improvement).
* Do not try to make too many connections at once, for example functions, equations, co-ordinates and graphs.
* Do not teach pupils anything which will have to be untaught at a later stage, for example algebra "codes" such as a = 1, b = 2.
Gerry Wearden is head of sixth form at Pent Valley Comprehensive, Folkestone, Kent and was formerly head of maths at the school E-mail:email@example.com