The National Curriculum specifies that in upper KS2 – years 5 & 6, pupils should be introduced to the language of algebra as a means for solving a variety of problems, before becoming an intrinsic part of mathematics at KS3.
In the year 4 programme of study, pupils should be able to read Roman numerals from 1 to 100 and up to 1000 in year 5. With ‘x’ being the most commonly used variable in algebra, this is likely to cause a great deal of confusion and research has shown that pupils up to the age of 15 find difficulty in interpreting algebraic letters in general.
The importance of learning algebra is unquestionable and in 2016, Makonye and Stepwell suggested that one cannot be successful in mathematics without it. Leonard Katz deemed in 2007 that algebra was an essential concept for business, science and technology, and an indication that algebraic thinking is vital to all learners if they are to participate fully.
That being said, there is a general opposition to algebra by pupils and the National Curriculum’s own wording ‘introduced to the language of algebra’ may hold clues to some of that opposition. Questions are often verbose and require some comprehension and assimilation before even attempting to translate the information into an algebraic formula. This syntactic to semantic relationship reverberates throughout the topics of algebra).
Algebra is too broad a topic to specify individual errors and misconceptions (e.g. 2 + 3x = 5x) and so, the focus will be on generalisations of pupils’ errors and misconceptions and appropriate strategies to improve pupils’ understanding. Almost all errors can be classified as procedural or conceptual. There is even more confusion when procedural errors arise from conceptual beliefs.
A conference paper by Dr Hassan listed pupils’ errors and misconceptions and noted that the errors were either detachment errors (mostly order of operations, e.g. 23-8+3=12) or errors due to lack of technical vocabulary. The misconceptions noted were equalities, letter usage, operational symbols and finally bracket usage. There is an obvious overlap between some of the things listed as errors and those classified as misconceptions. His conclusion was that misconceptions are ‘not simply careless mistakes but rather intelligent generalisations’ and a detailed understanding of these misconceptions should form part of any initial teacher training, giving teachers the tools to present pupils with problems that will cause them to conceptualise new ideas.
When presented with an unfamiliar concept, pupils will often develop their own ideas and strategies. By encouraging pupils to try to solve unfamiliar problems, they become challenged and need to conceive a new mode of thinking. Perhaps they relate to some preconceived idea or try to logically conceptualise the problem to obtain an answer. A cognitive conflict is created if expectations and predictions do not conform to current reasoning, in other words, their answer is incorrect. This cognitive conflict enables pupils to discover that their preconceptions are inadequate and ‘show more curiosity and interest when the given phenomenon or information is not consistent with their expectations’. Other research has found cognitive conflict to be very effective in aiding conceptual change. ‘The initiating factor for conceptual change is disequilibrium, dissatisfaction or cognitive conflict’.
The idea of creating a conflict was shown to accelerate pupils’ ability to make these conceptual changes. The idea was to use misconceptions and errors for learning. Lessons were designed around particular misconceptions with the intent of making the pupils challenge their own way of thinking, thus creating the conflict for them to challenge their own misconceptions. It was suggested that care must be taken, when using this approach, to not compound or reinforce the misconception, rather to start with a misconception and have the pupils question and reflect upon it. This could be considered as a ‘why does it not work?’ approach as opposed to learning from making mistakes. For a teaching concept such as this to be successful, it is necessary to identify the misconceptions that will likely arise and phrase a task in such a way to bring it out. This will initially mean more work and continual research by teachers, but would also ensure that best current practices are observed as a result of their continuing professional development.
A report by Cambridge Assessment, in 2014, lists noted errors from examiners’ reports under the heading of algebraic fluency. Most common was found to be manipulating expressions, then rearranging equations and formulae, writing equations from descriptions, substituting and solving equations and lastly, solving equations. They noted that the higher tier GCSE students found difficulty with indices and interpreting worded questions to form their own equations.