Algebra Errors and Misconceptions in Mathematics Education

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 in society.

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.

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 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'.