'I hear, and I forget. I see, and I remember. I do, and I understand.' Confucius.
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.
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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 with many pupils not understanding the concept of a variable. For example, 'the previous answer was p=11, so p must always equal eleven.'
Learning the language of algebra needs to be expanded beyond the simplistic representation of an amount by a letter. This would be more an example of codifying and potentially lead to many new misconceptions being formed.
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. By definition, if a pupil has a misconception, they are not going to know why they are wrong. They need to develop a sense of unease that something might be wrong and challenge their current understanding, as well as feel the topic is worthy of the effort required to make a change. It is the teacher's role to motivate their pupils.
A conference paper by Dr Hassan (2014) 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'. Pupils who are taught to develop problem solving skills tend to be far better equipped to tackle more complex problems as they arise
The idea of creating a conflict was shown to accelerate pupils' ability to make these conceptual changes. The strategy 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.
Provoking a conflict has been shown to be very effective when combined with a plenary, requiring pupils to reflect on their findings. This feedback can also provide the teacher with a better understanding of where future misconceptions are likely to arise.
The use of manipulatives and diagrammatical representation is practiced extensively in Singapore. Research suggests that using a tangible object can be very effective with pupils that find mathematics difficult. With the Concrete Pictorial Abstract approach to problem solving, pupils are encouraged to use a manipulative object (the concrete) visualise the process diagrammatically (the pictorial) and create many versions of the same concept until the actual property of the concept may be grasped (the abstract). A classic example of the Concrete-Pictorial-Abstract is an extension of Bruner's enactive-iconic-symbolic, using AlgeCards to visualise the problem.
In a case study at Bolingbroke Academy, London, despite teachers and parents being very sceptical about the use of manipulatives, they were later referred to as a vital instrument in pupils understanding. They demonstrated that mathematics can be fun and exciting as well as giving pupils a deeper conceptualisation of different topics. The use of manipulatives has been developed over the last 200 years, with the most notable proponents being Jean Piaget, Zoltan Dienes and Jerome Bruner.
Booth's research in 2014 highlighted that the most prevalent error involved in algebra was the minus sign. It was misinterpreted operationally, by incorrectly applying order of operations (e.g. 4x-(2-2x)=2x-2), and quantitatively with pupils often not recognising a negative quantity (e.g. -x-3x=-2x). Their conclusion was to introduce more complex concepts at an earlier age (Booth et al 2014). Many large-scale studies carried out in the 1970's and 1980's also discovered that the same types of error kept occurring. The curriculum needs to be set out in such a way that basic ideas and concepts learnt at KS1 and KS2 can be expounded upon to enable a higher level of thought and understanding to develop in later years. It effectively has to provide a series of stepping stones as ones education progresses. Algebra is essentially a new language for pupils and the foundations for learning it should be laid as early as possible. This contrasts with the opinion of many educators that algebra should be considered a high ability skill with less able pupils being judged unable to grasp the concepts involved.
To identify the errors and misconceptions that are likely to occur, a reasonable starting point could be the examiners' reports. 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. Higher tier students also had difficulty in solving equations containing fraction and factorisation. It was noted that the higher tier students had different problems in factorisation than core and foundation tier students. Foundation students commonly used trial and error to solve equations rather than taking an algebraic approach. When attempting to use algebra core and foundation students made substitution and order of operations errors, most notably, omission of a minus sign. Higher students also found difficulty with substituting negative numbers into equations. Higher and extension students had difficulty manipulating formulae, especially involving fractions. When solving quadratic equations, higher students did not generally use the quadratic formula, defaulting to a trial and error approach, and extension students made errors within it. Extension students often failed to fully understand the question being asked and lost marks by giving incomplete answers.
Whilst this list is not exhaustive, it does provide an insight into where procedural errors occur and how misconceptions have developed. An increased focus on method and procedure in earlier years may be beneficial in avoiding many careless errors when it comes to sitting GCSE examinations.
Recent research seems to favour a combination of pedagogical strategies. Creating a cognitive conflict or targeting specific misconceptions that are expected to arise, causes pupils to 'think!', 'ask why?' and 'learn by making mistakes!'. Reinforcing basic principles, algebraic rules and procedures, 'do it this way!' is a more traditional approach to teaching but perhaps discourages a more questioning nature. Using manipulatives enhances the visualisation of processes, which is a fundamental part of the Concrete-Pictorial-Abstract. And teaching concepts at an earlier age. Teachers should be encouraged to develop their own style of teaching and be aware that their style can affect different groups with varying degrees of success.
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