Common Errors & Misconceptions in Mathematics
In our work as teachers we will often encounter errors in students work and thinking. Trying to unravel what has gone wrong is so important as this opens a door into our understanding of where students are in terms of their mathematical development and what possibly has gone astray in their thinking and deductions. Making errors is not always a bad thing – it can also be a step in learning. As researchers at Warwick University have said, “Errors are essential when developing understanding.” 
Errors can be just simple arithmetic errors or a lack of accuracy (which we probably all succumb to from time to time) and they can be due to lapses in concentration or mistyping/miscopying a calculation or value. But they can exhibit more serious problems showing misconceptions about a topic. It is these errors that as teachers we need to gain some insight into. The errors may display muddled thinking in a theory, for example using Pythagoras or Trigonometric ratios but no right-angle is present, or more often using an incorrect strategy, such as multiplying when dividing is required. This latter type of error is common in Key Stage 3 (KS3). In higher level GCSE we will see errors in logical reasoning, where a student has tried to develop reasoning but part of a result is mis-applied. This is quite common in a two-stage problem where students need to find one result and use this to determine a second result (for example using the sine/cosine rule and then with this answer, finding the area of a scalene triangle). Misconceptions can occur when for example a student has started to learn Calculus but in developing a deep knowledge of the subject such as at KS5, they begin to integrate functions such as:
as algorithm, rather than as a function to the power -2.
Does it matter if these errors or misconceptions pass without comment or correction? As Swan has said in his seminal work, one of the guiding principles of effective teaching is to “Expose and discuss common misconceptions.” . I would argue that in Mathematics our knowledge and progress in the subject is built as in the foundations of a house or as links in a chain. If we leave parts of students’ knowledge incomplete, with slight holes or with gaps, as a whole the structure of thinking doesn’t quite develop fully. Mathematical thoughts and processes are so inter connected, that not knowing Topic A will have a debilitating effect on learning Topic B. For example we use algebraic manipulation in many aspects of problem solving, to help us to clarify a problem and set it into mathematical context and language. Thus, if algebraic thinking isn’t securely developed many other future aspects will not quite establish themselves in a students’ repertoire.
As teachers how we help? Feedback to help improve learning is a first point of call. We can make sure feedback is corrective in nature; tell students how they did in relation to specific levels of knowledge. Rubrics are a great way to do this. Keep the feedback timely and specific is also helpful as is encouraging students to lead feedback sessions – peer-teaching.  Other teaching tools will also help, for example, questioning wrong thinking, addressing logical errors, addressing a lack of knowledge or a mis-learnt knowledge.
Finally as research shows, “Teacher Orientation towards Student Performance has some influence on improving learning & school improvement.”  The fact that we as teachers exhibit some care for our students’ progress can affect that progress.
 Ingram, Baldry, Pitt, “What role do errors have in the Learning of Mathematics?” ATM Magazine, May 2014
 PD1 & PD2 National Stem Centre e-documents, Standards Units – “Improving Learning in Mathematics”
 Marzano R., Pickering D., & Pollock J., (2001) “Classroom Instruction That Works”
 Honingh M & Hooge E, (2014) “The effect of school-leader support and participation in decision making on teacher collaboration in Dutch primary and secondary school”, EMAL journal, Vol 42, pp75-98. Http://Educationendowmentfoundation.Org.Uk/Toolkit/About-The-Toolkit/