Something Dan Meyer said at the MEI Conference 2021 really struck a chord with me, he said “There are no mistakes or misconceptions, just takes and conceptions.” Dan Meyer mentioned WW Sayer who said:
Most remarks made by children consist of correct ideas very badly expressed. A good teacher will be very wary of saying ‘No, that’s wrong’. Rather he will try to discover the correct idea behind the inadequate expression. This is one of the most important principles in the whole art of teaching.
For a starter addressing common misconceptions try the excellent Classic Mistakes resources by Nigel Hopley.
A superb resource to use in class (or for students to use at home) to address misconceptions is of course Craig Barton’s and Simon Woodhead’s Diagnostic Questions site. The site has many thousands of questions with carefully designed multiple-choice responses to address common misconceptions.
The Insights feature is so helpful for learning about misconceptions, suppose we look at a White Rose Quiz on Algebraic Notation, for example, looking at the Insights we can see for any question the number of responses for each option from the many students who answered this question.
From NCETM, these videos and resources for teaching Key Stage 3 maths topics include common misconceptions and pitfalls; looking at Directed Numbers for example we find slides and a pdf support document including as illustrated here, “What things typically go wrong?”
Some years ago a website, counton.org which is now no longer available published a very useful document on misconceptions. In 22 sections, in each section misconceptions are given along with the correct version. Further explanations are also provided and also follow up exercises with answers.
The above pdf document includes all 22 sections. The first 8 of these documents, by Ilan Samson & David Burghes, are on the CIMT website.
Malcolm Swan’s excellent ‘Improving Learning in Mathematics‘, includes a section (5.3) on exposing errors and misconceptions. An activity suggested there is to let your students become examiners and mark the work of others, this works very well, I have highlighted some excellent resources for this on the ‘Spot the mistake!‘ page.
On the SERP website see MathByExample and AlgebraByExample which is a set of Algebra 1 assignments that incorporate worked examples and prompt students to analyze and explain. These resources can provide prompts for discussing common misconceptions.
The collections of problem sets have been developed to help students dislodge misconceptions–both small errors and fundamental misunderstandings. Students can analyse the problems and then try a similar problem.
From this page you can access all the resources.
From Michael Pershan, see his Math Mistakes site, to quote the Author:
The purpose of this site is to collect, organize and make sense of the mistakes that students make while doing math. I’m also increasingly interested in using mistakes to help us create worked examples that students can learn from.Michael Pershan
Thinking about misconceptions in Algebra I came across “2b or not 2b: Misconceptions in algebraic reasoning” from Barbara J. Dougherty.
All the examination boards publish helpful material which addresses common misconceptions, such resources can promote very useful class discussion as can examiners’ reports.
From Cambridge University, see Common Errors in Mathematics.
Having started this post with a quote from Dan Meyer it seems fitting to end it with his thought-provoking December 2022 post, “…students don’t have misconceptions.”