Our regular feature highlighting an article or research paper that will, we hope, have a helpful bearing on your teaching of mathematics
The equation, the whole equation and nothing but the equation by Susan Pirie gives us the theme for this issue. In the article, Dr Pirie reflects on approaches to solving linear equations. The paper was presented at the British Society for Research into Learning Mathematics day conference held at the University of Birmingham, on Saturday 25 March 1995. The article talks about the balance method and then the alternative inverse operation method, stating that “Both these methods lead to the pupils "solving" an equation that is not the original given one, but some altered, in some sense equivalent, equation”. At the start, the article is insightful about how learners read “=”: it’s well worth asking your pupils / students what they think the difference is, if any, between 2x + 7 = 29 and 29 = 2x + 7.
It is worth persevering with the slightly odd symbolisation in this print version of the article to uncover how the pupils use reasoning to solve the equations by thinking of the idea of a fence (rather than a balance), and how this extends into solving equations with unknowns on both sides. In particular, on page 5 of the paper there is some dialogue between a ‘low ability’ (to use what is now out-dated language) child (Joan) and her teacher about solving the equation 11 + 5b = 3b + 25. The conversation reveals that Joan is clearly thinking through the logic of the equation and not relying on a formulaic method. How could we help our pupils develop the confidence of their mathematical reasoning so that more – all – of them have this insight.
We think that you will, having read this paper, have a deeper understanding of why some pupils find equations difficult, and also have some ideas how to respond to this in order to help your pupils develop more secure conceptual understanding and procedural fluency. The paper is now 20 years old: how much of it has stood the test of time, do you think? Would the findings be the same if the research were carried out today? Have you taught successfully a method for solving equations that is not based upon the very common “balance” model? If so, what was it and why was it effective? Let us know, @NCETMsecondary or firstname.lastname@example.org.