It’s the symbol you put before the answer Laura Swithinbank discovers that pupils often ‘see’ things differently “It’s the symbol you put before the answer.” 3. A focus on both representing and solving a problem rather than on merely solving it; ave you ever heard a pupil say this in response to being asked what the equals sign is? Having 4. A focus on both numbers and letters, rather than H heard this on more than one occasion in my on numbers alone (…) school, I carried out a brief investigation with 161 pupils, 5. A refocusing of the meaning of the equals sign. Year 3 to Year 6, in order to find out more about the way (Kieran, 2004:140-141) pupils interpret the equals sign. The response above was common however, what really interested me was the Whilst reflecting on the Kieran summary, some elements variety of other answers I received. Essentially, it was immediately struck a chord with me with regard to my own clear that pupils did not have a consistent understanding school context, particularly the concept of ‘refocusing the of the equals sign across the school. This finding spurred meaning of the equals sign’. The investigation I carried me on to consider how I presented the equals sign to my out clearly echoed points raised by Kieran. For example, pupils, and to deliberate on the following questions: when asked to explain the meaning of the ‘=’ sign, pupils at my school offered a wide range of answers including: • How can we define the equals sign? ‘the symbol you put before the answer’; ‘the total of the • What are the key issues and misconceptions with number’; ‘the amount’; ‘altogether’; ‘makes’; and ‘you regard to the equals sign in primary mathematics? reveal the answer after you’ve done the maths question’. So, how can we define the equals sign? The equals sign Whilst studying for a Masters in mathematics education, is a symbol that indicates that a state of equality exists I encountered the work of Kieran who emphasises and that the values on either side of the equals sign are the importance of algebraic thinking in relation to the same (Mann, 2004:65). A seemingly straightforward understanding the equals sign. definition, yet why is it that so many pupils have a I reflected on the following question: multifaceted understanding of the symbol? What exactly • Could more opportunities for algebraic thinking are the key misconceptions that are holding pupils back? aid my pupil’s understanding of the equals sign? Key misconceptions with regard to the equals sign in The wide range of interpretations of the equals sign, primary mathematics and the lack of explicit opportunities for algebraic Ginsburg (referenced in Fischbein, 1989) found evidence thinking highlighted a gap in the curriculum at my school, that primary pupils saw ‘+’ and ‘=’ as actions to be prompting me to develop an overview for ‘Algebraic performed, therefore would not accept: □ = 3 + 4 as thinking: equations - Y1 to Y6’. Here I present this it is ‘written backwards’. He found that 3 = 3 is also overview, alongside an example lesson plan, resources, meaningless as it contradicts the idea that the equals and lesson outcomes. The aim of the overview is to help sign expresses a process by which, combining some enable pupils to engage more in algebraic thinking early ingredients, one produces a certain result. Pupils on in learning, and thereby gain a broader understanding changed it to 6 – 3 = 3 to ‘make sense’. 4 + 5 = 3 + 6 was of the equals sign before they move on to secondary also deemed wrong as pupils believed after ‘=’ should school and are confronted with formal algebra. come the answer. How can we define the equals sign? Fischbein (1989) summarises: What is algebraic thinking? The conclusion is that pupils tend to interpret the Kieran (2004) summarises the difference between equals sign not in equivalence terms, but rather in arithmetic thinking and algebraic thinking. Arithmetic the light of an input-output model from which the thinking is predominantly number focused and answer properties of reflexivity and symmetry are absent. orientated. The equals sign is seen as a separator In such a model one has, on the left side, the initial between problem and solution, with the solution always ingredients, while on the right side one represents seen to be on the right of the ‘=’ sign. Pupils, when doing their product. a sequence of computations, often treat the equals sign as a left-to-right directional signal. Algebraic thinking, I started to wonder whether pupils at my school followed on the other hand, focuses on the relational aspects of similar trends. Influenced by the examples above, I put operations: together some questions, primarily to gain an impression of my pupils’ interpretation of the equals sign, a snapshot 1. A focus on relations and not merely on the so to speak. 161 pupils, Year 3 to Year 6, were asked to calculation of a numerical answer; complete a range of questions independently, including: 2. A focus on operations as well as their inverses, • What does this symbol mean? ‘=’ and on the related idea of doing/undoing; • Complete this number sentence: 17 + 26 = ____ + 8 July 2015 www.atm.org.uk 43 It’s the symbol you put before the answer • What does 7 = 7 mean? Fischbein, and others, I began to wonder how these errors develop in the first place and what we could do, • 22 + 12 = 34. What other calculations can you as a school, to address them. One of the most significant write using only these numbers and symbols? factors that influence possible misconceptions is how • What do you notice about this? 7 + 8 = 3 x 5 children are introduced to the equals sign in the first place. According to Linchevski, when children learn to Below is a detailed breakdown of answers to the first read the equals sign in arithmetic, the symbol denotes question, where the same 161 pupils (Y3 to Y6) were simply asked to identify the symbol: see Table 1 below. a left to right directional signal. The formal view of the expression 2 + 3 = 3 + 2, for example, is quite different. What does this symbol mean? ‘=’ The equation gives the same amount of information on Year Number Range of answers – Equals sign as a relational/operator No. of % of both sides of the equality sign. Equality is an equivalence group in cohort symbol pupils cohort relation and, from a psychological point of view, the Year 3 41 Answers displaying some ‘equals’, ‘the same as’ 21 51.21% learner is required to move from a unidirectional mode understanding of equals sign as relational symbol: of reading to a multi-directional way of processing Answers displaying some ‘the answer ’, ‘the symbol you 17 41.46% information, (Linchevski, 1995:113). If pupils are taught understanding of equals put before the answer’, ‘the total sign as operator symbol: of the number’, ‘the amount’, to ‘read’ the equals sign in a multi-directional way from ‘altogether’, ‘makes’, ‘you reveal the answer after you’ve done the the beginning, would this then solve the problem? maths question’, ‘that it equals a number for a sum’. Furthermore, children are usually introduced to ‘equals’ in Pupils unsure of meaning indicated by ‘?’ or blank answer 3 7.31% the context of adding, in the format 1 + 1 = ?. Year 4 42 Answers displaying some It means ‘ the same as’ or 20 47.61% understanding of equals ‘equals’, ‘ equals which means Primary school textbooks and workbooks often reinforce sign as relational symbol: balance’ this format and children become accustomed to ‘equals’ Answers displaying some ‘the total ,‘to sum it up’, ‘equals 21 50% understanding of equals which means answer, so you implying ‘adds up to’. (Baroody and Ginsburg, 1983:200) sign as operator symbol: write the answer there’, ‘It goes at the end of a number Therefore it would seem sensible to conclude that, if sentence and it also means the same as’, ‘It tells you the teachers consider the resources they are using more number after you’ve added all carefully, and introduce opportunities for pupils to view the the numbers’, ‘altogether’,‘ the answer’ or ‘what you put before equals sign flexibly early on, then this would presumably the answer’ Pupils unsure of meaning indicated by ‘?’ or blank answer 1 2.38% counter the idea of children becoming accustomed to one Year 5 43 Answers displaying some ‘the same as’, ‘equals’, 18 41.86% particular way of ‘reading’ the sign. understanding of equals ‘equivalent’ sign as relational symbol: In discussion with secondary colleagues it became Answers displaying some ‘This symbol is the conclusion 24 55.81% understanding of equals which equals your sum up’, clear that this is not just an issue in primary education. sign as operator symbol: ‘the answer to the sum’, ‘equals which means you put Colleagues shared examples of students misusing the it before the answer’, ‘the total equals sign throughout secondary education, even in of something‘, ‘what all the numbers add up to’, ‘when A-level classes. Kieran states that these misconceptions two other numbers are added together this symbol creates a in primary school can lead to a tenuous grasp in new one’, ‘more than the other number’, ‘it means that one secondary school of equivalence, equal value, sameness, number goes to another’. and the use of equations. A common error is using the Pupils unsure of meaning indicated by ‘?’ or blank answer 1 2.32% equals sign to keep a ‘running total’, for example: Year 6 35 Answers displaying some ‘the same as’, ‘equals’ 28 80% understanding of equals sign 1063 + 217 = 1280 – 425 = 1063.