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. (Kieran, 1981:318-320) as relational symbol: This is something I have often observed in my own Answers displaying some ‘The symbol you put just 7 20% understanding of equals sign before the answer’, ‘the total practice. Could this sort of ‘error’ simply be a part of a as operator symbol: of the sum’, ‘the summary of the sum’. child’s experimentation and development? Or is it crucial to correct this error early on? Mann (2004) emphasises I was interested to note that pupils gave a wide range the need for early intervention: of interpretations, particularly in lower Key Stage 2. By An understanding of the concept of equality is vital Year 6 the range of interpretations had decreased and to successful algebraic thinking and is one of the big the majority of pupils were able to identify the symbol ideas of algebra about which students should reason. as meaning ‘the same as’, or ‘equals’. This may be The concept of balance, or equivalence, is the basis due to the fact that Year 6 had received some explicit for the comprehension of equations and inequalities teaching on the equals sign some months prior to this (Greenes and Findell, 1999). Exposing students to investigation. Across the school, many pupils were this important algebraic concept in the lower grades able to relate the symbol in some way to the idea of is essential to develop an understanding of equality. equivalence or balance; however, a substantial number (NCTM 2000) of pupils interpreted it as an ‘operator’ symbol. I would also query the pupils’ understanding of the word ‘equals’. So, a new question emerged: What opportunities Even though some pupils identified the symbol to mean for algebraic thinking can be included in the primary ‘equals’, when they elaborated and stated that ‘equals mathematics curriculum in order to enable pupils to gain a means answer’ further misconceptions were highlighted. broader understanding of the equals sign? The key impression I gained from the investigation was the lack of consistency across the school with regard to Curriculum overview for ‘Algebraic thinking: symbolic interpretation, and the need for more explicit equations – Year 1 to Year 6’ teaching on the equals sign. My overview is based on the ‘Progression map for How do these misconceptions occur and why is it algebra’, taken from the National Centre for Excellence important to address them in primary school? in the Teaching of Mathematics (NCETM) website. The original progression map is based solely on the new Once I had established that many pupils at my school national curriculum objectives and divided into three do indeed follow the misconceptions outlined by Kieran, sections: equations, formulae, and sequences. I have

44 July 2015 www.atm.org.uk It’s the symbol you put before the answer

focused on the ‘equations section’ of the progression Numeracy Planning Sheet – Equations Investigation Year group: 5/6 map and have amended it to include more opportunities Mental starter/ Key objectives Teacher Input Differentiated Steps to respond to marking for main input activities success for algebraic thinking and explicit teaching of the equals including adult sign. Venenciano and Dougherty (2014) outlined the support Maths attitude How to formulate Whole class input: Bring Mixed ability Select the educational priorities that support algebraic thinking, dance to get ready algebraic whiteboards to carpet pair work: required number for investigation equations of rods work! Start with quick review of Work with your primarily referring to the Davydov curriculum, in their Context: ‘Cuisenaire’ notation and key partner on the Form two trains ‘Balance Challenge’ Finding all the vocabulary. Key vocabulary: sugar paper of equal length recent article. The following outlines a clear starting starter: possibilities Equation, formulate, systematic, provided. Cuisenaire rod, Cuisenaire ‘train’. Write the Pupils are presented Key questions: Use these equation point for algebraic thinking with regard to the concept of with a series of On ‘Cuisenaire Environment’ Cuisenaire Rods: weights and balances. What do you know 2 reds, 2 light Check your The activity about..? http://nrich.maths.org/content/ greens, 1 purple equation has the equivalence, which I have referenced in my overview to id/4348/cuisenaire.swf same values on is designed to How are -- and 1 yellow. stimulate discussion each side of the support teaching in Year 1, and Year 2. related? Model building a ‘train’ using 3 rods. Start by equals sign and highlight Then build another ‘train’ of equal the concept of Do you agree? choosing three length and place it underneath. Ask of the rods to Repeat above equivalence needed Why? pupils to describe how the trains steps until Davydov (1975b) advocated that instruction begin to build equations form 2 trains of How many are related. equal length. you think have later in the lesson. found all the equations can On whiteboards guide pupils to with comparisons of physical attributes of objects Pupils answer key you formulate Write as many possibilities question: What do write an equation to match the equations as you using..? TOP TIPS: and collections. Consideration of this approach, you know about the ‘train’, using correct Cuisenaire can to describe notation. weights of the_____ Once you found how the two Work as mentioned above, attends to the way in which and _____? Pupils an equation, could E.g. w + g + p = y + g. Rearrange trains are systematically feedback on findings you change it rods (interchange and reverse) and related. to whole class. to come up with ask children to write an equation Record your children make sense of their everyday world. These another one? Extension: findings carefully Differentiation: which reflects new arrangement. How many comparisons can be described without using numbers How? E.g. y + g = w + g + p and clearly Non-secure prior equations can Do you think Ask pupils what ‘steps to success’ you formulate Record any learning: Scales A you have found patterns – shorter, longer, heavier, lighter, more than, less + B + C they need to formulate an using 3 rods? all the possible equation. KEY POINT – CORRECT How many you spot or than, and equal to – and represented in relational Secure prior equations? Why? USE OF EQUALS SIGN. The equations can observations you make learning: Scales C How did you work values on either side of the you formulate statements, like G > L, that use letters to represent + D + E systematically? equals sign must be the same. using 4 rods? Test your How many Steps to success: Quickly go through investigation results – are you the quantities being compared. Fundamentally, and children get on. equations can sure you have Read the information you formulate Plenary: In pairs pupil explain found all the and questions next using 5 rods? students realize that if G > L, then it is also the case their findings to the class. possibilities? to each balance Encourage and model use of Pupils to keep an How do you carefully. Write reasoning starters. eye on: know? that L < G. First graders can write these relational any workings out/ observations on I think this because… statements and understand what they mean, because your whiteboard. Discuss the balances If this is true then… the statements describe the results of physical actions with your partner. I know that the next one is… Use this type of because… that the children themselves have performed in the language: I think this because…If this is The pattern looks like… true then… comparison process. (Venenciano & Dougherty, Because…then I think… Table 2 2014:20) My school investigation showed that some pupils found An overview of algebraic thinking: equations – Year 1 writing these relational statements difficult, so giving them to Year 6 opportunities not only to make physical comparisons Note: Any content in black print is the original content but also to write the equivalent relational statements taken from NCETM ‘Progression map for algebra: will hopefully counteract this difficulty. The Measure Up equations’ and refers to the new national curriculum. Any programme, based on the Davydov curriculum, follows content in red print has been added by me with original this problem-solving format: sources referenced in brackets. See Table 3 Children were able to solve problems with concurrent ALGEBRAIC THINKING: EQUATIONS representations: (a) a physical model (e.g. two Year 1 Year 2 Year 3 Year 4 Year 5 Year 6 amounts of liquid), (b) an intermediate model Give pupils Prior to teaching Prior to teaching Prior to teaching Prior to teaching Prior to teaching opportunities carry out quick carry out quick carry out quick carry out quick carry out quick (e.g. paper strips or a line segment diagram to to make diagnostic to diagnostic to diagnostic to diagnostic to diagnostic to comparisons using highlight any highlight any highlight any highlight any highlight any represent the comparison of the quantities), and (c) everyday objects misconceptions misconceptions misconceptions misconceptions misconceptions – shorter, longer, and review and review and review and review and review symbolization (e.g. algebraic-looking statements of heavier, lighter, if necessary if necessary if necessary if necessary if necessary more than, less (Fischbein, Kieran, (Fischbein, Kieran, (Fischbein, Kieran, (Fischbein, Kieran, (Fischbein, Kieran, than, and equal Ginsburg) Ginsburg) Ginsburg) Ginsburg) Ginsburg) equality or inequality). (Venenciano & Dougherty, to – and write using relational Give pupils Teach pupils a Teach pupils a Use Give pupils 2014:22) statements e.g. G opportunities flexible approach flexible approach manipulatives, opportunities to > L (Venenciano, to make to reading the to reading the such as complete tasks Dougherty) comparisons using equals sign e.g. 1 equals sign e.g. 1 Cuisenaire, which focus on I have included elements of this problem-solving format everyday objects + 1 = 2, 2 = 1 + 1, + 1 = 2, 2 = 1 + 1, to formulate both numbers and Ensure any – shorter, longer, 2 = 2 , 2 + 3 = 3 + 2 = 2 , 2 + 3 = 3 + equations. letters, rather than ‘missing number heavier, lighter, 2 (Linchevski) 2 (Linchevski) (Watson) on numbers alone. in lesson plans developed for Year 5/6 (see example problems’ or more than, less (Kieran) similar include than, and equal Give pupils Give pupils Give pupils plan). Pupils use cuisenaire rods as the ‘physical model’ explicit teaching to – and write opportunities to opportunities to opportunities to Give pupils on meaning of using relational complete tasks complete tasks complete tasks opportunities to to build trains of equivalent length, then draw their own the equals sign. statements e.g. G which focus on which focus on which focus on complete tasks (Kieran, Baroody > L (Venenciano, operations as well operations as well both numbers and which focus on ‘intermediate model’ on sugar paper and finally use & Ginsburg, Dougherty) as their inverses, as their inverses, letters, rather than both representing Mann) and on the related and on the related on numbers alone. and solving a Ensure any idea of doing/ idea of doing/ (Kieran) problem rather ‘symbolization’ such as y + g = w + g + p to express Solve one-step ‘missing number undoing. (Kieran) undoing. (Kieran) than on merely problems that problems’ or Give pupils solving it. (Kieran) equality. The following example lesson plan (using the involve addition similar include Give pupils Give pupils opportunities to and subtraction, explicit teaching opportunities to opportunities to complete tasks Use standard school planning format) which I developed for using concrete on meaning of complete tasks complete tasks which focus on manipulatives, objects and the equals sign. which focus on which focus on both representing such as Y 5/6 puts this idea into practice. It purposefully includes pictorial (Kieran, Baroody both representing both representing and solving a Cuisenaire, representations, & Ginsburg, and solving a and solving a problem rather to formulate problem rather problem rather than on merely equations and missing Mann) a ‘low threshold, high ceiling’ (McClure, NRich, 2014) than on merely than on merely solving it. (Kieran) (Watson) number Recognise and solving it. (Kieran) solving it. (Kieran) finding-all-the-possibilities-style investigation so that it problems Use the properties Express missing such as use the inverse relationship Use manipulatives Use manipulatives of rectangles to number problems can be adapted easily for other year groups. See Table 2. 7 =  - 9 (copied between and balances to and balances to deduce related algebraically. from Addition and addition and develop concepts develop concepts facts and find (NC) Subtraction, NC) subtraction and of equals, of equals, missing lengths equivalence and equivalence and and angles. Find pairs of use this to check numbers that Represent and calculations and balance. (Watson, balance. (Watson, (copied from use number Mann) Mann) Geometry: satisfy number missing number bonds and related Properties of sentences problems.(copied Solve problems, involving two subtraction facts from Addition and Shapes, NC) within 20 (copied including missing unknowns. (NC) Subtraction, NC) number problems, from Addition and Enumerate all Subtraction, NC) Recall and use using number facts, place possibilities of addition and combinations of subtraction facts value, and more complex addition two variables. to 20 fluently, and (NC) derive and use and subtraction. related facts up (copied from to 100 (copied Addition and from Addition and Subtraction, NC) Subtraction, NC) Solve problems, including missing number problems, involving multiplication and division, including integer scaling. (copied from Multiplication and Division, NC) Table 3

July 2015 www.atm.org.uk 45 It’s the symbol you put before the answer

Example resources Some pupils had started to formulate predictions on how many possibilities there would be for seven rods, eight These ‘Balance challenge’ starter activities used in rods etc., moving towards making a statement. the example Year 5/6 lesson plan were adapted from Rebecca Mann’s article ‘Balancing act: the truth behind Conclusion the equals sign’ (2004) See Figure 1 below. At the beginning of this article I stated that pupils at my school do not have a consistent understanding of the equals sign. I propose that creating more opportunities for algebraic thinking and explicit teaching on the equals sign will begin to address this issue, and enable pupils to gain a broader understanding. Perhaps some questions for us all to consider are: • How can we begin to develop a more consistent understanding of the equals sign? • What are the key issues and misconceptions in regards to the equals sign in our setting? • What opportunities for algebraic thinking can be included in our curriculum in order to enable pupils to gain a broader understanding of The interactive ‘Cuisenaire environment’ used in the the equals sign? example lesson can be found on the NRich website. I believe it would also be invaluable to simply See Figure 2 below. take a moment to reflect on how you present the equals sign to your pupils and how you yourself ‘read’ the equals sign, and whether or not the resources you use inadvertently cause misconceptions.

Laura Swithinbank teaches at St John’s CE VC Primary School, Clifton, Bristol.

References Baroody, A. Ginsburg, H. (1983) The effects of instruction on children’s understanding of the ‘equals’ sign. The Example lesson outcomes Elementary School Journal, 84, No. 2, 198-212. Pupils used Cuisenaire rods as the ‘physical model’ to Fischbein, E. (1989) Tacit models and build trains of equivalent length to start off with. The mathematical reasoning. lesson outcomes below demonstrate how pupils then For the Learning of Mathematics, 9, No. 2, 9-14. drew their own ‘intermediate models’ on sugar paper and Kieran, C. (1981) Concepts associated with the equality finally used symbolization‘ ’ such as y = r + g to express symbol. Educational Studies in Mathematics, 12, 317-326. equality. As part of the steps to success for this lesson I had reminded pupils to: work systematically; record their Kieran, C. (2004) Algebraic thinking in the early grades: findings carefully and clearly; record any patterns they what is it? spot, and to test their results. These examples show the The Mathematics Educator, 8, No.1, 139-151. variety of ways in which pupils attempted to structure Linchevski, L. (1995) Algebra with numbers and arithmetic their work systematically. Some have started to identify with letters: a definition of pre-algebra, Journal of patterns and recorded their findings in such a way that Mathematical Behaviour, 14, 113-120. has enabled them to test their results. Pupils were asked to share their findings with the class at the end of the Mann, R. (2004) Balancing act: the truth behind the lesson. equals sign. Teaching Children Mathematics, September, 2004, 65-69. Venenciano, L. and Dougherty, B. (2014) Addressing priorities for elementary school mathematics. For the Learning of Mathematics,34 , No. 1, 18-23. Websites McClure, L. (2014) Using low threshold high ceiling tasks in ordinary primary classrooms (www.nrich.maths.org.uk) ‘Progression Map for Algebra’, National Curriculum 2014. (www.ncetm.org.uk)

46 July 2015 www.atm.org.uk The attached document has been downloaded or otherwise acquired from the website of the Association of Teachers of Mathematics (ATM) at www.atm.org.uk Legitimate uses of this document include printing of one copy for personal use, reasonable duplication for academic and educational purposes. It may not be used for any other purpose in any way that may be deleterious to the work, aims, principles or ends of ATM. Neither the original electronic or digital version nor this paper version, no matter by whom or in what form it is reproduced, may be re-published, transmitted electronically or digitally, projected or otherwise used outside the above standard copyright permissions. The electronic or digital version may not be uploaded to a website or other server. Any copies of this document MUST be accompanied by a copy of this page in its entirety. If you want to reproduce this document beyond the restricted permissions here, then application must be made for express permission to [email protected]

ATM is a not for profit professional teaching association. The majority of funding used to produce and prepare the MT journal is procured through our membership subscriptions.

Mathematics Teaching does not seek to conform to an ‘official’ view on the teaching of mathematics, whatever that may be. The editorial board wishes to encourage contributors to express their personal views on the teaching and learning of mathematics.

ATM is an association of teachers in which everyone has a contribution to make, experiences and insights to share. Whether practical, political, philosophical or speculative, we are looking for articles which reflect on the practice of teaching mathematics. We aim to publish articles that will be of interest to the breadth of our membership, from the Foundation Stage to Higher and Further Education; as well as a balance between those derived from research and from practical experience. Submitted articles are accepted for publication based on their clarity, topicality, the extent to which they reflect upon knowledge and understanding of mathematics teaching and learning, and their contribution to inspiring further development and research.

Join ATM at any time and receive twelve months of membership, including instant access to member discounts and resources. Spread the cost and pay in ten monthly instalments. Membership Includes:

 Five copies of the ATM journal Mathematics Teaching (MT)  A 25% discount on all shop items  Considerable discounts at the hugely popular annual ATM conference  Electronic access to thousands of online MT journal articles  Access to all online member-only resources  Professional support and enrichment – being part of a community where ideas are generated and shared  Regular ATM e-newsletters, containing current news and activities  A network of local branches offering regular meetings  Accreditation - ATM is proud to offer members the opportunity to apply for the CMathTeach Designation, making ATM membership the route to Charted Mathematics Teaching status  Influence and having a voice - eligibility to vote on resolutions that shape the direction of ATM