The objective of this thesis was to investigate pre-service teachers and student misconceptions of the equal sign, and then offer suggestions to pre-service teachers, teachers, university programs, and schools to prevent common misconceptions from occurring in classrooms. Some students do not realize the equal sign can have two different functions, operational and relational. There are several different reasons for this misconception, beginning with the lack of defining what the equal sign is and what it means in the classroom. In the first study, eighteen participants were interviewed to explain their responses when evaluating student work to gain an in-depth knowledge of pre-service teachers' perceptions of the equal sign and their ability to evaluate a students' response to a specific math task. Results showed that pre-service teachers have a better understanding of the equal sign and may be ready to teach the equal sign as a relationship between numbers. Furthermore, pre-service teachers would benefit greatly from evaluating students' work and looking for common misperceptions that students may have. In the second study, six fifth grade classes were studied to determine if there was a positive relationship for teaching atypical type equivalence statements to students and performing better on equivalence questions. Three classes from Spring 2011, were administered a test; two of the test items were used to analyze their understanding of the equal sign. In Fall 2011, another three fifth grade classes participated in lessons, which required students to analyze atypical type equivalence statements, and then they were given the same two test items. Results from this study supported the use of atypical type equivalence statements because more students in the experimental group correctly responded to the two items and were also able to justify their responses with work that exemplified good understanding of the equal sign as being a relationship. Both of these studies support increasing student and pre-service teachers understanding of the equal sign and the misconceptions students have regarding the equal sign. University programs and schools should utilize these results to require preservice teachers and teachers to evaluate student work to identify common misconceptions and teach the equal sign as a relationship between both sides and not as an operation.

One of the main reasons that students struggle as they transition from arithmetic to algebra in the middle grades is that they fail to develop the appropriate understanding of the equal sign. Previous research has suggested that students need to move past an operational understanding and develop a relational understanding of the equal sign in order to work with algebraic equations successfully. Other research has suggested that the way that we interpret and utilize the equal sign is based on three main factors: multiple meanings of the equal sign, equation types, and structural conventions. This study extends both areas of research by analyzing two middle grade curricula and looking for what meanings, equation types, and structural conventions appear in both teacher and student materials. The study confirms that students are exposed to three main meanings of the equal sign in the middle grades. The study also describes which meanings of the equal sign are associated with particular equation types and the frequency with which these equation types appear throughout the 7th and 8th grade curricula. Study findings can be used to inform instruction, as they delineate the factors that are attended to while making sense of the equal sign in the middle grades.

The main purpose of this study is to investigate eighth-grade students¡¦ understanding of the equal sign and analyzed error types of quadratic equation in one variable. To achieve this purpose, the investigator did a survey and development instruments. Participants were 215 eighth-grade students who formed a convenient sample. There are three results. First, participants with a relational definition of the equal sign added to about 80% of the sample. Second, the performance of students with relational definitions is higher than the performance of students with operational definitions. Third, students¡¦ understanding of the equal sign was related to their respective problem-solving performance on quadratic equation in one variable. In this study, participants with an operational definition of the equal sign tended to guess randomly or leave a blank. Problem-solving performance of participants with a relational definition of the equal sign involved multiple strategies. The researcher suggested that teachers should strengthen students¡¦ understanding of equal sign and related students¡¦ prior algebraic knowledge.

Studiens syfte är att bilda kunskap om elevers förståelse av likhetstecknet och hur undervisning om likhetstecknet i årskurs 3 kan varieras för att bli så effektiv som möjligt. Syftet ska besvaras genom frågeställningarna: vilka variationsmönster som de studerade eleverna kan möta och vilka kritiska aspekter som elever kan identifiera. Kritiska aspekter är det som elever behöver identifiera för att förstå det som ska läras. Variationsmönster innebär att det som är kritiskt i undervisningen ska varieras för att synliggöras. Studien har sin ansats i variationsteorin, där variationsmönster och kritiska aspekter är centrala begrepp. En lektion observerades i årskurs 3 och sex elever valdes ut och intervjuades om likhetstecknet. Resultatet av studien visar att eleverna mötte sex kritiska aspekter i undervisningen. I varje kritisk aspekt mötte eleverna ett eller flera variationsmönster. Studien visar också att alla de intervjuade eleverna har en relationell och instrumentell förståelse av likhetstecknet. Eleverna har förståelse av en kritisk aspekt, som den observerade lektionen inte hade berört, nämligen att alla tal måste ingå i en uppgift för att lösningen ska vara korrekt. Fyra av sex elever har förståelse av den kritiska aspekten att det ska vara ekvivalens i en kedja av likheter. Resultatet visar också att elevernas förståelse av likhetstecknet inte är beroende av att de möter variationsmönster i undervisningen, men att de möter de kritiska aspekterna på något sätt. / The aim of the study is to explore the understanding of the equal sign and how the teaching about the equal sign among third grade students can be varied to be as effective as possible.  The aim will be answered trough the questions: what patterns of variation can the studied students meet and what critical aspects have the students identified. Patterns of variation means that what is critical in the teaching should be varied to become visible. Critical aspects is what students need to identify to understand what should be learned. The foundation of the study is the Variation Theory, where patterns of variation and critical aspects are central concepts. The observations were accomplished during a third grade lesson and six students were selected for interviews about the equal sign. The result of the study shows that the students met six critical aspects during the lesson. For every critical aspect there were one or several patterns of variation that was exposed to the students. The result also states that the students who were interviewed have a relational and instrumental understanding of the equal sign. The students also have understanding of a critical aspect that they did not meet in the observed lesson, namely that all numbers have to enter in a task. Four out of six students have understanding of the critical aspect that there should be equivalence in a chain of similarities. The result also show that the students understanding of the equal sign is not dependent of that they meet patterns of variation in the teaching, but that they meet the critical aspects somehow.

The purpose of this study is to investigate students¡¦ understanding of the equal sign, problem-solving strategies of equations with one unknown, and the strategies of solving equations with one unknown under different understanding types of the equal sign. To achieve this purpose, the investigator did a survey and development instruments. The participants were 203 seventh-grade students in a convenient sample. Descriptive statistics were used to analyze data in frequency and percentages. The main results was that participants with a relational definition of the equal sign were the most (close to 50%), and an operational definition of the equal sign was approximately 1/4. There was a higher successful performance associated with a relational definition than an operational definition. The primary strategy of operations on the left-hand side of equal sign is the mathematical operations; the main strategy of an unknown quantity on the right-hand side of the equal sign was by going to the parenthesis-reverse and bringing different denominators into a common denominator; the principal strategies of one number on the right-hand side of the equal sign, equations with operations on the right side of the equal sign and equations with operations on both sides of the equal sign are cover-up and transposing. To use the strategies of trial and error substitution and undoing is minority in a linear equation with one unknown. The strategy of an operational definition participant in five equal sign topics is similar to the strategy of one with a relational definition. However, those with a relational definition apply multiple strategies and exhibited varying particular and algebraic property. On the other hand, participants with an operational definition used arithmetic strategies more frequently than participants with a relational definition. From the above results, the researcher suggested instruction to include strategies with algebraic property to help learners to develop stable understanding of the equal sign in Algebra. In addition, the recommendation is to have teachers to encourage students to apply multi-dimensional thinking and different strategies in algebraic problem-solving.

Syftet med studien är att studera hur lärare introducerar och arbetar med likhetstecknet för att inkludera alla elever och ge dem förutsättningar i sin undervisning. Studiens urval är fyra verksamma lärare som genom semistrukturerade kvalitativa intervjuer belyser sina tankar, erfarenheter och åsikter om hur de arbetar och introducerar likhetstecknet, samt vilka svårigheter som kan uppstå. Resultatet visar att lärarna i studien har ett entydigt arbetssätt men att det finns faktorer som skiljer sig åt. Att det finns faktorer som skiljer sig åt tydliggörs genom att lärarna introducerar likhetstecknet med bland annat konkret material medan introduktionen av likhetstecknet kan se olika ut, lärarna använder olika arbetsmetoder där en lärare väljer att introducera addition och talkamrater innan likhetstecknet, medan en annan lärare uttrycker att likhetstecknet bör introduceras först. Studiens problem är att forskning och myndigheter visar på att elever missuppfattar likhetstecknets betydelse. Den slutsats vi kan dra utifrån syfte, problem och resultat är att lärare behöver anpassa undervisningen för att ge eleverna goda kunskaper för att kunna bemästra matematiken. / The purpose of the study is to observe how teachers introduce and work with an equal sign in order to include all students. This is to give them the prerequisites in their teaching. The study sample involves four active teachers who, through semi-structured qualitative interviews, shed light on their thoughts, experiences, and opinions on how teachers work and introduce the sign of equality as well as what difficulties may arise. The results show that the teachers involved in the study have an unambiguous way of working, however, there are factors that differ. This is shown by the teachers introducing the equals sign with, among other things, concrete material while the introduction of the equals sign can look different. The teachers use different working methods where one teacher chooses to introduce additions and speech mates before the equal sign, while another teacher expresses that the equal sign should be introduced first. The problem with the study is that the research and authorities show that students misunderstand the meaning of the equal sign. The conclusion we can draw based on purpose, problems, and results is that teachers need to adapt the teaching to give students sufficient knowledge to be able to master mathematics.

Den här studien har som syfte att bidra med kunskap om elevers förståelse av likhetstecknet. Studien besvarar två frågeställningar: På vilka kvalitativt olika sätt uppfattar elever likhetstecknet? Vilka nyanser av statisk och dynamisk uppfattning av likhetstecknet framträder av elevernas uppfattningar? Datan består av semistrukturerade intervjuer med 14 elever som har genomförts i par. Studien är begränsad till årskurs 1–3. Studien visar att elever kan ha olika uppfattningar av likhetstecknet både en statisk och dynamisk. Däremot visar studien också att ”statiskt” och ”dynamiskt” inte är tillräckligt för att beskriva hur elever uppfattar likhetstecknet. De nyanser av den statiska och dynamiska uppfattningen som presenteras i resultatet är: likhetstecknet indikerar ett svar, likhetstecknet som ett ”stopp”, likhetstecknet indikerarar att det måste finnas en uträkning och likhetstecknet indikerarar att det måste vara jämvikt. / The aim of this study is to contribute with knowledge about students' understanding of mathematical similarities. Two questions are answered: In what qualitative different ways do students perceive the equal sign? What nuances of static and dynamic perception of the equal sign emerge from students´ perceptions? Data are based on, semi-structured interviews with students 14 students. The interviews have been conducted in pairs. The study is limited to grades 1-3.  The study show that students have different perceptions of the equal sign both static and dynamic. The results also show that “static” and “dynamic” are not sufficient to describe details in students perception of the equal sign The nuances of the static and dynamic perception presented in the result are: the equal sign indicates an answer, the equal sign as a "stop", the equal sign indicates that there must be a calculation, and the equal sign indicates that they must be equilibrium.

Syftet med studien är att fördjupa kunskapen om hur eleverna introduceras till likhetstecknet och vilka svårigheter som ligger till grund för att eleverna brister i sin förståelse kring likhetstecknet. Studien utgår från två forskningsfrågor. I studien används en kvalitativ ansats i form av semistrukturerade intervjuer. I resultatet framkommer det att eleverna i viss mån introduceras till symbolerna för större än och mindre än samtidigt, dock är det få som även får kunskap om olikhetstecknet. Svårigheterna hos eleverna är att de utgår från en operationell förståelse för tecknet, att ett resultat ska räknas fram. Denna svårighet beror på bristande förståelse för likhetstecknet vilket kan påverka eleverna när de introduceras för ekvationer. Vår slutsats är att för att motverka detta bör lärarna arbeta med att utveckla den relationella förståelsen genom varierande räknehändelser och bör även introducera alla relevanta symboler samtidigt som större än, mindre än och olikhetstecknet samt undvika symboler för de olika räknesätt under denna introduktion för att undvika att eleverna skapar en operationell förståelse. / The purpose of the study is to deepen the knowledge about how students are introduced to the equal sign and what difficulties underlie the students' lack of understanding of the equal sign. The study is based on two research questions. In the study uses a qualitative approach in the form of semi-structured interviews. In the result appears that the students are to some extent introduced to the symbols for greater than and less than at the same time, however, few also gain knowledge of the sign of dissimilarity. The difficulty of the students is that they start from a operational understanding of the sign, that a result should be calculated. This difficulty is due to a lack of understanding of the equal sign, which can affect students when introduced to equations. Our conclusion is that to counteract this, teachers should work on developing the relational understanding through varying counting events and should also introduce all relevant symbols while being greater than, less than and the sign of difference and avoiding symbols for the different counting methods during this introduction to avoid students creating an operational understanding.

Syftet med denna empiriska studie är att undersöka hur lärare kan erbjuda möjligheter för 6-åringar att vidga sin förståelse av likhetstecknets innebörd. Undersökningen är inspirerad av Learning Study metoden vilket är en metod som har sin utgångspunkt i variationsteorin. Barnens initiala förmågor analyseras före planeringen av undervisningen och det ger en uppfattning om barnens kunskaper. Utifrån det planeras lektionerna och de aspekter som är de mest kritiska för barnen lyfts och på så sätt utvecklas deras kunskap. Vår slutsats är att det är viktigt att arbeta utifrån ett variationsinriktat lärandeperspektiv. Barnen lyckas bra i de uppgifter där de får laborera med konkret material. / The purpose of this empirical study is to examine how teachers can provide opportunities for children aged 6 to extend their understanding of the equal sign. This study is inspired by the Learning Study method which is a method that has its origins in variation theory.  By analyzing the children's initial abilities before planning our lessons, it gives us an idea of the children's knowledge. From there the lessons are planned and the aspects that are the most critical for the children are lifted and thereby develop their knowledge. Our conclusion is that it is important to work with variety oriented learning. The children are successful in tasks in which they are allowed to experiment with concrete materials.

Syftet med den här studien är att belysa faktorer som möjliggör eller hindrar förståelsen av likhetstecknet hos elever och dess övergång mellan aritmetik och algebra. Med en systematisk litteraturstudie som metod söktes vetenskapliga artiklar som sammanställdes i fem olika kategorier. Svårigheter i tidig algebra, matematikens språk, relationell/instrumentell förståelse samt relationella tolkningar och till sist lärarperspektivet. Resultatet från den systematiska litteraturstudien bekräftade att elever inte har en relationell förståelse av likhetstecknet. Artiklarna visade framför allt att traditionell aritmetikundervisning hindrar elevers utveckling i algebraiskt tänkande. Uppgifter med operationer skrivna i vänsterled visade sig stärka elevers instrumentella förståelse av likhetstecknet. Ett instrumentellt och relationellt språk (både skriftligt och verbalt) framträdde också som en avgörande faktor för elevers förståelse. Utifrån artiklarnas resultat undersöktes vilka faktorer av likhetstecknet som elever får möjlighet att öva i digitala spel med hjälp av en innehållsanalys. Det visade sig att de digitala spelen i den här studien inte är utformade för att stödja elevers relationella förståelse av likhetstecknets betydelse utan fortsätter att stärka den instrumentella förståelsen hos elever. / The purpose of this study is to illustrate factors concerning the concept of the equal sign and its impact on the transition from arithmetic to algebra. Using a systematic literature study as a method, scientific articles were compiled into five different categories. Difficulties linked to early algebra, The language of Mathematics, Relational/instrumental understanding as well as relational interpretations and finally the Teacher perspective. Our systematic literature review confirmed that students do not have a relational understanding of the equal sign. Indeed, it highlighted that traditional arithmetic teaching methods tended to hinder student development in algebraic thinking through tasks, such as operations being on the left side of the equal sign and the answer as an outcome on the right side of the equal sign. This has been shown to strengthen students' instrumental understanding of the sign rather than dismantling it. Further, instrumental, and relational language (both written and verbal) proved to be decisive factors in students’ learning. Based on the results of our scientific articles, we undertook a content analysis of digital resources and examined whether the equal sign strengthens students instrumental or relational understanding of the equal sign. It turned out that the digital games analysed in this study are not designed to support students’ relational understanding of the equal sign but continues to strengthen their instrumental understanding.