TEACHING THE SOLVING OF LINEAR EQUATIONS – WHAT IS AT STAKE?

Proceedings of the annual meeting of the Georgia Association of Mathematics Teacher Educators, 2014

Many teachers have trouble transitioning their students between natural recursive thinking about the data and algebraic notation for representing linear functions (Zazkis & Liljedahl, 2002). In this study, we interviewed eighteen middle school students to see how they used prior instruction to think about a geometric pattern and construct its corresponding linear equation. All students were given the same task to complete and were questioned about their thinking during the interview. We found that the recording of pattern recognition plays a substantial part in helping students recognize and write explicit patterns. By having students decompose the total perimeter into how they saw the pattern growing, students were more successful in making the connection to the numeric representation of growth. In addition, they were better able to explain how they set up the equation, and the connection of each part of the equation to the original pattern. As teachers work with their students in developing a conceptual understanding of linear equations, it is critical that students are exposed to geometric patterns. The results of this study will help mathematics teacher educators better prepare teachers to develop their students' develop rich and connected mathematical understanding.

The Journal of Mathematical Behavior, 2012

In this paper we compare how three teachers, one from each of Finland, Flanders and Hungary, introduce linear equations to grade 8 students. Five successive lessons were videotaped and analysed qualitatively to determine how teachers, each of whom was defined against local criteria as effective, addressed various literature-derived equations-related problems. The analyses showed all four sequences passing through four phases that we have called definition, activation, exposition and consolidation. However, within each phase were similarities and differences. For example, all three constructed their exposition around algebraic equations and, in so doing, addressed concerns relating to students' procedural perspectives on the equals sign. All three teachers invoked the balance as an embodiment for teaching solution strategies to algebraic equations, confident that the failure of intuitive strategies necessitated a didactical intervention. Major differences lay in the extent to which the balance was sustained and teachers' variable use of realistic word problems.

Pythagoras, 2019

Concerns have been expressed that although learners may solve linear equations correctly they cannot draw on mathematically valid resources to explain their solutions or use their strategies in unfamiliar situations. This article provides a detailed qualitative analysis of the thinking of 15 Grade 8 and Grade 9 learners as they talk about their solutions to linear equations in interviews. The article stems from a study that describes whether learners use mathematically endorsable narratives to explain and justify their solutions. Sfard’s theory of commognition is used to develop a framework for analysis of their discourse. The findings show that all learners use ritualised rather than explorative discourse, characterised by applying strict rules to operations with disobjectified entities. The only mathematical objects they produce endorsed narratives about are positive integers. Thus they do not meet the relevant curriculum requirements. Nevertheless, the analytic tools – adapted from Sfard specifically for the study of linear equations – give a particularly nuanced account of differences in the learners’ ritualised discourse. For example, some learners used endorsed narratives about negative integers, algebraic terms and the structure of an equation when prompted by the interviewer. There is not sufficient evidence to suggest that any learners are in transition to explorative discourse. However, the article shows that learner discourse is a rich resource for teachers to understand the extent to which learners are thinking exploratively, and offers suggestions for how their thinking can be shifted. This is an opportunity for teacher professional development and further research.

Pythagoras

Concerns have been expressed that although learners may solve linear equations correctly they cannot draw on mathematically valid resources to explain their solutions or use their strategies in unfamiliar situations.This article provides a detailed qualitative analysis of the thinking of 15 Grade 8 and Grade 9 learners as they talk about their solutions to linear equations in interviews. The article stems from a study that describes whether learners use mathematically endorsable narratives to explain and justify their solutions. Sfard’s theory of commognition is used to develop a framework for analysis of their discourse.The findings show that all learners use ritualised rather than explorative discourse, characterised by applying strict rules to operations with disobjectified entities. The only mathematical objects they produce endorsed narratives about are positive integers. Thus they do not meet the relevant curriculum requirements. Nevertheless, the analytic tools – adapted from...

Journal of Educational and Social Research, 2022

The study aims to examine the impact of using real life situation in solving linear equations by seventh graders. In order to achieve the designated aim of the study, the study was conducted in one of the Arab schools in Israel. A sample of 20 average students was deliberately chosen depending on their educational achievement. Two approached were employed within the study; the qualitative approach and its quantitative counterpart. Results of the study clearly exhibit that student have undergone three stages: in the first stage, a development in the concept of “similar terms” was noticed. The second stage showed a development in the concept of “quantity comparison.” In the final stage of the study, the students became familiar with the concept of parity in the linear equations. In light of the researchers’ findings, the study could be concluded with some important recommendations and suggestions. Namely, to review the mathematics curriculum for the middle school students and recreati...

International Journal of Trends in Mathematics Education Research

The objective of this study is to enhance students’ understanding of solving linear equation with one variable through teaching using balancing model. To achieve this research objective, design-based research approach was chosen. The target population of the study was grade five students and their mathematics teacher. From this population, the participants of the study were grade five section A students and their respective mathematics teacher who were selected using simple random sampling and comprehensive sampling techniques, respectively. The data were collected through observation, teacher made tests and interview. The test results were analyzed using paired samples t-test and percentage, and the interview and classroom observation data were analyzed through thematic description. Findings showed that most students performed better in the post-test as compared to the pre-test, and most students came to use more flexible strategies to solve linear equation after a series of learni...

In K. Krainer, N. Vvondrová (Eds.), Proceedings of the Ninth Congress of the European Society for Research in Mathematics Education (pp. 419-425). Prague: Charles University., 2015

The goal of this paper is to contribute to the research on the introduction of solving linear equations. Subsumed in the “Comparing and Contrasting” category introduced in Prediger, Bikner-Ahsbahs and Arzarello’s (2008) networking strategies, we contrast two episodes informed by two distinct theories and offer an insight into the teacher’s role in introducing new knowledge in the classroom and the meaning-making narratives of hands-on didactic approaches to algebra. We examine the teachers’ gestures and hints and what appears to be unsayable in the teacher-students’ interaction.

Universal Journal of Educational Research, 2019

This article aims to describe the actions taken by the teacher to bring out students' thoughts so that students are able to make strategies in solving mathematical problems. Data obtained from direct observation of the research subject when giving actions to students. The subject of the study was the mathematics teacher who was chosen based on special characteristics that fit the purpose of the study. In this study, there are 2 different characteristics when the teacher gives action to students so that students are able to make strategies in solving problems with the story. First, the teacher gives four actions as follows: (1) the teacher asks students to rewrite the problem using their own words, write down what is known and what is asked in the problem; (2) the teacher asks students to turn problems into symbolic problems; (3) the teacher asks students to create symbolic mathematical problems and use operations and procedures in mathematics appropriately to solve problems; (4) the teacher directs students to check the results of problem solving. Second, the teacher gives the following actions: (1) the teacher asks students to write down what is known and what is asked in the problem; (2) the teacher asks students to turn problems into symbolic problems; (3) the teacher asks students to create symbolic mathematical problems and use operations and procedures in mathematics appropriately to solve problems.

Educational Studies in Mathematics, 2010

This study examines eighth grade students' use of a representational metaphor (cups and tiles) for writing and solving equations in one unknown. Within this study, we focused on the obstacles and difficulties that students experienced when using this metaphor, with particular emphasis on the operations that can be meaningfully represented through this metaphor. We base our analysis within a framework of referential relationships of meanings . Our data consist of videotaped classroom lessons, student interviews, and teacher interviews. Ongoing analyses of these data were conducted during the teaching sequence. A retrospective analysis, using constant comparison methodology, was then undertaken in order to generate a thematic analysis. Our results indicate that addition and (implied) multiplication operations only are the most meaningful with these representational models. Students also very naturally came up with a notation of their own in making sense of equations involving multiplication and addition. However, only one student was able to construct a "family of meanings" when negative quantities were involved. We conclude that quantitative unit coordination and conservation are necessary constructs for overcoming the cognitive dissonance (between the two representations-drawn pictures and the algebraic equation) experienced by students and teacher.

This study examines the views of mathematics tutors on the benefits and challenges of using algebra tiles in solving linear equations with one variable. The qualitative research paradigm was used with a case study design. The sampling technique used was purposive with four mathematics tutors as the sample. The instrument was semi-structured open-ended interview guides. The findings revealed, among others, that the benefits of algebra tiles in solving linear equations with one variable includes (1) helping students to distinguish between coefficients and constant values, (2) the best way for teachers to explain the neutralization of equal positive and negative values. While some of the challenges of algebra tiles are (1) teachers lack the knowledge to use the algebra tiles, (2) non-availability, and inadequacy of algebra tiles. KEYWORDS Algebra tiles, benefits, challenges, college Mathematics teachers RÉSUMÉ Cette étude examine les points de vue des tuteurs en mathématiques sur les avantages et les défis de l'utilisation des tuiles d'algèbre pour résoudre des équations linéaires à une variable. Le paradigme de la recherche qualitative a été utilisé avec une conception d'étude de cas. La technique d'échantillonnage utilisée était intentionnelle avec quatre professeurs de mathématiques comme échantillon. L'instrument utilisé était un guide d'entretien semistructuré à questions ouvertes. Les résultats ont révélé, entre autres, que les avantages des tuiles d'algèbre dans la résolution d'équations linéaires à une variable comprennent (1) l'aide aux étudiants à faire la distinction entre les coefficients et les valeurs constantes, (2) la meilleure façon pour les enseignants d'expliquer la neutralisation des valeurs égales positives et négatives. Certains des défis posés par les tuiles d'algèbre sont (1) le manque de connaissances des enseignants pour utiliser les tuiles d'algèbre, (2) la non-disponibilité et l'inadéquation des tuiles d'algèbre. MOTS-CLÉS Tuiles d'algèbre, avantages, défis, professeurs de mathématiques au collège