Investigation of eighth-grade students' understanding of the slope of the linear function

Abstract

This study aimed to investigate eighth-grade students' difficulties and misconceptions and their performance of translation between the different representation modes related to the slope of linear functions. The participants were 115 Turkish eighth-grade students in a city in the eastern part of the Black Sea region of Turkey. Data was collected with an instrument consisting of seven written questions and a semi-structured interview protocol conducted with six students. Students' responses to questions were categorized and scored. Quantitative data was analyzed using the SPSS 17.0 statistical packet program with cross tables and one-way ANOVA. Qualitative data obtained from interviews was analyzed using descriptive analytical techniques. It was found that students' performance in articulating the slope of the linear function using its algebraic representation form was higher than their performance in using transformation between graphical and algebraic representation forms. It was also determined that some of them had difficulties and misunderstood linear function equations, graphs, and slopes and could not comprehend the connection between slope and the x-and y-intercepts. BIRGIN, O. diferentes modos de representar a inclinação de funções lineares. Os participantes foram 115 estudantes de oitava série de uma cidade situada na parte leste da região do Mar Negro, na Turquia. Os dados foram coletados por meio de questionários escritos e complementados com um protocolo de entrevista semi-estruturada realizada com seis alunos. Os dados quantitativos foram analisados estatisticamente e para os qualitativos optou-se por uma análise descritiva. Verificou-se que o desempenho dos alunos em articular a inclinação da função linear usando sua forma de representação algébrica foi maior do que o desempenho nas questões relativas à articulação entre as formas de representação gráfica e algébrica. Outras dificuldades foram detectadas, como as relativas ao conceito de função linear e gráfico de funções; e a falta de entendimento sobre a conexão entre a inclinação e os pontos de interseção com os eixos.

Linear Functions and Slope: How Do Students Understand These Concepts and How Does Reasoning Support Their Understanding?/Linearne funkcije i nagib pravca: kako učenici shvaćaju te pojmove i kako razmišljanje djeluje na njihovo razumijevanje?

“…In short, these results suggest that our students could not understand the meaning of symbols in line equations, but they merely memorized formulas without developing a deeper understanding of concepts. The presence of similar results in the literature (Anderson, 2008;Barr, 1981;Birgin, 2012;Moschkovich, 1996;Stump, 1999;Teuscher & Reys, 2010) unfortunately points to the continuous presence of difficulties experienced with these concepts.…”

“…For example, in a study in which seventh grade students in primary school assessed their ability to solve questions about graphs of linear equations, it was observed that students were constrained to plot graphs of linear equations that intersect axes, to construct value tables, and to show coordinates of a given point in a Cartesian coordinate plane (Tekay & Doğan, 2015). In addition to this, in another study on understanding the slopes of linear functions of primary eighth grade students, Birgin (2012) determined that the performance of the students in expressing slope in the result of transformation between algebraic and graphical representations is lower than the performance of slope expressions using only algebraic representation. In this study, it has been seen that some students have difficulties with equations, graphs and slopes of linear functions and that they have misconceptions.…”

In Pursuit of a Connected Way of Knowing: The Case of One Mathematics Teacher

“…Barr (1981), Birgin (2012) and Stump (2001) have documented students' difficulties considering slope as a ratio. This difficulty is more apparent when, in a given equation of the form y = mx + b, m is an integer (Moschkovich, 1996).…”

“…Stump (2001) found that a group of high school precalculus students demonstrated understanding of slope in functional situations but many of them had difficulty interpreting slope as a rate of change. Students find linear functions, and slope as one of its defining attributes, quite difficult to understand; they have particular trouble identifying connections between different representations of slope (see Birgin, 2012;Stump, 2001). Research on teachers' knowledge for the teaching of slope is scarce; however, the studies of Stump (1997) and Zaslavsky, Sela & Leron (2002) provide important insights on the topic.…”

Dificultades para Codificar, Relacionar y Categorizar Problemas Verbales Algebraicos: dos estudios con estudiantes de secundaria y profesores en formación

“…Tal es su importancia, que la búsqueda y activación de problemas relacionados con el propuesto ha sido considerada por algunos autores como una fase diferenciada del resto, por considerar que en toda resolución hay que activar conocimiento previo en forma de esquemas resolutivos previamente elaborados (QUEIROZ;LINS, 2011) El propósito del presente trabajo es estudiar cómo diferentes características definitorias de los problemas verbales, afectan a su codificación y a las analogías entre ellos percibidas por estudiantes de distinto nivel académico. Consideraremos problemas cuya resolución exige el uso de estructuras algebraicas correspondientes a dos funciones lineales, que tienen múltiples aplicaciones en el mundo real y desempeñan un papel relevante en la comprensión de temas matemáticos y científicos (BIRGIN, 2012). ISSN 1980-4415 DOI: http://dx.doi.org/10.1590/1980 Transferir significa aplicar el conocimiento aprendido en determinadas situaciones y contextos, a nuevas situaciones o contextos diferentes.…”