Procedural embodiment and magic in linear equations

How do students think about algebra? Here we consider a theoretical framework which builds from natural human functioning in terms of embodiment – perceiving the world, acting on it and reflecting on the effect of the actions – to shift to the use of symbolism to solve linear equations. In the main, the students involved in this study do not encapsulate algebraic expressions from process to object, they do not solve ‘evaluation equations’ such as by ‘undoing’ the operations on the left, they do not find such equations easier to solve than , and they do not use general principles of ‘do the same thing to both sides.’ Instead they build their own ways of working based on the embodied actions they perform on the symbols, mentally picking them up and moving them around, with the added ‘magic’ of rules such as ‘change sides, change signs.’ We consider the need for a theoretical framework that includes both embodiment and process-object encapsulation of symbolism and the need for communication of theoretical insights to address the practical problems of teachers and students

Problems that the Students Face while Solving 1st Degree Equations with Two Unknown, During their Prepare to the High School Entrance Exam (SBS)

AbstractThe purpose of the research is identified and manufactures a solution to the problems of the students who attended the High- School Exam (SBS) while solving the 1st Degree Equations with two unknown. Sample of the research is 30 students from 2 different schools in 8th grade of the 2011-2012 academic years in Fahir Ilkel Primary School and Inönü Primary school. The study used both qualitative and quantitative methods. To gather the results of the research, three verbal and three numerical, in total 6 classical methods, questions are given. The data was analyzed by researchers jointly. After analyzing the results we discovered that some of the problems the students face are; calculation errors, students cannot remember the subjects efficiently, lack of information about the related subject, ability to solve test method rather than classical method questions and to establish the relationship between them because of summarize without understanding the conceptual and operational knowledge

A Systematic Analysis of Errors in the Simplification of a Rational Expression

Exploring the errors that mathematics students frequently make is a means by which teachers can gain a better understanding of students’ difficulties. Reported here are the process by which the algebraic working of 95 undergraduate students who incorrectly simplified a rational expression was analysed and the results of the analysis. Initially, a deductive approach to analysing the errors was planned, categorising students’ mistakes using the error types identified, named and described in the literature. In reviewing the literature, however, it became clear that this would be no simple task. The large body of literature, while rich in examples of “typical errors” that could be expected in students’ working, had two limitations. Firstly, the error types lacked precise descriptions and were mainly described by example only. Secondly, insufficient details of the procedures used to categorise the errors prevented replication of the categorising process. Consequently, a mainly inductive approach, that categorised the errors by their location and inferred student operation was devised. This systematic approach resulted in generating descriptions of three error categories

Diagnosing student errors in e-Assessment questions

© The Author 2015. We demonstrate how the re-marker and reporter facility of the DEWIS e-Assessment system facilitates the capture, analysis and reporting of student errors using two case studies: logarithms and indices for first-year computing students at the University of the West of England, and Sturm-Liouville problems for second-year mathematics students at Leeds University. The differences in approach needed for error capture for commonly used numerical or algebraic answer inputs are discussed and shown to facilitate efficient capture and reporting of student errors. Not only does such information provide away to tailor question feedback to address these errors for use by future students, but can be made available to current students by re-marking their answers using the newly identified errors and hencemaking the improved feedback available to them too

Persistent and Pernicious Errors in Algebraic Problem Solving

Students hold many misconceptions as they transition from arithmetic to algebraic thinking, and these misconceptions can hinder their performance and learning in the subject. To identify the errors in Algebra I which are most persistent and pernicious in terms of predicting student difficulty on standardized test items, the present study assessed algebraic misconceptions using an in-depth error analysis on algebra students’ problem solving efforts at different points in the school year. Results indicate that different types of errors become more prominent with different content at different points in the year, and that there are certain types of errors that, when made during different levels of content are indicative of math achievement difficulties. Recommendations for the necessity and timing of intervention on particular errors are discussed

Dialogic Teaching Model For Ninth Class Students To Conceptualize Inequalities

It is known that difficulties are often experienced in conceptual learning of mathematics, which is an abstract lesson. For this reason, it is difficult for students to conceptually learn inequalities, one of the difficult subjects of mathematics. The aim of this study is to investigate the effect of dialogic teaching to overcome the general mistakes and difficulties of 9th grade students in deepening the conceptual teaching of inequalities. This study was designed as an action research. The answers and solutions given to 7 open-ended questions prepared to determine students’ misconceptions and mistakes were scored between 0 and 2 points. When a detailed analysis of solutions written by the students was done, it was determined that the students had difficulty in establishing the concept of numbers, that they ignored the real numbers in a defined range and only focused on integers, that they ignored zero when finding the square of the inequality in a defined range, and that they had difficulty in understanding the principle of reversing when the inequality was multiplied by a negative number and also had difficulty in the solution of inequalities when two inequalities were combined into a single inequality. According to the results of the research, dialogic teaching played a supporting role for the students to reach the conceptual learning of inequalities. It was also seen that high school students were able to reconstruct the concept of inequality conceptually in the learning process. Keywords: dialogic teaching, inequalities, conceptual teaching, reconstructin

Equações quadráticas e a fórmula de Bhaskara: sucesso garantido?

Neste artigo, apresentamos uma análise do trabalho de 77 alunos de ensino médio com uma situação não-familiar: a solução de uma equação quadrática escrita na forma fatorada. Os dados são analisados à luz de um quadro teórico que considera três diferentes mundos da Matemática e a influência dos “já-encontrados” derivados de experiências anteriores. Concluímos que ter a fórmula de Bhaskara como o único já-encontrado para resolver equações quadráticas pode não ajudar os alunos a trabalharem situações que as envolvem. Conjecturamos que o aluno deve se envolver com situações relacionadas a pelo menos dois mundos, o corporificado e o simbólico, mas de formas que também permitam considerar características do mundo formal, sem o qual alunos podem criar suas próprias técnicas inapropriadas

Algebraic errors in the productions of university students from Costa Rica and Mexico

En este trabajo se analizan comparativamente los errores frecuentes que cometen los estudiantes universitarios, al realizar tareas algebraicas. Se aplicó un cuestionario centrado en los niveles de entendimiento del uso de las letras y también se abordó desde los enfoques del álgebra. Este estudio se realiza desde el enfoque cuantitativo con carácter diagnóstico y descriptivo. La muestra está conformada por 54 estudiantes de Costa Rica y México. Como resultado se obtuvo que la mayoría de los estudiantes cometen errores de cálculo o tienen un aprendizaje deficiente, estos recurren a procedimientos aritméticos y se evidencia incapacidad de establecer modelos matemáticos.In this work, the frequent errors committed by university students when performing algebraic tasks are comparatively analyzed. A questionnaire focused on the levels of understanding of the use of letters was applied and it was also approached from the algebra approaches. This study is carried out from the quantitative approach with diagnostic and descriptive character. The sample is made up of 54 students from Costa Rica and Mexico. As a result, it was found that most students make calculation errors or have poor learning, they resort to arithmetic procedures and there is evidence of an inability to establish mathematical models