Solving a Diophantine problem using different expressions of the difference of two squares

Guided discovery learning can be included in the teaching methods supporting independent and active learning. This paper presents ways how to create and implement guided discovery lessons focused on the investigation of some properties of natural numbers. The subject of divisibility can offer many interesting problems which the teacher may use to develop students' mathematical competencies and the creative thinking. The key elements of the paper are the discovery and proof of the relationship for the sum of the first consecutive odd natural numbers and its application to the investigation of the differences of powers of natural numbers. Numerical and graphical spreadsheet tools are used to increase the students´ interest and speed up calculations in certain stages of learning

Remediation of first-year mathematics students' algebra difficulties.

Thesis (M.Sc.)-University of KwaZulu-Natal, Pietermaritzburg, 2009.The pass rate of first-year university mathematics students at the University of KwaZulu-Natal (Pietermaritzburg Campus) has been low for many years. One cause may be weak algebra skills. At the time of this study, revision of high school algebra was not part of the major first year mathematics course. This study set out to investigate if it would be worthwhile to spend tutorial time on basic algebra when there is already an overcrowded calculus syllabus, or if students refresh their algebra skills sufficiently as they study first year mathematics. Since it was expected that remediation of algebra skills would be found to be worthwhile, two other questions were also investigated: Which remediation strategy is best? Which errors are the hardest to remediate? Five tutorial groups for Math 130 were randomly assigned one of four remediation strategies, or no remediation. Three variations of using cognitive conflict to change students’ misconceptions were used, as well as the strategy of practice. Pre- and post-tests in the form of multiple choice questionnaires with spaces for free responses were analysed. Comparisons between the remediated and non-remediated groups were made based on pre- and post-test results and Math 130 results. The most persistent errors were determined using an 8-category error classification developed for this purpose. The best improvement from pre- to post-test was 12.1% for the group remediated with cognitive conflict over 5 weeks with explanations from the tutor. Drill and practice gave the next-best improvement of 8.1%, followed by self-guided cognitive conflict over 5 weeks (7.8% improvement). A once-off intervention using cognitive conflict gave a 5.9% improvement. The group with no remediation improved by 2.3%. The results showed that the use of tutorintensive interventions more than doubled the improvement between pre-and post-tests but even after remediation, the highest group average was 80%, an unsatisfactory level for basic skills. The three most persistent errors were those involving technical or careless errors, errors from over-generalising and errors from applying a distorted algorithm, definition or theorem

Albanian upper secondary students´ ways of working with equations : a case study based on task-based interviews

Equations are an important part in algebra in the Albanian school mathematics curriculum. This case study focuses on the way six Albanian students approach and solve equations. The students that took part were in the first year of upper secondary school (10th school year), and they have chosen to have more mathematics than the knowledge required. Students are motivated to work with mathematics, and their mathematics background is approximately the same. They came from different lower secondary schools but the programs that they have followed are the same. The research question that has directed this research is: How do the Albanian students in the study approach and solve equations? I had in focus a class with 34 students who firstly did a test dealing with equations. According to their performance on the test, I divided the students into three groups: high performing, middle performing, and low performing. I have picked two students from each group: one girl and one boy, and I have interviewed them. The method that I have used to interview the students is the task-based interview where the questions are based on the five requests of Newman‟s technique. The students were presented with four cards, where they had to choose two of them: the most difficult and the easiest, and to solve one of them. The cards contain two equations of these kinds: linear, quadratic, rational, and irrational. On the cards was also a word problem which was the same for all the cards. During the interview, I have made questions that are related with their reasoning in the solution process, and many times I have also tried to give hints to them. The interviews have lasted for approximately 45 minutes, where some of the time was used to ask some additional tasks, but I have analysed only the part where the students have solved tasks on the cards. The students in general showed that they knew how to solve equations, even if they lack some parts of the complete solution. They have shown that the word problems (simple ones) are not a problem for them, and the translating from word problem to equation is not a difficult for them. They have demonstrated to have a general knowledge on equations, even if sometimes this knowledge seems to be more of an instrumental understanding than a relational understanding. The students have shown difficulties in dealing with domain and checking part, because for some of them these are not seen as part of the solution of an equation. They have also shown some difficulties with the quadratic equations, and the main difficulty that the students have shown is the solution of an irrational equation. Most of the students have said that irrational equations are more difficult than the other kinds of equation or they have shown lack of understanding during their solutions

Engaging multiple representations in grade eight: exploring mathematics teachers' perspectives and instructional practices in Canada and Nigeria

This study was inspired by and utilises representations, one of the mathematical learning processes (NCTM, 2000), which is currently acclaimed as one of the reform-based instructional approaches to teaching and learning algebra. This concurrent mixed methods research project explored elementary in-service teachers’ goals for, beliefs about and knowledge of representations, both in Ontario and Lagos. Data were collected through an online survey completed by 91 middle school in-service teachers concurrently with interviews with ten of them. Findings from the survey indicated that teachers from the Lagos subsample had weaker understandings about representations compared with their counterparts from Ontario. In the interviews, participants described to varying degrees their goals for and use of representations as opportunities for students to show connections, relationships, and reasoning, supporting students’ confidence in problem-solving, and facilitation and opportunities for questioning and discussion. This research suggests that teachers generally, but particularly in Lagos, need a deeper understanding of representations and need to further develop the specialized mathematics content knowledge related to patterning and algebra. Other findings showed that: planning and sequencing instruction, use of contextual learning tasks, opportunities for students to generate their own representations, linking students’ prior knowledge to new situations, and translation among multiple representations were reported as critical to teachers’ use of representations. Recommendations are made to create more awareness among teachers, of the value, use and knowledge about representations. These findings would be relevant to school boards, teacher educators, researchers, and professional development providers wishing to improve teachers’ use of representations, via enhanced beliefs, and knowledge

La enseñanza de la resolución algebraica de problemas en el entorno de la hoja de cálculo

Nuestra investigación pretendía dar respuesta a: 1) ¿Cuáles son las actuaciones de los estudiantes cuando resuelven problemas en la hoja de cálculo después de haber sido instruidos en la resolución algebraica de problemas en dicho entorno? 2) ¿Cómo influye la enseñanza de la resolución algebraica de problemas en la hoja de cálculo en la competencia de los estudiantes cuando resuelven problemas verbales con lápiz y papel y, en especial, mediante el método cartesiano? Sobre el armazón de los Modelos Teóricos Locales construimos un marco teórico y metodológico que tenía por intención: A) Describir las características de la hoja de cálculo prestando especial atención a la sintaxis y la semántica de su lenguaje para poder compararlo con el lenguaje del álgebra. B) Determinar qué se considera una resolución algebraica competente en la hoja de cálculo (en adelante, MHC) y establecer similitudes y diferencias con la resolución mediante el método cartesiano (en adelante, MC). C) Diseñar una secuencia de enseñanza sobre la resolución algebraica de problemas verbales en la hoja de cálculo. D) Establecer un método que nos permitiera determinar lo próxima que una resolución se hallaba de la que realizaría un sujeto competente. La población de nuestro estudio experimental estaba formada por un grupo natural de 24 estudiantes de segundo curso de ESO. Para dar respuesta a la primera pregunta de investigación, analizamos la actuación de cinco parejas de estudiantes resolviendo seis problemas típicamente algebraicos tras haber sido instruidos en la resolución de problemas mediante el MHC. Las actuaciones observadas se puede pueden agrupar en: la tendencia a evitar el uso del MHC y en las dificultades y errores a la hora de usarlo. Dentro de la tendencia a evitar el uso del MHC podemos distinguir: 1) El uso de cantidades variables en lugar de cantidades determinadas que se manifiesta en el uso de lo que hemos llamado líneas de vida (donde se sustituye la operación con la cantidad desconocida tiempo transcurrido por el cálculo de la edad el año siguiente de manera recursiva) cuando se resuelven los problemas de edades y en la modelización de un posible proceso que une la situación descrita en el enunciado con otra situación hipotética en la que todas las cantidades son conocidas o se pueden calcular a partir de éstas. 2) El recurso a la aritmética. 3) El recurso al MC de manera verbal. Entre el catálogo de dificultades y errores al usar el MHC destacamos: la dificultad para operar con lo desconocido; la dificultad para invertir las relaciones obtenidas tras la lectura analítica; la falta de atención a las restricciones del problema para centrarse en la verificación de la ecuación; la tendencia a seguir calculando hasta "cerrar los cálculos" y la necesidad de que la igualdad se construya sobre dos expresiones de una cantidad conocida. Para dar respuesta a la segunda pregunta de investigación, comparamos las actuaciones de los estudiantes al resolver dos cuestionarios formados por ocho problemas isomorfos típicamente algebraicos: uno administrado previamente a la instrucción y el otro, al acabarla. El análisis de los resultados nos lleva a concluir que tras la enseñanza del MHC se incrementa el número de lecturas algebraicas, pero disminuye el uso del lenguaje del álgebra, produciéndose un aumento significativo del uso de valores provisionales para las cantidades desconocidas. También se observa que disminuye significativamente la competencia de los estudiantes al afrontar de manera algebraica problemas de la subfamilia edades, pero que aumenta en el resto de problemas; lo que parece plausible atribuir a las estrategias espontáneas, correctas o incorrectas, que usan los estudiantes en la hoja de cálculo cuando resuelven los problemas de edades.Our research aimed to give an answer to: 1) Which are the student performances when trying to solve word problems on the spreadsheet after having been instructed in an algebraic problem solving method we call Spreadsheet Method (SM)? 2) How teaching SM has influence on student competence when they solve problems on paper and pencil, and, especially when they use the Cartesian Method (CM)? Our experimental study population consisted of a natural group of twenty-four secondary students (13-14 years old). The observed performances when trying to solve a problem on the spreadsheet are classified either according to the tendency to avoid using the SM or according to the difficulties and mistakes when using it. Regarding the tendency to avoid using the SM we can distinguish: 1) Using variable quantities instead of determined quantities. 2) Resorting to an arithmetical solving procedure. 3) Resorting to CM in a verbal way. Regarding the difficulties and mistakes we can distinguish: difficulties dealing with the unknown; difficulties reversing the connections obtained after an analytical reading; lack of attention to the problem restrictions in order to focus on the equation verification; the tendency to go on calculating until “closing the calculation”, and the need to build the equation using two expressions of a known quantity. Analysis findings lead us to conclude that after teaching SM, when students solve word problems using paper and pencil, the number of algebraic readings increases, but the use of algebraic language decreases, and a significant rise in the use of provisional values for the unknown quantities occurs. It is also worthwhile to notice that the student competence to face age problems in an algebraic way decreases significantly, but this algebraic competence increases when dealing with problems from other problem types