On Mathematical Conjectures and Counterexamples

This article provides an overview of the limitations of checking out a few cases to prove conjectures in mathematics. To that end, I present a purposeful collection of number-theoretic conjectures where extensive checking of cases has found counterexamples, with emphasis on the historical backgrounds. Historical examples of long-term attempts to prove or disprove such conjectures could help individuals to realize more deeply that a limited number of observations does not guarantee the correctness of a conjecture, even though there may be many examples in its favor

Students’ Equation Understanding and Solving in Iran

The purpose of the present article is to investigate how 15 year old Iranian students interpret the concept of equation, its solution, and studying the relation between the students’ equation understanding and solving. Data from two equation-solving exercises are reported. Data analysis shows that there is a significant relationship between understanding and solving equation. The results indicate that students’ understanding of equation has, basically, been shaped by their experiences in arithmetic and students usually have not any perception of equations and real world problems. Moreover, the study shows that students rarely paid any attention to the equality sign and the use of operators in both sides of the equation

Modélisation de la relation entre les composantes de connaissances et les compétences d’élèves pour l’apprentissage de l’expression algébrique dans les écoles secondaires à l’aide de la méthode AHM

The purpose of this study is to apply the Attribute Hierarchy Method (AHM) in the cognitive domains of algebraic expressions to find cognitive inferences about students’ mathematical problem-solving skills. Initially, cognitive content techniques were developed to determine the knowledge and skills needed to solve mathematical assignments. Then, items were written specifically to assess skills in cognitive models. Finally, confirmatory psychometric analyses were used to evaluate students' response information by estimating the proportionality of the data model, attribute probabilities to report the diagnostic score and attribute validity. The first domain is concerned with the cognition and diagnosis of general polynomials and algebraic expressions and encompasses other areas. Therefore, the focus is on the precise definition of the basic concepts of the recognition of polynomials such as the polynomials and the number of very important terms and similar monomials and incorrect learning of algebra. Nevertheless, in the second domain, which focuses on simplification and related concepts, less emphasis has been placed on the seventh to ninth grades. The defect in the expression and practice of this field leads to weakness in solving and analyzing relevant mathematical problems. The third domain is related to the second domain and directly to the first domain. Factorization and distributive properties are often used without considering the rules of simplification by students. The weakness associated with the second domain causes the students not to be able to easily analyze and solve the problem in difficult polynomials in which the rules do not apply easily.L’objectif principal de cette étude est d’appliquer la méthode de hiérarchie d'attributs dans les domaines cognitifs d’expressions algébriques pour trouver d’inférences cognitives concernant les compétences d’élèves en résolution de problèmes mathématiques. Initialement, des techniques de contenu cognitif ont été développées pour déterminer les connaissances et les compétences nécessaires pour résoudre des problèmes de mathématiques. » Ensuite, l’on a écrit d’articles spécifiquement pour évaluer les compétences en modèles cognitifs. Enfin, des analyses psychométriques de confirmation ont été utilisées pour évaluer les informations sur la réponse d’étudiants en estimant la proportionnalité du modèle de données, les probabilités d’attributs pour rendre compte du résultat diagnostique et la validité d’attributs. » Le premier domaine concerne la connaissance et le diagnostic des polynômes généraux et des termes algébriques, et comprend d'autres domaines. Conséquemment, l’accent est mis sur la définition précise de concepts de base de la reconnaissance des polynômes tels que les polynômes et le nombre de termes très importants et de monômes similaires et l’apprentissage incorrect de l’algèbre. Cependant, dans le deuxième domaine, qui met l'accent sur simplification et concepts connexes, moins d'attention a été accordée aux septième à neuvième années. Le défaut dans l'expression et dans la pratique de ce domaine entraîne une faiblesse dans résolution et analyse de problèmes mathématiques pertinents. Le troisième domaine est lié au deuxième domaine et directement au premier domaine. La factorisation et les propriétés distributives sont souvent utilisées sans tenir compte des règles de simplification appliquées par les étudiants. La faiblesse du deuxième domaine empêche les étudiants d'analyser et de résoudre facilement le problème dans des polynômes difficiles dans lesquels les règles ne s'appliquent pas facilement

A Study of the Relationship of Academic Achievement and Impulsivity in Children with Specific Learning Disabilities to Parenting Strain and Sense of Competence in their Mothers

The purpose of this research was to study the relationship of academic achievement and impulsivity in children with learning disorders with parenting strain and self-competence of their mothers. A sample of 70 students with learning disorders along with their mothers was recruited for the study. Data were collected using the UPPS Impulsive Behavior Scale, a modifi ed version of Zarit Burden Interview, Parent Sense of Competence Scale and each student’s grade point average as an index of academic achievement. Data were analyzed using Pearson’s correlation coeffi cients and multiple regression analysis. Findings revealed that academic achievement correlates signifi cantly and positively with mothers’ sense of competence but negatively with their experience of parenting strain. Furthermore, children’s impulsivity was negatively associated with mothers’ sense of competence but showed no relationship to parenting strain. Regression analysis indicated that while children’s impulsivity predicted mothers’ sense of competence but not their parenting strain, children’s academic achievement predicted both. Findings imply that when children with learning disorders are highly impulsive, their mothers only report an incapability to cope with stress. But when children’s academic achievement is low, mothers tend to not only experience increasing parenting stress but also report feeling overwhelmed by the stress. Findings imply that features of children with learning disorders have a differential impact on the mental health of their mothers

Students’ Equation Understanding and Solving in Iran

The purpose of the present article is to investigate how 15 year old Iranian students interpret the concept of equation, its solution, and studying the relation between the students’ equation understanding and solving. Data from two equation-solving exercises are reported. Data analysis shows that there is a significant relationship between understanding and solving equation. The results indicate that students’ understanding of equation has, basically, been shaped by their experiences in arithmetic and students usually have not any perception of equations and real world problems. Moreover, the study shows that students rarely paid any attention to the equality sign and the use of operators in both sides of the equation