Irit Peled

Inhibiting Factors in Generating Examples by Mathematics Teachers and Student Teachers: The Case of Binary Operation.

The main objectives of this study were to identify difficulties encountered by mathematics teachers and student teachers associated with the concept of binary operation regarding the associative and commutative properties and to reveal possible sources for them. Thirty-six in-service mathematics teachers and 67 preservice mathematics teachers participated in the study. All participants were presented with a task calling for the generation of a counterexample, namely, a binary operation that is commutative but not associative. Responses to the task were analyzed according to four categories: correctness, productiveness, mathematical content, and underlying difficulties. The findings point to similarities and differences between the two groups. Both groups exhibited a weak concept by failing to produce a correct example and by using a limited content search-space. These findings suggest two main inhibiting factors: one related to the overgeneralization of the properties of basic binary operations and the other related to pseudo-similarities attributed to these properties, which seem to be created by the recurring theme of order. Teachers were superior to student teachers on the categories of correctness and productiveness. This study is part of a larger study the aim of which was to investigate ways in which mathematics teachers and student teachers generate counterexamples in mathematics. Two main goals of the study were, first, to identify difficulties encountered in generating examples related to a number of mathematical topics that teachers and student teachers are expected to teach and, second, to reveal possible sources of difficulty for them. The possible sources of difficulty in generating such examples were presumed to include the following: incomplete knowledge, inability to process existing knowledge, misconceptions, and insufficient logical knowledge. The case of binary operation represents a case in which the relevant knowledge is assumed to be available to secondary mathematics teachers and student teachers; however, that knowledge requires processing in order to produce such examples. The current work is designed to probe for factors inhibiting the processing involved in generating counterexamples, factors that in turn may shed light on the limited binary operation concept. The search for these limiting factors was carried out by an extensive analysis of the processes and difficulties involved in generating the required examples. As indicated by Bratina (1986), the task of generating an example is considered powerful in terms of revealing strengths and weaknesses. The notion of a binary operation is dealt with in various stages and courses at different levels and contexts throughout the mathematics education that prospective mathematics teachers experience. However, in many cases prospective teachers do not manage to integrate their various encounters into one comprehensive, abstract

Shifts in reasoning

The process of transition from a novices state to that of an expert, in the constrained domain of decimals, is described in terms of explicit, intermediate, and transitional rules which are consistent, yet erroneous. These rules can be traced to former rules already established in earlier knowledge domains. Empirical data from children at grades 6, 7, 8 and 9 will demonstrate the evolution of an experts knowledge through an elaborated learning path.

Journey to the Past: Verifying and Modifying the Conceptual Sources of Decimal Fraction Knowledge

The present study follows earlier research on fraction unit conception in learning decimals and shows that children with this difficulty also exhibit a similar problem in fractions. Specifically, the focus on the fraction unit observed in decimals (e.g., evaluating 0.4 as larger than 0.68 because tenths are larger than hundredths) is preceded by a similar deficient coordination between the size of the fraction unit and the number of units in determining fraction value. Approximately three quarters of 59 students in seventh and eighth grades identified as having a fraction unit conception in decimals were also identified as having a similar fraction unit focus in fractions. The impact of fraction knowledge was further demonstrated by showing that “changing the past” through meaningful instruction in fractions affected the “present” and transferred to decimals.RésuméCette étude s’inscrit dans la lignée d’études précédentes sur la conception des fractions dans l’apprentissage des décimales et montre que les enfants qui présentent des difficultés dans ce domaine ont également des difficultés du même ordre avec les fractions non décimales. Plus précisément, les difficultés centrées sur les fractions décimales (par exemple, le fait d’estimer que 0,4 est supérieur à 0,68 parce les dixièmes sont plus grands que les centièmes) sont précédées d’une déficience de coordination similaire entre le dénominateur d’une fraction et le nombre d’unités dans la détermination de la valeur de cette fraction. Environ les trois quarts des 59 étudiants de 7e et 8e années chez qui on a observé ce type de problèmes avec les décimales avaient également des difficultés similaires pour ce qui est des fractions non décimales. Nous avons aussi confirmé l’impact des savoirs préalables concernant les fractions en démontrant que le fait de «revoir le passé» grâce à un enseignement sérieux des fractions affecte le «présent» de fac¸on positive et permet un transfert des connaissances sur les décimales.

(Fish) Food for Thought: Authority Shifts in the Interaction between Mathematics and Reality

This theoretical paper explores the decision-making process involved in modelling and mathematizing situations during problem solving. Specifically, it focuses on the authority behind these choices (i.e., what or who determines the chosen mathematical models). We show that different types of situations involve different sources of authority, thereby creating different degrees of freedom for the problem solver engaged in the modelling process. It also means that mathematics plays different roles in these problems and situations. This epistemological analysis on the meaning of modelling implies that we should reconsider the mathematical status of realistic solutions and raises questions on the validity of some traditional choices of mathematical models and their use in diagnosing children’s conceptions. It also suggests constructing modelling tasks by choosing a certain variety of situations that might lead to a better understanding of the roles of mathematics.

Difficulties in know ledge integration: revisiting Zeno's paradox with irrational numbers

The study investigates sources of difficulties exhibited by student teachers in tasks involving the construction of an irrational length segment, and other irrational number tasks. The results show that student teachers know the definitions and characteristics of irrational numbers, yet fail in tasks that require a flexible use of their knowledge and in tasks that involve making connections between different representations. Thus, for example, students say that irrational numbers are real numbers, yet many think they have noplace on the real number line. Their explanations indicate that misconceptions about the limit concept, that relate to the dilemma in one of Zenos paradoxes, are a main source of difficulty. These findings stress the importance of creating tasks that facilitate the integration of different knowledge pieces.

Using Modeling Tasks to Facilitate the Development of Percentages

This study analyzes the development of percentages knowledge by seventh graders given a sequence of activities starting with a realistic modeling task, in which students were expected to create a model that would facilitate the reinvention of percentages. In the first two activities, students constructed their own pricing model using fractions and then extended the model while experiencing reinvention and extension of their knowledge of percentages. In the last two activities, they coped with a realistic changing reference situation. A control group used a traditional instructional unit. A pretest and two posttests showed between-group differences.RésuméCette étude analyse l’acquisition de la connaissance des pourcentages par des élèves de 7e année qui doivent effectuer une suite d’activités débutant par une tache de modélisation réaliste dans laquelle ils doivent créer un modèle qui faciliterait la réinvention des pourcentages. Au cours des deux premières activités, les élèves ont construit leur propre modèle de tarification à l’aide de fractions, puis ils l’ont étendu au fur et à mesure qu’ils faisaient l’expérience de la réinvention et de la progression de leurs connaissances en matière de pourcentages. Dans les deux dernières activités, ils ont dû gérer un changement de situation de référence réaliste. Un groupe de contrôle a utilisé, pour sa part, une unité d’instructions traditionnelles. Un test préliminaire et deux tests postérieurs ont mis en évidence les différences entre les deux groupes.

Emergence of tables as first-graders cope with modelling tasks

In this action research, first-graders were challenged to cope with a sequence of modelling tasks involving an analysis of given situations and choices of mathematical tools. In the course of the sequence, they underwent a change in the nature of their problem-solving processes and developed modelling competencies. Moreover, during the task sequence, the first-graders spontaneously discovered the power of organizing problem data in a table. They did not merely use their existing mathematical knowledge, but also ‘reinvented’ tables as a new mathematical tool. This paper describes the gradual development of this tool as the children moved along the task sequence. Notably, the first-graders exhibited this progress in spite of having relatively little mathematical knowledge.

Investigating new curricular goals: what develops when first graders solve modelling tasks?

ABSTRACT Children in a first-grade class who were taking their first steps in formal mathematical knowledge were given a sequence of five modelling tasks during their regular mathematics lessons. These tasks were different in nature from the problems they encountered in their textbooks. They are more complex and challenging, requiring analysis of the problem situation and decisions about the mathematics that could be used. The nature and implementation of the tasks involved new curricular goals and new problem solving norms aimed at developing children’s modelling competencies. The findings show that the children were able to accept the new norms that promoted situation analysis and realistic considerations. They increased their use of argumentation, utilised existing additive strategies and displayed some use of multiplicative structures which the children had not encountered formally yet.

Commentary on Chapters 8 to 10: Teachers’ Knowledge and Flexibility—Understanding the Roles of Didactical Models and Word Problems in Teaching Integer Operations

The crucial role of teachers in introducing integers to children is highlighted in chapters 8– 10, comprising this section. The three chapters discuss (prospective) teachers’ conceptions of integer equations, of children’s thinking about integer expressions, and of the role of some didactical models used in teaching integer addition and subtraction. These different aspects of teacher knowledge and conceptions draw an important picture of characteristics and issues that should be taken into account by teacher educators in preparing teachers for teaching integers. In the first part of our commentary we highlight the main contributions of each of the chapters, focusing on the central findings and on important issues brought up by each chapter. The second part offers a meta-perspective of some of the issues by discussing more general educational implications. In this part we also take the opportunity to express our own insights emerging and associated with the ideas presented in the three chapters.

Using Variation of Multiplicity in Highlighting Critical Aspects of Multiple Solution Tasks and Modeling Tasks

We urge the reader to solve these problems before reading the paper. Then we suggest that the reader asks himself whether there is a different route or a different analysis of the given situation that leads to a different solution to any of the problems.