Development Of Mathematical Knowledge Research Articles

La relación: matemática-semiosis-argumentación, en la elaboración de diseños didácticos

Este artículo presenta resultados de investigación sobre las relaciones entre los componentes curriculares, cognitivos, comunicativos, matemáticos, que se requieren para los diseños didácticos que buscan la formulación de realciones geométricas como la pitagórica. El diseño investigativo general se estructut´o desde la ingeniería didáctica.Se presentan aquí los análisis a priori y la posteriori de uno de los talleres del diseño didáctico; y en la dimensión matemática, el desarrollo semiótico-argumentativo. De manera que los resultados proporcionan un marco interpretativo de la interacción entre procesos didácticos y desarrollo de conocimiento matemático en un contexto escolar.

El ánalisis de tareas para el desarrollo de competencias argumentativas en geometría

Este artículo presenta resultados de investigación desarrollo de lenguaje y desarrollo de conocimiento matemático, desde una perspectiva argumentativa. La propuesta parte de la hipótesis fundamnetal de que el desarrollo de un argumento matemático depende de una ralación básica entre la lengua natural como sitema semiótico como algebraico y el figural que se privilegian en los razonamientos matemáticos en la elaboración de conceptos matemáticos como los geométricos. Desde este punto de vista, en el presente estudio está el interés por comprender, describir, analizar y cualificar los proceso de aula, tanto en la dimensión de la enseñanza como en la del aprendizaje, en los campos particulares del lenguaje y las matemáticas. Así, en este documento se presenta, como resultado de la investigación, una estructura de análisis a priori como el a posteriori, que se realiza desde el modelo de la ingeniería didáctica (Guy Brousseau, 1998, 2004 y Régine Douady, 1984, entre otros).El diseño de tareas se encaminó al desarrollo de competencias argumentativas en geometría, en un grupo de estudiantes de 10 grado de educación media. Se realiza una estructuración especifíca para el análisis de una secuencia de actividades orientadas al desarrollo de procesos de razonamiento (como el argumentativo) y su relación geométrica:la relación pitagórica.

Understanding Early Mathematical Competencies in American and Chinese Children

Most cross-cultural research on Chinese and American children’s early mathematical competencies has focused on their understanding of number and number operations. The present study broadened the range of tasks assessed to include geometric shapes, problem solving and logical reasoning, as well as number and numerical operations, in an effort to determine if: (a) there are within- and cross-culture differences in the development of mathematical knowledge in all domains and (b) the rate/order of development of the four different mathematics skills are comparable across cultures? One- hundred-and-sixty Chinese and US first grade students participated in this study one month after they entered first grade. The results indicated that Chinese and US children’s mathematical knowledge in the domain of numbers and operations is better developed than their mathematical knowledge in the other three domains, Chinese children outperformed American children on almost all tasks in all of the domains and the level of difficulty of mathematics concepts, across culture, seems to be universal (e.g. children’s conceptual knowledge within each of these domains seems to develop in the same order across cultures). The theoretical and educational implications of these results are discussed.

Mathematical modelling in the early school years

In this article we explore young children’s development of mathematical knowledge and reasoning processes as they worked two modelling problems (the Butter Beans Problem and the Airplane Problem). The problems involve authentic situations that need to be interpreted and described in mathematical ways. Both problems include tables of data, together with background information containing specific criteria to be considered in the solution process. Four classes of 3rd-graders (8 years of age) and their teachers participated in the 6-month program, which included preparatory modelling activities along with professional development for the teachers. In discussing our findings we address: (a) Ways in which the children applied their informal, personal knowledge to the problems; (b) How the children interpreted the tables of data, including difficulties they experienced; (c) How the children operated on the data, including aggregating and comparing data, and looking for trends and patterns; (d) How the children developed important mathematical ideas; (e) Ways in which the children represented their mathematical understandings.

How Informal Out-of-School Mathematics Can Help Students Make Sense of Formal In-School Mathematics: The Case of Multiplying by Decimal Numbers

This article presents a teaching experiment on the relationship between informal out-of-school and formal in-school mathematics, and the ways each can inform the other in the development of abstract mathematical knowledge. This study concerns the understanding of some aspects of the multiplicative structure of decimal numbers. It involved a series of classroom activities in upper elementary school, using suitable cultural artifacts and interactive teaching methods. To create a substantially modified teaching/learning environment, new sociomathematical norms (Yackel & Cobb, 1996) were also introduced. The focus was on fostering a mindful approach toward realistic mathematical modeling, which is both real-world based and quantitatively constrained sense-making (Reusser & Stebler, 1997). In addition, procedures not commonly used in ordinary teaching activities, such as estimation and approximation processes, were also introduced.

Students' Coordination of Knowledge When Learning toModel Physical Situations

In this article, I present a study in which 12 pairs of 8th-grade students solved problems about a physical device with algebra. The device, called a winch, instantiates motions that can be modeled by pairs of simultaneous linear functions. The following question motivated the study: How can students generate algebraic models without direct instruction from more experienced others? The first main result of the study is that students have and can use criteria for judging when 1 algebraic expression is better than another. Thus, students can use criteria to regulate their problem-solving activity. The second main result is that constructing knowledge for modeling with algebra can require students to coordinate criteria for algebraic representations with several other types of knowledge that I also identify in the article. These results contribute to research on students' algebraic modeling, cognitive processes and knowledge structures for using mathematical representations, and the development of mathematical knowledge.

Contributions and Obstacles of Contexts in the Development of Mathematical Knowledge

A framework for discussing (through the concepts of ‘model’ and ‘field of experience’) the nature of the relationships between contexts and the formation of mathematical knowledge is proposed, and some educational, cultural and cognitive problems of situated teaching-learning are tackled within it. This framework is supported and illustrated with references to the curricular innovation and educational research carried out by the Genoa group, of which the authors are members.

Call for Manuscripts on Professional Development

The Editorial Panel of Mathematics Teaching in the Middle School is committed to the ongoing professional growth of educators, both as teachers of students and as students of mathematics. Professional growth is multifaceted, including the development of mathematical knowledge; growth in knowledge about teaching, learning, and curriculum; and the development of self-assessment strategies.

Concept mapping in research on mathematical knowledge development: Background, methods, findings and conclusions

The theoretical background and different methods ofconcept mapping for use in teaching and in research on learning processes are discussed. Two mathematical projects, one on fractions and one on geometry, are presented in more detail. In the first one special characteristics of concept maps were elaborated. In the second one concept mapping allowed students' individual understanding to be monitored over time and provided information about students' conceptual understanding that would not have been obtained using other methods. Regarding the students' individual concept maps in more detail there were some additional findings: (i) The characteristics of the maps change remarkably from fourth grade to sixth grade; (ii) There is some evidence that prior knowledge related to some mathematical topics plays a very important role in students' learning behaviour and in their achievement; (iii) Concept maps provide information about how individual students relate concepts to form organised conceptual frameworks; (iv) Long-term difficulties with specific concepts are able to be traced. These findings are discussed with regard to results of other studies as well as to their implications for the teaching of mathematics in the classroom.

The stability of children's mathematical behavior: When is a bug really a bug?

This article describes an empirical investigation of the extent to which incorrect arithmetic algorithms persist over time. The use of subtraction procedures by children aged 9–11 was charted over time periods of 1–3 months. The ways in which several bugs might be related to a single misconception were considered and a set of “bug classes” was identified. The results indicated that the prevailing tendency to attribute stable procedural knowledge and systematic behavior to children of this age is misguided. They provide some insight into the development of mathematical knowledge under the influence of conventional instruction.

More about mathematical paradoxes in preservice teacher education

This is a third report on a naturalistic study in Israel of the role mathematical paradoxes can play in the preservice education of high school mathematics teachers. The study examined the potential of paradoxes as a vehicle for: (a) sharpening student-teachers' mathematical concepts; and (b) raising their pedagogical awareness of the constructive role of fallacious reasoning in the development of mathematical knowledge. Course material development and data collection procedures are described. Results obtained through written responses and class-video-tapes are analyzed and discussed. The findings indicate that the model of dealing with paradoxes as applied in this study has relevance to such aspects of mathematics education as cognitive conflicts, motivation, misconceptions, and constructive learning. Generalizations are discussed.

Preservice education of math teachers using paradoxes

This is a report on a naturalistic study of the role mathematical paradoxes can play in the preservice education of high school mathematics teachers. The study examined the potential of paradoxes as a vehicle for: (a) sharpening student-teachers' mathematical concepts; (b) raising their pedagogical awareness of the constructive role of fallacious reasoning in the development of mathematical knowledge. Course material development and data collection procedures are described. Results obtained in parts of the study through written responses and class-videotapes are analyzed and discussed. The findings indicate that the model of dealing with paradoxes as applied in this study has relevance to such aspects of mathematics education as cognitive conflicts, motivation, misconceptions and constructive learning.

Implementing The Standards: The Vision of Problem Solving in the Standards

Problem solving has been espoused as a goal in mathematics education since the late 1970s, with focused attention ansmg from NCTM's An Agenda for Action (1980). But problem solving should be more than a slogan offered for its appeal and widespread acceptance. It should be a cornerstone of mathematics curriculum and instruction, fostering the development of mathematical knowledge and a chance to apply and connect previously constructed mathematical understandings. This perception of problem solving is presented in the Curriculum and Evaluation Standards for School Mathematics (Standards) (NCTM 1989, 23, 75). See table 1. Indeed, as noted in the Standards, “students need to work on problems that may take hours, days, and even weeks to solve. Although some may be relatively simple exercises to be accompplished independently, others should involve small groups or an entire class working cooperatively” (NCTM 1989, 6).

Inductive problem‐sequences for independent study of linear algebra

Three chapters of the syllabus of a basic mathematics course for engineering students at the Technion† were adapted for the student's independent development of mathematical knowledge by means of solving a chain of inductive problem‐sequences: (i) vectors and scalars in the plane; (ii) vectors in the three‐dimensional space; (iii) planes and lines in the 3‐dimensional space. The main principles which guided the adaptation of course materials for this purpose are presented. In particular an inductive problem‐sequenceis defined as a sequence consisting of problems which lead the learner to investigate particular cases, discover a common regularity, make a conjecture about a unifying rule, and finally prove or refute the conjecture. The major part of the paper is an annotated analysis of an excerpt of the materials. † Technion is the oldest and largest school of engineering in Israel.

The Development of Mathematical Knowledge in Two West African Societies

POSNER, JILL K. The Development of Mathematical Knowledge in Two West African Societies. CHILD DEVELOPMENT, 1982, 53, 200-208. The research investigated the development of mathematical concepts among children from 2 groups in West Africa, an agricultural tribe (Baoule) and a merchant society (Dioula). Baoule and Dioula children at 3 age levels participated: preschoolers, 5-6 years (study 1); 7-8-year-olds with 1 year of school (study 2); and schooled and unschooled 9-10-year-olds (studies 1 and 2). Experimental tasks covered several systems of mathematical thought from the perception of relative quantity and the identification of static equivalence (study 1) to the ability to solve practical addition problems (study 2). Results provide evidence that the ability to judge numerosity rapidly may be a universal capacity. All subjects by middle childhood performed equally well on this task. On the latter two tasks, which required the activation of other than perceptual strategies, schooling improved performance significantly among Baoule subjects while there were no reliable differences between schooled and unschooled Dioulas. The advancement of quantitative understanding appears to be dependent on certain kinds of experiences which both schooling and a merchant culture afford.

The Development of Mathematical Knowledge in Two West African Societies

The Development of Mathematical Knowledge in Two West African Societies