Development Of Mathematical Competencies Research Articles

Didactic and theoretical-based perspectives in the experimental development of an intelligent tutorial system for the learning of geometry

This paper aims at showing the didactic and theoretical-based perspectives in the experimental development of the geogebraTUTOR system (GGBT) in interaction with the students. As a research and technological realization developed in a convergent way between mathematical education and computer science, GGBT is an intelligent tutorial system, which supports the student in the solving of complex problems at a high school level by assuring the management of discursive messages as well as the management of problem situations. By situating the learning model upstream and the diagnostic model downstream, GGBT proposes to act on the development of mathematical competencies by controlling the acquisition of knowledge in the interaction between the student and the milieu, which allows for the adaptation of the instructional design (learning opportunities) according to the instrumented actions of the student. The inferential and construction graphs, a structured bridge (interface) between the contextualized world of didactical contracts and the formal computer science models, structure GGBT. This way allows for the tutorial action to adjust itself to the competential habits conveyed by a certain classroom of students and to be enriched by the research results in mathematical education.

Notion of mathematical competence

The article introduces the notion of mathematical competence as an example of subject-orientated competence. It presents a unique way of identifying a competence - competence test - and its specifications for mathematics activities. The features and reasons of a tendency to simplify the notion of mathematical competence in secondary and higher education are investigated. The analysis of the process-based and situational qualities of mathematical competence, which helps raise awareness of the mechanism of mathematical competence development, is presented. The future research will focus on the study of life stages dynamics and age-appropriate forms of competence tests.

New Avenues for History in Mathematics Education: Mathematical Competencies and Anchoring

The paper addresses the apparent lack of impact of ‘history in mathematics education’ in mathematics education research in general, and proposes new avenues for research. We identify two general scenarios of integrating history in mathematics education that each gives rise to different problems. The first scenario occurs when history is used as a ‘tool’ for the learning and teaching of mathematics, the second when history of mathematics as a ‘goal’ is pursued as an integral part of mathematics education. We introduce a multiple-perspective approach to history, and suggest that research on history in mathematics education follows one of two different avenues in dealing with these scenarios. The first is to focus on students’ development of mathematical competencies when history is used a tool for the learning of curriculum-dictated mathematical in-issues. A framework for this is described. Secondly, when using history as a goal it is argued that an anchoring of the meta-issues in the related in-issues is essential, and a framework for this is given. Both frameworks are illustrated through empirical examples.

The developmental relations between conceptual and procedural knowledge: A multimethod approach.

Interactions between conceptual and procedural knowledge influence the development of mathematical competencies. However, after decades of research, these interrelations are still under debate, and empirical results are inconclusive. The authors point out a source of these problems. Different kinds of knowledge and competencies only show up intertwined in behavior, making it hard to measure them validly and independently of each other. A multimethod approach was used to investigate the extent of these problems. A total of 289 fifth and sixth graders' conceptual and procedural knowledge about decimal fractions was measured by 4 common hypothetical measures of each kind of knowledge. Study 1 tested whether treatments affected the 2 groups of measures in consistent ways. Study 2 assessed, across 3 measurement points, whether conceptual and procedural knowledge could be modeled as latent factors underlying the measures. The results reveal substantial problems with the validities of the measures, which might have been present but gone undetected in previous studies. A solution to these problems is essential for theoretical and practical progress in the field. The potential of the multimethod approach for this enterprise is discussed.

Exploring the impact of phonological awareness, visual–spatial working memory, and preschool quantity–number competencies on mathematics achievement in elementary school: Findings from a 3-year longitudinal study

This longitudinal study explored the importance of kindergarten measures of phonological awareness, working memory, and quantity–number competencies (QNC) for predicting mathematical school achievement in third graders (mean age 8 years 8 months). It was found that the impact of phonological awareness and visual–spatial working memory, assessed at 5 years of age, was mediated by early QNC, which predicted math achievement in third grade. Importantly, and confirming our isolated number words hypothesis, phonological awareness had no impact on higher numerical competencies (i.e., when number words needed to be linked with quantities [QNC Level II and above]) but predicted basic numerical competencies (i.e., when number words were isolated from quantities [QNC Level I]), explaining the moderate relationship between early literacy development and the development of mathematical competencies.

Sound-based Number Facts Training in a Private Speech Internalization Perspective: Evidence for Effectiveness of an Intervention in Grade 3

The overall purpose of the present study was to examine whether an intervention program (50 weeks), modeled as training in accordance with the developmental course of private speech (from audible private speech to silent inner speech), itself would add positively to the development of mathematical competence. The sample comprised two comparable groups of 82 third-grade children. One group followed the intervention program, the other group followed a traditional teaching program. The main concern was whether private speech and task-specific strategy use differences would arise as a function of the different teaching programs carried out in the two groups. Two separate laboratory investigations were performed for each child to explore their levels of strategy use and private speech internalization. The results indicated successful influence of the intervention program reflected in more internalized private speech and more internalized task-specific strategy use. Given the results of the study, classroom teachers should be encouraged to utilize private speech internalization as a valuable and potentially powerful tool for efficiently retrieving number fact information.

Secondary School Students’ Construction and Use of Mathematical Models in Solving Word Problems

This study focussed on how secondary school students construct and use mathematical models as conceptual tools when solving word problems. The participants were 511 secondary-school students who were in the final year of compulsory education (15–16 years old). Four levels of the development of constructing and using mathematical models were identified using a constant-comparative methodology to analyse the student’s problem-solving processes. Identifying the general in the particular and using the particular to endow the general with meaning were the key elements employed by students in the processes of construction and use of models in the different situations. In addition, attention was paid to the difficulties that students had in using their mathematical knowledge to solve these situations. Finally, implications are provided for drawing upon student’s use of mathematical models as conceptual tools to support the development of mathematical competence from socio-cultural perspectives of learning.

The development of mathematical competence in Flemish preservice elementary school teachers

A large-scale longitudinal study was conducted in which the elementary mathematical knowledge and skills of a large group of Flemish preservice elementary school teachers from 15 different institutes was assessed by means of a paper-and-pencil test that was administered both at the beginning and at the end of their 3-year training. The 30-items test covered the new standards for mathematics in the elementary school curriculum in Flanders. The test was divided in six subsets differing in terms of the curricular subdomain and of the cognitive operations being addressed by the item. The results confirmed the frequently heard concern that at the beginning of their training preservice elementary school teachers have rather weak mathematical competencies. At the end of their 3-year training, the overall test performance had become substantially better, although there were still reasons to be seriously concerned about the readiness of some student teachers to teach mathematics to elementary school children. A number of more specific comparisons helped to identify the relative role of two different factors—selection and instruction—in student teachers’ gain from pretest to posttest. Besides documenting the development of elementary mathematical competence in preservice elementary school teachers, the study also resulted in an instrument for (self-) assessment of elementary school mathematical competence. This instrument is now being used in many institutes for elementary school teacher training in Flanders.

The development of mathematical thinking in school: a comparison of the action- psychological and information-processing approaches

The learning and teaching of mathematics can be analyzed from different psychological points of view. Information-processing theories focus on the various information-processing mechanisms underlying mathematical competence and try to foster the development of these mechanisms in order to stimulate the growth of mathematical competence. In the action-psychological approach of the cultural-historical school, mathematics is viewed as a kind of culturally developed human activity governed by the rules that mathematicians themselves follow while doing their job. Consequently, the development of mathematical competence is regarded as the formation of a system of meaningful mathematical actions that constitute that activity. At a general theoretical level these approaches can be shown to be basically different and even incompatible. With regard to mathematics education the differences are illustrated with respect to several themes such as task-analysis, automatization, and the learning of elementary arithmetic. Considering insight and meaningful and sophisticated problem-solving as the core of mathematical thinking, the action-psychological approach appears to be the more promising candidate as an aid in the design of future mathematics education.