Epistemic Complexity of the Mathematical Object “Integral”

The literature in mathematics education identifies a traditional formal mechanistic-type paradigm in Integral Calculus teaching which is focused on the content to be taught but not on how to teach it. Resorting to the history of the genesis of knowledge makes it possible to identify variables in the mathematical content of the curriculum that have a positive influence on the appropriation of the notions and procedures of calculus, enabling a particularised way of teaching. Objective: The objective of this research was to characterise the anthology of the integral seen from the epistemic complexity that composes it based on historiography. Design: The modelling of epistemic complexity for the definite integral was considered, based on the theoretical construct “epistemic configuration”. Analysis and results: Formalising this complexity revealed logical keys and epistemological elements in the process of the theoretical constitution that reflected epistemological ruptures which, in the organisation of the information, gave rise to three periods for the integral. The characterisation of this complexity and the connection of its components were used to design a process of teaching the integral that was applied to three groups of university students. The implementation showed that a paradigm shift in the teaching process is possible, allowing students to develop mathematical competencies.

Review of research on monitoring mathematical competencies of students

The article analyzes foreign research on monitoring the mathematical competencies of students. The key issues related to the use of the results of assessment of students' academic achievements in conducting national monitoring of the quality of education are considered. The specifics of using the results in each of the countries are discussed, including the interpretation and presentation of evaluation data for different user groups.

Quality of attachment relationships and frequency of mathematics- and science-related activity offers in kindergarten as predictors of girls' and boys' mathematics-related motivation

During the kindergarten years and until shortly before school start, there are no gender differences in (precursors of) mathematical competencies or mathematics-related motivation. Shortly after school entry, however, boys are already superior to their female peers in mathematics-related competencies and motivation. We investigated in a cross-sectional study two aspects of process quality in kindergarten that can favorably influence the development of mathematics-related motivation, especially of girls: the frequency of offers of mathematics- and science-related activities and a high-quality attachment relationship with the teacher. In 135 independent dyads, the quality of attachment between kindergarten teacher and child was assessed by a one and a half-hour standardized observation (Attachment Q-Set). The teacher provided information on how often she provides mathematics- and science-related activities. The children were asked about their mathematics-related motivation and precursors of mathematical competencies were measured using a standardized test. Results show, in line with existing studies, that girls and boys did not yet differ in their precursors of mathematical competencies and mathematics-related motivation at the end of kindergarten. Girls were involved in significantly higher quality attachment relationships with their teachers than boys. While girls' mathematics-related motivation increased with the frequency of the provision of relevant activities, it did not play a role for boys' motivation. We discuss (a) how teachers can be encouraged to offer mathematics-and science-related activities more often and (b) whether a comparable quality of attachment would be shown for boys as for girls if the kindergarten teacher were male.

THESAURUS APPROACH: ON THE FORMATION OF MATHEMATICAL COMPETENCE AND COMPETENCE OF FUTURE ENGINEERS

In the article, the question of improving the formation of mathematical competencies and competencies of future engineers on the basis of a fast-paced approach is developed, in particular, the main concepts, the main issues of the topic, the importance of building a thesaurus, which includes the methods of activity for solving these issues.

Quantitative and Qualitative Differences in the Canonical and the Reverse Distance Effect and Their Selective Association With Arithmetic and Mathematical Competencies

Understanding the relationship between symbolic numerical abilities and individual differences in mathematical competencies has become a central research endeavor in the last years. Evidence on this foundational relationship is often based on two behavioral signatures of numerical magnitude and numerical order processing: the canonical and the reverse distance effect. The former indicates faster reaction times for the comparison of numerals that are far in distance (e.g., 2 8) compared to numerals that are close in distance (e.g., 2 3). The latter indicates faster reaction times for the ordinal judgment of numerals (i.e., are numerals in ascending/descending order) that are close in distance (e.g., 2 3 4) compared to numerals that are far in distance (e.g., 2 4 6). While a substantial body of literature has reported consistent associations between the canonical distance effect and arithmetic abilities, rather inconsistent findings have been found for the reverse distance effect. Here, we tested the hypothesis that estimates of the reverse distance effect show qualitative differences (i.e., not all participants show a reverse distance effect in the expected direction) rather than quantitative differences (i.e., all individuals show a reverse distance effect, but to a different degree), and that inconsistent findings might be a consequence of this variation. We analyzed data from 397 adults who performed a computerized numerical comparison task, a computerized numerical order verification task (i.e., are three numerals presented in order or not), a paper pencil test of arithmetic fluency, as well as a standardized test to assess more complex forms of mathematical competencies. We found discriminatory evidence for the two distance effects. While estimates of the canonical distance effect showed quantitative differences, estimates of the reverse distance effect showed qualitative differences. Comparisons between individuals who demonstrated an effect and individuals who demonstrated no reverse distance effect confirmed a significant moderation on the correlation with mathematical abilities. Significantly larger effects were found in the group who showed an effect. These findings confirm that estimates of the reverse distance effect are subject to qualitative differences and that we need to better characterize the underlying mechanisms/strategies that might lead to these qualitative differences.

An Analysis of Students’ Mathematical Competencies: The Relationship between Units

This study aimed to investigate students’ mathematical competence in learning relationships between units according to the students’ performance in a SUKEN test of Level 6. A total of 139 students were selected as our target group and involved as examinees. Research instruments include students’ answer sheets, test item analyses, and textbook analysis. SUKEN test is a mathematical proficiency test used to identify related issues to improve teaching practices. The results from the first phase showed that there were 24 examinees or 17.27 percent of them had been successfully passed the passing criterion as 70 percent of the total marks 100. However, there was a lowest percentage (33.21%) of examinees showed that they were able to answer correctly in the questions related to the content domain of Quantities and Measurement compared to other content domains. On top of that, only 38.49 percent and 31.09 percent of the examinees possessed their competencies in content knowledge and the method of application respectively while they answered the Quantities and Measurement questions. Besides, the results of in-depth analyses from students’ answer sheets revealed that there were seven different groups of answers by analyzing examinees' responses in terms of their reasoning skills to support their responses. Examinees were found to have problems finding the relationship between cm3 and m3, whenever they have to use a relationship in three dimensions. Therefore, teachers are suggested to use the geometry model to assist students in understanding the relationships between the units.

University Mathematics-Laden Education, Competencies and the Fighting of Syllabusitis

Syllabusitis is a name for a disease that consists of identifying the mastering of a subject with proficiency related to a syllabus. In this chapter I argue that using a set of mathematical competencies as the hub of mathematics-laden education can be a means to fight syllabusitis. The introduction and thorough exemplification of this idea was the main outcome of the Danish KOM Project. Furthermore, a two-dimensional structuring of the relation between subject specific competencies and subject matter was suggested. As the analytic core of this chapter I argue that such a two-dimensional structure has proven to be a crucial element when attempting to put the competency idea into educational practice, and exemplify how that can be done when it comes to mathematics-laden education at university level.