Mathematical modelling in science and mathematics education

Scientific research involves mathematical modelling in the context of an interactive balance between theory, experiment and computation. However, computational methods and tools are still far from being appropriately integrated in the high school and university curricula in science and mathematics. In this paper, it is discussed the relevance of mathematical modelling and illustrated how a computer modelling tool (Modellus, a free tool available on the Internet and developed at FCTUNL) can be used to embed modelling in high school and undergraduate courses. Modellus allows students to create and explore mathematical models using functions, differential and iterative equations, and visualize the behaviour of mathematical objects.     

The WebQuest model effects on mathematics curriculum learning in elementary school students

Since its inception in 1995, the WebQuest instructional model has received substantial attention from educators who have applied it to teaching activities. However, WebQuest has seldom been applied to mathematical teaching. Therefore, exploring curriculum development and learning achievement in mathematical teaching that integrates the WebQuest model is necessary. The objective of this study is to explore using the WebQuest model for teaching the concepts of proportion in elementary school mathematics, and the subsequent effects on student learning achievements. Furthermore, this study endeavors to understand student learning attitudes and learning satisfaction regarding the WebQuest model. This study employed the quasi-experimental research method, and the participants were two sixth-grade classes with 52 students at a Taiwanese elementary school. The experimental group was taught using WebQuest, while the control group was taught by employing the traditional IT-integrated curriculum. The two groups completed a 5-week course on proportion concepts. The study results indicate the following: (a) Regarding learning achievements, students who learned with the WebQuest model demonstrated superior learning performance than students who were taught using the traditional IT-integrated curriculum; (b) no significant difference existed between the learning attitudes of students who were taught using the WebQuest model and those who learned with the traditional IT-integrated curriculum; (c) on the learning-satisfaction survey, the students who learned with the WebQuest model showed high learning satisfaction, selecting between agree and strongly agree for the 6 dimensions, including WebQuest teaching, IT capability, the assistance of mathematics curriculum, collaborative learning, learning reflection, and learning feedback. This indicates that the students were largely satisfied and identified with the task-oriented WebQuest learning curriculum that was designed in this study. Finally, specific recommendations are made based on the results, providing teachers and future researchers with a reference for integrating the WebQuest model in the mathematics field.     

Review of Piero Matematico (CD)

Orietta Pedemonte reviews the CD Piero Matematico by Daniela Gentilin and Ennio Bettanello.The work merits an ample diffusion among teachers and students at the high school level and the first years of architecture at the university level, as well as among those who are interested in history, art, architecture and mathematics.     

Logo building blocks: Student teachers using computer-based mathematics apparatus

We describe experiments in which student teachers, who have been identified by their college of education as needing mathematical help, undertook a computer-based elementary maths course. Our objectives were to assess the effect of this work on the students' mathematical performance and attitude.The students were given hands-on experience of the programming language LOGO, which included both writing simple programs and running pre-defined procedures, such as those for illustrating the multiplication of fractions pictorially. The computer was used as a piece of mathematics apparatus with which the students could experiment: it did not take any tutorial or managerial role.A variety of mathematics performance and attitude tests were administered pre and post, and the results are described and analysed. Some evidence for the success of the approach is shown, though the experiments underline the enormous difficulty of helping this kind of student.     

Using CAS to solve a mathematics task: A deconstruction

I investigate how and whether a heterogeneous group of first-year university mathematics students in South Africa harness the potential power of a computer algebra system (CAS) when doing a specific mathematics task. In order to do this, I develop a framework for deconstructing a mathematics task requiring the use of CAS, into its primary components. This framework is based on the semiotic notion of diagrammatic reasoning whereby reasoning consists of construction of signs, transformation of signs, and observation and interpretation of signs. I use the framework to distinguish between the activities of students who were computer literate on entry to university and those who were not computer literate. The analysis suggests that formerly non-computer literate students are no worse than computer literate students in using CAS to construct various representations of signs, but that they are less able to interpret these signs. I propose that, in the South African context, this is largely due to inequities in prior mathematical education, rather than a lack of computer literacy per se.     

Action to process: constructing functions from algebra word problems

Abstract

The ability to build a mathematical model of a situation described in ordinary language has always been an important goal of school mathematics teaching. This paper gives one explanation for the difficulties many students experience with certain stylized modeling tasks (solving algebra word problems) and puts these difficulties in the context of a theory about the acquisition of the concept of mathematical function. It also describes a visual programming language that can act as a bridge, and in some sense an alternative, to formal algebraic notation as a language for describing situations. Tracing the author's classroom experiments and his reflections on those experiments, the paper suggests a method for helping students.     

Predicting performance in computer programming courses

Currently available methods for selecting computer programmers are reviewed. The academic achievement, in high school and in the university, of over 3000 students is reviewed and correlated with their performance in the different computer programming courses. The results revealed that performance in high school and college mathematics and science courses account for up to 50 per cent of the variance in computer programming course performance. The implications of these results for training and selection are discussed.     

Teaching A level physics through microcomputer dynamic modelling: Evaluation of teaching

Abstract  This paper describes an evaluation of teaching A level Physics through dynamic modelling. This teaching was carried out with a group of 12 lower sixth Nuffield A level Physics students and was evaluated with three tests internal to the school and with the external public Nuffield A level Physics (NAP) examination which those students later took. The internal school tests showed that the treatment group performed better than an equivalent non-treatment group in both general ability and in a few particular skills. The public NAP examination showed no superiority of the treatment group, but indicated a significant preference for an optional question that involved mathematical skills, even though none of the treatment group took mathematics at A level. Comments are made about the educational significance of dynamic modelling and about computers in education generally.     

The Integration of Mathematics in Physics Problem Solving: A Case Study of Greek Upper Secondary School Students

This article presents a case study that examines the level of integration of mathematical knowledge in physics problem solving among first grade students of upper secondary school. We explore the ways in which two specific students utilize their knowledge and we attempt to identify the epistemological framings they refer to while solving a physics problem. Participant observation was used for data collection, and the students' verbal interactions were video-recorded. The analysis shows that they tend to use a wide spectrum of epistemological framings that entangle mathematics and physics but at the same time face significant practical difficulties in modulating the two subjects.     

Students’ Understanding of Loops and Nested Loops in Computer Programming: An APOS Theory Perspective

Abstract

The purpose of this study is to explore students’ understanding of loops and nested loops concepts. Sixty-three mechanical engineering students attending an introductory programming course participated in the study. APOS (Action, Process, Object, Schema) is a constructivist theory developed originally for mathematics education. This study is the first attempt to use the APOS framework in the context of programming education. Results showed that APOS theory is a useful framework for examining engineering students’ understanding related to loops and nested loops. The results of this study have specific implications for researchers and practitioners when designing programming instruction.     

论函数编程技术在计算机数学教学中的应用

计算机数学作为计算机学科一门重要专业基础课程在计算机的教授与学习中起到承上启下的作用.随着网络时代的到来,企业、政府、个人都看到了计算机在生产生活中必不可少的作用,计算机成为生活中必不可少的东西.网络时代的发展也带动了计算机学科的发展,成为热门学科.计算机学科和数学是分不开的,因此也称为最有难度的一个学科.虽然很多学生选择计算机学科,但是在实际课堂教育中,学生的听课效率却不如预期,主要因为计算机数学内容理论性很强,而且一些概念知识很抽象,让学生很难理解,从而对学科内容不感兴趣.函数编程技术在教学中应用,对计算机数学的学习起到了辅助作用,让学生可以利用这种编程语言了解计算机数学的一些抽象性概念,更好的在实践中运用计算机数学.函数编程技术在计算机数学中应用,使得死板的理论学科变得稍显活泼,这回让学生对该学科更感兴趣,带动学生积极主动地学习.     

The computer as a Trojan Horse

Abstract This paper reports on a study of boys in an Australian secondary school who were under achieving in school mathematics. The subjects were asked to develop computing software to teach fellow students about aspects of Ancient Egyptian mathematics. The paper reports four example case studies of the experience of the subjects during the course of the intervention. The paper examines the use of computing software development as a vehicle for the incidental learning of mathematical technique and for changes in attitude towards mathematics. Four different mathematical activities are reported in the case studies with a screen-dump of the subjects work included as an illustrative example.     

A Proof Environment for Teaching Mathematics

The EPGY Theorem Proving Environment is a computer program used by students to write mathematical proofs in a selection of computer-based, proof-intensive mathematics courses at the high-school and university level. The system allows easy input of mathematical expressions, application of standard proof strategies and logical inference rules, application of mathematical rules, and verification of logical inference. Each course has its own language, database of theorems, and mathematical rules. The system uses a combination of automated reasoning and symbolic computation to verify individual proof steps. The proof environment has been used by over 170 students who have taken the EPGY high-school geometry course. In addition to providing a general overview of the system, we describe what we have learned from student use of the Theorem Proving Environment in the EPGY geometry course.     

The Role of Teachers’ Knowledge of Functions in Their Teaching: A Conceptual Approach With Illustrations From Two Cases

Abstract

We investigate whether and how personal mathematical knowledge at an advanced level impacts on teaching at a lower school level. We study this in the context of functions because understanding them permeates secondary and advanced mathematics. Textbook treatment of these can be patchy, implying a need for knowledgeable teachers to rectify weaknesses. A small observational study of two mathematically well-qualified teachers, teaching at a range of levels, leads to a deeper understanding of how their personal mathematical knowledge is manifested in their teaching. Finally, we present a theory of pedagogy combining dual development, intellectual necessity, and repeated experience of reasoning, for which these teachers’ practices provide illustrations.     

WISEngineering: Supporting precollege engineering design and mathematical understanding

Introducing engineering into precollege classroom settings has the potential to facilitate learning of science, technology, engineering, and mathematics (STEM) concepts and to increase interest in STEM careers. Successful engineering design projects in secondary schools require extensive support for both teachers and students. Computer-based learning environments can support both teachers and students to implement and learn from engineering design projects. However, there is a dearth of empirical research on how engineering approaches can augment learning in authentic K-12 settings. This paper presents research on the development and pilot testing of WISEngineering, a new web-based engineering design learning environment. Three middle school units were developed using a knowledge integration learning perspective and a scaffolded, informed engineering approach with the goal of improving understanding of standards-based mathematical concepts and engineering ideas. Seventh grade math students from two teachers in a socioeconomically diverse and low-performing district participated in three WISEngineering units over the course of a semester. Students significantly improved their mathematical scores from pretest to posttest for all three projects and on state standardized tests. Student, teacher, and administrator interviews reveal that WISEngineering projects promoted collaboration, tolerance, and development of pro-social skills among at-risk youth. Results demonstrate that informed engineering design projects facilitated through the WISEngineering computer-based environment can help students learn Common Core mathematical concepts and principles. Additionally, results suggest that WISEngineering projects can be particularly beneficial for at-risk and diverse student populations.     

The Influence of Programming Learning on Mathematics Scores of Middle School Students

Programming teaching plays an important role in improving students' computational thinking and practice. However, due to the fact that programming has not been included in the entrance examination, the rate of having programming lessons in primary and secondary schools is low. At present, there is still a lack of empirical research on how programming affects the performance of other subjects. This paper studies the influence of programming on students' mathematics achievement for the students of an experimental middle school. The results show that programming has a significant impact on students' math scores, and the students who had not learned code were more likely to show a significant decline in math scores. In addition, learning programming makes it easier for middle-grade students to improve their math scores.     

Nouvelles Brèves

Abstract

Proof understanding, production, and appreciation (PUPA) are important parts of a mathematician's repertoire. Many university students in the US, however, have difficulty with proof. One intent of this study was to examine the development of such students’ PUPAs and possibly to identify significant influences on that development, through interviews with the students throughout their studies in mathematics. Three case studies show a great variance in the development of the students’ proof skills. Some students come to university with excellent PUPAs and continue to thrive in a proof environment. Others enter university with poor PUPAs and unfortunately graduate without a significant change in their proof skills and attitudes. Still others come with poor proof skills but do show some growth during their undergraduate mathematics programs. Results of teaching experiments suggest that making proofs tangible is a means of helping those with poor PUPAs to grow in their proof understandings and abilities.     

离散数学课程建设(英文)

Many students find great difficulty in the formal programming modules taught in almost every computing science degree.These modules,often taught in later years,rely on some mathematical abilities that seem beyond many students.This paper describes two new modules in years I and II of the curriculum that are helping students to prepare for the formal programming courses.These modules are algorithmic problem solving and discrete mathematics for computing science and are now running in two third-level institutions in Ireland.This paper also shows the impact that these new modules have had on the students in question.