Mathematical modelling in mathematics education and instruction

This paper aims at giving a concise survey of the present state-of-the-art of mathematical modelling in mathematics education and instruction. It will consist of four parts. In part 1, some basic concepts relevant to the topic will be clarified and, in particular, mathematical modelling will be defined in a broad, comprehensive sense. Part 2 will review arguments for the inclusion of modelling in mathematics teaching at schools and universities, and identify certain schools of thought within mathematics education. Part 3 will describe the role of modelling in present mathematics curricula and in everyday teaching practice. Some obstacles for mathematical modelling in the classroom will be analysed, as well as the opportunities and risks of computer usage. In part 4, selected materials and resources for teaching mathematical modelling, developed in the last few years in America, Australia and Europe, will be presented. The examples will demonstrate many promising directions of development.

A Integração curricular da demonstração

Neste artigo, são discutidos os conceitos de demonstração e de esquema demonstrativo, bem como a relevância curricular da demonstração e os contextos favoráveis à sua aprendizagem. Alguns dos resultados do estudo, relativamente ao papel da demonstração no currículo, bem como ao modo como se desenvolveu o processo demonstrativo, são apresentados e discutidos. A metodologia adotada no estudo teve uma natureza interpretativa e os participantes no estudo foram uma turma de 9.º ano e a respetiva professora de Matemática. As conclusões do estudo apontam para o facto de os alunos tenderem a usar exemplos particulares para validar as suas afirmações matemáticas. Apontam ainda para as múltiplas funções da demonstração nas tarefas em que esta surgiu como um meio de descoberta da solução do problema. Sugerem também que a introdução e a negociação da importância da demonstração implicam uma intervenção curricular, na qual a professora detém um papel fundamental.

Meteorology and climate inspire secondary school students

As part of its National Science and Engineering Week activities in 2009 and 2010, the University of Reading organised two open days for 60 local key stage 4 pupils. The theme of both open days was ‘How do we predict weather and climate?’ Making use of the students’ familiarity with weather and climate, several concepts of relevance to secondary science were investigated. The open days also provided an opportunity for more than 30 research staff from the university to interact with the students. Feedback from students and teachers was extremely positive. This article shows how meteorological science can be used to illustrate elements of the secondary science and mathematics curricula.

The background impact in HEI math lectures: an experience in math in the 1st year of Accounting and Management Degree

It is a fact, and far from being a new one, that students have been entering Higher Education courses with many different backgrounds in terms of secondary school programs they attended. The impact of these basic skills is a general and worldwide challenge, fundamentally when facing some specific “constructive” subjects like foreign languages and Mathematics. Working with students with an extensive variety of Math qualifications is an outrageous challenge when they enter an advanced Math course, leading to an almost generalized expectations’ failure - from students enrolled in course and from their teachers, who feel powerless in trying to monitor knowledge construction from completely different “starting points”. If teachers’ "haste" is average, more than half of the students do not “go along” and give up, even before experiencing any kind of evaluation procedure. On the contrary, if the “speed” is too low, others are discouraged (feeling not progressing at all) and the teacher runs the risk of not meeting the minimum objectives (general and specific) of its course, which may have a negative impact on students’ future training development. Failure in Mathematics, despite being a recurrent and global issue, does not have any “magical solution”, however, in general, teachers in this area seem untiring, searching, investigating, trying and implementing new and old “recipes” to tackle and demystify this subject. In this article we describe a project developed in a Math course, with the first year students from an Accounting and Management bachelor degree, and its outcomes since it was brought to practice, revealing its impact in students’ success, from approval to dropout rates, in this course. We will shortly describe students’ differentiated Math backgrounds, their results in a pre-assessment analysis and how we try to deal with these differences and level them up, having in mind the same “finish line”. One should never forget that all these students where officially accepted in higher education institutions, so they are ones’ reality, the reality of institutions whose name one should value and strive to defend.

Financial Education Through Mathematics and IT Curricula: Pocket Money Management

Report published in the Proceedings of the National Conference on "Education in the Information Society", Plovdiv, May, 2013

Developing Financial Expertise Through Mathematics and IT School Curricula

Report published in the Proceedings of the National Conference on "Education and Research in the Information Society", Plovdiv, May, 2014

New challenges in mathematics for the european higher education

In this paper we will talk about a math project submitted to the Lifelong Learning Programme. European higher education needs a reform in order to play its full role in the Europe of Knowledge. Modernisation of higher education is necessary in the areas of curricula (Bologna process), funding and governance so that higher education institutions can face the challenges posed by globalisation and contribute more effectively to the training and retraining of the European workforce. On the other hand Mathematics is an essential component of all educational systems. Mathematical literacy is being scrutinized in assessment efforts such as the OCDE Programme for International Student Assessment (PISA). This showed a low level in Europe. Due to the Bologna Process, which brought several didactical implications for Higher Education (HE) institutions, there is the need of lifelong learning. This evolution is in conflict with the earlier mentioned lack of competencies on basic sciences, such as Mathematics. Forced by this duality, efforts are combined to share expertise in the Math field and the integration of pedagogical methodologies becomes a necessity. Thus, several European countries have proposed an International Project to the Lifelong Learning Programme, Action ERASMUS Modernisation of Higher Education, to make institutions more attractive and more responsive to the needs of the labour market, citizens and society at large. One of the main goals of the project is to attract students to math through high-quality instructional units in an understandable, exciting and attractive way.

Crònica d'una reforma: Curricula de matemàtiques

L'any 2000 -Any Internacional de les Matematiques- ha provocat i provocara multitud d'actes, publicacions, jornades i congressos promoguts pels qui les valoren i les estimen, amb la intenció d'apropar-Ies a la major quantitat de gent possible. Per aquest col·lectiu, les matematiques són una manifestació cultural i, com a tal, patrimoni de la humanitat. Si qualsevol moment és bo per meditar sobre el passat, el present i el futur, ara és un moment ideal per fer-ho sobre I'educació matematica en el nostre país, amb una reforma educativa en marxa. El present de I'ensenyament de les matematiques és extremadament divers, cosa que fa impossible una descripció acurada d'on som i cap a on anem. Pero tot present té un passat proper que ajuda a entendre'l i que dóna eines per prendre decisions de cara al futuro La meva implicació en el procés de la reforma educativa que s'esta duent a terme m'impossibilita de fer-ne una descripció plenament objectiva, així que donaré la meva visió personal d'aquest projecte tan complex amb la intenció de proporcionar, d'una banda, informació sobre alguns fets rellevants i, de I'altra, elements de reflexió sobre el que pot representar per a l'estament educatiu el camí cap a la renovació professional. En aquest itinerari m'aturaré en les causes de la reforma, les intencions i les condicions per a I'éxit. Les consideracions sobre el passat que no es vol que torni i I'analisi que farem del present portara de manera natural a les propostes per al futuro.

Teaching mathematics and programming : new approaches with empirical evaluation

Programming and mathematics are core areas of computer science (CS) and consequently also important parts of CS education. Introductory instruction in these two topics is, however, not without problems. Studies show that CS students find programming difficult to learn and that teaching mathematical topics to CS novices is challenging. One reason for the latter is the disconnection between mathematics and programming found in many CS curricula, which results in students not seeing the relevance of the subject for their studies. In addition, reports indicate that students' mathematical capability and maturity levels are dropping. The challenges faced when teaching mathematics and programming at CS departments can also be traced back to gaps in students' prior education. In Finland the high school curriculum does not include CS as a subject; instead, focus is on learning to use the computer and its applications as tools. Similarly, many of the mathematics courses emphasize application of formulas, while logic, formalisms and proofs, which are important in CS, are avoided. Consequently, high school graduates are not well prepared for studies in CS. Motivated by these challenges, the goal of the present work is to describe new approaches to teaching mathematics and programming aimed at addressing these issues: Structured derivations is a logic-based approach to teaching mathematics, where formalisms and justifications are made explicit. The aim is to help students become better at communicating their reasoning using mathematical language and logical notation at the same time as they become more confident with formalisms. The Python programming language was originally designed with education in mind, and has a simple syntax compared to many other popular languages. The aim of using it in instruction is to address algorithms and their implementation in a way that allows focus to be put on learning algorithmic thinking and programming instead of on learning a complex syntax. Invariant based programming is a diagrammatic approach to developing programs that are correct by construction. The approach is based on elementary propositional and predicate logic, and makes explicit the underlying mathematical foundations of programming. The aim is also to show how mathematics in general, and logic in particular, can be used to create better programs.

Recreational Mathematics - Only For Fun?

In this paper, I explore recreational mathematics from two perspectives. I first study how the concept appears in educational policy documents such as standards, syllabi, and curricula from a selection of countries to see if and in what way recreational mathematics can play a part in school mathematics. I find that recreational mathematics can be a central part, as in the case of India, but also completely invisible, as in the standards from USA. In the second part of the report, I take an educational historical approach. I observe that throughout history, recreational mathematics has been an important tool for learning mathematics. Recreational mathematics is then both a way of bringing pleasure and a tool for learning mathematics. Can it also be a tool for social empowerment?

Physics and Mathematics in the Engineering Curriculum: Correlation with Applied Subjects

This paper presents a study in which the relationship between basic subjects (Mathematics and Physics) and applied engineering subjects (related to Machinery, Electrical Engineering, Topography and Buildings) in higher engineering education curricula is evaluated. The analysis has been conducted using the academic records of 206 students for five years. Furthermore, 34 surveys and personal interviews were conducted to analyze the connections between the contents taught in each subject and to identify student perceptions of the correlation with other subjects or disciplines. At the same time, the content of the different subjects have been analyzed to verify the relationship among the disciplines.Aproper coordination among subjects will allow students to relate and interconnect topics of different subjects, even with the ones learnt in previous courses, while also helping to reduce dropout rates and student failures in successfully accomplishing the different courses.

Mathematics notebook : manuscript, 1782

With sections on numeration, surveying, trigonometry and other topics, accompanied by diagrams and hand-colored illustrations.

Mathematics notebook : manuscript, 1785-1789

Sections on numeration, interest, square root, geometry and surveying with accompanying diagrams.

Changes and Shifts in Computing Curricula

Report published in the Proceedings of the National Conference on "Education and Research in the Information Society", Plovdiv, May, 2015

The impact of standards -based practices in mathematics on the achievement of low -performing students

This study examined standards-based mathematics reform initiatives to determine if they would improve student achievement on the part of low-performing students. New curricula, the Carnegie Learning Cognitive Tutor®, were provided for algebra and geometry students. The new instructional strategy relied on both the teacher-led instruction and the use of computers to differentiate instruction for individual students. Mathematics teachers received ongoing professional development to help them implement the new curricula. In addition, teachers were provided with ongoing support to assist them with the transformation of the learning environments for students using standards-based practices. This quasi-experimental (nonrandomized) study involved teachers in two matched urban high schools. Analyses (ANCOVAs) revealed that the experimental group with an appropriately implemented program had significantly higher learning gains than the comparison group as determined by the students' 2007 mathematics Developmental Scale Score (DSS). In addition, the experimental group's adjusted mean for the second interim mathematics assessment was significantly higher than the comparison group's mean. The findings support the idea that if the traditional curriculum is replaced with standards-based curriculum, and the curriculum is implemented as intended, low-performing students may make significant learning gains. With respect to the teaching practices as observed with the Classroom Observation Protocol (COP), t-tests were conducted on four constructs. The results for both the algebra and geometry teachers on the constructs were not significant. The COP indicated that teachers in both the experimental and comparison groups used traditional instruction strategies in their classrooms. The analyses of covariance (ANCOVA) on the use of technology revealed no significant main effects for computer use.