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Modern mathematics

Project to take 20th century mathematics research progress into the classroom

PETER M. FISHER / CORBIS/CORBIS (DC) / LATINSTOCKAn ambitious project, led by scientific societies connected with research and with mathematics instruction in Brazil, aims to create a new paradigm for teaching the subject.

The Klein Project in Portuguese is to mobilize the Brazilian mathematical community to prepare teaching material that incorporates the progress achieved in this subject over the last 100 years. A series of workshops will debate cutting-edge themes in basic areas, such as algebra, geometry and topology, as well as in other areas that have arisen recently, such as applications in the field of computing. The first workshop was to be held in early July, in Belo Horizonte. “These workshops will focus on specific themes, with a lot of objective discussion, leading to producing texts, products and innovative ideas,” says Marcelo Viana, a researcher from Impa, the National Institute of Pure and Applied Mathematics, who is the project’s coordinator. The material discussed in these workshops will be subsequently submitted to pilots, with the help of high school teachers and mathematical education undergraduates, in order to test them in real classrooms. Mathematicians from Portugal are also working on the project; it is hoped that the project’s contribution will reach other Portuguese-speaking countries.

The key objective is to make schoolteachers familiar with achievements in mathematical research, helping them to understand them and to establish connections with the traditional contents. “We don’t want to re-invent the instruction system, but to improve it, making it more interesting and effective,”  states Viana. The project organizers include representatives from SBM (the Brazilian Mathematical Society), SBEM (the Brazilian Mathematical Education Society), SBHMat (the Brazilian Mathematics History Society) and SBMAC (the Brazilian Society of Applied and Computer Mathematics), as well as from OBMEP (the Brazilian Mathematics Olympics of Public Schools). According to them, strengthening the training of teachers is crucial in order to introduce into the classrooms the wide scope of basic mathematics, whose contents, to their minds, are being taught in a fragmented and mechanical fashion. “In general, mathematics is taught mechanically, based on the introduction of abstract concepts but without a clear understanding and on the repetition of methods, without encouraging creativity and discovery. This makes the instruction both inefficient and disliked by students,” says Yuriko Yamamoto Baldin, a professor at the Mathematics Department of the Federal University of São Carlos (UFSCar) and one of the project’s coordinators. The absence of recent mathematical achievements in the school curriculum results in the defective education of students. “The students, when they get to university, are faced with a way of working with mathematics with which they did not become familiar during school,” she states.

Poincaré’s Conjecture
According to the professor, only rarely are mathematics teachers able to address research subjects in the classroom, as these are generally very complex and abstract, unlike that which happens with physics or biology teachers, who are generally happier about providing responses to student curiosity on cutting-edge themes. One example, she says, concerns the recent resolution of Poincaré’s Conjecture, a feat achieved by the Russian mathematician Grigory Perelman. Formulated in 1900 by Jules Henry Poincaré, a Frenchman, it concerns a core issue of topology, an area of mathematics regarded as an extension of geometry and that studies the geometric properties that remain unchanged when objects are distorted, stretched or shrunken. “Many teachers are embarrassed because they are unable to explain things to their students. And topology is one of the areas that has advanced the most in the recent past,” she states.

This project is the Brazilian part of an initiative launched in 2008 by ICMI, the International Commission on Mathematical Instruction, and by IMU, the International Mathematical Union, called the Klein Project for the 21st Century, celebrating the 100th anniversary of the publication of the texts of the German mathematician Felix Christian Klein (1849-1925). In his writings, Klein challenged secondary school teachers to transmit to students the wealth of contemporary mathematics. “My task will always be to show them the mutual connection between problems in a range of areas. Thus, I hope to make it easier for them to acquire the skill to derive from the great mass of knowledge a living stimulus for instruction,”  wrote Klein. The project resorts to the same inspiration stated in Klein’s original texts, proposing this time to include the twentieth century research advances in the curricula. The international project’s intent is to produce a book in understandable language that transmits the link, the growth and the relevance of mathematics, from its major ideas to the frontier of research and applications.

IMAGE SOURCE / IMAGE SOURCE / LATINSTOCKThe twentieth century witnessed progress in several fields of mathematics that helped the appearance of new specializations, such as computer science and statistics, and that aided the development of technologies. According to Mario Jorge Dias Carneiro, a professor at the Federal University of Minas Gerais and one of the project’s coordinators, the fields of probability and statistics are two striking examples of progress in research. “Statistics is used in almost all areas and progressed substantially in the twentieth century. The same can be said about probability, which is also part of our day-to-day life,”  he states. According to him, in the late twentieth century and early twenty-first century, there was a lot of success employing probabilistic methods in the study of deterministic problems. “This had made probability a very important theme, an almost central issue, in mathematics. To learn probability properly, it is important to be familiar with counting methods, a topic that in basic education we call combinatory analysis, but this is covered very superficially in school, often becoming a very tough area for both students and teachers,” he explains.

Computer
Marcelo Viana, from Impa, mentions another example: dynamic systems, a mathematical discipline that studies the types of phenomena that evolve over time. The field, which is relatively new in mathematics, having arisen about one century ago, aimed to solve problems connected to astronomy and celestial mechanics, in an attempt to assess planets’  future behavior and predict whether they were going to crash into each other. However, over the last few decades, an increasingly large number of phenomena are being regarded as a complex dynamic system, such as the climate, chemical reactions and ecological environments. “Dynamic systems can help students develop concepts in various fields of knowledge,” says Viana, an expert on this subject.

Besides the content, there are other research aspects that the mathematicians would like to address, such as using computers in classrooms. “It is an acknowledged fact that the use of computers for mathematical research was a landmark of the late twentieth century, but the incorporation of computers or even of calculators has not been consensually established in curricula, probably because it is desirable for the student to become properly familiar with the four operations and their properties,”  comments Dias Carneiro. According to Marcelo Viana, from Impa, the use of computers is still very limited and this could be used as a mathematical experiment laboratory. The way in which the research is being conducted also steers away from the classroom, according to Dias Carneiro. “The Internet allows us to develop scientific collaboration at a distance. For a mathematician, it is essential to exchange ideas with other mathematicians. If this was achieved at the beginning of the last century via letters and scientific meetings, now we are witnessing the development of joint scientific projects that are “open” to collaboration. This means that not only have the themes advanced, but the way in which mathematics is being produced changed significantly at the end of the century,”  states the UFMG professor.

Terezinha Nunes, a senior professor at the Education Department at Oxford University, in England, states that the updating of the approach to mathematics in the classroom, as set forth in the Klein Project, is an excellent and very welcome initiative, but she admits that it will prove insufficient as a means of dealing with one of the major bottlenecks in the teaching of this subject, namely, people’s difficulty understanding some of the content. “It isn’t enough to have a mathematics teacher with better training in the subject. The teachers must also know how to teach mathematics and this includes an awareness of students’ difficulties,” states Terezinha, who focuses her research on the writing, reading and mathematics learning process. The research into the process of teaching and learning mathematics is the target of mathematical education, another field of study that is quite well developed in Brazil. As an example, she mentions learning fractions. “Many teachers teach fractions following the same rationale used for integers, which causes huge confusion among the students. It’s very common for students to say that one fifth is more than one third, because the number five in the denominator is greater than three,” she explains. “It isn’t enough to use the allegory of pizza slices to teach fractions; one must show that it is a relation between two numbers, a complex concept that many teachers avoid,”  she states. According to Terezinha, this type of problem is the subject of study not of mathematics, but of mathematical education, hence the fact that this cannot be solved by merely improving the training of teachers and by modernizing curricula. That author of the book Na vida dez, na escola zero [Ten in life, zero at school], Terezinha states that many schools still fail to fully make use of their students’ practical knowledge of mathematics, submitting them to traditional teaching practices. “What we see is that the mathematical performance of students at school is worse than in real life,” she states.

The Klein Project may lack the scope to deal with all the problems of teaching mathematics, but initiatives of other scientific societies are trying to handle them. One of these, headed by SBM, the Brazilian Society of Mathematics, is focusing on creating a professional distance master’s degree course geared toward improving the mathematics teacher training, with the backing of public institutions connected with the Ministry of Education’s Open University of Brazil. The idea is to select one thousand teachers this very year in order to form an initial group that should graduate in March 2011. They will have a certain number of hours of distance training, as well as attending some lectures. SBM is also engaged in re-discussing the curriculum directives for mathematics teachers’ undergraduate courses. “We have the obligation to hold this discussion in order  to improve the undergraduate mathematics teachers’ course,” says Marcelo Viana.

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