Mathematics: Subject Assessment

By Eszter C. Neményi (Grade 1–4) and Zsuzsa Somfai (5–12)

The tradition of mathematics education in Hungary and the expectations of the modern world with regards to schools both stand as testimony to the important role this field of study plays in public education. All students who complete their secondary school studies with final exams study mathematics throughout the entire 12-year duration of their school career.

Different methods of development, problems of various complexity that arise along the way, and last but not least, teacher training and the school structure itself are all reasons that justify the creation of two studies by two separate authors to evaluate the current situation of mathematics.

The study herein deals with grades 5–12, and focuses on issues of mathematics training for students 10–18 years of age. In addition to the professional sources indicated, the author bases her conclusions on many years of personal experience in the field of mathematics training. A questionnaire and two professional conferences were also of assistance in preparation for this work. Two supplementary studies were created for the purpose of subject assessment.¹

The situation of mathematics in the process of subject modernisation: Changes made in the course of content regulation

Renewal in the methodology and content of mathematics training in Hungary since the 1960s can be attributed to the work of Tamás Varga. The essence of his didactical approach – a training concept named the Tamás Varga method in his honour – is this: up-to-date mathematics learning is a process that entails the development of thinking as a whole via the active participation of students as opposed to merely drilling mechanical instruments of knowledge into their heads. In this way, knowledge is expanded through experience, with a strong focus on the age-specific characteristics of the learner. An appropriately guided process of discovery ensures the freedom to make mistakes and leaves room for students to develop their creativity and problem-solving capabilities.

These concepts were realised with the introduction of the primary school mathematics curriculum in 1978. The teaching community, however, was not ready to welcome its application. Many instructors felt a contradiction between the amount of teaching material and the time-consuming discovery method. They believed that new topics included in the subject (basic set theory, elements of combinatorial analysis, the transformational approach in geometry) could only be taught to the certain detriment of arithmetic skills-development. Subsequently, a correction procedure was launched in the interest of successfully implementing the curriculum, which resulted in new recommendations taking into account the experiences described above. The modified curriculum was established in 1986 and in most cases remained in effect until the introduction of the National Core Curriculum (NAT) in 1998.

Preparation for the NAT system had already begun during the previous decade (1989), and a considerable amount of professional and educational policy debate took place prior to its approval in October 1995. At the time, mathematics was one of the only subjects in which a rapid professional consensus was achieved regarding its conception and content. Most schools today operate according to the Framework Curriculum introduced in 2000. Also worth mentioning is the curriculum for special classes beginning in the 7^th grade to assist in the training of pupils who are especially gifted in mathematics.

Characteristic approaches and new features of the mathematics curriculum

The most important goals of mathematics in the upper grades of primary school include familiarising students with the relationship between quantity and space in their environment, to provide up-to-date and applicable knowledge, and to develop their thought processes. On the secondary school level, its responsibility is to facilitate the continued development of independent and systematic thinking as well as the improvement of applicable skills. The crucial basic principles of current mathematics education in Hungary are best reflected in a study created for practicing teachers when the Framework Curriculum was introduced. The following are quoted from this study:²

Primary characteristics of the Framework Curriculum in mathematics

• The most important task of training-education is the formation and development of skills and capabilities, an ongoing responsibility fostered by activation. This approach plays a decisive role in the curriculum, which is why the first column contains material under the heading “Development tasks and activities”.
• Knowledge and content are necessary for development – see Column 2. If we read columns 1 and 2 of the curriculum “horizontally”, we will see the tools and content corresponding to each development task.
• The encouragement of logical thinking is part of a fine Hungarian tradition. (Our mathematics training is strongly influenced by the professional methodology of György Pólya, Rózsa Péter, Tamás Varga and other outstanding Hungarian mathematicians.) Development takes place via concentrated activity, games, demonstration, the appropriate use and handling of instruments, intensive study, well chosen exercises etc. We must work to preserve these teaching methods.
• At the same time, we must adapt to the expectations of the outside world. Nowadays, there is a need for more practical mathematics (economics exercises, graphs and charts, statistics, probability etc.). Most students are interested in these areas, which is a motivating factor. This means that we must be prepared to change the emphasis in our teaching process.
• Mathematics is a part of cultural history, represented in the work of Hungarian and foreign mathematicians, unique mathematics phenomena, and the presentation of easy to understand, yet unresolved questions. This is a motivating factor for students interested in the humanities.
• Libraries, media stores and the Internet can greatly assist in dealing with the previous two points.
• Mathematics is an extremely abstract subject. Students must be exposed to many concrete facts in the course of their study if they are to reach the stage of general abstraction by the age of 12–14. Much representation and concrete demonstration are necessary for the development of an internal view of mathematics. Students reach the stage of deductive reasoning only after finding solutions to many problems in an inductive manner, and in some cases, there is also an important need for factual explanation, demonstration and inductive reasoning in upper-level classes.

The Framework Curriculum retained the main principles stated in the context of the NAT system, as well as content. Teaching materials are structured according to the 5 main themes presented in the NAT curriculum, and methods for the development of logical thinking are incorporated in other topics throughout the entire teaching process. Grades 11-12 are devoted to arranging and summarising individual details.

Recent developments

The process of curricular modernisation in mathematics over the last decade was followed after a slight delay by the improvement of teaching aids, primarily in connection with textbooks. Publishers with experience in creating textbooks for mathematics training organised teams of writers to prepare new textbook series. A common characteristic among these series was that the material put a strong emphasis on the motivation of students’ thinking skills.

To my knowledge, no large-scale methodological experiments are taking place in mathematics at the moment. Tender applications announced by the Foundation for Modernisation in Public Education (KOMA) are one way in which current innovation in the field of mathematics can obtain support.

Challenges in the subject of mathematics

The most important global issues

Some of the essential challenges facing mathematics today are rooted in the current conditions for public education and the consequences of educational policy decisions:

In comparison to curricular lesson plans in effect prior to the framework curriculum, the timeframe for mathematics training on the upper levels of secondary school has suffered a slight decrease. Instructors have difficulty in accepting this since they fear that the secure development of skills will be threatened by the loss of time. This issue is primarily of concern to mathematics instructors teaching in higher grades.

A rise in the age limit for compulsory education and the expansion of secondary school training means that secondary schools now have large numbers of students who are less prepared and less motivated to learn. Naturally, this is mainly a situation facing secondary education, but in addition to questions regarding curricula, it also gives rise to general issues of pedagogy and didactics.

In the course of finding solutions to these problems, we must not neglect the fine results achieved by several decades of care and dedication given to the training of gifted students who participate in mathematics competitions. From the standpoint of training, the search for answers to the challenges described above brings issues of methodology and approach to the forefront. At present, research in this area is sporadic, and there is widespread professional support for the establishment of a Mathematics-Didactics Centre to suitably coordinate this task in light of its seriousness.

Another critical challenge is the changes taking place in education world wide, and depends on whether these are acknowledged, followed or rejected. This issue is more general than the problems facing one particular subject, but directly touches upon the essential concerns of mathematics teaching.

The fundamental question today is this: What constitutes “sound knowledge”? In our case, the question can be taken one step further: Once we have determined what genuine knowledge is, how can this contribute to its formation in mathematics? If we accept that the developed world today prefers the kind of knowledge demonstrated by, for example, the PISA 2000 assessment, then we must regard this study not only as a comparison of international results, but also that of concepts in education – including mathematics.³ There is no doubt about the validity of the idea that schools must do everything possible to educate young people who are capable of succeeding in the labour market and in society. The key lies in the answer to the second question i.e. determining what a school can do in the interest of the latter and how. A great deal of further research and analysis is needed on this issue, and also in the field of mathematics education.

The role of mathematics in the course of learning and life

Certain school subjects can be linked from the aspect that effective learning in these areas strongly determines the achievement of students in other subjects and in different areas of life. In light of the above, it is my belief that languages, mathematics and information technology belong to a single category, and in terms of efforts at systematisation, this relationship is reflected in the situation and experience of mathematics.

Taking into consideration the traditions of the Hungarian school system, basic cultural knowledge and the modern need for the development of various competencies, mathematics is a fundamental, well-respected and important subject. The problem of skills-development, theory vs. practical orientation, and the transfer of applicable knowledge is an issue that requires a careful and refined approach if we are to avoid the pitfalls of oversimplification and schematisation. It is a mistake to assume that mathematics provides applicable knowledge only by teaching formulae to solve practical problems. If someone is able to utilise the systematisation skills and logical reasoning they have learned in mathematics to solve problems that arise in any area of life, then this can most certainly be regarded as a case of knowledge applied.

The relationship between socio-cultural issues and mathematics

The analysis of domestic and international surveys suggests that there is also a strong cultural determination behind the performance of students in mathematics. This may seem surprising in comparison to the previously held traditional view, but in light of the modern interpretation it is a perfectly natural experience. At any rate, we must accept it as fact and take it into account when designating areas for development.

Phenomena mentioned here have much less to do with the unique nature of mathematics than with serious difficulties prevalent in the entire school system. The fundamental issue is that schools and the achievement of students also reflect the high degree of economic and cultural polarisation currently taking place in society. This conclusion is supported by various studies in connection with mathematics or others that simply touch upon the subject. In observing the results taken from these surveys, our experience has been that the children of parents with low qualifications show the weakest performance.⁴, ⁵ Where students live also plays a significant role in their level of achievement; schools located in small communities have the lowest rate of performance in mathematics. The important conclusions of an international comparison study are necessary to call attention to this problem, and the results should be taken very seriously.⁶

An investigation conducted within the framework of the Third International Mathematics and Science Study (TIMSS) aimed to find out more about how mathematics and natural science education depends on culture (or perhaps how it does not). The widely accepted belief until then had been that in contrast to history, languages and social studies, mathematics and natural science education are relatively independent of culture. The study showed this hypothesis to be false: the quality of training varies tremendously in different countries, and mathematics as well as natural science training, along with languages and history education are measurably influenced by national culture. This has far deeper implications than merely the emphasis placed on various elements of training. Throughout the course of their educational history, individual countries have all developed unique methods to guide their students towards the essence of mathematics and natural sciences – and strong cultural factors contribute to these methods. Therefore, when we wish to adapt various concepts of mathematics teaching in the interest of modernisation, we would do well to seriously consider the findings above.

Teaching aids

The development of teaching aids over the past decade can be traced to the analysis of changes in school structure and curricula as well as the establishment of a market for supplementary tools. Printed materials – textbooks and exercise books – continue to play the biggest role in education; approximately 15 years ago, perhaps one or two textbooks were available for each school grade, whereas today we can say that there is an abundant supply of textbooks for both primary and secondary schools. Within certain limits, this is a very good thing.

In addition to textbooks, various electronic teaching aids have also appeared in mathematics training. One goal of the Framework Curriculum was to expand the use of such tools, although current practice shows that only initial steps have been made in this area. Teachers indicate that the available computer equipment in most schools now primarily serves the needs of training in the field of information technology, and so mathematics instructors have no regular access to these tools. We found no schools that had technicians on hand to assist mathematics teachers in their work.

There are different types of interactive software and CD-ROMs for mathematics training, but teachers are largely uninformed in connection with their use and find it difficult to choose from the selection available. Although a list of compulsory teaching aids for mathematics also exists, schools can only afford to acquire various tools of demonstration on a gradual basis.

International comparison

Based on a comparative assessment conducted by the European Mathematics Association, the following data in reference to mathematics training in Hungary provides a brief look at our situation compared to that of other countries:⁷

Positive phenomena:

• Mathematics is taught by qualified professionals
• All students taking final exams are also tested in the subject of mathematics
• The training of gifted students has a long-standing tradition of good results
• We teach verification and pay close attention to the activation of students in the training process
• Bi-lingual schools also function with bi-lingual mathematics teachers

Negative phenomena:

• Our view of mathematics is one-sided and demonstration of how mathematics skills can be utilised in many different ways is lacking
• Teaching practice does not stay abreast of technical development appropriately
• Teachers leave the profession for financial reasons

Teacher training and post-graduate study

One of the most important conditions for effective mathematics teaching is a well-informed, qualified instructor. Despite its valuable and positive tradition, the training of mathematics teachers in Hungary also reflects problems in connection with content and the financial circumstances of educators as well as their level of respect in society.

Careful and detailed study of classic themes (analysis, algebra, geometry) can be regarded as one of the positive aspects of professional training for mathematics instructors. Training institutions handle more up-to-date themes (finite mathematics, statistics, probability theory) with varying degrees of intensity. The amount of time devoted to pedagogical and methodological subjects also differs greatly from institution to institution. The general tendency can be described as follows: the degree of emphasis a given institution places on the study of pedagogical issues and methodology to prepare teachers for the realities of everyday life in schools drops in direct proportion to the grade level teachers are being trained for. In other words, the higher the grade level, the lower the emphasis on these topics. Generally speaking, it would be necessary for teacher training to address challenges in connection with the activities that take place in schools much faster and with more flexibility than it does at present.

Practicing teachers, professionals in educational service institutions and instructors active in the field of teacher training all sense a continuous decline in the financial compensation provided to educators and in respect for the teaching profession in society as a whole. This is also directly reflected in declining professional standards among those who apply for teacher training in mathematics at the university level. The alarmingly low minimum scores needed to gain entrance to universities foreshadow this phenomenon. Everything possible must be done to reverse this tendency; after all, only well-trained, creative teachers are capable of educating creative students.

The orientation and methodological training of teachers in the field is provided in part on the basis of compulsory services offered by pedagogical institutions competing on the market, and partly by private enterprises. Accreditation procedures are meant to guarantee appropriate standards, but experience has shown that this mechanism does not adequately perform its function. It is often hard to distinguish between programs that offer genuine content and those that conceal a low standard of quality behind promising titles.

Practical lessons and recommendations

Rapid changes taking place in society and their direct and indirect influence on the educational system give rise to numerous questions that can only be dealt with appropriately the precise coordination of research and development in education. Both conceptual issues and minor details need to be addressed, tracking and analysing the results of development already in progress, and making the necessary adjustments. In addition, a much stronger and more effective link should be established between research, development and teaching practice. These tasks must also be coordinated and executed in connection with the field of mathematics training. Activity in this area is irregular at the moment, hence the need to establish an effective educational development centre to facilitate the process from the standpoint of both content and organisation, and I suggest this in light of widespread professional support. This kind of Mathematics-Didactics centre would act as an overview of the training process in its entirety, taking into account the mutual effect different areas have on one another, assisting in the establishment of missing points of contact, and ensuring the free flow of professional information.

The prevailing tasks of such an organisation include:

• Tracking the present situation of curricula; maintaining the criteria stipulated under section 1.3 of the Framework Curriculum
• Keeping records on the available selection of textbooks and teaching aids, and ensuring the opportunity for consultation among developers of curricula, textbook authors and practicing instructors
• Establishing contact between professional researchers dealing with methodology and teachers in the field
• Developing concepts for evaluation and measurement, creating tools of measurement and tracking measurement procedures

Based on the analysis of subject observation, short-term research and development tasks are:

• In connection with theoretical topics, varied and thorough examination is necessary to create solutions for the following issues:
  • A comparison of the situation in Hungary with the characteristics of mathematics education in countries that have achieved good results in mathematical competencies, focusing on curricula, methodological procedures and educational traditions
  • What themes should be taught and what methodology should be used to strengthen competency in mathematics?
  • What are the results of a practical orientation in mathematics training, how do these serve to develop students’ progress and how can this be adapted to the tradition of education in Hungary?
  • Describing, assessing and measuring competency in mathematics
• Concerning special subjects and methodology:
  • The creation of exercise sheets and suggestions for methodology in order to assist the differentiated teaching of various materials
  • Producing methodological materials for statistics and probability theory
  • The development of methodological procedures suited to the expansion of training at the secondary school level.