The global perspective

Peter Bender

Paderborn, Germany

Plenary lecture: Walter Oberschelp	Chances and limits for teaching in the information age — Human mind models and society demands
John Berry Roger Fentem	Investigation into student attitudes to using calculators with CAS in learning mathematics
Stefanie Krivsky	The potential of the Internet for innovations in didactics of mathematics
Ewa Lakoma	On the impact of hand-held technology on mathematics learning — From the epistemological point of view
Tatyana Oleinik	A project on the development of critical thinking by using technology
Tadeusz Ratusinski	The role of the computer in discovering mathematical theorems
Monika Schwarze	Self-guided learning — Scenarios and materials from a German pilot project
Angela Schwenk Manfred Berger	Mathematical abilities of university entrants and the adapted use of computers in engineering education
John Searl	Of Babies and Bath Water

For this strand I had prepared the following programme:

"In this strand we want to look beyond mathematics education. Without paying too much attention to the educational system, information technology has and will have far reaching effects on society, i.e. economics, science (and technology), administration, politics, military on a large scale as well as on the vocational, private, and social life of the individuals and smaller groups, possibly in the countries of the first, the second and the third world in rather different ways, and, of course, influencing the educational system and, in particular, mathematics teaching.

The global perspective of information technology does not only consist in these social matters (in a wide sense), but also includes epistemological, ethical, psychological and pedagogical etc. aspects.

The lectures and discussions could be concerned with

the future nature of learning and teaching;

the future contents of school and university education;

the future organization of the educational system;

the influence of information technology on everyday life (with regard to education in school, in particular to mathematics teaching);

information technology as a subject matter (in mathematics, technology, philosophy, social sciences instruction?);

the potential as well as the limits and dangers of information technology with respect to epistemological, ethical, psychological and pedagogical aspects;

the state of the art of information technology (e.g. in the fields of Artificial Intelligence, life sciences, networks etc.) with respect to practical, philosophical, ethical, cognitive points of view;

the overall effects of information technology on societies, and how governmental and non-governmental organization can exert influence.

Contributions to strand 7 can be utopian (with a minimum of realistic background) as well as sceptical. They need not relate explicitly to the field of education, but if they have some relevance to mathematics teaching, it is all the better."

Mainly the plenary lecture given by Walter Oberschelp and John Searl's talk fit the global, more theoretical character of this programme, whereas most of the other talks, originating in the contributors' middle and long term work, could be subsumed under the topic 'the future contents', and they reported more or less intensively about concrete examples of mathematics teaching (which is no disadvantage at all) on high school or university level. After all, they often laid emphasis on educational ideas with their own importance independent from the use of technology in the first place, thus being more affine to strand 7 than to other strands, namely: student attitudes to using calculators (John Berry and Roger Fentem), modules with mathematical contents (Stefanie Krivsky), epistemology (Ewa Lakoma), critical thinking (Tatyana Olejnik), discovery learning (Tadeusz Ratusinski), self directed learning (Monika Schwarze), mathematical abilities (Angela Schwenk and Manfred Berger). In view of this diversity of topics, it goes without saying that the discussions concentrated on the single talks, and nearly no cross connections were drawn.

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Chances and limits for teaching in the information age —

Human mind models and society demands

Walter Oberschelp

Aachen, Germany

1. Introduction

2. General issues

3. Mathematics teaching in the Information Age

IT makes new general forms of learning and teaching possible. Most of the effects are consequences of multi-medial influences, e.g. of TV, which are usually organised by non-educational institutions, often by a commercial enterprise, without an official vision about what people should learn. They learn by watching any kind of entertainment or by communicating via devices like the Internet. Usually the learning process is indirect and implicit; it leads to a certain multiple-choice skill rather than to creative activities. Goal-oriented and IT-based teaching activities have to take into account general principles of successful instruction and of support for mnemonics. I discuss the specific chances of modern technology like programmed interactivity, visualization and animation and pick up classical experiences like learning by playing. A personalized learning context is desirable but hard to get. Fundamental principles of human perception have to be observed – the main temptation is to overwhelm the learner with abundant trash. Dangers and shortcomings of the new technologies are pointed out, especially uncertainty about semantic reliability and loss of reality. We discuss anthropological issues in connection with new fitness requirements. Some capabilities, like to formulate texts or to design interactive menus, are investigated. The most challenging questions for the future will be whether we can conceive convincing learning environments using IT.

Turning to mathematics teaching I first state that mathematics for the future is not necessarily alien to the human mind. Unnecessary formalization, mystification and spasmodic ritualization were partially due to „New Math“. Thus mathematical teaching was made a training based discipline like teaching reading and writing and lost its intellectual charm. I point out that mathematics school teaching usually does not bear the potential that the pupils create original research results: Teaching rather takes place in a garden of pre-paved paths, where the scholar has to learn how to move and how to use available tools appropriately. This means that applying mathematics is prior to finding and proving theorems. A short sketch of the P-NP theory distinguishes different modes of mathematical cognition. The scientific background and the advantages of CAS are outlined; they work like a skateboard, which brings the learner over boring passages. The curriculum of the information age has to abandon obsolete material; candidates are such popular fields like conics, curve discussions and even parts of geometry, but definitely not the exponential function. The meaning of „meaning“ is changing. The availability of CAS at school encourages experimental mathematics and offers new perspectives for assessment. IT-based projects about how to present mathematics to the public are sketched.

Finally I deliver some good reasons why the society should support modern mathematics teaching. The mathematics of daily life becomes complicated, long range predictions are not intelligible for everybody, but society needs many people who are able to control and to explain economic and ecologic decisions on the firm background of our science.

1 Introduction

Successful IT-based teaching requires motivation, understanding, training and didactic sugar. The main problem is to adapt the problem structure to the intellectual structure of the learner and to his individual needs. And there must be results, which are useful for the society.

We experience that the charm of having huge information resources e.g. via internet is only temporary: The present IT scratches only the surface of the human and social demands: The main need of man is not the consumption of news, but production of and interaction with personal signals on a reliable basis in order to be sure of one’s own uniqueness.

Surfing for information through open and heterogeneous nets will lose importance compared with new types of devices, which guarantee, e.g., security of transmission, legal control of transactions and semantic reliability of information. The task to keep the society in good order is incompatible with unrestricted informational liberalism, and the society needs more than a netiquette without obligations. New problems for jurisdiction arise: Information crimes cannot be judged by simply counting bits like peas. Some epistemological problems, which are connected with the concept of information, have to be discussed. And the realization of a global justice is one fundamental basis for the global society.

In particular, I investigate, how math-learning will have to develop: The special problems of math-teaching are the alienation by formalism, the lack of personal appeal and the somewhat metaphysical nature of mathematics, whereas its pragmatic value is often invisible. Since mathematical ideas are often very compact, the abundant information of the Internet is hard to combine with mathematical thinking. And yet, mathematical techniques establish useful tools for mastering the complexity in the global society.

We exemplify problems in private and global economy and in our real physical world and discuss essential and obsolete material. I sketch, how methods for self-guided instruction may be improved. But I emphasize that, due to the anthropological situation, personal instruction and care will become even more important in the future. The satisfaction of really understanding an argument from the scratch and the experience of responsibly solving problems without the assistance of non-transparent tools will become a source of creativity and a well-accepted motive in the education of independent and mature citizens.

2 General issues

What we learn by Information Technology

Teaching and learning are complementary aspects of one and the same thing.

There is a new quality of learning and teaching: While the classical learning modes were induced by family education, school instruction, everyday reality, experiences in profession, leisure and social activities, the availability of multimedia brings a new dimension into the resources from which we learn. New kinds of visual impressions are used and new multi-medial techniques create virtual reality. These innovations lead to some questions:

What is the vision of the society of the teaching endeavours? Which type of education will be favoured or inhibited by politics? The answer is somewhat mixed: The first and usual answer is: Education and teaching have to produce mature humans and responsible citizens.

But there are more specific aims under the surface: One important target is the consumer, who is aimed at by commercial advertising and who is able and willing to spend money. Another one is the ecological citizen, who keeps the earth and its resources in good balance; or the engineer, who intends to change the world and who is ready to spend resources – even by ruinous exploitation – in order to optimise profit; or the scientist, who is ... etc. etc.

The greatest common divisor of these intentions is the agreement to educate human beings who are able to develop under certain liberal conditions.

Surely, the main intention of the different publicly accessible medial influences is not education. It might be described as entertainment; but in the background the goal is to attract attention, to gain authority, to stimulate desires and to satisfy the human demand for entertainment for cash. The omni-presence of new media bears –among others – one irritating consequence: We observe a certain habituation and loss of sensitivity, by which, e.g., the inborn thrill of mystery and crime becomes trivial, such that the educative challenge, e.g., to solve a secret by one’s own thinking usually gets lost.

Another human demand is to get access to communication. But the quality of medial communication is often disappointing: Sending SMS messages lacks the comfort of a personalized context; expressing one’s feelings via smiley is only a joke; and in transmitting a love letter via net we realize that „between male and female there is a little bit more than e-mail“.

The educational essence of these new technologies with respect to the aims of teaching is somewhat poor. The preferential behavioural pattern is voyeurism, the habit is to surf and to zap into suggested directions, and the result is often a feeling of idleness and of a vacuum, which should be filled up. At best we get some multiple choice routine, which might improve the quality of personal decision making a little. But it is not enough to specialize in the four alternatives of a quiz show and to decide by rejecting those cases, which are obviously wrong. These primitive behavioural patterns are very different from those which judge the possible outcomes of a decision by content and which in addition argue on a logical basis. Usually in multi-medial learning we stay at the know-what-level and don’t reach the know-how‑, the know-why- let alone the creativity-level. Learning is much more than compiling facts in a memory. With modern technologies it is possible to collect data on a problem within a quarter of an hour; but reading and selecting the essential information requires much higher abilities. Good support in this direction is given by using high quality search engines. The techniques of applying hints, of link-evaluation and of pattern recognition often produce satisfaction and surprise in the thoughtful user. It may be that those stimulations bring about a new capability to carry out systematic combinations as well as to look for and to pick up hints.

Of course, there are some arrangements, which, at least in principle, do not only turn us on or make us watch them, but also enable us to become active. In using the new technologies we can choose our own goals of action: Watching a cookery- or a gymnastics-course in TV might lead to own activities. Contact groups in the Internet and visual conferences give even chances. But communication is often spoiled by the desire to stay anonymous or – on the contrary – to exhibit oneself. From the viewpoint of the net-management the ideal activity is anyway to fill an order form and to increase a quota.

The way how we learn from all these techniques is an implicit one.

One of the most efficient modes of implicit learning is playing. Teachers as well as the media can learn a lot from the nature of homo ludens, as we know already from J. Huizinga ^[1].

One fundamental experience with children playing shopkeeper is that they insist on selling things to instead of buying them from other children. Man is producer by nature and not consumer; he is rather speaker than listener. Why? This is a way to make oneself sure of one’s own uniqueness. I believe that there are serious misunderstandings about the principal human motives. Neither is Siegmund Freud’s theory of complexes convincing to me, nor do I share the obvious belief of information technology that curiosity is the main attraction. Big-brother-voyeurism is only temporarily dominant. In my opinion man needs a tremendous amount of self-confidence and of self-confirmation, and from this fundamental observation many features of successful learning can be deduced. Therefore teachers should not forget to award a good performance with sensitive empathy; this is more efficient than to fill learnware into the Nuremberg funnel. Usually, people who claim to know everything are disappointed about their low degree of acknowledgement in the society.

One has also to care for the negative aspects of this observation: The psychology of envy and jealousy and of mobbing colleagues also has its sources in the need for self-confirmation. Here obstacles for learning do arise, and they have to be recognized. The desire to get influence and power, a high status in the society, money and luxury and even perversions like finding satisfaction in the horrible glory of torturing and terror fit into this explanation pattern, which should be familiar to the experienced teacher. We live in a culture of glamour and of sparkling appearance. Educators have also to guide responsibly those students, who sink into despair because of their seeming imperfection; teachers have to appeal to those (hidden) qualities of their pupils which the competitors don’t have.

Thus we come into the field of ethics of teaching with the problem to balance knowledge and know-how against emotional and spiritual values. Playing is an exemplary model for that. A playing individual with her or his inborn activity has the chance to become creative: It is a good advice to keep the play free from ritualization, mere gambling and hacking. You better support the experimental trait of playing; change rules, support associations and the observation of similarities, encourage fantasy, and you are at the very source of high quality implicit learning and teaching!

The role of implicit learning becomes more and more prominent, and explicit teaching gets a competitor. In the past instruction was the partner of explicit learning; today instruction has to cover also implicit learning.

This does not mean that our schools have to become playgrounds – even though former attempts in this direction (e.g. M. Montessori) were in parts successful. In my opinion today’s teaching has to make serious attempts to incorporate the experiences with the techniques of implicit teaching, in particular from IT, into the organized learning processes.

The principle of explicit learning will keep its importance. But we have to take a fresh look at the differences in success of distinct classical media for explicit teaching: Why has a zoo much more teaching effect than a botanic garden? The trivial answer should be kept in mind: Plants are fixed and static, but animals are attractive because they are animated.

Structures of successful teaching

Why are Internet conferences not that successful?

They could save time and other expenses. But the charm of a region like Wörther See or Côte d’Azur, of real personal relations, to see and to contact humans who consist of flesh and blood is a crucial motor of the learning success. Why aren’t you sitting in your home office and watch my speech via Internet or, even worse, load down the video, which I send to you from my office in Aachen by e-mail? Obviously, we feel the importance of fresh air, of mobility and, more generally speaking, of the authenticity of an event. Everybody who ignores this and plans to change school instruction by substituting real teachers by e-learning software should be condemned to spend his further life in front of a big multi-media screen with air condition. We must claim at any time that we need human contacts and that therefore it is legitimate in principle to expend money in order to bring the teacher and the learner together. The teacher has to be a living help function in addition to his usual job of teaching some material. This is even more important in our time, when the computer help function still often does not do what it should. The task to analyse shortcomings of automatic help devices and to develop remedies is still unsolved.

This is partly due to the trend of producers to implant unnecessary gadgets into systems and then to claim that their products are universal. But universal systems will be discarded more and more because of their complexity. Since the ordinary human being is a creature of habit and cannot master many different recipes at the same time, the development of invariant and specialized user interfaces is most likely to be successful.

In general, we are losing direct contact to devices. Therefore we often regard those systems as enemies, and we are exposed to a helpless call centre philosophy. The interdependence is growing and de-motivates independent creativity.

Although we are aware that the greatest drawback of learning via IT is its failure to transmit something like a life context and that therefore only the surface of some human demands is scratched, we should not resign: IT is good for something: It provides a partial substitute for personal interaction: – computer interactivity, visualization and animation. It is worthwhile to develop models and applications, which fit in some way the structures of the human mind. Being far from underestimating those tools we have to improve the existing models, always keeping in mind the human nature and looking carefully at the anthropogenic constants, which are in danger.

Of course, the elementary anthropogenic needs like eating, sleeping and loafing around are not affected. But progress in technology has very often changed intellectual and other capabilities of man: Hand writing and later on Gutenberg’s printing technique decreased the importance of memory; transportation techniques made it unnecessary for men to be good walkers. Making music by one’s own is somewhat unusual today; the pocket calculator has made multiplication algorithms and looking up logarithm-tables obsolete. And the iconisation of user interfaces in connection with the drag and drop paradigm gives illiterate people (almost) equal chances in using IT. So will literacy also become obsolete in the long run? Will first writing and then reading become outdated?

If we consider these historical facts without emotions, some change and loss of human capabilities which is to be expected as a consequence of information technology needs not to be looked upon as a signal to restrict or even to stop the utilization of IT; life goes on anyway. But we have to observe those changes and have to decide responsibly whether some developments have to be checked critically.

I also mention an apparently positive example: Action games on the computer are attractive for the youth. They train and develop the qualification to react watchfully, quickly and appropriately in changing situations. Beyond all pedagogic criticism a new anthropologic quality seems to develop here, and it might be useful for mastering specific challenges in the modern world, e.g. in traffic situations.

Surely, in the long run a subsequent Darwinian selection will change the genetic material of mankind according to the changing fitness criteria.

According to my own observations, the most obvious change-taking place right now is – with respect to young people – the lack to conceive and to design a written text. Working with menus on the screen, operating with icons and filling data into forms is a counter-indication to conceiving an own textual design.

I know this from my own experiences with university students: It is not their main problem to understand a difficult scientific paper (even though this remains, of course, a problem): A lot of students simply fail – which older people would not expect at all – writing down their own exposition (protocol) of their experiences with the problem. Today the ability to develop structures, forms and menus (with tools like Visual Basic) with all those features like interaction points, fields, switches, buttons, slide bars etc. can be found rather rarely. Good programming languages are an invaluable help for the programmer. Surely, at the same time they press him into a developmental corset, which is limited in structure with respect to many creative features.

However, the existence of those design tools is a real progress, and a specific teaching how to use them may be a general goal of higher education, which, of course, divides mankind as ever into masters and servants. Of course, it is an additional problem to explain the structure and the ideas behind those products and to teach people to use them appropriately.

Here I recur to my own experiences with students of a new vocational field called „Technical Editor“ (Technischer Redakteur) at the Technical University (RWTH) of Aachen: Look at books which claim to explain e.g. MS Word. Usually we find a more or less complete enumeration of the user features, which is sorted along the order of menu points, icons and fields. Texts are rare which support a natural learning process, starting with the beginner’s problems and extending the actions in concentric circles with pointers for building bridges of associations and of comprehension. From my own experience in the well established German system of adult education (Volks-Hochschule) I know that students in the computer department usually have to take three or four courses until they are really fit in a field: Often the instructors are unable to teach their courses well (partially thanks to bad teaching material or, even worse, thanks to the poor quality of the software itself). The instructors throw us into the scene of a complete menu, and you don’t actually see what you could see: We do notice only those objects which are moving, flashing or have a striking warning colour – this is a residue from our genetic heritage, from the time when our ancestors were climbing a tree when a danger occurred.

Attention is the mother of learning. If a flock of words or a speedy scrollbar is pouring into your perception, you better close your eyes and hide your mind behind that curtain. And, since flushing and moving commercial demons on websites can distract our attention, the important information is very often camouflaged so that our mind cannot detect the essentials.

For the group of "technical editor"-students mentioned above it was a painful experience to recognize that it takes more than to learn a certain science (e.g. computer science) and to learn a good style of (e.g. German) language in order to write successful textbooks or to conceive good e-learning software. The ability to form a convincing learning environment from all this stuff is the real qualification, and we need additional interdisciplinary efforts in psychology, didactics, pedagogy, information- and communication-science etc., which are rather rare at the moment.

The main components of successful learning are showing and imitation, combination, exercise and repetition, training and reward, didactic sugar and redundancy; and last not least interaction with a human partner who is involved into the same problem. In addition a clear problem structure is helpful, aided by mnemonic features. There are well-established classical mnemonics: The rhythm in memorizing Latin examples or mathematical theorems has turned out to be very efficient.

But the most efficient mnemonics are given by animated motions in space, colour or intensity, since our brain, in connection with the visual apparatus, is trained by evolutionary selection to pay special attention to those actions. Fortunately, modern IT-technology is able to take care of these needs: The advanced standard of visualization and animation techniques in 2D and in 3D which has been reached in the meantime makes cheap productions in these areas possible. Now the main problem for the producers of e-learning material remains to conceive programs, which fit the learning structure of the mind. An immense field of work is ahead; IT-technique alone will not do the job.

Dangers and the future

I am not sure, whether the distributed and free internet in its present realization will survive: Its marvellously successful openness and heterogeneity might dig its own grave: The possible invasion of chaotic worms, the inundation with commercials, unsolved problems with filters for the protection of children, illegal and annoying pornography are obvious dangers. In addition, the uncertainty about responsibilities about whether the rights and the secrets of a person or a group are violated and about how to persecute the culprit is virulent. There is another somewhat reverse aspect: Granting world-wide patents for innovations might affect the mental and intellectual development of people who simply want to free other people from the shackles of monopolistic over-copyrighting.

Under the expectation that the world will be a global village I doubt that a fair global jurisdiction can be established in the near future. We had to learn painfully, that the global civilisation yields polarisation, radicalism and terrorism.

The most serious problem of the Internet is the absence of semantic reliability and a lack of certification to distinguish values from trash. A remedy seems only possible for (password‑) protected technologies, which are – of course – not available free of charge.

Uncertainty about the authority and truth of information has another unpleasant consequence: It is a source of tremendous superstition and of a turn to esoterics in our society.

It seems evident that in the information age the user of any net technology has to learn much more about the nature of information. But: There are important lessons, which are not yet realized by the public. An example: The value of information cannot be estimated by counting bits like peas. Shannon’s mathematical information theory is not the last cry concerning information, because this theory only takes into account context assumptions about probabilities for the events, which have to be considered.

There are much more epistemological and cognitive categories which characterize an information context. In the same way as there are star moments of history there are star pieces of information – maybe only binary hints – from which we can learn more than from a data dump of terabytes. Being an insider does not mean to have more stuff in the brain than other people, but to know facts which are relevant for a problem-context. We must note that the simple spelling of a sequence like the genetic code is not necessarily the key to understanding. And that the alleged progress in the field of artificial intelligence, staged very skilfully, does not bring nearer the solution of the still unsolved problem how to realize and to understand natural intelligence. Here we are confronted with a limit for mastering information issues with the help of mathematics and information technology.

Another crucial shortcoming of IT-based learning is that this technology is – as I mentioned above – inappropriate for transmitting real life contexts and thus scratches only the surface of human demands. We have to accept that, in addition to the classical competencies (which are of a serving, structuring and designing nature), a very general communication competence, namely the ability to generate real life contact and content, becomes important and highly estimated in our automatised world. One can expect that jobs, which give human support – like teachers, nurses and social workers – will rise in their social rank.

Another aspect of the distance of IT towards real life is a risk to which in particular non-experienced young people are exposed: The confusion of virtual with true (real) reality and the danger to become addicted to a life in a dream world. Modern visualization technology allows nearly every trick: Why should it be impossible to develop a perfect camouflage tool to make a man invisible? Why shouldn’t it be possible to make a man shrink and enlarge at will? What about having virtual sex with a famous movie star?

Even though I concentrated on dangers and shortcomings of IT in this section, the future of IT-teaching is not only dark. I also mentioned several positive experiences and chances of IT. And, furthermore, we can learn from our experiences, may they be positive or not, and can conceive and develop responsible means for the explicit and the implicit teaching mode by adapting our IT-teaching tools to the subject matter structure of the problem and to the intellectual structure of the learner.

3 Mathematics teaching in the Information Age

Is mathematics an alien to the human mind?

Now I turn to mathematics as the main subject of our paper.

A common opinion says that mathematical thinking and the natural feeling of man are strangers to each other. Mathematics as a formalistic system may be fine for egg-headed intellectuals who, from their ivory tower, try to reduce the many facets of human life to certain abstract principles about numbers, space and time. From that point of view doing mathematics is in the end only an annoying activity which admittedly might be good for intellectual training, as a source of undoubted statements, obviously as a toolbox for engineers and scientists, and for no other reason a compulsory field for school instruction.

The high standing of mathematics in the timetables of schools worldwide is explained partly by this quality of mathematics: to be indispensable like reading and writing. But, as a consequence of pocket calculators and CAS, marching in step with the decline of the literal skills, mathematical skills – in particular calculation skills – could also become obsolete. Then there might only remain the selection argument: Someone, who survives mathematics instruction, must belong to an intellectual elite, because her or his natural mind resisted the attacks of the strange world of abstract thinking.

Consider the leading people in our society who can look down at the class of technical servants for our world: Many of them are not ashamed to confess that their attitude to mathematics is that of a stranger. A typical example happened in a German quiz show: The candidate had to select from several answers the right one to the question „how much is 30 divided by ½?“ When he chose "60", the renowned quizmaster looked into his paper: „The answer is correct, but please don’t ask me why“ – and the audience approved this confession silently.

Is mathematics really that strange to natural thinking? Of course, only humans are able to do mathematics; but since the earliest days of our culture man has used mathematics not only as a tool to deal with daily life problems like measuring farmland, managing trade problems or predicting remarkable astronomic constellations. Man also very early abstracted the fundamental structures behind these ever repeating tasks. In the course of history some mathematical disciplines became obsolete, e.g. computistic techniques to determine the date of Easter, spherical trigonometry for navigation, or the slide rule, logarithms and interpolation techniques for calculating product terms. The focus of mathematics has changed very often during history. In ancient times mathematics was also needed for a first understanding of nature, astronomy and geography. While the eighteenth century invested much mathematics into the kinematics and the dynamics of the new astronomy with the abstract concepts of gravitation, energy and mechanics of motion, the nineteenth century did this to physics and engineering. The computer age has brought new highlights into mathematics, but school mathematics has not yet incorporated much of it.

Most important for our situation at school is the second part of the last century, which has experienced a formalistic era. In my opinion the so-called New Math (the overrating of set theory with axiomatics and Bourbakism in the background as an elaboration of the important theories of G. Cantor, Dedekind, Peano, Hilbert etc.) was a disaster for the public acceptance of mathematics. Felix Klein’s wonderful geometry as a theory of transformations – especially in vector notation – caused at school a lot of spasmodic features. I just mention the battle against the congruence theorems for triangles (which are one of the most ingenious inventions of early geometry). And the permanent confusion between a vector fixed to the origin and an arrow class. I know this from my own time as a schoolteacher years ago. The important achievements of these great scientists, which were essential for the clarification of mathematics and for fundamental research, were pushed into school rather brutally. There they induced mathematical activities, which looked quite mysterious to the parents of our students and gave mathematics the image of being incomprehensible and harassing, – an image which is lasting up to today, even though the New Math is now out of school.

What did we learn from that? Mathematics instruction becomes an annoyance if it only has to apply and to confirm structural ideologies. It gains more acceptance by the involved learners, if it is in contact with familiar facts, if it works in a transparent manner and if it challenges imagination.

Another criticism has to be expressed with respect to the over-emphasis of the notions of limit and continuity. The Cauchy-Weierstraß purification of the formerly somewhat vague ideas about the continuum – (as they were inherent in Euler’s and even in Gauß’s and Laplace’s analysis for physicists and astronomers) – was the basis for material for the school, which can hardly be justified in my opinion. I remember a note in one of my schoolbooks, saying that the proof of the equation lim {(a_n+b_n)} = lim {a_n} + lim {b_n} is „a matter of university math“. It is this type of mystification, which made mathematics such a terrifying field.

A lot of today’s mathematics looks deterrent to the non-specialist, because there is involved much knowledge which is good for the design of assessment tests, but nevertheless obsolete or at least not in the core of the present mathematical interest: I mention the splendid theory of conics, which was fundamental for Kepler’s, Newton’s and Gauß’s astronomy and which I personally like very much (since conics form a very nice chapter concerning the mathematics of sundials).

I also mention – with some hesitation – the algebraic discussion of curves: The charm of this theory is based on the fact, that the degree of a polynomial decreases under differentiation and makes the calculation of zeroes easier for degree 3; and that the basic exponential and trigonometric functions essentially reproduce themselves under differentiating once or twice. Those features are in some sense accidental and not applicable to curve discussions with more general functions. And the notorious p-q-formula for the zeros of quadratic polynomials is of no practical help for people, who want to find zeros for arbitrary polynomials. The simplest CAS-devices, which belong to the normal equipment of a pupil in our time, solves all special and general problems in the same way as the pocket calculator works on numerical problems, and they achieve much more. Let’s do this material with a CAS!

And facing the danger to be condemned for that, I predict that the importance of geometry will decline even more than it did already. Geometry is not the only context to give meaning to abstract algebraic concepts (unless one considers geometry as a proper, even small – maybe tiny – part of graphics and of real world space exploration). What „meaning“ really means depends on the context, which one is used to. I claim that a manager has a much more concrete understanding of the success in his company, if the figures are presented to him as an algebraic vector, which carries meaning for him in itself. Thus for many people an ordered tuple of real numbers is better to understand than any visualization in 2D or 3D space. Maybe my mind is spoiled by my close contact to informatics students; but these people connect a clear and concise meaning with data structures like matrices and vector columns and with notions like linear independence, and they refuse to pay attention to any geometric interpretation. One may say that this is a pitiful loss; but the importance of geodesic surveys for territorial claims in a land-rush does no longer exist. Certain losses in the course of changing technological paradigms are inevitable. In the future the ability of geometric imagination might be not much more than a wonderful hobby, which not everybody can afford. Four or more dimensions are conceivable for modern people without the need of plunging into the mysteries of geometry!

On the other hand many really important mathematical notions are quite unknown to the modern man: Most well educated people don’t have the slightest idea of the practical significance of powers of integers like 2ⁿ, of factorials n! and of binomial coefficients (let alone the normal distribution). But these functions are the standard basis for the ordinary discrete mathematics in daily life, as far as organizing, planning and systematic combinatorial testing of possible situations is concerned. I remember with amusement an erroneous statement by a very influential German mathematician some 30 years ago (H. Behnke), who claimed that combinatorics (and discrete mathematics as a whole) had had a great history, but would never again get mathematical importance. But today we feel that the spectacular Y2K- (year-2000) sequencing of the human genome is – parallel to the accompanying technological progress – the beginning of a triumph for discrete mathematics affected by the computer. The subsequent problem of finding out what these sequences mean again poses challenges to discrete mathematics and to data manipulation techniques (like finding the longest common substring of several strings or recognizing shortest fitting superstrings), which in a simpler form occur also in daily life.

Most of us are not really aware of what is new in the mathematics of the information age: Of course – the classical theories with their impressing network of definitions and of elegant and deep theorems will advance further. But the era of discrete mathematics has begun, of number theory, graph theory, in competition with algorithms for data structures and techniques of data mining. Some of these fields have been incorporated by informatics. More generally speaking, the field of experimental mathematics is expanding. In this field we are looking for facts, which are not covered by elegant super theorems.

All this is or will be the mathematics of the information age, and it will be concrete mathematics very close to the experience of everyday life. We will emphasize the „exciting“ invitation to „add 1+1“ and to argue, e.g., with simple considerations on finite sets rather than to look for complicated theories and techniques in artificial or over-specialized application contexts. We shall have to teach, that heuristics is a creative technique of mathematics, if we are aware of the limits of this practice. In particular the most important paradigm of greedy heuristics ^[2] as the strategy at the very beginning of optimisation has to be discussed in some of its many applications. And we have to build a bridge to those non-mathematical paradigms like the evolutionary optimisation of nature, which can also be modelled mathematically in their basic structure. There is no Rubicon between qualitative and quantitative reasoning! Logic, common sense, our emotions and many other qualities of our human kernel provide transitions between the „exact“ sciences and the full wealth of our life.

The discouraging results of the TIMSS concerning German pupils are not a national catastrophe with respect to our school system, but a clear indication that the developers of those tests and some other national curricula are well ahead of the future development.

And future mathematics will be accompanied by algorithms, which have been in the very centre of mathematics since the times of Euclid. While informatics occupied algorithmics as one of its basic fields, mathematics also has to go back to algorithms. Since in my opinion algorithms are common to both sciences, there is no reason to quarrel about where algorithms do legally belong to.

Nota bene: Doing important school mathematics is a natural, although intellectual, activity for our mind. Most of school mathematics in the information age is – or should be – comprehensible to the normal intellect. Of course, in addition we need motivation and some devotion in teaching it. In a later section I will explain some more aspects of the mathematics, which is required in the information age.

Mathematics is modelling, describing and controlling the behaviour of systems and of activities, which occur in our life and our society. It is trivial that not everything can be explained by mathematics, but on the other hand there is no complicated action, which can be readily understood without mathematics.

There are important first steps in realizing these claims: In his nice touring exhibition „Mathematik zum Anfassen“ (touchable mathematics) A. Beutelspacher (Gießen) gives good examples for the observation that mathematics is near to daily life. One can discuss about some details of his exposition: In my opinion Beutelspacher should give more hints for explanations of what is going on in the solutions of the problems which are posed. Mathematics is not self-explaining. In general it is not enough to pose problems and to suggest devices for the solution. Usually we have to explain solutions and to reduce them to simpler, well-known facts or to provide the learner with hints for further investigations. But nevertheless Beutelspacher’s basic idea – that mathematics can be interesting, near to everyday-experience and that it is not alien to the human mind – is fascinating and convincing, at least partially.

The idea of self-guided learning in contrast to cooperative learning needs to be discussed again for mathematics. The ability to work in a team is only one position; autonomy in thinking is another one. I, personally, am tired of hearing again and again that only cooperative learning effects progress in learning mathematics. There are ways for mathematical creativity, which differ essentially from cooperative work in daily life contexts; the lone wolf is also an important figure in the mathematical scene. A cooperative mathematical call centre is a nice idea, but not the ultima ratio for future mathematics. It seems to be just a dream of teachers, that in the difficult process of learning mathematics our understanding relies only on short-step communication between different people.

The impact of IT on mathematics teaching

I will sketch some fields, where IT can support mathematics instruction. If we look for internet-aided mathematics instruction, the present situation seems quite disappointing: The internet might be good to support those fields of school-instruction, where a lot of empirical data has to be provided, e.g. history, geography, politics and biology. But does there really exist some interesting mathematical material, which can be taken from the Internet or from other media better than from books or from personal expositions by the teacher?

The answer is Yes. But it is not a bulk of theorems or other facts, which is relevant, but a great variety of e-learning material, of programs in well-conceived software, which are and will be of interest. We get to know Cabri-Java and Internet based DGS as tools to represent geometric functionalities with vivid dynamisations.

The facilities of CAS are extended by more general and publicly available techniques for graphic processing. The main benefit of those systems consists of the power to transform abstract situations into images and functional dependencies into animated objects like curves with motion, i.e. into paradigms preferred by our sensorial apparatus. There exist a lot of other tools, which support animated presentations. The teacher can produce Power Point presentations with some nice animation facilities. Furthermore, spreadsheets for the visualization of facts and structures as well as mathematical movies can be produced with the help of non-expensive technologies.

But the animation potential has not yet been bidden out: Future techniques are: Morphing as a technique to transform arbitrary objects into each other by continuous (non-linear) transformations; an input tool in form of a generalized mouse to navigate in three dimensions; the parallel management of several parameters in generating geometric objects in order to change the weight and the position of points in Bézier- and spline-curve generation: While the traditional computer mouse is a device to play on the parameter keyboard with only one finger, we shall be offered in the future a parameter organ with several manuals. It is my dream that there will be executable software, which displays the zeros of a high degree polynomial in the Gaussian complex plane and allows to change the coefficients continuously.

Most of us underestimate the present and the future chances in managing „unassigned variables“. Based on the powerful theory of unification in term rewriting systems ^[3] with methods of universal algebra, the potential of a CAS to recognize, e.g., algebraic identities will be improved further. And what is much more: Based on editor techniques which are in principle available today, those systems will care for comments, which not only give us theorems, but hints to prove true identities with the help of available references. And which, instead of simply bringing in the verdict of „wrong“, will show us where our mistakes are and which type of confusion might have caused them.

Thus the present CAS are good for control tasks, for doing calculations and presentations and for recognizing formal situations; but they are somewhat weak in aiming at „interesting“ features. Therefore CAS require an intelligent user who searches into a fruitful direction.

Of course, usually there is also an environment of inputs for pre-fabricated features like the percent- or the rule-of-three-button on the pocket calculator. But a naive user will not profit much from those aids, since an ill-prepared press on such a button doesn’t really confirm correctness of the application, but looks a bit like gambling with an accidental success.

Presently there are still many wishes open: Specialized people would like to have systems with some really essential tools like a comfortable editor for mathematical formulas or helpful software to create an index of subjects for a written text.

On the other hand I think that many computer algebra and office systems have incorporated too many playful automatic options, which annoy and disturb the user. In the long run this overloading will affect the willingness to use these systems. The producers of pocket calculators did already learn their lesson: Since at present only few customers are able to use non-elementary features like the exponential or the trigonometric functions appropriately, most calculators are only equipped with basic arithmetic.

I am working on the design of a science centre (science museum) project at Cologne, where in the computer division the possibilities of IT to teach mathematical processes via visualization will be presented in several exhibits. I mention a few of them – we also treated some topics in seminars for beginners at the RWTH Aachen, which deal with the visualization of algorithms.

One example is about primes – a mathematical story over millenia in its surprising modern context: We first look at a primality test for a number with at most 12 decimal digits and then for a product decomposition of such a number. The second question turns out to be much more difficult than the first. But this is only a prelude to the design of a RSA crypto-system for data security. The learners can gradually complete their knowledge about this most fascinating topic, – and it is really interesting in every detail. The special charm of prime numbers consists in the easy accessibility of many statements about them (although they often are very hard to prove). Instead of giving a proof the learners can – with the aid of a carefully programmed software – make their own empirical experiences and thus get respect for the mysterious structure of (prime) numbers. Statements like Dirichlet’s progression theorem can be investigated for different residue classes, which can be selected interactively by the learners.

Another exhibit of our project is concerned with the idea of a random sequence and Monte Carlo techniques. The problem to generate a random sequence is the first major challenge. As a good technique it turns out to use the decimal expansion of the circle number p=3.14159...We know, that the transcendence of p was proved more than 100 years ago. But more interesting for us: It is more or less empirically known that p seems to be a perfect random generator. p is known up to more than 10¹¹ digits, these can partly be loaded down from the Internet. If we take the first 10⁷ digits, we can control the randomness of this sequence empirically: Take your own birthday in the 6-digit format, e.g. 290383, and look for the first occurrence of this sequence in the decimal expansion of p. In doing so for different data one can get a good feeling for randomness (this is also one of Beutelspacher’s exhibits).

This scenario can also be used to produce random discrete structures step by step. We can construct random graphs edge by edge in that way, and the learners can exercise different algorithms on such an example (which cannot be manipulated by some invisible instance!), e.g. Kruskal’s algorithm ^[4] to find a minimal spanning tree in a (valuated) graph.

Other possible visualization/animation projects, which may be realized, are the „Tower of Hanoi“^[5]or the famous sorting algorithm „Heapsort“^[6]. In these examples the attention can be attracted by moving objects being processed during the runtime of the algorithm. Also attractive is a project, which shows in a dynamic fashion the possibilities to transfer data in permutation nets (these are basic structures for the internet). Many more examples could be given.

There are well-established techniques for the production of those animations ^[7]. A useful technique takes external Java applets, which are transferred to the learner via Internet and then are processed by the personal browser. The most important remaining problems concern us, the teachers: How should a good visualization and animation be conceived?

I must admit that all these tools do not contribute to the promotion of mathematical creativity. Here are definite limits of IT-support for mathematics teaching, and there is also a limit for classical instruction. The disillusioning truth is that in mathematics instruction the proportion of genuine research creativity is only small. It is a self-deception to believe that pupils could really find a theorem on their very own. The world’s very rich fund of mathematical discoveries is the result of a history lasting for millenia of real progress, which collected all the essential discoveries. The research results permanently filled up the ever-growing stock of mathematical knowledge to its present level.

We repeat: To claim that school or university students during mathematics instruction really re-invent theorems which were discovered by a mathematical genius centuries ago is a mere self-deception.

So the question arises: How can we expect that a normal mathematics student masters within a semester a bulk of material which has been developed by mathematicians over centuries? There must be a different quality of passive learning. And what is the role of the teacher in this process?

Fortunately, abstract complexity theory tells us what is really going on: The P-NP theory ^[8] has detected the fundamental difference between finding a new truth and verifying it. Is there really a complexity gap between the deterministically polynomial and the non-deterministically polynomial problems? This is the most challenging problem to theoretical computer science in our new century. Since 30 years a proof or a disproof of this conjecture has been overdue. One equivalent problem is, e.g., whether finding a so-called Hamiltonian cycle in a given graph is essentially harder than to verify that a certain sequence of edges is a Hamiltonian tour (which is easy). There is an overwhelming abundance of material that indicates the truth of the conjecture P¹NP. And as long as it is not disproved – what nobody really believes to happen – we feel authorized to behave as if the conjecture was true.

This situation entitles us to say that those teachers who claim that they can really teach their pupils how to find a theorem are chasing an illusion. What they really do is mousetrap-induction. In the best case the teacher puts baits on the way in order to lure the students into a region (which was prepared in advance) and finally to let them verify everything. I am not ironic – this is the appropriate way of mathematics instruction, and it can be great fun for a teacher to have good students, i.e. to have a big fish at his rod. And this indeed is something.

The so-called Markgraf-Karl-Projekt of artificial intelligence ^[9] at Karlsruhe (which claimed that a computer would be able to prove and even to find automatically all the theorems of a normal mathematical textbook) has been buried silently. In my opinion the state of the art in AI is disappointing ^[10], and this has been the case since several decades: Automatic finding of nontrivial theorems is impossible by now; proving a conjectured theorem is possible to a certain extent, but it is not really done by a CAS. If such a system answers that a certain series like Sn^-1 does not converge, it didn’t find the (very nice) proof, but took this fact from a database, which contains information about series under substitution techniques for terms. It will answer, that the sum of reciprocal squares is p²/6, if the theory of Riemann’s Zeta-function is built in. And a good CAS will give you – presently without any comment – the assertion that the sum of the reciprocal third powers is irrational since it knows that this was proved by Apéry ^[11] some 15 years ago.

There is a big difference between teaching mathematics and teaching empirical sciences: Mathematical instruction cannot be viewed as a geographic expedition into an unknown country where the research results fall into our bag like ripe apples from a tree. It is rather a travel into a country where many people have already paved our way and where – in Newton’s words – we must stand on the shoulders of giants if we really can see something new – unless we are giants ourselves.

But all this is no reason for despair or for blaming IT to be an insufficient aid for mathematics instruction: IT can help us to create application problems of high practical significance and to support our solution steps without hiding the key for understanding what’s going on. There are differences about the mathematical preconditions of the learning process: Should we fix our final goal in advance, or should we – like Jean Flower ^[12] on this conference – decide in an online democratic decision process where and how we want to proceed and then approach a solution via DGS? The latter attempt will fail in complex situations. Can we, e.g., start with a vague notion of the best fitting ellipses for the orbit of the moon in space? This problem is complicated and has occupied the endeavours of many people for a long time. Here only a recourse to astronomy as a specialized field (and no general discussion) can help us: In a simultaneous process of optimisation one has to find the main axis, eccentricity and knot localization of the best „osculating“ ellipse which is consistent with local and global quantities (e.g. maximal distance in the apogee) in our empirical data!

It is the task of mathematics teaching to enable the student to walk by himself and to find signposts in this pre-paved world of mathematics. This is much more than only to memorize what is in there or to stare at the dreadfully spectacular landscape of mathematics with the eyes of a voyeur or of a terrified deer. This is the methodology for the design of teaching mathematics, which goes far beyond the naive attempts to push mathematical stuff into the student’s brain by watering him with the e-learning shower. In other words: We have to equip the students with walking techniques and to assist them in distinguishing different paths and in recognizing the best way to reach a destination.

Just walk your way to your destination in secured steps! And be sure: Walking in the garden of mathematics needs not always to be troublesome, but might be delightful and even relaxing. And it is worthwhile to practise passing through sequences of algorithmic and logical argumentations. Such a sequence is not as rigid as a technical module, which can be fit into a device like a work-piece. It has to be designed, varied and polished with appropriate and problem-specific sensitiveness, and this way to proceed is typical for mathematics.

The role of CAS in mathematics instruction is that of a skateboard which brings us quickly over boring passages – but IT helps us to really move and not to stay at some point as spectators. The acceptance of CAS is usually good on the learners’ side, and some of them feel that they have become addicts, especially in exams when they are not allowed to use it. On the other hand, investigations of R. Fentem ^[13] show that even after a long time of using CAS people are convinced that they have not lost their handicraft abilities in mathematics. One can doubt about that; in any case it seems desirable that a regular fitness training in mathematical skills should produce „mathematical sweat“ in order to keep us fit for our movements in the garden of Eden of mathematics. The advice is: Move, and don’t forget your handicraft abilities!

We are aware of the problems which arise in connection with assessment issues, but I only say a few words here: We shall have to develop many new methods for testing. The main problem will be to convince the public that these new techniques are adequate to assess a student’s performance in mathematics in this environment. We have to make clear that IT is not good only for organizing and evaluating multiple choice tests and thus for saving teaching personnel. The human learner is the most valuable object we are dealing with in our culture, and it is a shame if we apply economic optimisation and rationalization efforts to our human capital only in order to save money. Assessing mathematical performance means judging the style of the walker (in a 6 point scale) like that of a figure skater. This does not exclude local performance tests with the help of the stopwatch and the measuring tape and a computerized evaluation of those tests. But it is of crucial importance that the assessment refers to personalized features; and in any case an editor for the localization of the occurring mistakes has to be integrated into the test software.

With this assessment philosophy, doubts about the objectivity seem to be inevitable in the end. But no resignation: There are qualified arguments in favour of our recommendations. Mathematics education is expensive, – it is particularly expensive to assess the individual quality of a student.

Society demands concerning future mathematics education

Finally I will discuss the demands of society on mathematical instruction. Recalling our discussion on general society demands with respect to instruction, let me ask some slightly nihilistic questions: Why should in our time society demand a good level of mathematical education? This seems to be in some contrast to the plausible demand for keeping people in good physical condition, since otherwise the expenses for health care would be too big. But what the heck makes us insist on a broad mathematics education for the public? Isn’t it true that we need only very few specialized mathematicians who keep our civilization intact!

Picking up this line of argumentation we could, of course, also ask, why the government supports instruction in writing and literal style, as the iconised IT-technology will make literacy obsolete in the future. Maybe it is much more comfortable for a government to have calm citizens who are unable to articulate themselves by writing and who simply vote once every four years by throwing their ballot into a slot.

The crucial argument in this discussion is – of course – that a civilized society has a general interest and responsibility to care for the cultural integrity of its citizens and to enable them to base their social decisions on an independent judgement about what is necessary for them and for the society.

If we extend this argument from literacy to mathematics it seems plausible at first glance that a citizen should be able to understand the conditions under which the world is running. Thus some basic knowledge about the techniques, chances and limits of IT and in particular of net-technology should be commonplace¹¹. However, teaching and learning IT as a subject is not a matter of mathematical instruction, but a general and basic task.

The specific justification for mathematics instruction is much more delicate. It is of course illusory to claim that everybody should work out his own models and the global behavioural patterns mathematically. But at the same time it is defeatist to say: You won’t understand it anyway, let experts and their software do the mathematics, which is good for you and for the society, and keep your fingers from that!

We already indicated that at least in our personal environment a certain mathematical know-how and know-why might improve our quality of life. There are many more mathematical topics, which are relevant for oneself and for the society:

Many people are unable to understand the different components of their personal burden of taxation; they cannot estimate the importance of the progression part of the tax table. The important concepts of a discounted value or of an effective interest rate, of the growth of a fortune by applying iterated interest rates, of annuities and mortgages go far beyond the usual and poor techniques of percentage calculations.

In this field still a lot of work has to be done. Most of the public is deeply fascinated by global percentage statements about the annual inflation rate or the employer’s contribution for health insurance. Mysterious curves explaining the actions at the stock exchange and esoteric time series prognostics by analyst gurus cover and conceal the general ignorance about what is really going on behind the surface of the market. The citizen should at least be able to pursue a rational strategy in order to save his own money from the temptations produced by property consultants. It is worrying that in this situation mathematicians¹² who are trying to reduce the abundance in mathematics curricula think about cancelling the exponential function from the list!

Moreover, the mathematical ideas behind the trends in global economy should be familiar to a considerable proportion of our population. In that case, politicians being sometimes unaware of the rigor of system theory themselves, could not treat us any longer like innocent babies with populist prognostics and with demagogic postulates, which cannot withstand the simplest calculations. The numbers about unemployment or global tax estimations often appear as the output of a divine oracle. Usually, even manipulations with indecent numerical tricks do not give rise to an outcry of the cheated public. The feeling for big numbers is under-developed today. The public, e.g., does not realize psychologically the difference between 500 millions Euros for arms supplies and 900000 Euros for a catastrophe fund.

System analysis, the behaviour of discrete processes in economy and ecology with a periodic (e.g. annual) balance account is usually done with the help of (systems of) difference equations with easily recognizable parameters. By now there are only few contributions¹³ providing us with material how CAS give the chance to pursue those dynamics with all their instabilities, delays etc. The demographic development of supply-, demand- and price-issues, the population of whales and other biological resources, the balance of CO₂, the different characteristics of exothermic processes, of alternative sources of energy – all these are topics of importance in this context. While the relevant mathematical theories are usually quite charming – but rather deep and hardly known to teachers at the present time – it is a matter of a few lines of code to plot those things in a CAS and to observe in an experimental mode mathematical phenomena like growth and decrease. A good example is the famous hog cycle: If the price of pork is high, hog-raisers intensify the production, which, again, makes the price drop. When the production reaches its peak, the price is at its bottom. If the peasants respond to this situation by throttling the production, the price will rise again. Etc. And last not least, a CAS can yield a look at chaos, which is especially attractive for young people, if there is no mathematical attractor at all.

Global policy today often produces the impression that it is impelled by an abstract force (the market or the public opinion), and recommendations how to boost or to dampen the economic cycle can very often be unmasked as the activities of a helpless medicine man. Mathematicians and especially mathematics teachers should therefore insist on the outstanding importance of mathematics of economics, which in our information age is even higher than mathematics for engineers and physicists. We should not hesitate to work out and to propose curricula for different levels, which fill this regrettable gap. Here is an example illustrating the importance of mathematics in economics: Several years ago, profit oriented companies found out that the modern technique of data mining which takes into account different external conditions provides qualified prognostics for their business. Analogously, it is a necessity for future times to base economy on sound plans (which has nothing to do with a non-manageable socialist control of other people).

Governmental policies often accept mechanisms like the mentioned hog-cycle as natural laws. We as mathematicians claim: Demographic numbers for different distinguishable groups of our society (these are obtainable from the governmental data-bases) have to be processed appropriately. Is it really impossible, e.g., to predict the number of required and of available teachers or computer scientists for more than only very few years?

Here are some more nuisances: In Germany there are huge differences between the prognostics and reality in the current green card policy. Only few of the well-established newspapers run the risk to present simple calculations about our social system. Usually, only tables with numbers and no context are presented; the usual diagrams showing market prices at the stock exchange cut off the base and thus exaggerate small changes. But we would like to have more and better explanations and to understand how the figures were assembled by the experts.

This deplorable state of the public understanding must and will change in the future. Today the data sources for information are usually available to or reconstructible by the public, – what we only still need is understanding the data and the courage to work with them. Usually only a minimum of mathematics is required, and we can and must urge the public to discuss the facts appropriately.

Stay cool in view of alleged votes of no confidence against mathematics: In ecologic prognostics people deplore that present climate models are too hypothetic to be judged appropriately; and they often contradict each other. It might look as if mathematics is unreliable. But actually, these models are based on various assumptions, which are not transparent to the public. We have to make clear that the arbitrariness of the assumptions and not mathematics is responsible for this confusion!

On the other hand, be fair and careful not to intimidate the public by recurring to hidden mathematical reasons which cannot be explained at least in rough ideas; mathematicians will then get the image of crazy gurus with no contact to real life. It is in the interest of the society – not necessarily also of the government whose members often just strive to preserve their power until the beginning of the next election period – to develop a wide public attention to well founded long range prognostics. Otherwise we will be surprised by catastrophes.

A last example: In the tough discussion about future health policy it is easy to argue that there must be a balance between input and output contributions and that the government cannot ignore this fact. Therefore very often the economic silverware is wasted hastily for short-term effects, and the problems are postponed into a far future. But we are responsible also for our children: The mathematical content of discussions which cover a longer period is not so simple and needs more than elementary arithmetic and the observation of fundamental economic trade-off relations. We have to discount the future – this is not beyond the capability of a mathematician – and we have to teach politicians and the general public what will be going on. Society has a fundamental interest that many citizens observe and control these principles and care for a sustainable future.

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Investigation into student attitudes to using calculators with CAS in learning mathematics

John Berry and Roger Fentem

Plymouth, UK

5. Brief outcomes and conclusions

1 Introduction

Hand-held technology incorporating graphical, numerical and symbolic algebra capabilities has much to offer the teaching and learning of mathematics at all levels. A number of studies have investigated the use of graphic calculators in schools and many mathematics education researchers and teachers assert that graphic calculators positively influence the way that mathematics is taught as well as the content and emphases of the mathematics curriculum (for a critical review of research on the graphic calculator in mathematics education see Penglase and Arnold, 1996). In many countries curriculum designers, educators and examiners receive mixed messages about the role that should be played by graphic calculator technology and computer algebra software. In the USA the National Council of Teachers of Mathematics Standards states that at the high school level, graphic calculators must be available to students at all times (NCTM, 1989). This has led to syllabuses, assessment and teaching approaches that actually encourage the use of hand-held technology. In France, all types of calculators, including those with computer algebra software, have been freely used in secondary school examinations. However contrary to the accepted opinion that such technology are tools for teaching, Trouche and Guin (1996, 1999) report that although calculators are much used by students the French academic system has not properly integrated their use. In England, Wales and Northern Ireland the upper secondary school mathematics curriculum was modularised so that in mathematics there are six modules in which the examinations are externally set and marked. In two of these modules students are restricted to the use of a scientific calculator and in four of the modules students may use a graphic calculator provided it does not have computer algebra software. In Finland, where the research study described in this paper was carried out, the matriculation examination is taken in students’ final year in school and the use of graphic calculators without computer algebra software is permitted. With such a mixed message there is a need for international research collaboration to identify the appropriate use of hand-held technology and to share results with curriculum developers.

The study reported in this paper explores some of the implications of teaching and learning mathematics in an environment where the teachers believe that the use of computer algebra software can help to develop understanding and skills but in which the technology is not allowed in the final examination.

Much of the research in the use of computer algebra software has involved the use of software such as Derive on a computer in a laboratory activity. Although the activities have often been integrated into the courseware the technology itself has not been fully integrated in the sense of homework tasks etc. The inclusion of computer algebra software into hand-held technology such as the TI-92 and TI-89 changes the availability of such software in teaching and learning. No longer do students need to leave ‘the computer behind at school’. We can now fully integrate the software as a cognitive tool to develop concepts and skills in mathematics. But what do students believe will be the effect on their knowledge as measured by the performance indicators of matriculation examinations?

It is one thing to use hand-held technology in the teaching and learning of mathematics but quite another to use it appropriately and effectively. The full potential of the technology will manifest itself only if teachers and students have positive perceptions of its use. The relationship between the nature of the use of the graphing calculator and teachers’ beliefs was explored by Tharp, Fitzsimmons and Ayers (1997). Their results suggested that teachers’ beliefs in mathematics as a subject, how it should be taught and the potential to produce the desired learning outcomes using graphic calculators is likely to influence their classroom practice. Doerr and Zangor (1999) found three aspects of the teachers’ role, knowledge and beliefs that contribute to the development of support of their students’ learning of mathematics. These were (1) the teachers’ confidence and flexible use of the tool, (2) the teachers’ awareness of the limitations of the technology, and (3) the teachers’ belief in the value of the calculator to support meaningful investigations. In our study the teachers who were chosen to participate in the research satisfied these criteria.

There is a need for extensive investigations of the effective and appropriate use of hand-held technology in upper secondary school mathematics courses and to identify the barriers that still exist. The teacher is one important factor, however, the student must not be forgotten as we recognise that learning should be student centred. Little research has been carried out on students’ attitudes and beliefs when hand-held technologies are embedded into the teaching and learning of mathematics. In this paper we describe the findings of a research project on students’ beliefs after they had studied a course in mathematics in which the TI-92 was fully integrated into the resource materials.

2 Research aims

The overall purpose of the research study was to investigate the cognitive development of students learning mathematics with computer algebra fully integrated into the school lessons and homework activities. In particular, we were interested in investigating the personal attitudes of students towards the use of the technology, their perceived training needs and in their ability at mathematics when they learn using a computer algebra intensive environment.

The topics chosen for the study formed two separate eight week courses: functions and equations (polynomial and rational functions) with grade 11 students (aged 16 years) in one school and an introduction to differential calculus with grade 12 students (aged 17 years) in a second school. The choice of different grades was to explore the effects of mathematical maturity and graphic calculator familiarity. The teachers in consultation with the researchers prepared the course materials. The Texas Instruments TI-92 calculator with built in computer algebra software was integrated into the course as a platform for investigational activities for introducing new concepts to the students and as a tool for problem solving. In each school there was an experimental group using the TI-92 and a control group using the TI‑85 graphic calculator.

The aims of each course were to teach the concepts, skills and applications associated with the syllabus. One important constraint for the pupils is the Finnish matriculation examination taken by pupils in their final year in school, the performance on which decides their choice of higher education. In this examination the use of computer algebra software is not allowed so that it was important for the students in the TI-92 groups to develop the same mathematical problem solving skills for assessment as the other school pupils in Finland.

The research question investigated in this study was: what are the feelings and attitudes of the students in their ability at doing mathematics with and without the support of the TI-92? This paper reports on the findings of this question.

3 Research design

The research reported in this paper seeks to investigate the personal attitudes of students in their ability at mathematics when they learn using a computer algebra intensive environment compared with their peers who are using a graphic calculator without this facility. The subjects in the research were students in two upper secondary schools in Finland.

The teaching approach used by the two teachers of the TI-92 groups assumed a socio-constructivist epistemology in which the students actively constructed mathematics through investigations, group work and discussion rather than the transmission model by the teachers. The research methodology adopted in this study was both quantitative in that the students answers to a Likert-type rating scale was used to collect their attitudes to their personal use of the hand-held technology and to their mathematics skills and qualitative in that students were asked to write statements about their feelings and were interviewed at the end of the experimental period.

Both experimental groups used course materials written for the project whereas the control groups used a traditional textbook, a more didactic teaching style, and the TI-85 graphic calculator was available as a problem-solving tool. The experimental group resources in each school embedded the TI-92 as an integral part of the learning of mathematics in that the technology was used in investigational activities to introduce new mathematics concepts and as a tool for doing mathematics.

A questionnaire on student attitudes was completed by all four groups after the post-test. For our study it was important that we collected information about students’ attitudes as soon as possible after the course was delivered so that the influence of the next courses did not influence their judgement. The questionnaire consisted of 25 statements that the students had to agree or disagree with in various strengths and an open space inviting further comments. The statements are of four types and the classification is shown in Figure 1. Three statements have been omitted from the analysis. Statement 9: “I would not like to do calculus without a TI-92.” and Statement 18 : “I do not understand what differentiation is about.” were included only in the Rovaniemi and not in the Keminmaa questionnaire and Statement 15: “I only learn mathematics well if the teacher is good.” is not associated with the attitudes towards the use of technology.

	Attitudes associated with the use of calculator technology:		Attitudes associated with pupil performance:
1.	I can think more clearly when I use a calculator	3.	Using the calculator makes me less good at doing mathematics by hand
2.	Using a calculator makes me lazy	4.	Using the calculator makes me better at solving mathematics problems
7.	Using a calculator is cheating	8.	Using a calculator helps me understand my mathematics
10.	I rely too much on my calculator	11.	Using a calculator makes me less good at mental mathematics
12.	Only pupils who are good at mathematics should use a calculator	14.	I will do better in the matriculation exam because I have learnt with a calculator
25.	I wish to study all mathematics courses with a calculator	17.	I think that the pupils who have not used a TI-92 will do better in the examination
	Attitudes associated with the amount of use of the calculator		Course and pedagogy statements
5.	I used a calculator in most of my mathematics lessons	13.	I enjoyed this mathematics course
6.	I used a calculator in most of my mathematics homework	16.	The course materials were good
21.	The calculator has been used too much in this course	19.	I feel better about doing mathematics after this course
22.	I need more time to learn about the calculator before using it in mathematics classes	20.	I have more fun in this mathematics course than other mathematics courses
		23.	This mathematics course is different from other mathematics courses I have taken
		24.	The role of the teacher is different in this course compared with other courses

Fig. 1: Classification of attitude statements

4 Data analysis

The goal of the data analysis was to identify any recurring themes in students’ attitudes regarding their use of the hand-held technology in their course. The degree of agreement or otherwise to the statements presented on the questionnaire instrument provided data on a Likert scale which was coded 1 to 5. These data were treated as ordinal in nature.

A reliability analysis using Cochran’s alpha was undertaken to investigate the internal reliability of the attitude instrument. For the reduced questionnaire a value of a = 0.845 was obtained, this is an indication of a sound instrument.

Non-parametric statistical procedures were used to analyse the data. A 5% significance level was employed (1% was used to indicate a highly significant outcome), adjustment for ties was used throughout and exact critical values were employed where technology permitted.

The initial analysis to determine if there was evidence that the students as a whole held a view on a particular item was undertaken using the 1-sample Wilcoxon test with the null hypothesis H₀: h=3, against the alternative hypothesis H₁: h¹3. Evidence of differences in opinion between control and experimental groups were explored using the Mann-Whitney U statistic. The same analytical tool was employed in determining differences between students in the two schools. The Kruskal-Wallis statistic provided evidence of differences between the four groups of students. Both the Wilcoxon and Mann-Whitney U statistics featured in any subsequent analysis at this level in the data structure.

*Statement*		*All*	*Rovaniemi* (grade 12)		*Keminmaa* (grade 11)
*Statement*		*All*	*Control*	*Experim*	*Control*	*Experim*
1	I can think more clearly when I use a calculator	A*	A*	U	U	A
2	Using a calculator makes me lazy	D*	D*	D	U	D*
7	Using a calculator is cheating	D	U	U	U	D
10	I rely too much on my calculator	D*	D	U	D	D*
12	Only pupils who are good at maths should use a calculator	D*	D*	D*	D*	D*
25	I wish to study all maths courses with the calculator	U	U	U	D	U
5	I used a calculator in most of my maths lessons	A*	A*	A*	U	A*
6	I used a calculator in most of my maths homework	A*	A*	A*	U	A*
21	The calculator has been used too much in this course	D*	D	D	D*	D*
22	I need more time to learn about the calculator before using it in maths classes	D	U	D	U	D
3	Using a calculator makes me less good at doing maths by hand	U	D*	U	U	U
4	Using a calculator makes me better at solving maths problems	A*	A	A	U	A
8	Using a calculator helps me understand my maths	A*	A	U	U	A
11	Using a calculator makes me less good at mental mathematics	D*	D*	U	D	D*
14	I will do better in matriculation because I have learnt with calculator	U	A	U	U	D
17	I think that the pupils who have not used a TI-92 will do better	D*	D*	U	U	D*
13	I enjoyed this maths course	A*	A	A*	U	A*
16	The course material was good	A*	A*	A*	A*	A*
19	I feel better about doing mathematics after this course	U	U	U	U	U
20	I have more fun in this maths course than other maths courses	U	U	A*	U	U
23	This maths course is different from other maths courses I have taken	A*	U	A*	U	A*
24	The role of the teacher is different compared with other courses	A*	U	A	U	A*

Fig. 2: Outcome of statistical analysis at group level, * indicates significance at 1% level –

A Agree - D Disagree - U Unsure

5 Brief outcomes and conclusions

The outcomes suggest that integrated technology teaching impacts in a complex fashion on student attitude. Items 12, 21, 16 and 19 indicate that the ‘treatment’ has had no effect. Items 22, 14, 23 and 24 indicate a positive effect – with maturity influencing the extent of the effect for item 14. Items 1, 8 and 17 reveal an interaction between treatment and level. Items 2, 25, 5, 6, 4 and 13 expose the young control group with a negative attitude. Items 10 and 11 expose a negative attitude in the mature experimental group, whereas item 20 provides the opposite impression. Item 7 reveals a particularly positive attitude from the young experimental group as does item 3 with regard to the mature control group.

An important research question that needs further investigation through a longitudinal study is ‘as students mature mathematically do they see technology as a barrier to performance?’

On the issue of maturation: the two control groups offer the best prospects to gauge student opinion and how it may change over time. Here is a summary of the significant differences in their responses. An ambivalence about whether use of calculators allow them to become lazy turns into a firm conviction that it does not. They make greater usage in both lesson time and for homework. They are more inclined to want to study other mathematics courses with calculator technology. Their enjoyment of mathematics increases. They record an improvement in their clarity of thought, in their traditional mathematical skills, in problem solving and in their understanding of mathematics. Their opinion of their own examination prospects improves significantly.

The process of transforming any artefact, such as a graphing calculator, into an instrument to support learning is a complex process. Such a process is called instrumental genesis and is described in Drijvers & Herwaaden (2000) and Guin & Trouche (1999). The instrumentation of hand-held technology tools refers to the relation between the development of the mathematical concepts and skills of using the technology. We would argue that the students in our study achieved instrumental genesis because the technology was available to them both in school and at home, in other words, throughout the mathematics module for schoolwork and for homework.

The views of the control and experimental groups are significantly different here with the CAS groups disagreeing (p<.01) with the need for more training in the use of the technology.

The results provide some interesting insights into student perception and attitudes to the use of calculators in learning mathematics.

Students perceive calculators as a cognitive tool which helps their understanding, and clarifies their thinking as opposed to giving them an edge in examinations.

Familiarity and personal competence with the technology is an important part of building mathematical confidence. All four groups of students have used the technology in most of their mathematics lessons and homework and only the control group at Keminmaa felt that they needed more time to learn about using the TI-85. One reason for these results is that in both schools time was given to develop TI-92 skills, as it was the ‘new’ technology. The TI-85 control group in Rovaniemi was second year upper secondary and had used their calculator for one year longer than the Keminmaa students.

The importance of mathematical maturity is demonstrated in the responses to several statements. The Rovaniemi students are one year further on in their studies so that they have studied and been assessed in four more courses than students in Keminmaa. Thus it might be too early for the Keminmaa students to deduce whether their performance in examinations and in doing mathematics by hand will be reduced. Where the maturity effect begins to show itself, is whether they are using a calculator too much (item 10) in which the difference between the two schools is statistically significant (p < 0.05) (note that there is a difference in their views on whether using a calculator is cheating however this is not statistically significant). The Rovaniemi students have studied courses in algebra, trigonometry, pre-calculus and differentiation and so are beginning to see advanced mathematics and the power of modern hand-held technology at being able to do mathematics at the press of a button. An important research question that needs further investigation through a longitudinal study is ‘as students mature mathematically do they see technology as a barrier to performance?’

Both teachers who took part in the research study expressed feelings of pleasure and disappointment during the teaching of the courses. We finish with a paragraph from the diary of one of the teachers towards the end of the course:

On Friday afternoon after the lesson of the experimental group. I happened to ask one girl attending the course how she feels studying like this. She said that actually it has been fun. She said that usually in the mathematics lessons she is looking at her watch and waiting for the lesson to end. But now she had forgotten the watch. She continued: "If I am saying that mathematics is fun it is really something extraordinary". This is almost enough for me after six months work with the material!

Acknowledgements

The authors of this paper presented at ICTMT5 acknowledge the research team John Berry, Roger Fentem, Anna-Maija Partanen (Lyseonpuiston Lukio, Rovaniemi, Finland) and Sirkka Tiihala (Keminmaan Lukio, Keminmaa, Finland) who carried out the research on which this paper is based.

References

Doerr, H.M. and Zangor, R. (1999) The Teacher, the Task and the Tool: The Emergence of Classroom Norms. The International Journal of Computer Algebra in Mathematics Education 6 (4), 267–279

Drijvers, P. and van Herwaarden, O. (2000) Instrumentation of ICT-tools: the Case of Algebra in a Computer Algebra Environment. The International Journal of Computer Algebra in Mathematics Education 7 (4), 255–275

Guin, D. and Trouche, L. (1999) The Complex Process of Converting Tools into Mathematical Instruments: The Case of Calculators. International Journal of Computers for Mathematics Learning 3, 195–227

National Council of Teachers of Mathematics (1989) Curriculum and evaluation standards for school mathematics. NCTM, Reston, VA

Penglase, M. and Arnold, S. (1996) The Graphics Calculator in Mathematics Education: A Critical Review of Recent Research. Mathematics Education Research Journal 8 (1), 58–90

Tharp, M.L., Fitzsimmons, J.A. and Ayers, R.L. (1997) Teachers’ perceptions of the impact of graphing calculators in the mathematics classroom. Journal of Computers in Mathematics and Science Teaching 16 (4), 551–575

Trouche, L. and Guin, D. (1996) Seeing is Reality: How graphic calculators may influence the conceptualisation of limits. Proc. 20^th Conf. of the Intern. Group for the Psychology of Mathematics Education, Valencia, 323–333.

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The potential of the Internet

for innovations in didactics of mathematics

Stefanie Krivsky

Wuppertal, Germany

1. The project MathePrisma

2. Internet software

3. How to make use of the technical features of the Internet

4. Results

Originally, the Internet was designed by scientists for the purpose of exchanging information, but since then, it has been more and more adopted by entertainment and commercial use. The Internet project 'MathePrisma' (math prism) tries to combine these two objectives with the aim to simplify the learning of complex mathematics using multimedia and entertainment aspects of Internet. MathePrisma is a collection of modules posing several mathematical questions on different educational levels. Technical and didactic possibilities of Internet pages are presented by means of some examples of MathePrisma-modules.

1 The project MathePrisma

The Internet project 'MathePrisma' consists of a collection of modules, which is continuously growing. Each module is dedicated to a mathematical problem, e.g. the four-colours-problem, the bridges of Königsberg, number walls, magic with numbers or paradoxes from statistics. For each of these problems, different mathematical methods are introduced and directly applied. The modules focus on users' activities like discovering problems and on developing solution strategies for solving them. By working with the modules the users can

form mathematical ideas on their own,

perceive interrelations between mathematical concepts,

become acquainted with applications inside and outside mathematics.

One more aim is to help users to (re)discover the fascination and benefits of mathematics and to recognize how interesting and attractive this subject can be. MathePrisma is written in German and can be found in the Internet under the URL http://www.matheprisma.uni-wuppertal.de/

It is intended to be used by pupils, university students and people with a general interest in mathematics. It was launched in 1998 as a joint project of the Group of Didactics of Mathematics and the Institute of Applied Computer Science in Wuppertal. One important feature of the project is to include the university students in the process of building MathePrisma. Within practical exercises as part of their studies, mathematics and informatics students, in particular prospective teachers, can write texts, work out questions, create small pieces of animation, or develop more ambitious programs like simulations.

2 Internet software

One great advantage of the Internet is to be an open and freely accessible system. So users can draw upon offers from many sources, they need no special equipment and can make use of the offers in an easy and intuitive way. On the other hand Internet pages can be developed without too many sophisticated tools. In general, the following three types of teaching software can be identified:

simulation software for presentation of specific topics, e.g. the Internet project 'Mathe online' (www.univie.ac.at/future.media/mo/galerie.html): Math online offers a lot of java-applets for experimentations with tasks or hints for mathematics including problems and information, all in a very impressive way.

knowledge software, i.e. encyclopaedia or reference books, which are mostly text oriented, e.g. the Internet software (www.educeth.ch/mathematik/leitprog/lingl/): This is a text with a lot of information which is organized like in a book and which contains many links to other internet addresses for direct access.

training software for the improvement of techniques which one has already acquired, e.g. with the help of the Internet program (www.zum.de/ma/fendt/math/md10.htm): By using it one can gain more self-confidence in the execution of several algorithms.

In each module of MathePrisma all these aspects are taken into account. Working at a precise mathematical problem, users can learn mathematical methods and working techniques of mathematics, e.g.

''playing'' with problems in order to understand their mathematical content,

introducing mathematical tools and applying them to actual problems,

developing problem solving strategies and comparing them with each other.

3 How to make use of the technical features of the Internet

We use the Internet as a platform to offer texts, pictures, films, animation, experiments, and text-fields for inputs, links and sounds. (For compatibility reasons we did not include sound effects.) The following three features can be identified:

The fundamental structure

The main path: Because reading a hypertext is not as easy as reading a book, it is very important to have a clear structure in the documents. The modules consist of a main path containing all-important information about the mathematical topic and a path with optional background information. Historical remarks are often interesting, but they can draw the attention away from the subject matter. To each module belongs a site map of all pages and their topics.

Links: Now and then we include different links with different paths to follow, in order to support individual ideas and ways of thinking.

Exercise sheet: Each module contains a printable exercise sheet, as it is well known that some people feel unsure when they shall pass through learning processes with the help of material presented at a computer screen. The exercise sheet underpins the main aspects of the module by recapitulations and transfer exercises.

Interactions

Experiments: Each module starts with an experimental part under the slogan "playing with the problem". Such an interactive and intuitive introduction can considerably diminish the inhibitions, which many people have towards mathematical topics. Problems can be understood better, and there is created a spirit of curiosity. The module "The four colouring problem", e.g., starts with a map and the users can try and colorize it with less than four colours on the basis of the trial-and-error-principle or she or he can try to find a strategy how to colorize it.

Questions: There are questions, which recapitulate the contents of the module and thus allow the user to control her or his progress. But more important than these controlling-questions are questions with hints to new ideas. The method of gradually approximating a solution by this kind of questions is much more effective than presenting the solution directly. The module "secrets of primes", e.g., deals with the sieve of Eratosthenes. For many students it is often surprising that they have to cross out non-prime numbers, when they want to find out the primes. In order to get them on their way to meet this strategy, we start the module with the task, to find out two primes out of a set of six consecutive numbers. The numbers are chosen quite large. Thus students make the experience that it makes sense to reverse the question and to find out the composite numbers beforehand.

Task generator: One talent of the computers is the power to produce an infinite number of new tasks with random parameters. Thus, users can do a lot of simple exercises in order to become fit in easy tasks, and gain motivation for new ones. In the module "number walls", e.g., students are trained in adding and subtracting numbers as well as in solving small systems of linear equations.

Visualization

Pictures: Another supporting element is the possibility of visualization. Pictures are more than nice descriptions of mathematical ideas. The process of modelling a problem often starts with designing a picture with all necessary information, e.g. representing the map of "Königsberg" as a graph.

Sequences of pictures: Many formulas and notations are easier to understand and to store in one's memory if they are visualized by a sequence of pictures. So called "switch images" are adequate for the description of mathematical ideas. For instance, the sum of the first n integers equals n(n+1)/2. This expression can be visualized by the area of a stair-case (viewed from the side) or by the half area of a rectangle.

Animation: Animated pictures, with the possibility to change one or more parameters, e.g. by controlling them with scrollbars, are often more instructive than still pictures. The module "recursive series", e.g., shows an impressive application of the Fibonacci numbers and their connection to the golden section: Sunflower grains grow like spirals. Counting these spirals clockwise and counter-clockwise results in two consecutive Fibonacci-numbers. The reason lies in the connection between these numbers and the golden section. So the grains grow optimally by observing the so-called golden angle. The user can study this interesting phenomenon by changing the angle or the number of grains with the help of a scrollbar.

4 Results

MathePrisma is addressed to two target groups: The first one is, of course, the group of users or learners. These are pupils, university students or people with general interest in mathematics. Studies with pupils at their schools, or at open days at the university as well as with university students in their lectures show that understanding mathematics can be facilitated with multi-media approaches like MathePrisma.

The modules help pupils to overcome their resistance against mathematics (e.g.: Children want to continue the lessons even in their free time).

Successful experiences make the users more self-confident with respect to their own mathematical competence (e.g.: They do not want these solutions to be shown to them).

The easy handling of a module helps to overcome inhibitions towards multimedia techniques (e.g.: Most users do not read the written instructions for the modules, but just work with them).

Different learning types can be addressed by the same module (e.g.: All users can work on the modules with individual speed and intensity and sometimes with individual paths).

It is much easier to concentrate because of this individualization (e.g.: Films and sequences of pictures can be operated individually).

Due to the lively presentation and the possibilities of trying out ideas the contents of the modules can be memorized in a better way. They can be recalled more intensively because of many associations (e.g.: Each module is characterized by a picture which shows the mathematical idea of the main topic in some situation of everyday life).

MathePrisma deals with mathematical subjects apart from school mathematics (e.g.: Graph theory, number theory, numerics).

Interdisciplinary aspects of mathematics can be recognized (e.g.: statistics in genetics or strategies in economics).

The fascination and benefits of mathematics can be discovered.

The second target group of MathePrisma is constituted by itself: the authors. By developing modules students learn to find new forms for the description of mathematical ideas. They are forced to work out the basic essentials of a subject. In particular the prospective mathematics teachers have to learn to plan teaching scenarios. By interactions and animation they can find out new forms of motivation and can transfer their experiences in learning to later users of their modules. For students of mathematics or informatics it is a good exercise to learn to cooperate with students of other disciplines. Creating a module can encourage students to use expectantly, and at the same time critically, other multi-media tools.

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On the impact of hand-held technology on mathematics learning —

From the epistemological point of view

Ewa Lakoma

Warsaw, Poland

1. General education in the digital era

2. Mathematical education for the future

3. Graphing calculators in the mathematics learning process – Potential and obstacles

4. Final remarks

In many countries mathematics is regarded as one of the most important components of general education and general culture, and therefore it is extremely important to enable students to develop their own mathematics as a language for communication. To do so, mathematics learning processes have to be taken into consideration from an epistemological perspective and the students’ ways of mathematical thinking, especially when using information technology, must be investigated. Today I would like to report about my research in this field. I will focus on several examples where students at secondary and tertiary level develop the most fundamental statistical and probabilistic concepts, using graphing calculators as supporting tools.

1 General education in the digital era

At the beginning of the 21st century, in most countries an essential transformation from industrial society to information society can be observed. This change causes a growing demand for people who are able – not only at work but also in everyday life – to deal with new challenges rather than to just behave in routine situations. General aims of education – at every stage of the educational system – are also subject to fundamental changes. Knowledge, which people gained a few years ago, is today – in the digital era – no longer sufficient for them to function effectively during their whole life. There is a deep need to prepare today’s students to live and to work in a future world – a world of rapid development and wide application of information and communication technology (Noss 1997). Thus, the main goal at every level of today’s education is to provide students with an operative knowledge and key competencies, as a basis for further learning – according to actual social and professional demands and potentialities. Current viewpoints on general education indicate a clear shift of social and educational expectations: developing creativity and cognitive abilities instead of mastering simple skills, which are only useful in typical situations.

2 Mathematical education for the future

Great progress in many disciplines of science and technology, rapid extension of information technology (IT), widespread access to information, changes of style and standards of everyday life – all these transformations imply new expectations to education. Current general demands to education entail mathematics, as a learning subject, to become even more indispensable for all than it used to be in the past. However, in all main categories of behaviour towards mathematics: using, doing and learning, the new technologies require new mathematical literacy for all. Instead of exposing the formal structure of mathematics – as it was the use in the past, it is necessary today to treat mathematics as a language for communication and as a tool for predictions and explanations of reality (Freudenthal 1983, Sierpinska 1998).

In order to understand real phenomena, students must be able to describe them, to simplify them and to reduce them to their essential features (mathematical modelling). Then, on this basis, students can pose hypotheses, make predictions and conclusions, generalise and justify them (mathematical reasoning), and – apply them to practice or present and explain their ideas to other people (mathematical communication). In the field of mathematics it is extremely important to enable students to develop a deep – relational rather than instrumental – understanding of mathematical concepts (Skemp 1976) and to create their own mathematics in the three fundamental categories pointed out above. This is a conditio sine qua non for consciously using mathematics as a language for communication in every domain of life. Thus, learning mathematics means acting in these three categories – in a necessary balance –within didactical situations, involving students’ initiative and creative activities related to real contexts, which are interesting for students and which take into account their nature of cognitive development (Freudenthal 1983, Sierpinska & Kilpatrick 1998).

The active style of mathematics teaching – promoting active methods, involving interactions among students and co-operation in small groups, stimulating students’ initiatives – becomes more and more expedient at every stage of education (Lakoma 1990, 2000). New technologies like computer programs, graphing calculators, video, films support this style of teaching (Laughbaum 2000). It seems to be particularly effective for students who will be users of mathematics in their professional life, where relations of mathematics to reality are extremely important and natural, as such relations have to be exposed in the learning processes (Lakoma 1990, 2000a).

In order to create appropriate conditions for effective learning it is necessary to understand students’ ways of mathematical thinking and to recognise symptoms of grasping mathematical concepts. This epistemological perspective is very important when we use didactical tools, which are expected to be helpful in making the mathematics learning process more effective (Lakoma 1990, 2000, 2001).

Usually, when solving a (mathematical) problem, students have to make some trials and to gather experience as staring point for mathematical reasoning. Using computers, graphing calculators or algebraic calculators can enhance this important stage of the process of mathematics learning (Laughbaum 2000).

Living in the information society requires the ability to solve problems, which arise in connection with handling large amounts of data, and which lead to posing and verifying hypotheses, and also to critical reasoning. Thus, in today’s mathematics education, statistics should play a more and more significant role. In most countries elements of statistics are included in the written curricula. Social changes in the recent decade enlarged the group of users of statistics dramatically. Thus, statistical literacy as a component of mathematical literacy “for all” has become extremely desirable. The statistical mode of reasoning has some specific characteristics: on the basis of a small sample, general conclusions concerning the whole population are made. Reasoning in a more responsible way involves relating to a theoretical model and to confront it with the raw data. This is a probabilistic model. So, probabilistic education becomes very useful, also as a tool for statistical investigations. This change of attitude towards statistics and its place in mathematics education for all – especially when using technology – brings about the following key questions:

How does statistical thinking develop? By which symptoms can the understanding of fundamental statistical concepts be recognised? How to organise statistics teaching with respect to students’ cognitive development? What is the impact of information technology on the learning of statistics? How can hand-held technology improve learning statistics according to the cognitive development?

3 Graphing calculators in the mathematics learning process – Potential and obstacles

In this chapter I will give some answers to these key questions, on the basis of my research on statistics & probability learning processes of students at the secondary and the tertiary level (Lakoma 2000, 2001, Lakoma, Zawadowski 1996-2001).

I will analyse examples of students’ ways of thinking, i.e. developing fundamental statistical concepts and methods, such as: sampling, mean value and dispersion, drawing statistical conclusions, data distributions. In all these examples graphing calculators (TI-83 and TI-92) are used, and I will analyse their role in the process of mathematical thinking. Are they really useful for students?

As the most basic statistical competencies at secondary level we identify the following: to read and to understand information from graphical presentations of statistical data; to present statistical data distributions graphically and to interpret them; to calculate and to interpret numerical characteristics of statistical data: mean, median, mode, and measures of dispersion: range, variance, standard deviation; to pose simple hypotheses and to verify them by means of statistical reasoning; to gather data by creating a statistical sample; to use basic statistical tools in everyday life and in professional life.

Now I will focus on observing how students develop the most fundamental competencies, as posing simple hypotheses and using common sense reasoning in order to verify them.

Example A: Who is better, in comparison to the whole group?

Each of two girls participated in school competitions in mathematics. The maximum number of points to be obtained was 60. The results of the first stage were the following:

39 40 40 41 42 42 44 46 47 48 48 48 52 54 55 58 59 59 59 59 60 60 60

The results of the second stage were as follows:

39 39 40 41 43 43 44 44 44 45 45 46 46 46 47 47

The students’ ranks in both lists determined the final position at the whole competition.

Anne obtained 55 points at the first stage and 44 at the second stage. Betty obtained 48 points at the first stage and 47 points at the second stage. Which girl was better (in comparison to the whole group of participants)?


Fig. 1	Fig. 2

This task is one of the most fundamental statistical problems. The common sense idea of solving it is just to sort data, to find the positions of the two girls’ results and to compare them with the whole collection of data. It is reasonable to display several characteristic numbers in a sequence of data – those, which are typical, and those, which are exceptional. The minimal and the maximal value are exceptional data; it is natural to point out in the middle of them the “central” number – the median. The medians of both halves of a sequence – distinguished by the median – are the quartiles. Those data, which are placed between the two quartiles, are “defined” as typical. These considerations lead to the conclusion that Betty was better: her position seems to be better than Anne’s position.

This kind of thinking – which is natural for students at the elementary level of learning statistics – is a variant of statistical data analysis, which was introduced by John Tukey with his famous box-and-whiskers diagram. Hand-held technology, which is equipped with the power of drawing such diagrams, is a natural tool for working with data distributions at the elementary level. Using this kind of graphical presentation of data is also extremely important, because it helps to analyse dispersion of data in a very simple way and to develop the concept of dispersion before creating elaborate concepts like variance and standard deviation. Although box-and-whiskers diagrams (Fig. 2) can easily be obtained with graphing calculators, we must keep in mind that the students should also be equipped with that kind of techniques which offer them ways of working with statistics, also if there is no calculator at hand. For instance, stems-and-leaf diagrams (Fig. 1) support quite well the analysis of data.

Example B: Diversity of uniform distributions – Is it possible?

The graphing calculator enables students to develop some of the most fundamental concepts: the concept of relative frequency and the concept of probability. Let us consider the following example:

Generate 200 natural numbers between 1 and 10 at random, and draw their bar chart. What shape do you expect for this diagram? What do you expect, when you increase the number of generated numbers?


Fig. 3

Looking at examples of these diagrams (Fig. 3), students can easily notice that distributions of random numbers are various, but theoretical distribution should be uniform. This is one of the most difficult epistemological obstacles for students. In order to overcome this obstacle, they really need time and opportunity to gather rich statistical experience.

Example C: From statistical data to fundamental probabilistic concepts

Another exercise connected with the concept of probability and its relations to relative frequencies is as follows:

Throw a coin 200 times, and notice the sequence of heads and tails. How does the number of heads after n throws (n= 1,2,...,200) develop? Draw a graph of this relation. What is your hypothesis?


Fig. 4	Fig. 5

Looking at graphs of relative frequencies (Fig. 4), students conclude that in the long run they can expect that this frequency is 1/2. By analysing many experiments, students will have the opportunity to understand the key feature of random mechanism in a proper way, which is theoretically confirmed by the law of large numbers (Fig. 5).

Example D: Gathering rough data and confronting them with a theoretical model

Let us consider the following problem: we throw a coin 6 times and count the number of heads. How many heads in average do you expect?


Fig. 6	Fig. 7

The programmable graphing calculator allows the students to repeat this “game” many times in order to observe the random mechanism and to pose a hypothesis. Such a hypothesis, on the basis of a long observation (Fig. 6), can be verified on the basis of a theoretical model. At a more advanced level students are able to discover the theoretical distribution of possible outcomes (Bernoulli distribution), by considering the properties of the model (Fig. 7).

On the other hand, we must keep in mind that using the graphing calculator may have some shortcomings. One of them is the overestimation of a visual representation of the situation in consideration.

Example E – Searching for a central tendency of a set of grouped data

After a series of examples of graphical presentations of data by means of graphing calculators, students were asked to solve the following problem:

Calculate the arithmetic mean of the following data:

*Data*	*frequency*
<500,1000)	14
<1000,1500)	20
<1500,2000)	12
<2000,2500)	4

Fig. 8

It turned out that the students were not able to calculate the mean, although they perfectly remembered the formula for rough data. They tried to remember what they had been doing with the calculator in similar situations, but they were not able to use their common sense to find the right solution. The only progress they made was to draw a bar chart of the data (Fig. 8) – but this did not help to find a solution. This example suggests that several students, though provided with some experience in working with data with the help of calculators, do not understand the concept of average in the relational mode – their understanding was still an instrumental one (Skemp 1976) and failed in a case given in a non-standard way. Moreover, it seems that visualisation of data given by means of the calculator can become an obstacle for their thinking naturally.

4 Final remarks

All examples presented above show that hand-held technology can serve as a tool, which enables students to observe properties of great amounts of data, of repeating random experiments, random mechanisms in action. When we observe the students’ “natural ways of thinking” when solving statistical problems, we can distinguish the following steps of their reasoning: to discover and to formulate a problem; to construct a model of the real phenomenon; to analyse the model; to confront the results obtained from a model with “real” situations. These are local models which have a great explanatory value (Lakoma 1990, 2000). Natural mathematics learning processes consist of creating and successively developing mental objects, which anticipate mature mathematical concepts (Freudenthal 1983). Students do not acquire mature mathematical concepts if they do not become acquainted with numbers, circles, adding numbers etc. in various real contexts and create and use mental objects. Mental objects are built spontaneously. It is not possible to see them “by eye” but only “by mind”. Various mental objects anticipate mentally mature mathematical concepts. Therefore in statistical education students must be given the opportunity to gain experience in creating and using mental objects connected with the main statistical and probabilistic concepts. Using hand-held technology – when searching for various representations of a problem and confronting experimental results with a model – can support this indispensable stage of learning. The examples show that using hand-held technology implies also a need of new mathematical literacy – in mathematics education we can observe a shift from emphasis on traditional functional thinking to algorithmic thinking. The main present-day task for mathematics educators is to identify those new contents and mathematical competencies, which are indispensable in the information society, and to include them into curricula.

References

Freudenthal, Hans (1983) Didactical phenomenology of mathematical structures. Reidel, Dordrecht.

Lakoma, Ewa (1990) The local models in probability teaching (in Polish). Doctoral thesis, Warsaw University, Department of Mathematics, Informatics and Mechanics.

Lakoma, Ewa (2000) Stochastics teaching and cognitive development. In: Fauvel, John and van Maanen, Jan (eds.): History in Mathematics Education. The ICMI Study. Kluwer, Dordrecht.

Lakoma, Ewa (2000a) Using graphing calculators in active methods of mathematics teaching for future engineers. In: Berry, John (ed.): Proceedings of the ICTMT 4, August 1999 (CD version), Plymouth, UK.

Lakoma, Ewa (2001) How to teach statistics at school? (in Polish) WsiP, Warsaw.

Lakoma, Ewa, Zawadowski Waclaw and others (1996-2001) Mathematics 2001. (in Polish) Curriculum documents, series of textbooks, teachers’ guides, films, programs for students of age 10–19. WsiP, Warsaw.

Laughbaum, Edward (ed.) (2000) Hand-held technology in mathematics and science education. A collection of papers. The Ohio State University.

Noss, Richard (1997) New Cultures, New Numeracies. Inaugural Professoral Lecture. Institute of Education, University of London.

Sierpinska, Anna (1998) Whither mathematics education? Proc. 8th International Congress in Mathematics Education. Seville 1996, S.A.E.M. ’Thales’.

Sierpinska, Anna and Kilpatrick, Jeremy (eds.) (1998) Mathematics Education as a Research Domain: A Search for Identity. An ICMI Study. Kluwer, Dordrecht.

Skemp, Richard (1976) Relational and Instrumental Understanding. Mathematics Teaching 70, 20–26.

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A project on the development of

critical thinking by using technology

Tatyana Oleinik

Kharkov, Ukraine

This paper presents the results of special courses given to undergraduate mathematics and computer science teacher students. A general problem for these students is to understand the possibilities of technologies for the realization of the ideas of our Project on the Development of Critical Thinking. The main goal of these courses is to change the participants’ level of social performance: only a widely educated person with SOFT SKILLS is able to restructure the direction and essence of her or his activities flexibly. This means that people have to be enabled to analyse new information comprehensively and deeply, to formulate original ideas, to select ideas from competing ones rationally, to solve non-standard problems, to have constructive dialogues, to orient towards self-diagnosis in respect to the degree of formation of different skills by comparing one’s own results with extraneous standards, etc.

During the existence of the Soviet Union good mathematical and natural science curricula were in existence, and in the post-Soviet countries the educational level in general was higher than in western countries. However, this advantage lost its significance with the increasing occurrence of non-standard situations in which students had to use their factual knowledge. Thus, on the current stage, education must have a certain degree of flexibility, i.e. education has to meet new requirements and, at the same time, it should keep its strong features.

In our opinion critical thinking technology (CTT) has to be one of the leading modern technologies, and its adaptation in the Kharkov State Pedagogic University was to promote constructive discussions concerning the utilization of the peculiarities of the Ukrainian scientific-pedagogical school for the training of teachers who are able to master the challenges of the future. As a consequence, our university takes part in the international project RWCT (Reading and Writing for Critical Thinking).

Educators who want to take part in this project are expected to acquire the following abilities:

to be master teachers, able to serve as instructional models and resource people within their own professional setting, i.e. school, university, methods centre, etc.

to conduct staff development workshops for other educators who do not have access to the same intensive staff development.

Further, by implementing the program in their own professional setting they gain credibility as experts. Thus, the goals of the RWCT project are:

to develop open, collaborative, collegial, long term relations between educators from various cultures and circumstances which will expand the understanding of teaching and learning for all students and lead to a free flow of ideas between peoples;

to increase the capacity of students to think critically, engage in critical reflection, take responsibility for their own learning, form independent opinions and show respect for the opinions of others;

to present practical methods of teaching based on philosophically consistent and theoretically sound ideas;

to place teaching within a comprehensive instructional framework which guides instructional decision-making;

to empower the participants to take responsibility for becoming model teachers of RWCT, able to reflect on students’ thinking and learning and refine methods based upon those reflections;

to engender confidence in the participants based on a successful implementation of the program in their own educational setting;

to prepare participants to deliver the program to their peers.

Among the peculiarities of the implementation of RWCT in our region is the attempt to adopt CTT also for mathematical subjects, including the use of IT. It’s worth mentioning that within the past 10 years the Kharkov State Pedagogic University has formed a good tradition in the development and investigation of the most important trends for using different software in studying mathematics and working out courseware.

We think that among numerous IT programs, first of all, algebra software packages (Derive, Gran) and dynamic (“live”) geometry (Tragecal, DG) are needed to carry out deeper investigations. In other words, the non-productive (data processing) part of work (calculations, making diagrams, storing, etc.) is left to IT, while the productive part (data evaluation and organization, planning and decision making, etc.) is carried out by students.

The basic ideas and philosophical fundamentals of CTT activate students’ participation in the development of learning projects and pithy problem solving (e.g.: What to produce and for whom?; or: What would happen if?), mastering research methods, finding efficient solving algorithms, etc.

As for the IT application in the development and implementation of students’ projects, solving integrated problems (e.g.: mathematics and economics) is rather easy today. We are convinced that IT is an effective intellectual tool in the development of educational technologies in compliance with cooperative learning and collaborative work. This is mainly due to the fact that IT enables the user to orient in the endless “sea” of constantly changing and updating information.

Compared with the tasks of the traditional school curriculum these learning projects are more practice oriented: Firstly, their description part can be of a rather different character (story, diagrams, calculations, etc.). Secondly, projects should have an immediate practical value (searching methods for improving work efficiency, forecasting product outputs and costs). The core of the work in projects is to always get an actual result, i.e. solving a theoretical problem should produce a concrete solution; and working on a practical task should result in a concrete outcome. Thirdly, the employed data should be real ones, taken from live situations, which, again, requires integrated knowledge (knowledge of different subjects, fields of science, technologies, cultures).

When working in learning projects students have to master new skills, e.g.: 1) purpose oriented building and analysing mathematical models of actual problems; 2) selecting data needed for problem solving (if statistical data have to be used); 3) selecting investigation methods which are not given; 4) formulating and solving problems by using previous analytical considerations; 5) examining the solutions whether their outcome could be used in practice; 6) using guidebooks, tables, layouts, diagrams, drawings, software (e.g.: symbolic mathematics packages); 7) applying the calculus of approximations, calculations with different values; 8) evaluating the error and the order of values; 9) using solution-validity check methods.

During the workshops we use to pay big attention to the introduction of meta-cognitive processes, i.e. mastering of “thinking strategies”, “implementation rules” for cognitive actions since, if they are ignored, false and formal views and conceptions can emerge. In this context it is important to overcome stereotypes like “right” and “wrong” responses. We are sure that the students, who mastered the CTT (including additional rhetoric workshops) are interested in listening to different opinions of their colleagues; they understand the importance of joined work and the development of a more argumentative status of their ideas, creating an atmosphere in the classroom that favours free acceptance or argumentative rejection of the others’ views.

In the projects students, firstly, work individually and in groups, solve different tasks connected with the application of knowledge from the different fields of science; organize brainstorming, expert group meetings etc., learn how to use and utilize technology in their projects, etc. Secondly, they turn from the presentation of ready-made knowledge and from memorizing to self-searching activities and grouping of forces. The teacher acts as coordinator assisting the students on the complicated stages of the projects. Thirdly, it’s also very important to shift the attention from the “average” student to the strong ones and to those who are lagging behind. Thereby the teacher pays more attention to the students’ individual peculiarities and to their uniqueness, which provides possibilities of individually solving educational problems.

One more trend of our investigations should be mentioned: the acquisition of skills for making the right decisions on the basis of a rational choice. It is evident that people today, as never before, are facing different aspects of this process, in particular: the need to check reliability and to understand new information, to develop alternative options and to choose optimal solutions, to use probability theory when thinking about problem solving, to forecast and analyse consequences, to support their point of view in discussions with the help of logical conclusions, etc.

Among the most spread mental stereotypes that attracted our interest are the under-estimation of the power of visual information and occasional information (visual heuristics and representability) by personal experience with technology, the distortion of actual pictures and the lack of ability to recognize obvious contradictions (confirmation tendency).

By getting personal critical thinking experience, the teachers’ attitudes towards non-standard or unexpected students responses become the manifestation of their professionalism, readiness to revision, change and development of their own plans and visions. Besides, some of them have already acknowledged that they are involved in the creative search for critical thinking strategy applications. Thus, the teacher, when implementing the critical type of teaching, inevitably is no longer only the subject of teaching but its object as well.

Our experience shows that such a training allows the teacher to solve effectively the problem of choosing training objectives, developing training plans, designing a system of lessons, etc., thereby motivating the development of elements of situational pedagogy and situational methods which have to be created by the teacher each time anew, depending on the training situation due to specific features of both the students and the teacher, their creative potential, talent for improvisation, etc.

Depending on the difficulties and type of mistakes made by students, the teacher can use CAS and dynamic geometry software in order to encourage the students to follow their interests, to reflect, to pursue the principle of self-determination (analysis of the created situation, reflection upon the performed actions, selection of optimal implementation methods), and to train cognitive methods and their application to particular situations (formulation of questions, contradictions, generalizations, search for analogues, etc.). — I’m sure that all project participants will agree that the RWCT project was favourable to a training environment featuring, first of all, orientation towards investigation, a democratic style of communication, the development of self-reflection and the ability to understand each other.

Our belief in the feasibility of such developments is grounded on the peculiarities of the training process, which allows to solve a basically new didactic problem, i.e. to study local or global phenomena and processes in complex systems on the basis of modelling. Obviously it is necessary to modify curricula and the methodical framework, which should focus on the formation of successful learners.

Besides, new standards of mathematics education require understanding how meaningful classroom dialogues can stimulate the collaboration of the teacher, the students and the software. The realization of this direction is based on the idea that successful learning activities assume some features of research and that a lesson begins to resemble a research study or a joint project with IT being used as a tool of cognition.

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The role of the computer in

discovering mathematical theorems

Tadeusz Ratusinski

Cracow, Poland

Mathematical studies at the Pedagogical University are part of the education of future mathematics teachers. Prospective teachers should be prepared to work in a modern school, in particular, they have to be enabled to use modern study aids. However, so far the curriculum of my university does not include a subject that would deal with the application of computers to mathematics teaching. Up to now there has been only a computer course, where selected elements of mathematics have been used as illustration. In the next academic year a new course will be established: „the application of new technology in mathematics instruction.” Students will enter it after the computer course, so that they are supposed to be computer literate. Therefore, the classes will not emphasise the technicalities but rather the applications of the computer to the didactic process. The lack of that kind of course so far had been compensated by the individual lecturers within their lectures. Thus, for instance, within my lectures in mathematics didactics I had made my students familiar with the possibilities of using computers in class. In this paper I want to report my observations in such a lecture with Year IV students.

The class consisted of 16 students in the age of approximately 22 years, and it dealt with the properties of monotonic functions. In a traditional course the notion of the monotonic function is discussed and various attempts to investigate its properties are made using various kinds of exercises. However, in no course- or exercise book I ever came across a problem formulated as follows: what can be said about the sum, difference, product and quotient of two monotonic functions? This approach does not exist in a traditional course. It is rather tiresome, complicated, if not outright impossible to obtain a solution in the traditional way, step by step. Working on the solution the students encounter a number of problems that are hard to overcome. This is where the computer can help.

The students were given a spreadsheet which I had prepared like in Fig. 1. For the purpose of my investigations I invented some „free” monotonic functions in order to avoid interferences which could be effected by the use of (smooth) elementary functions. The spreadsheet presented two such functions, each depending on two parameters A and B, and their sum (difference, product or quotient). The functions were defined in the following way: Parameter A determined the value of the function at the starting point of the time interval [x_S, x_E]. Given a finite sequence (x_n) of arguments with x_S=x₀<x₁<...£x_E the value of the function for some argument x_n₊₁ was calculated as the sum of the value for the preceding argument and the product of a random number from the interval (0, 1) multiplied by the value of the other parameter, parameter B. By the sign of B was decided whether the function was decreasing (B<0), increasing (B>0) or constant (B=0); its value determined the tempo of monotonicity ( i.e. the higher the value of B>0 was, the faster the function grew). The students could choose the parameters freely, thus determining the shape of the graphs on the screen. Their task was to investigate the properties of the graph of the sum (difference, product or quotient) depending on the component functions, in order to find some regularities and to answer the question that had been asked in the beginning, i.e. about monotonicity.


Fig. 1

The importance of a situation where students discover a new theorem has been emphasised by Z. Krygowska (1977): „Provoking a student to formulate hypotheses for which he finds an empirical or intuitive basis is an essential element of the process of teaching. Regardless whether the hypothesis turns out to be true or not, and no matter how important a mathematical fact it describes, its formulation, rationale and an attempt to prove it, backed up with the proper approval of a teacher, develops an ,intellectual courage’ of a student, if only they are not reduced to thoughtless proposing of anything. Developing a sound ,intellectual courage’ – it is one of the important goals of teaching in general and of teaching mathematics in particular.”

One should think that the knowledge the students had acquired in former instruction would lead their investigation, influence the formulation of theorems and help to recognise the falseness of hypotheses faster. My observations do not support this assumption, though.

Here is an overview of the hypotheses that the students came up with (I omit a great number of examples because they are analogous to those which I present):

	(1)	The sum of increasing (decreasing) functions is an increasing (decreasing) function
	(2)	F1 constant; F2 increasing (decreasing) Þ F1 + F2 increases (decreases)
	(3)	If F1 increases faster than F2 decreases, F1 + F2 increases
	(4)	If F1 decreases faster than F2 increases, F1 + F2 decreases
	(5)	If F1 increases faster than F2 decreases, F1 – F2 increases
	(6)	If F1 decreases faster than F2 increases, F1 – F2 decreases
	(7)	F1 decreasing, F2 increasing Þ nothing can be said about the monotonicity of F1 + F2
	(8)	The sum of two monotonic functions is not a monotonic function
	(9)	The graph of a function being a sum of a decreasing function and a constant function is identical to a graph of a decreasing function
	(10)	F1 increasing, F2 decreasing Þ F1 + F2 is decreasing
	(11)	F1 increasing, F2 increasing Þ F1 + F2 is decreasing
	(12)	If F1 increases almost as much as F2 decreases one may say that F1 + F2 is almost constant, i.e. approaching constancy (... but is not totally constant since for various arguments it assumes various values as seen in the table).
	(13)	F1 constant and positive, F2 increasing (decreasing) Þ F1 F2 increasing (decreasing)*
	(14)	F1 constant and negative, F2 increasing (decreasing) Þ F1 F2 decreasing (increasing)*
	(15)	F₁ & F₂ increasing; F₁ > 0 & F₂ > 0 Þ F₁ F2 increasing*
	(16)	F₁ & F₂ increasing; F₁ < 0 & F₂ > 0 Þ F₁ F2 decreasing*
	(17)	If F1 increases faster than F2 decreases, F1 F2 increases*
	(18)	If F1 decreases faster than F2 increases, F1 F2 decreases*
	(19)	The product of two increasing functions is increasing

There is a great variety of phrasing in these statements. The students preferred logical shortcuts in the form of implications. Now and then there were more elaborate statements like 1, 8, 9, 12, 19. On several occasions students went beyond the concept of monotonicity and took into consideration certain interdependencies associated with the sign of the functions (13 – 16), or the velocity with which the graph increases or decreases (hypotheses 3 – 6, 12, 13, 17, 18). The analysis of the students’ work shows that they concentrated on the sum and difference of two monotonic functions. There were a few cases where the product was taken into account, whereas only one student worked, yet unsuccessfully, at the quotient.

The students did not manage to avoid false hypotheses, e.g. (12): „If F1 increases almost as much as F2 decreases one may say that F1 + F2 is almost constant, i.e. approaching constancy” and (19): „The product of two increasing functions is increasing” (with its relatives 17 and 18).


Fig. 2

Presumably, the students’ selection of examples affected their proposal of hypotheses, in particular, the authors of hypothesis (12) somehow „polished” the graph (Fig. 2). This way of reasoning is displayed by the practice of transposing the properties of the general outline of the graph to the function itself. For instance, when noticing that the graph shows an upward tendency students were inclined to claim that the function was increasing in its complete domain, disregarding the possibility that it could decrease locally. The reason for considering a wrong hypothesis to be true might lie in the students’ inclination to generalise too hastily or to draw conclusions on the basis of a particular choice of cases. Lack of analysis of a wider range of cases led to wrong conclusions. Similarly, the observation of the product of two monotonic functions in the interval where they only assume positive values led to the formulation of hypothesis (19) (for counter-examples see Fig. 3).


Fig. 3

To the majority of students who formulated false hypotheses the computer was so much persuasive that they thought they were legitimated to generalise the observed cases and were not at all disturbed by the question whether their conclusions were true. This is nothing new to mathematics educationalists. S. Turnau (1990) claims that at a certain level the analysis of an ostensibly sufficiently wide and diversified range of examples is so much persuasive that the truth of the hypothesis becomes evident. However, students ought to have an inner drive for proof, and I did not notice any attempts on their part to confront their findings with their already existing knowledge on the subject.

Hypothesis (8): „the sum of two monotonic functions is not a monotonic function”, which seems to result from a superficial analysis of a rudimentary number of cases, is another example of a peculiar „infatuation” with the computer.

The outcome of my experiment points to a few research problems which one can express in the following questions:

What made the pre-service teachers, who ought to have enough knowledge on the subject, so easy to be misled? What were the original causes of their mistakes?

To what extent could the mistakes have been caused by gaps in the body of mathematical knowledge that the students had acquired?

How were the results of the experiment related to computer skills and experience with computers?

Could the mathematical problem turn out to be too unorthodox? Did the fact that in the traditional approach to the subject these problems are rather non-existent affect the results of the experiment?

It appears that my observations could be a starting point for further research.

References

Krygowska, Zofia (1977): Dydaktyka matematyki. T.3. Warszawa.

Turnau, Stefan (1990): Wykłady o nauczaniu matematyki. PWN, Warszawa.

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Self-guided learning —

Scenarios and materials from a German pilot project

Monika Schwarze

Hamm, Germany

1. At a glance – Organization of the pilot project

2. Self-guided learning: Different scenarios and concepts

3. Scenarios and material: Use in the classroom and evaluation

How can new media make the teaching and learning of mathematics more exciting? This is one of the questions that the project ‘SelMa – Selbstlernen in der gymnasialen Oberstufe – Mathematik’ tries to answer. SelMa is supported by the Federal Republic of Germany and the state of North-Rhine-Westphalia. Different scenarios of self-guided learning have been created. Some of them and results of the first evaluations are presented.

1 At a glance – Organization of the pilot project

The four-year pilot project "Self guided learning in mathematics in senior high schools" –called SelMa started in early 1999. Its aim is to analyse the interdependencies and interactions between mathematics, learning in general and the use of new media.

Main questions focussing upon learning, mathematics and the use of media are

Which mathematical topics match the idea of self-guided learning?

How should classroom material be arranged and presented? In particular: online or offline?

How should students and teachers be supported?

How can the progress of learning be 'controlled'?

How can new media improve the quality of learning (mathematics) and stimulate life-long learning?

How can knowledge be consolidated by means of intelligent practice?

How can telecommunication (platforms for collaborative learning or a "teacher-on-demand" via email) support learning of mathematics?

A team of 3 to 4 teachers from 5 schools in North-Rhine-Westphalia, called 'authors', began – addressing the issues mentioned above – to create scenarios of self-guided learning and develop suitable classroom material.

In 2000, when the first projects were finished and successfully tested by the authors in their own classes, a second group of 10 schools, so-called 'evaluators', was established in order to evaluate the material and to systematically test whether it works in everyday usage. Other schools were invited to participate, too. The feedback of all evaluators will be incorporated in the on-going development of the material. The current state of affairs can be viewed online ( http://www.learn-line.nrw.de/angebote/selma/index.htm). This site offers a wide range of materials that can be trialed and adapted to the teacher's individual needs.

The aim of this web site is to be a platform for information, communication und co-operation between teachers working in the field of self-guided learning of mathematics.

Furthermore, authors and evaluators are going to disseminate their practice in in-service-teacher-training, in order to build up networks of schools in the different regions. We hope that this process will lead to more self-guided learning in everyday school lessons. Including publishers in our project at an early stage effected high-quality (offline and online) media for the work in the periods of self-guided mathematics learning.

2 Self-guided learning: Different scenarios and concepts

The first projects were based on rather different ideas of self-guided learning. The authors could not base their work on concrete concepts or learning arrangements because in German mathematics education there is a severe lack of research material which would be documented comprehensively so that the ideas could be transferred to other fields within mathematics education.

In one scenario of self-guided learning environments are intensively used. Students work on it in longer periods in mathematics lessons as well as at home. The material consists of a hypertext with exercises, contextual aid, a glossary, solutions and general ideas how to optimize individual and collaborative learning in school and at home. Another group of authors established an independent learning centre (for all subjects) at their school. Some parts of our mathematics curriculum have been set aside for self-guided learning, that means that these topics are not taught collectively in mathematics lessons but that the students have to study them on their own without any support of the teacher. These learning environments consist of a course with a rather linear structure – with graded aid, suggested solutions and a collection of problems, in particular real-life problems of different categories.

Another means for increasing student activity and self-guided learning is the method of the 'learning carousel' (in Germany we call it 'learning at stations'). Ten to twenty different stations (exercises, real-life problems depending on the subject) are exposed to the students. Some stations deal with a special task, a new mathematical context, others invite to exploration or investigation of mathematical problems using handheld computers. All stations offer special aids how to approach the task and other hints suitable to the students' needs and a paper with a complete solution. All students receive a 'to-do-list' which informs them about all the stations (number; title; topic; obligatory or additional station; individual, pair or group work, media). Students can choose the order of tasks and might individually (or in groups) choose their learning pace.

These two methods imply certain dangers. During periods of self-guided learning teachers automatically change their roles from acting as instructors to being supporters of individual learning processes. Usually, teachers cannot exactly point out how big the learning progress of the groups in general and the individual really is. Students must be capable of monitoring their learning processes on their own, but this ability has to be acquired in a similar way as subject matter has to be learned. Additional tools like learning diaries, mindmaps and electronical communication tools might support this process of self-control.

3 Scenarios and material: Use in the classroom and evaluation

Learning environments

Educational research tells us that learning and understanding mathematical concepts and using problem solving strategies work better if there are various approaches with real life problems of various levels accommodating different types of learners. As the understanding of mathematics requires knowledge of details, conclusions and relations between single topics, learning environments (based on hypermedia in a linked-up, not linear structure) enable the learners to create their individual mental network of mathematical knowledge.

The pilot study 'SelMa' offers two examples of learning environments, 'linear programming/ optimization' and 'matrices'.

'Linear optimization'

This learning environment has been created for the revision of concepts around linear functions. The students choose one of a range of problems (on the basis of brief descriptions of the problems) which make up the 'heart' of the learning environment, and then they are guided through the important steps to solve a mathematical optimization problem. At the same time they revise what they were taught about linear functions at lower secondary level. The learning unit links new contents and mathematical concepts with topics that the students had acquired in previous mathematics lessons (and possibly had forgotten in the meantime). One part of this learning environment deals with the learning process and the monitoring by the students themselves. In the learning environment students find, e.g., advice for self-control and hints how to optimize group work and their study at home.

'Matrices'

In this learning environment students are offered several real-life problems related to the same mathematical topic of the subject 'matrices'. They choose one problem that they are interested in, then they are 'guided' through the problem (which is posed rather openly), not step by step, but by more general questions concerning strategies of problem solving, by a glossary or by questions that prepare the formation of the mathematical theory behind the problem. New definitions and theorems etc. will be discussed whole-class teaching. Students can see that different problems lead to the same mathematical concepts.

Both learning environments, intended for the use in the classroom and at home, offer some more details that support orientation and self-guided learning in hypermedia:

survey of the subject to be learned

table of contents

glossary

review of the topic

some recommended paths

different modes of representation and visualization and interactions (as often as possible)

some interesting historical facts of the subject and real-life applications

exercises

contextual, graded aid

a chapter concerning learning strategies, problem solving and self-control of the learning process.

The hypertext based on HTML is open, can easily be modified (e.g. integration of other documents and visualizations – either static or dynamic – interactive and ready to be published in the internet), improved, developed, discussed ... The material includes practical advice for the teacher, who becomes an individual adviser during the work with this learning environment and act as moderator, when the results of the group work are presented and general methods to solve problems of linear optimization are discussed.


Fig.1a: Different parts and their role within the learning environment and important activities without the learning environment	Fig.1b: Main page of the learning environment 'matrices'

Learning carousel — Learning at stations

The project "Geometry of Circles" consists of two parts. In the first part the students have to investigate the equation of a circle and then create – using CAS or a graphing calculator – a mathematical description of a logo, a window of a church, a pattern or a model of an existing object containing several circles. Here, students can see the importance of geometry in real life. The students work in groups of two or three and have to present their results on posters or WORD-documents to the rest of the class.

The second part of the project is based on the method 'learning at stations', often practised in elementary schools. It focuses on the development of new aspects of coordinate geometry and poses problems which connect the new geometrical object 'circle' with other objects like parabola and lines (tangent, points of intersection, ...).


Fig. 2 a, b: Students working in groups at the same station or using CAS for exploration


Fig. 3: "To-Do-List" for the learning carousel about the subject 'circles'

The problems are presented on worksheets and files, first with the help of concrete exercises, then by generalizing the solution. Each station consists of the worksheet, some helpful questions and a complete solution. Some stations are more graphically oriented, e.g. including investigations of families of curves or a puzzle where descriptions, graphs and equations of circles have to be matched.

The fact that students work on different tasks at the same time enhances their individual learning process. The complete offer consists of 10 stations. The stations are accommodated to different background levels of learning, different speeds of learning and working, and different modes of working (individually, in groups of two or three).

Different media are used at different stations, e.g. the CAS DERIVE or the TI-89 calculator. The tasks are usually activity-oriented. The students normally work in groups of two or three and decide together at which station to work next. At each station the materials lie on a table during the whole lesson. Each station exists in 3 to 4 copies so that the students really have the choice what to do next. During the work the teacher answers the groups' questions. In our first evaluation we noticed that students only tentatively used the additional aid, which was put on a table further away from the exercises. First they tried to help each other, then they asked the teacher who had much more time to give individual advice than in traditional lessons. Collaborative working is highly supported by this method. No students looked at the solution provided without trying to solve the problem on their own.

Mindmaps, diaries and communication tools

After longer periods of self-guided learning weaker students sometimes do not know whether they have learned all topics and understood all relations between new and old subjects. Mindmaps can support the review of the main steps of the learning process in different ways.


Fig. 4: Mindmap of the learning subject "circles"

Firstly, a mindmap containing only main topics can be completed individually after the period of self-guided learning. So the individual automatically reflects the own learning progress. New facts are linked to details of the 'old' individual network of knowledge. Different mindmaps – that means different points of view – can be presented and discussed in class.

Secondly, a mindmap of the subject matter can be constructed whole-class teaching. Mindmapping tools like 'Mindmanager' or 'Inspiration' offer various features like, e.g., annotations, links and moving branches to other positions. Students must discuss in detail relationships between different branches and will become aware of the structure of and the connections between mathematical topics.

Diaries are a means to encourage students to continuously reflect upon their learning progress. In an introductory session students of Year 11 were informed about the aims of this method and the prospect contents of their personal diary. A personal diary – only read by the student and the teacher – should contain all important facts of a lesson (steps to a new topic, definitions, proofs, examples) and may include a personal review (What did I learn? What was difficult for me to understand? How can I memorize it?). The diaries were checked (annotations if there are mistakes) and assessed by the teacher every three months. Most of the students who kept a diary with a lot of personal annotations stated that they felt better prepared for the tests in comparison with the situation at the beginning of the school year because they had paid more attention to their weaknesses. The SelMa-web-site presents 25 diaries in the following six fields of reflections: about lessons, aha-effects, individual explanations, self-assessment, analyses of mistakes, and further issues.

In addition to this we have started to gather experience with collaborative online tools like BSCW or Web-CT that can be incorporated in longer periods of self-guided learning. All students have access to the Internet at school, most of them at home, too.


Fig. 5: BSCW-Workspace (BSCW= basic support for cooperative work); http://bscw.gmd.de

As an additional offer, we invite students of Year 11 to use a workspace in the Internet. They may put individual questions concerning the topics of the previous lessons, they get the answers from the 'teacher on demand' or from other students. Weaker students can find intelligent practice, links to interactive online-tools and visualizations, brighter students may be interested in more demanding tasks and experiments e.g. with CAS.

The tools mentioned above encourage the students to share information and initiate discussions with experts. 'Communication on mathematics' seems to be more intense when these new technologies are used.

Websites

BSCW-website:	http://bscw.gmd.de
BSCW-workspace with public access:	http://bscw.gmd.de/pub/german.cgi/0/27877615
Inspiration (software):	http://www.inspiration.com
Mindmanager (software):	http://www.mindjet.com
SelMa-website:	http://www.learn-line.nrw.de/angebote/selma/
project "Matrices":	http://www.learn-line.nrw.de/angebote/selma/foyer/02b_hammproj3.htm
project "Linear Optimization":	http://www.learn-line.nrw.de/angebote/selma/foyer/02b_hammproj1.htm
project "Geometry of Circles":	http://www.learn-line.nrw.de/angebote/selma/foyer/02b_hammproj2.htm
Personal web-site:	http://www.mathematikunterricht.de/

References

Fankhänel, Kristine and Weber, Wolfgang (2001) SelMa – New Perspectives for Self-Guided Learning in Teaching Mathematics at Senior High School Level. Paper prepared for WCCE, 2001, Kopenhagen.

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Mathematical abilities of university entrants and

the adapted use of computers in engineering education

Angela Schwenk and Manfred Berger

Berlin, Germany

1. The changing role of computers

2. Mathematical basic abilities of students

3. Consequences to the use of computers in education

4. Summary

Any enquiry into the future of the teaching of mathematics should include an assessment of the current situation: University entrants to engineering courses have poor knowledge of mathematics. In this article the results of tests taken by university entrants and high school students in the years 1995 and 2000 are compared. As one result of our investigation more suitable uses of computers in education are presented.

1 The changing role of computers

Since the first appearance of personal computers in the early 1980's in the field of education dynamic developments of computer technology have taken place.

At first both teachers and students were fascinated by the new tool which had only poor facilities. As a teacher one was able to impress students by printing a simple table of values or even presenting a graph on the screen. The students tried to copy, usually they were successful. At that time the facilities of the computers and the students’ use were congruent. But there was a divergence concerning the teachers: They found themselves just at the beginning and hoped to seize their sphere of mathematics, which is not the students’ sphere. Both, teachers and students used the computers in close relation to the subject.

The 1990's were dominated by an increase of the facilities and features of computers. At that time the balance between the features and the use of computers by the students vanished: Nowadays nearly every student has a PC of his own and the medium lost its fascination. While more and more computers are available, the areas of application remain largely unchanged. The average student uses them mainly for entertainment purposes (games, internet), he/she is fascinated by perfect design and graphics and uses it as an ordinary tool in every day life, e.g. as a typewriter.

The teachers are still very fascinated by the improving facilities of the PC and the available software concerning mathematics, for example: Visualisation of mathematical ideas, notions, a tool for solving problems and doing mathematics. It is felt that in many cases this exceeds the intellectual capacity of average students.

There are two reasons for the described discrepancy: On the one hand there are the improving facilities of computers and on the other hand there are decreasing mathematical abilities of university entrants (Pfeiffle & Nairz-Wirth 2000). If one uses computers in education one should be aware of this fact and should look for an adapted use of computers. For this reason we divide our further explanations in two parts: In the following section 2 we report on investigations in the years 1995 and 2000 of mathematical abilities of university entrants at the TFH Berlin and of high school students of the Bertha-von-Suttner-OG (B.-v.-S.-OG) (Gymnasium) Berlin and compare the results. In section 3 we present some conclusions concerning a more adapted use of the computers as tools of education. We try to classify the different forms of usage and give some examples.

2 Mathematical basic abilities of students

The period of investigation, investigated groups and data

In the year 1995 we anonymously tested 329 university entrants and, five years later, 627 at the TFH Berlin. In 2000 students at the B.-v.-S.-OG of the 10^th year (135 testees) and 11^th year (108 testees) were tested, the same questionnaire was used.

Detailed comparison of TFH and B.-v.-S.-OG

Preliminary remark: All tasks had to be carried out without the use of a pocket calculator. If required, appropriate approximate values were given.

Question 1 – Skills in Arithmetic

1./2. Calculate as fraction and as decimal fraction:

3. How much is 0,225% of 3,2 billions?

Here we were confronted with the first unsatisfactory result (Fig. 1): More than 30% of the students could not handle the simple fraction of the first part. But, as shown below, this question generated one of the best results of the entire questionnaire.

Fig. 1: Results of question 1

Here the 10^th year students did best. We think that this is a result of the ban of pocket calculators until the 9^th year in this school. Everyone who is teaching mathematics should realize that using electronic tools only makes sense, if the underlying mathematical background is well understood. Moreover, dealing with fractions is also an important prerequisite for algebraic calculations like in question 2.

Question 2 – Algebraic calculation

1. Simplify:

2. Put together:

There is an alarming lack of abilities of both university entrants and high school students concerning algebraic calculations (see Fig. 2). Only 13 % of the university entrants and hardly anybody of the high school students (2%) were able to handle fractions with roots.

Fig. 2: Results of question 2

In the year 2000 even the more simple fraction in the second part was solved by only 19% of the entrants and less than 7% of the students. These results confirm the experience that the algebraic calculations taught in the 8^th year are not understood. Even complementary exercises in the 9^th year (calculations of roots) and the 10^th year (extension of power calculations) do not improve the abilities.

Question 3 – Power and logarithm

Determine x:

The results show the next gap in students’ knowledge (Fig. 3). These kinds of exercises are a topic in the 9^th and 10^th year of high school. But in Berlin the number of math lessons has been reduced to three per week. So these results might indicate that these topics were skipped.

Fig. 3: Results of question 3

Question 4 – Trigonometric functions


	Fig. 4: Results of question 4

1./2. Plot the graph of the sine and of the cosine function. Let in a given right-angled triangle the hypotenuse c=3cm and the angle a=59°.

Determine with the approximations sin a»6/7 and cos a»18/35:

3. the side a

4. the height h_b

5. the side b

Here we see that there is a wide gap between students’ knowledge and their actual ability: The students have some knowledge, but they are not yet able to apply it properly.

Between 60% and 70% of the students were able to draw the graph of the sine and cosine function, but asked to apply these findings in a practical way, the solving rate drops to less than 40% and in the case of the high school students to less than 34% (see fig. 4). This is remarkable since trigonometric functions were presented just before the investigation.

Question 5 – Linear functions

1. Draw a straight line through the points A(-6|9) and B(3| ⁹/₄ )

Let the corresponding equation be given by y=mx+n.

2. Determine m

3. Determine n

4. Determine the intersection with the x-axis


	Fig. 5: Results of question 5

At least 15 % of the university entrants and even a larger part of the high school students are not able to draw a line through two given points. At the beginning of the 11^th year of the high school the linear functions were intensively repeated as a preparation for the differentiation of functions. All necessary quantities like the slope m and the intersection with the axes were discussed in detail. This will explain why the students of the 11^th year are the best, but this will not explain why more than 50% of them are not able to determine the essential quantities of a line. This raises the question how far these students have understood the concepts of the differential calculus, which was a topic just before the investigation.

Question 6 – Quadratic polynomials / Quadratic equations

1. Draw the parabola y = x²/4 in the given coordinate system of question 5.

2. Determine its intersections with the edge of the plot.

3. / 4. Determine the intersection of the parabola and the line of question 5.


	Fig. 6: Results of question 6

More than 45% of the university entrants of the year 2000 and more than 60% of the high school students are not able to draw the given simple parabola. Here also the results worsen as the problems get more applied. Still, quadratic polynomials are continuously a topic in the 11^th year under different points of view. These repetitions to the subjects of the 8^th and 9^th year seem to be of no effect. It looks like students store their knowledge in pigeonholes and do not find it if they are not prepared for searching it.

Question 7 – Linear systems

1./2. Solve the linear system: I) 4x+7y=16 II) ‑4x‑3y=24

3./4. Determine the line equations of I) and II).

5. Draw the lines.

6. Get their point of intersection.


	Fig. 7: Results of question 7

These bad results (Fig. 7) were expected due to the bad results concerning the linear functions.

Looking at all these results one should take into account that this investigation compares a (special) university entrant of a technical program (who should be familiar with mathematics) with the average student of an ordinary German high school, where only a small part of them is going to study technical subjects.

University entrants of the years 1995 and 2000

In Fig. 8 the results of the university entrants of the years 1995 and 2000 are compared by box plots. Both white boxes at the left show the results of all participants: the results of 2000 are worse than 5 years before. The mean value, indicated by a cross, dropped from 43% to 39%, and there is also a significant decline of the 75% mark by 10% points. The long term investigation (over 20 years) by Pfeiffle & Nairz-Wirth (2000) in Austria shows the same trend.

Fig 8: University entrants 1995 and 2000, comparison of „Fachoberschule“ (F) and „Gymnasium“ (G)

In Germany there are two kinds of university entrants into “Fachhochschulen” (universities of applied sciences): Students after 12 years of school (the so-called “Fachoberschule”) and after 13 years of school (the so-called “Gymnasium”). While the decline of the 13 years students (G) is quite small there is an alarming decline of the 12 years students (F) especially a heavy dropping of the 75% mark. The number of participants double within these 5 years due to two reasons:

1 In general the number of students of technical programs is increasing.

2 Few more colleagues and their classes joined the investigation. The ratio of the “F”- and “G”-entrants changed a lot: while in 1995 the ratio of “F”:”G” was about 2:1 it was 1:1 in 2000.

3 Consequences to the use of computers in education

The foregoing remarks imply some demands on an adapted use of computers. We will describe these demands, classify the use of computers and give some examples.

No use at any price

The results of the second section in 2. concerning the skills of arithmetic demonstrate clearly the necessity to weigh the use or no use of computers: If there is no training there will be a loss of skills; the less the practice is established, the more the skill vanishes. The results of the questions concerning algebraic calculations, power and logarithm caution us to neglect students’ work of calculations. Here is a danger of using pocket calculators and computer algebra systems.

One of the most important guiding principle should be: if it is not necessary to use a computer, then it is necessary to use no computer (Bauer 1988).

The computer as a calculating machine – An example

Prerequisites: The students know definitely how to solve linear systems, they had proved it by solving several abstract problems on their own. This means that the problems stated with question 7 in section 2.2 are no more present.

Now we can follow a new aim:

Aim: The students are able to model a given practical economic system and should be able to discuss the solution.

Here the computer saves time for searching an appropriate approach for solving the practical problem. Since here the intention is modelling, the calculation is of minor importance. So the actual calculation can be left to the computer.

Another example is to trap students into rounding problems by extensive computer calculations in order to encourage critical discussions of the use of computers.

The computer for verification of results – Three examples

Example 1: Verification of hand made calculations

Prerequisites: The students are able to determine on their own the points of local extremum or of inflection of a family of functions.

Aim: The students are able to reconstruct their results by a computer algebra system, to plot the graphs of the family of functions and mark the special points. They verify their pencil and paper results.

Example 2: Verification of discussions

Prerequisites: The students have determined cubic splines through given points. Now the calculated curve together with the calculations is displayed on the screen.

Aim: The students discuss the effect of changing the tangents at the border points. After adapting the corresponding parameters they check their predictions with a computer algebra system.

Example 3: Critical reflection of hand made and computer made results

Prerequisites: The students know how to solve an ordinary differential equation by separation of variables. A computer algebra system confirms the results. Now they plot the slope field and the solution; in Fig. 9 one can see the slopefield

Its solution is:

Aim: The students recognize that the solution is not a solution in the whole range of the function and that computer solutions should be checked carefully. They check whether the validity of their calculation is limited by indirectly given conditions.

Fig. 9: A slopefield and its solution

The computer for visualisation of the invisible

Prerequisites: The students know how to differentiate a function and how to determine the slope of the tangent at a certain point. They have got a first idea of the meaning of the derivative.

Aim: After plotting the graph of the function and the tangent and after zooming into the graph the students recognize that the function becomes practically linear in a small neighbourhood and understand the concept of linearization.

The computer as a proof stimulus – An example

Prerequisites: The students know area-preserving transformations like shearing and rotation.

Then we animate the proof of Pythagoras’ theorem by shearing and rotation of the square into the corresponding rectangle (see Fig. 10).

Aim: The students discuss the question whether a picture does prove anything. They are motivated to prove correctly including all the careful checking whether the implicit assumptions are fulfilled.

Fig 10: Pythagoras’ theorem

4 Summary

By our examples we presented an adapted kind of use of computers which takes into account the poor knowledge of the students. We took as our starting point a careful checking of the abilities the students need for using computers and how we can verify that the students really have them. This phase of critical reflection is very important to working with computers. Otherwise there are two dangers using computers: 1. The teachers are too enthusiastic about the ‘nice’ facilities of the computers so that the students might not be able to follow their teaching. 2. The problems of or for the students may be covered. Before using computers, the teacher has to consider how to check afterwards (using operationalized aims) that his aims have been reached.

References

Bauer, Hans (1988): Was heißt und zu welchem Ende betreibt man ITG! Mitteilungen des Philologenverbandes Rheinland-Pfalz 2/88.

Pfeiffle, Horst, Nairz-Wirth, Erna (2000): Physikalische und mathematische Kenntnisse von naturwissenschaftlich orientierten Studienanfängerinnen und Studienanfängern. Robert Ruprecht (ed.): Unique and Excellent. Ingenieurausbildung im 21. Jh. Leuchtturm, Alsbach,176-181.

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Of Babies and Bath Water

John Searl

Edinburgh, UK

1. Introduction

2. Some historical examples

3. Concluding observations

1 Introduction

One of the phenomena of the teaching profession that I have observed since I joined it in the 1950's and still surprises me, is its reluctance to accept and welcome curricular changes. Here is an extract from a letter to a former colleague from his head of department in 1960:

‘.... As to £50 for hiring from time to time the Pegasus electronic computer now in Edinburgh, the question is: What use would you and your colleague make of it? . But it may be said that I am old fashioned. I don't think I am; but I do want to know where the Pegasus would come in this matter of teaching...’

It was another 10 years before elementary computer programming was introduced in our non-specialist first year courses (against much opposition from some colleagues who thought it inappropriate in a mathematics course). I have observed, and still observe, the same reluctance to accept the changes that are possible with the introduction of hand-held technology in the form of graphical calculators.

In the past I have sometimes uncharitably attributed this reluctance (to embrace change) to feelings of inadequacy in the teacher whose knowledge base is being challenged. More charitably I have understood their lack of motivation to be based on their sense of security. They know what they can achieve with the students using current methods and do not wish to put that at risk. Their first responsibility is to enable their students to pass the examinations. (Is it?) Of course, many teachers had been successful in mathematics without ever needing to use a calculator! They were quick at mental arithmetic and could perform all those tedious arithmetical tasks competently. If they had not needed such crutches, why should students now? A nice description of this feeling about the role of technology is given by Cosete Crisan (1999). Another reason, however, is possible. When introducing technology some previously important skills become obsolete and some knowledge becomes redundant. And the issue is whether that knowledge is redundant. Are we throwing out the baby with the bath water?

This has always been an issue in the learning and development of mathematics. At its very simplest, the change in representation of the number three from III to 3 puts at risk the concept of the cardinal number three and we have to compensate for the deficiency of the symbology by classroom activities which involve counting the numbers of objects in sets of different sizes. The concept of cardinality is needed for a proper understanding of the process of multiplication (in contrast to rote learning of number facts). A more substantial loss of a holistic view of a subject can occur when there is a change from the methods of pure geometry to those of algebraic geometry. That change is necessary for an understanding of what was originally described as organic geometry which formed the basis of the theory of differentiation. (Now many texts portray differentiation as tangent finding rather than rates of change so that the original theoretical methods have overtaken the meaning of the concept.) There is a danger now that such dynamic geometry will, with the attractiveness of its computer implementation, completely displace the ‘theorem and proof’ approach of pure geometry even for the simplest results. In geometry, as in trigonometry, there are static and dynamic images to be assimilated for a proper understanding of the concepts. And both images are needed for the student to make effective use of those concepts.

This issue was addressed by Zalman Usiskin (1992). He suggested that ‘You can tell that a concept/skill * is obsolete for ALL when:

a) Even adults who could use * and know * do not use *

b) When you ask questions involving * you need to know the answer first

c) The historical motivation for * is no longer and no other motivation has arisen

d) * is found only on tests, on contests, or in puzzle books

e) The only reason for teaching * is that it is needed next year.'

Usiskin's criteria usually produce feelings of discomfort among mathematics teachers. They will usually agree the syllabus is over-burdened but are reluctant to reduce the content.

At a popular (populist?) level, the use of calculators for basic arithmetic continues to attract controversy in the UK (and elsewhere to judge from the e-mails I receive). In the UK the situation has been confused by other social changes.

The change from £sd with its mixed number bases to a decimal system and the loss of coins like the farthing (1/4d) resulted in a loss of mental arithmetic practice and consolidation in everyday financial dealings. The change from imperial measures (pounds, feet, inches, etc) to the metric measures meant that common fractions became redundant giving way to decimal fractions. (We used regularly to do calculations like 1¾ of a yard at 2s6½d.) Of course performing mental arithmetic with decimal fractions is different from that of common fractions but a consequence of the changes is that most pupils have less difficulty than previously with decimal fractions and have considerably more difficulty with common fractions.

Fig. 1: Hogben (1936) 'Mathematics for the Million'

But does it matter? At the skills level I do not think it does, but there are consequences at the conceptual level. Much of the concept actions use common fractions, for example, sharing 2 pizzas between 3 people. When multiplying fractions, conceptually it is easier to image the process with common fractions.

The idea may not be lost if a 10 ´ 10 square is used to illustrate 0.2 ´ 0.7, but 0.23 ´ 0.74 begins to raise the level of technical demand of drawing accurately and obscure the imaging of the concept. The loss of the arithmetical skills associated with common fractions suddenly becomes a stumbling block in algebraic manipulation. It has become commonplace to find students who fail to perform simple manipulations with algebraic fractions competently.

Here we have a clear illustration of a situation in which the baby has been thrown out with the bath water, but what is it that has been lost? Previously many students experienced problems with the process of finding partial fractions but now more elementary mis-skills seem common. In a second year engineering class at Edinburgh University, students were asked (as part of a larger task) to solve the equation

(These students had all passed their first year course in Mathematics so we could not apportion blame to poor school teaching!) Fifteen different ‘answers’ were produced. The most common sources of error were the mis-skills or mis-concepts

We often describe these blunders as ‘trivial’ (they are not high level skills) and so the students do not apply adequate attention to them. Clearly something has been lost from the mathematical experience of these students and it is at a higher level than the skills of algebraic manipulations. It is not mere numeracy but a sort of comfortableness about mathematics that I call mathemacy, that gives students a ‘feel’ for the subject rather than a list of tricks.

I have written elsewhere that the introduction of technology raises the level of intellectual demand on both pupil and teacher (Searl 2000). I have no doubt that the use of technology can make a huge range of mathematics accessible to many students at every level. But I thought it would be worthwhile to examine situations where the introduction of technology has diminished the learners’ mathematical experience. The purpose of such an exercise is to ensure that appropriate compensation is made elsewhere in their experiences (cf III and 3), not to reject the use of technology.

2 Some historical examples

‘The Grounde of Artes’, Robert Recorde, 1543

This was a very popular text on arithmetic, written in English. It ran to 29 editions. Here is its algorithm for multiplication of numbers between 6 and 9, for example 8 ´ 7. The ‘units’ are given by (10 - 8) ´ (10 - 7) = 2 ´ 3 equals 6 and the ‘tens’ are given by either 8 - (10 - 7) or 7 - (10 - 8) equals 5, so that 8 ´ 7 equals 56. In these early Arithmetics, there is a plethora of algorithms for dealing with each “special” case. We can see here the difficulties people have always experienced in remembering the multiplication tables. Multiples of the lower cardinals, 1, 2, 3, 4 and 5, occur quite often in everyday usage and so get frequent refreshment. The higher cardinals are used less often and consequently their multiples are not remembered so well. In this example we can see the construction of an algorithm appropriate to the knowledge resource that was available. The widespread introduction of printing led to the production of multiplication tables and ready reckoners. This resulted in a loss of the sense of ownership of the mathematics. It became somebody else's mathematics that was being used and being used without necessarily understanding how that mathematics had been developed. Relational learning gave way to instrumental learning. Mathematics was to become the property of an elite.

‘Arithmetica’, unknown author, 1478

This was the first popular arithmetic book, published in Treviso, Italy. It was in Latin. It included the gelosia method of long multiplication, a form of which was taught in Italy until the 1950's. It is thought that it was borrowed from Arabic-Hindu mathematics. Thus, for the multiplication 365 ´ 293, we have:

Each of the partial products, 3 ´ 2; 6 ´ 2;...;5 ´ 3, is written in the appropriate square, the tens digit (if there is one) being written above the diagonal, the units below. The diagonals are then summed successively from the bottom right-hand corner, the sums being recorded at the edge of the square (with the usual carrying to the left).

Fig. 2

John Napier made his “Bones” to be used with this algorithm (‘Rabdologiae libri duo’, 1617) and added to it a simple checking mechanism (because of ‘slippery errors’). In addition to calculating 365 ´ 293, he suggested calculating 634 ´ 293 = 185762 and his “Bones” automated this process by their construction. Then he obtained the check calculation

1 0 6 9 4 5

1 8 5 7 6 2

2 9 3

2 9 3 0 0 0

Fig. 3

(This is described in a little more detail in Searl, 1998.) The “Bones” remained in popular use until the beginning of the nineteenth century in Britain. When they were abandoned, this checking method was ‘lost’ but perhaps the understanding of the need for a check by an independent method was also forgotten.

‘Rechentaflein’, A.L.Crelle, edited by O.Seeliger, 1907 (reprinted 1954)

This is a table of multiples of the whole numbers up to 1000, that is, a thousands times table. It was first published c. 1830. It was used until the 1960's for high precision calculations. Thus 26,457,081 ´ 247,183 is calculated:

26,457,081

247,183

20,007

112,879 14,823

6,422 83,631

4,758

6,539,740,652,823

Fig. 4

Here the digits to the left (in latin numerals) are calculated from 247 ´ 81, 247 ´ 457 and 247 ´ 26 and the digits to the right (in italic) are calculated from 183 ´ 81, 183 ´ 457 and 183 ´ 26. The tables give efficient methods for calculating square-, cube-, and fourth-roots of large numbers. This method was never taught in schools as far as I know, but underlying it is an experience of the algebra of arithmetic which facilitates understanding of the algebra of unassigned variables.

‘Chamber's Four-Figure Mathematical Tables’, L.J. Comrie, 1947

These tables (and the corresponding six-figure editions) represent the pinnacle of achievement in the production of tables for practical computation. The sixty-four pages are amazingly comprehensive. Many teachers bemoan the abandonment of such tables for multiplication and division using logarithms. That, they claim, gave a rich experience of the properties of logarithms before they met the logarithm function in a calculus context. But much more was lost: the ideas and practice of interpolation, the use of critical tables, the relationship between log^ex and log¹⁰x and much more. Perhaps most importantly, the need to set calculations out correctly and neatly to enable checking etc was consolidated by the use of these tables in school mathematics. The tables contain some gems of good practice, my favourite is the choice of type-face with old-style figures for easy legibility (demonstrated by de Morgan many years before), using ‘ascenders’ and ‘descenders’ rather than uniform heights.

Fig. 5

This result may have consequences for teachers of pupils suffering from dyslexia. The font is available electronically from Monotype. It is the ‘imprint expert’ font and was designed in 1913.

Locating the roots of an equation

The need for root-finding occurs in many practical problems and consequently the topic appears in many syllabuses. The criteria for convergence of fixed-point iteration are discussed. In the better texts, the criterion for stopping an iteration for true accuracy is investigated. It is all ‘good’ mathematics! There are some subtleties that are educative and emphasize important issues in numerical mathematics. (See, for example, Modern Engineering Mathematics, edited by G James, pages 602-606.) But, of course, all that material is simply replaced by using a graphic calculator. In terms of 'Mathematics for ALL', however, that mathematics is obsolete.

Elementary calculus

Here the word elementary is used in the sense of fundamental. The processes of integration and differentiation are quickly turned into a special kind of ‘algebra’. To integrate xⁿ you raise ‘it’ to the power n + 1 and divide by the new power AND add an arbitrary constant. To differentiate sin(x) you add p/2 to the argument x. There is no relational understanding and many students think that integrals are areas ‘under curves’ and derivatives are gradients of tangents (but do not know why ). Applications of the calculus to other situations cause great difficulties. Often students do not know whether to integrate or differentiate an expression they have found! In the same way as there was an elite in the fifteenth century who could perform arithmetical tasks, now there is an elite who understand the calculus. The use of Derive or Maple or Mathematica will not de-mystify the calculus. It will be neither worse nor better than teaching without the aid of CAS. The development of relational understanding depends on the modelling of different applications of the calculus.

3 Concluding observations

The introduction of technology within mathematics and mathematics teaching leads to gains and losses in the learning experiences of students. It is often easy to identify the former. Indeed the technology has been introduced to effect those gains. But it is important to identify the losses so that appropriate compensation can be made elsewhere. There are, in addition, other (perhaps more important) issues to address. The introduction of technology changes the classroom ethos. For example, teaching packages tend to a transmissionist view of education, seeing the learner as bringing nothing to the learning situation. They are often concerned with skills training and provide a sharp contrast with a constructivist view of knowledge. They emphasize an individualistic approach to the learning and neglect shared learning within the classroom’s mathematical community. They create or reinforce the view that the ‘teacher’ is the authoritative source of knowledge as opposed to a fellow learner working at a different level. At the same time there is a danger of the loss of that special relationship between students and their teachers. Students often respond better to materials produced for them by their own teacher (for example, see Milkova & Turcani 2001) as opposed to ‘imported’ but more ‘professional’ materials.

In the modern world much of the lives of ordinary people depends on the (often invisible) use of mathematics. Effective mathematics education is essential for the long-term health of democracy. Universal suffrage without effective education leads to tribalism and populism. Technology widens the range of mathematics accessible to pupils and students at every level but it is important that mathematics educators sharpen their awareness of the gains and losses of its introduction. By careful thought the losses can be redeemed, by inappropriate use of technology the gains can be squandered. In a powerful argument for mathematics education ‘for all’ based on democratic need, Mogens Niss (1994) observed that ‘Mathematical competence tends to contribute to the creation of expert rule if it remains a scarce resource’. The challenge to all teachers of mathematics at every level is to ensure that it does not remain a scarce resource.

References

Crisan, Cosete (1999): Reflecting on my first experience with new technology. Micromaths 15, 2, 20–23.

James, G. (ed.) (2001): Modern Engineering Mathematics. Third edition, 602–606. Prentice Hall.

Milkova, Eva and Milan Turcani (2001): Integration of ICT into teaching and learning Discrete Mathematics. ICTMT 5, Klagenfurt.

Niss, Mogens (1994): Why do we teach mathematics in school? Mathematics Teaching 1994, Conference Report, The University of Edinburgh.

Searl, John W. (1998): Mathematics, Technology and Mathematics Education: Some Reflections. Mathematics Teaching 162, 24–28.

Searl, John W. (2000): Changing mathematical demands: a consequence of teaching with technology. Micromaths 16, 1, 23–28.

Usiskin, Zalman (1992): From ‘Mathematics for Some’ to ‘Mathematics for All’. ICME-7, Quebec.

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[1] See Johan Huizinga (1994): Homo Ludens. Vom Ursprung der Kultur im Spiel. Rowohlt, Reinbek.

[2] A good introduction is given by Martin Aigner (2001): Diskrete Mathematik. 4th Edition. Vieweg, Braunschweig, in Chapter 9.3.

[3] See Tobias Nipkow and Franz Baader (1999): Term Rewriting and All That. Cambridge University Press, Cambridge, Mass., for an overview.

[4] Kruskal’s algorithm is to be found in Aigner (see note 2); for the towers of Hanoi see Ronald L. Graham, Donald E. Knuth and Oren Patashnik (1994): Concrete Mathematics. 2nd Edition. Addison Wesley, New York, Chapter 1.1.; Heapsort is explained in Donald E. Knuth (1998): The Art of Computer Programming. Vol. 3. 2nd Edition. Addison Wesley, New York.

[5] See John Stasko (ed.) (1997): Software Visualization. MIT Press, Boston, Mass.

[6] The basic text on the P-NP problem is the following book: Michael R. Garey and David L. Johnson (1979): Computers and Intractability. Freeman, San Francisco.

[7] An overview is given by Michael M. Richter (1992): Prinzipien der Künstlichen Intelligenz. Teubner, Stuttgart.

[8] See p. 238 in Graham, Knuth and Patashnik (see note 4).

[9] See Jean Flower (2002): Interactive web-based resources and a new perspective on algebra and geometry. This Volume.

[10] See John Berry and Roger Fentem (2002): Investigation into student attitudes to using calculators in learning mathematics. This Volume.

[11] See, e.g., Walter Oberschelp and Gottfried Vossen (2000): Rechneraufbau und Rechnerstrukturen. 8th Edition. Oldenbourg, München , as a guide to many of these questions.

[12] See Hans-Werner Heymann (1996): Allgemeinbildung und Mathematik. Beltz, Weinheim.

[13] See, e.g., Mazen Shahin (2002): Modelling with difference equations using DERIVE. Discrete delayed population models with DERIVE. This Volume

The global perspective

Paderborn, Germany

Chances and limits for teaching in the information age —

Aachen, Germany

1 Introduction

2 General issues

What we learn by Information Technology

Structures of successful teaching

Dangers and the future

3 Mathematics teaching in the Information Age

Is mathematics an alien to the human mind?

The impact of IT on mathematics teaching

Society demands concerning future mathematics education

Investigation into student attitudes to using calculators with CAS in learning mathematics

John Berry and Roger Fentem

Plymouth, UK

1 Introduction

2 Research aims

3 Research design

4 Data analysis

5 Brief outcomes and conclusions

Acknowledgements

References

The potential of the Internet

Wuppertal, Germany

1 The project MathePrisma

2 Internet software

3 How to make use of the technical features of the Internet

The fundamental structure

Interactions

Visualization

4 Results

On the impact of hand-held technology on mathematics learning —

From the epistemological point of view

Warsaw, Poland

1 General education in the digital era

2 Mathematical education for the future

3 Graphing calculators in the mathematics learning process – Potential and obstacles

Example A: Who is better, in comparison to the whole group?

Example B: Diversity of uniform distributions – Is it possible?

Example C: From statistical data to fundamental probabilistic concepts

Example D: Gathering rough data and confronting them with a theoretical model

Example E – Searching for a central tendency of a set of grouped data

4 Final remarks

References

A project on the development of

Kharkov, Ukraine

The role of the computer in

discovering mathematical theorems

Cracow, Poland

References

Self-guided learning —

Scenarios and materials from a German pilot project

Hamm, Germany

1 At a glance – Organization of the pilot project

2 Self-guided learning: Different scenarios and concepts

3 Scenarios and material: Use in the classroom and evaluation

Learning environments

'Linear optimization'

'Matrices'

Learning carousel — Learning at stations

Mindmaps, diaries and communication tools

Websites

References

Mathematical abilities of university entrants and

the adapted use of computers in engineering education

Angela Schwenk and Manfred Berger

Berlin, Germany

1 The changing role of computers

2 Mathematical basic abilities of students

The period of investigation, investigated groups and data

Question 1 – Skills in Arithmetic

Question 2 – Algebraic calculation

Question 3 – Power and logarithm

Question 4 – Tri­gonometric functions

Question 5 – Linear functions

Question 6 – Quadratic polynomials / Quadratic equations

Question 7 – Linear systems

University entrants of the years 1995 and 2000

3 Consequences to the use of computers in education

Question 4 – Trigonometric functions