Author: Elena F. de Carrera
Mathematics Department Principal
Faculty of Biochemistry and Biology Sciences
Universidad Nacional del Litoral, Santa Fe, Argentina
Address: Necochea 2932 (3000) Santa Fe, Argentina
As an answer to the ICMI request we send the present work, which is a summary of the activities the research group on MATHEMATAICS EDUCATION under my leadership has been carrying out since 1986.
The difficulties are: high deficiency on Mathematical knowledge the new students brought from High School, demotivation, lack of training of the students for problem solving.
To solve these problems a research group on Mathematics Education at the School of Biochemistry designed a Distance Education Course. This Course is written with an historical view, with Heuristic method and it is based in the methodology of problem solving.
Then, they looked for the problems intimately connected with the discipline chosen. The motivation is very important. The visual impact plays important roles in motivation because the sensitive experiences, are the background upon which deeper knowledge is built.
Finally in this moment, the problem solving is the methodology with we are working.
The resolution of mathematical problems supplies students with techniques, which can be used in different areas, even to every day problems. Mathematical thinking is logical and strict, intuitive and creative, dynamic and changing.
The difficulties our students have when learning Mathematics became more and more evident. We detected several facts which made the problem worse and darker. So in our University borned the research group on Mathematics Education under my leadership in 1986. This group was initially motivated by the low percentages of success and the high percentages of failure and desertion of the students as well as the increasing demand of mathematics knowledge among the graduates together with the better quality of that knowledge.
The difficulties we will mention below were the most evident and we are going to deal with them independently. They are the difficulties, which made us change the standards of Mathematics Education at the school of Biochemistry:
All the facts mentioned above made us present a really new approach for teaching Mathematics at the University this approach is based on four main facts:
The ever increasing demand of Mathematical Knowledge and the problem to carry it out.
It has always been accepted that Mathematics is an essential part in most of the University Careers. The diversity of science actually needs Mathematics but the amount of knowledge the students need today is so big and changing that it immediately becomes old.
Test, surveys and individual studies were made among the new university students to check their personal abilities and the amount of knowledge they had when joining the university, They proved the poor intellectual condition students brought year after year.
To solve these problems we designed a five - week course to provide students with the necessary elements high school had neglected. We used a student - centred approach because considered them not a mere receptor of knowledge already processed.
At the same time we decided to come closer to high school and help them by means of a distant education course to be attended by those who intended to study a university career.
This new course is aimed for students finishing their secondary school and wishing to enrol in College. Its purpose is to adapt them better to the new environment, as well as to train them to use their highest thinking abilities and to give them a better approach to the formal operations period. The course is based in the methodology of problem solving, so as to let the student become acquainted with the surrounding reality and expressing it in mathematical terms.
The Heuristic method is used to stimulate in the students the search of their own solutions. Their own discovery of the mathematical concept involved and to incite in them the necessity of increasing their knowledge.
The course is written with an historical view, in order to bring the students closer to mathematics, while taking them away from inaccessible unfinished and hidden views of the subject.
The distance education methodology was used so as to come closer to secondary school students of the vast region of influence of the University. The course was structured in six modules, with the advice of mathematicians (Dr. Néstor Aguilera y Dra. Eleonor Harboure), psychologists an pedagogues. It uses the active language of a 17 years old teenager, a weekly radio program, featuring also music, it is used as a help, bringing closer the teachers voice to the distant student.
The distance education methodology, a democratic view on education, is an optimum means to achieve the goals we teachers aim at when teaching. Education as a means of social development, a way of bridging the ever increasing gap between the rich and the poor. Not only an economic/financial gap but also a gap in the access to knowledge and the ability of individuals to adapt themselves to new technologies invading society in an accelerating way.
This social as well as educational crisis, with its marked un-balances, is present and more intense at the beginning of the tertiary studies. This is not only because the entering student suffers conflicts produced by this fact, but also because each individual brings inherent differences of knowledge and emotional maturity among the in-coming group.
The differences arise because of the structural knowledge each student carries in when entering College. This effect is deepened when they are away from home, trying to take off from their families and get in to the bigger city they come to. It is an eradication problem.
This "get in" into a new social environment is less traumatic if the institution (College) becomes known beforehand, when they are ending their secondary level. Knowledge of social environment means, in this case, to take notice of College demands, of its staff and their way of working. This knowledge helps in dealing with fear towards the unknown, one of the possible drawbacks for achieving success at this level.
The distance education methodology is optimum in getting to students with higher quality proposals, aiming at a gentler pass between secondary school and college. It is presents as a step to development of critical thought as well as autonomy and self-control. With this methodology, the students must work alone. They have to apply heuristics to solving problems, use self-evaluation, and so up to reach metacognition which is the highest step in the learning process.
As a result of this experience we noticed a remarkable improvement in the students efficiency. A larger number of them succeeded in first examination they sat for, immediately after the course, compared with the ones who had attended the conventional course.
Lack of motivation.
Motivation is essential when teaching Mathematics. It is the trigger of the learning process. It is even more important when teaching Mathematics in a non - centred Mathematics career.
Although Mathematics is only one, the approach for a good learning varies according to the students it is aimed at.
It is usual to speak of Mathematics for engineers, biologists and architects and there is plenty of bibliography and wide teaching experience for each area but is also usual to forget that each discipline has its particular Mathematics with a common root but, with different aspects which have to be borne in mind for the sake of students motivation and the planing of the future students activities. 
The concepts mentioned above made us reflect on the necessity of changing of the Mathematics. Motivation became an outstanding aspect of the new teaching approach and in the search of problems having strong link with the career students chosen.
It has been taken into account that a present - time graduate has to be trained in such a Mathematical environment that it allows him to develop his intuition and creativity as well as to use different mathematical patterns in different biological and technological processes and to develop their own solving strategies at the same time. 
Motivation was built around a "problem" intimately connected with life and the discipline chosen. When a student "knows" what is Mathematics for, when problems have triggered his/her curiosity, it is his/her interest in solving them, which makes him/her apt to begin learning subjects that will help him/her in solving the proposed problems.
Visual impact plays an important role in motivation, not because an image is worth a thousand words", but because, mathematically, visualisation is vital for teaching and learning this discipline. It is fundamental to make work imagination and intuition and to abstract a concept or solve a problem, abilities which otherwise are difficult to develop. Sensitive experiences, those concerning with senses, are the background upon which deeper knowledge is built.
From a certain level, it will be interesting to mingle and identify different symbolisation and meanings of a concept: the object or real model, the graphic representation, the word, and the synthetic description or definition. 
This visualisation does not mean sophisticated resources, a piece of chalk and a board, a retroprojector, the P.C., or may be a simple picture is enough.
Routine calculus are rather frequent when learning and using certain mathematical topics. The teacher quite often prevents himself from dealing with certain problems because of the amount of calculus, which have to be carried out.
It is very stimulating to work with real problems when matrixes, equation systems, or general linear algebra are taught at university level. In Colleges dealing with Biology, Chemistry or Medicine it is interesting to present dietary, species-equilibrium, chemical reactions balance, and sickness-spreading problems, among others. While in Engineering, electric circuits, heat-spreading problems are as important as well.
These problems lead to matrixes with high number components and numbers not always easy to handle where inequalities and conditions may be hard to solve.
All these problems are put aside when not dealing with real facts and so calculus become meaningless. Thus, it is much simpler and more efficient, for learning, the possibility of access to a P.C. and so avoids routine calculus so as to work on the model, to vary conditions, and to arrive to conclusions. This is much worthier than if working in solving easy matrixes and with "good" numbers as are supposing to be the whole numbers.
We had to bear in mind that authentic problems are not of an only predictable, sound answer, but those whose results should be argued, whose solution bothers us and carry a deep cognitive conflict. For the type of problems mentioned before there is no better thing than a P.C.
It is exactly here where the computers become in a valuable tool. It does not mean it is a class on computing techniques, but it helps for a better understanding of mathematics concepts.
The computer and the other visual aids are very helpful in the process of teaching and learning Mathematics. It is necessary to make deeper investigation on all these teaching aids including the computer.
With the motivation and the diversity of problem the record of the pupils were better but there are either problem.
The teaching of Mathematics at the University follows an almost exclusive pattern: theoretical classes with the professor introducing and developing a particular topic and a workshop where students have to solve exercises and problems. It is our goal to change this pattern.
Learning Mathematics is not only learning rules, statements, definitions, to demonstrate theorems and use them in the resolution of problems. Learning Mathematics is solving a challenging problem, try different strategies and find a shorter and simples way to come to an exact conclusion. Mathematics at the University is deeply rooted in the building up of knowledge; it is necessary that the student understand he has to study it and to learn how to do it. According to Chevallard, studying is the lost link between teaching and learning.
The resolution of Mathematical problems supplies students with techniques, which can be used in different areas, even to everyday problems. Mathematical thinking is logical and strict, intuitive and creative, dynamic and changing. Students who are not studying to become mathematicians have to be taken into account and motivate them to be interested in this science.
Since a year, we have been carrying out an experience in teaching mathematics in the first courses of the careers mentioned. This experience before to try to compare and evaluate similarities and differences in the learning of mathematics between those students learning it in a traditional way and those who apply a methodology centred in solving problems. There are no conclusions about this experience yet because it has only a year so it is a very little period.
There exist many studies about the solving problems method and children or youngster behaviourism, but little exists about students in their near twenties, entering College, who need Mathematics but are not specifically inclined towards it.
May be some of them have chosen careers apparently away from Mathematics thinking that it has nothing to do with the subject chosen but there comes reality. They enter into College and the first discipline they bump into is Mathematics. Such is the case of Biochemistry, Biology, and some Social Sciences. And if we add Statistics, when speaking about Mathematics, the phenomenon widens and there is practically no career without this subject.
Research on Mathematics Education at the university has to be encouraged because of the importance it has in the different University careers. There are plenty of works on teaching Mathematics at primary and secondary levels but are scarce with students who are quite close to the world of the adults. It is necessary to explore the cognitive processes, which are triggered, and their relationships with problems of increasing complexity. It is necessary to help those researchers who are sometimes carrying out their works under rather unfavourable conditions.