In the beginning of 1980's, Ed Dubinsky were interested from a computer program called SETL (Set Language). Dubinsky and some of his fellow teachers started to use this program on the university lectures with good response. At first they tried to understand that what was so good about SETL that makes students to learn and understand better mathematical concepts and structures. Later, as the new version of SETL called ISETL (Interactive Set Language) were published and as the user group of this program expanded, the study expanded to a more general level, i.e., what really happened in a students mind when she or he learned - or didn't learn - mathematics. The first RUMEC had been formed. Afterwards, there have been founded other RUMEC groups, so that Dubinsky and the other fellows in the original RUMEC had to rename their group to RUMEC 1.

RUMEC 1 especially aims 1) to increase the understanding of researchers in how students learn mathematics, 2) to develop a theory based teaching method for higher level mathematics and 3) to develop the techniques in getting information and in evaluation of the information.

The first day of the course was theory based. Participants learned about the believes how people learn, something from two important subjects called the APOS theory (Action-Process-Object-Schema) and the ACE cycle (Activities-Class discussion-Exercises), among others.

Let us consider the basics of the APOS theory. An Action is a transformation of an object which is perceived by the individual as being at least somewhat external. The transformation is carried out by reacting to external cues that give precise details on what steps to take. When an action is repeated, and the individual reflects upon it, it may be interiorized into a Process. Every element or a concept (for ex. a number, a symbol, a function, etc.) that can be a target for an operation like an action or a process, is called an Object. A Schema for a certain piece of mathematics is an individual's collection of actions, processes, objects and other schema which are linked consciously or unconsciously in a coherent framework in the individual's mind and may be brought to bear upon a problem situation involving that area of mathematics.

ACE cycle consists of Activities, Class discussions and Exercises. In Activities, the students are having computer tasks in small groups. The tasks are abstract and usually so big that the students have to finish them on their own after the class. In Class discussions students use a pen and a paper to consider the tasks given in the Activities phase. Explanations and definitions are also given in Class discussions. Students get Exercises for a homework. These are meant to deeper the mathematics that has previously been studied and learned. After the Exercises, the Activities come to the picture, and the cycle has been formed.

Even though the first day was a bit tedious, it was good to have a theoretical start for the course. The theoretical material given beforehand helped a lot on the course to understand a big number of new definitions and concepts. Also, many cups of delicious coffee and tea on a numerous coffeebreaks had something to do with the non-sleepiness on the lectures and exercises.

The second and the third day of the course was totally in a computer lab. The participants were divided into a groups of two or three. Computer tasks for ISETLW (a Windows version of ISETL) were given. The second day was spent for calculus and the third day for abstract algebra. After every ten or fifteen minutes, the teacher Dubinsky interrupted the participants by asking some questions. All the groups had one or two minutes time to consider the given question and to prepare the answer. The important thing in teaching and learning mathematics called cooperative learning, was learned the hard way.

The beginning of the fourth and last day of the course was spent in a computer lab to finish up the given computer tasks. After the lunch there were a summary and future activities in the form of a lecture, and, of course, the final discussion.

A natural question now arises: How to use ISETLW, APOS theory, ACE cycle, etc. in the undergraduate university level? The author's opinion is that these methods should be applied in more than just one course. There has been an experiment in the University of Joensuu, department of mathematics, on a course of abstract algebra. It is a good start, but also insufficient. It takes a lot of time to study the basics of ISETLW if one doesn't have much experience on computers and programming. The best way to start might be to teach ISETLW on a very first university course on mathematics. This course usually consists much of the material already studied in high schools, so that recalling this mathematics with the aid of ISETLW and cooperative working might be a good idea. After the first course on mathematics, the students know how to use ISETLW and also they know how to work cooperative. It is then a lot easier to use these methods in the becoming courses (linear algebra, abstract algebra, etc.).

How many courses using these new methods there should be? Should all the courses at the undergraduate university level be teached like this? The author's opinion is that one course per semester for the first four semesters in the university might be enough. After the first two years in the university, most of the students have a picture in mind how mathematics works. Also, after the first two years the mathematics becomes maybe too abstract for ISETLW. Still, cooperative working among other theories is valid even for experienced researcher and professors. So, some methods could be applied after the first two years.

There are still problems if one begins to apply these new methods from an empty table. Should one use the new methods for all students or for a special group of students? What about the lack of experts on the matter? It may not be enough to study these methods a day or two from books and articles and then to try them in practice. Anyway, it is nice to have new theories and ideas in the field of teaching mathematics.