Abstract:

A case-study in mathematical teaching at the middle school level (9th degree - 11/12 years old) is presented with its provisional conclusions.
An experimental mathematics approach was implemented. It is, with the due adjustments, the one being used (with several variations) by the first author from 1990 until now, on teaching Applied Math’s at the undergraduate and graduate level at Porto University.
The experimental classes are taking place at High - Middle School “João Gonçalves Zarco” (Matosinhos, Portugal) to two groups of students of the 9th degree (11th to 13th years old students; they are supposed to progress only until the 9th degree. Some other classes are scheduled to take place at a Private High School (Porto, Portugal) using High School students (17/18 years old) that are supposed to access to the University level.
At the Middle School the students are separated in five different groups each one with presumably homogeneous learning skills. We worked with two of them: a first one with As and Bs at almost all subjects (12 students) - the "better students" group. We used for project assessment the "worst students" group: those that failed at almost all subjects (namely Mathematics and Mother Language) at the end of the 8th degree. Even if the second one was used for control purposes we had always the “secret desire” that they would to over-come the “better students”.
The first contact was done during a lecture period of 90 minutes using two teaching groups. The activity is in a voluntary basis using the student’s free time. We substituted the “paper and pencil” techniques by the usage of a spreadsheet. Deliberately we refused to use elaborated teaching packages. None of the students had any previous contact with the software application.
Some basic skills to produce straight line graphs and to compare them were taught. The graphic solution of a system of two linear equations was also easily found. They were able to fulfill all the proposed targets during the class: the “better” students did all the exercises in 90 minutes but, amazingly, the major part of the “worst” ones did them in only 60 minutes. Of course we need at least an extra class to consolidate the acquired knowledge.
The next two steps will consist on the solution by Gaussian decomposition of several two equations systems. Afterwards we will progress, by extension of the method, to the solution of larger systems of linear equations. We think we will be able to teach the students - in context - several of the concepts usually seen only in Linear Algebra at the University level. Lastly we hope to be able to instruct them on the concepts and techniques of the theory of perturbation.
The purpose of the project is to show that all the facilitation policies issued being applied only to produce “good success statistics” is a wrong one. It is possible even for those that are considered to be “bad students” to learn effectively at very early ages subjects that are usually taught only at the University, if appropriate approaches are used. The quest for quality in Math’s is still possible,, if the teaching paradigms are modified.

Keywords:

algebra, computer uses in education, experimental mathematics, secondary education.