'Characteristics of Task-Oriented Mathematical Thinking among Students in Upper-Secondary School'

The most important and perhaps the most difficult concept in the theoretical framework of the study is ´mathematical thinking´, which is defined many different ways in the litterature. The study examines approaches to the concept of mathematical thinking. The approaches are student´s beliefs, student's mathematical abilities, anthropological perspective, problem solving and information process. In the study the mathematical thinking is described as the information process monitored by one´s metacognition. The information process is understood in the same way as in the theory of connectionism. The role of student's beliefs as part of one´s metacognition is crucial in the monitoring of thinking processes. Student's mathematical knowledge is described as procedural, conceptual and strategic knowledge. The concepts of ´skill´ and ´understanding´ are defined by the three kind of knowledge. In the study attitudes and emotions are part of beliefs, which belong to student's view of mathematics.

In the empirical part of the study is examined upper-secondary school student's mathematical thinking in the longer course in mathematics from three perspectives, which are the perspective of society, teacher's and student's perspective. The main problem in the study is to describe features of student's testoriented mathematical thinking.

The describtion from the three studyperspectives consists of features of student's mathematical proficiency and one´s view of mathematics. In the subproblems are studied what kind of differencies exist in the mathematical proficiency and in the view of mathematics between gender, between students who studied according to different curricula and between students who had chosen compulsory test or optional test in the matriculation examination.

In the societal perspective the focus is the test of the longer course of mathematics in the matriculation examination and curricula. Teacher's perspective consists of studies of evaluation of mathematics learning in Tervakoski upper-secondary school. Student's view of mathematics is studied in the student's perspective. The view of mathematics includes student's beliefs about mathematics, beliefs about oneself as a learner and as a user of mathematics, beliefs about mathematics teaching and beliefs about mathematics learning.

The empirical data of the study were collected in the 1990s (except 1994). The empirical data consist of testresults (n=89804) of matriculation examinations in mathematics, students' (n=103) testresults in the longer coursies of mathematics and students' interviews (n=6) in Tervakoski upper-secondary school. In addition, students (n=384) in eight secondary schools answered to the questionnaire. The questionnaire included 46 Likert-scaled statements. The empirical data of the matriculation examination and of Tervakoski upper-secondary school were analysed using the usual statistical analysis. Especially using regression analysis there was formed the two-by-two frequency table. Students' view of mathematics was researched using frequence and factor analyses.

There are found four groups in the two-by-two frequency table. They are ''Successed students'', ''Matured students'', ''´Just doing´-students'' and it's subgroup ''Loosers'' and the last group ''Disappointed students''. Student's mathematical thinking is described through five features of mathematical proficiency. They are conceptual understanding, strategic competence, adaptive reasoning, procedural fluency and the view of mathematics. Students' mathematical thinking in the two-by-two frequency table is found in three categories: (1) contents bounded mathematical thinking (2) broad mathematical thinking and (3) narrow mathematical thinking. The group ''Disapointed students'' consists of students, who have had success in the specific substance of the longer course in mathematics, but who had failed in the mathematic test of the matriculation examination.

''Disapointed students'' have contents bounded mathematical thinking. The subgroup ''Loosers'' has become mostly after the reformig of the matriculation examination in 1996 when the longer course in mathematics became optional in the matriculation examination. The differencies in mathematical thinking between those who had chosen a compulsory test or who had chosen an optional test in the matriculation examination are observable in every one of the three perspectives. In the research about student's beliefs there was the result that students' perseverance has dimished in studies.

Keywords: Mathematical thinking, mathematical proficiency, view of mathematics, longer course in mathematics, upper- secondary school