Learning to prove by investigations: a promising approach in Dutch secondary education

The Mathematics Department of the University of Nijmegen in collaboration with the Graduate School of Education has developed a math course in the field of fractals and dynamic processes for volunteer students in the second phase of secondary education in the Netherlands. The students, of approximately 16 years of age, show a special interest in, and an aptitude for mathematics and informatics. One of the main goals of the course was to highlight the deductive aspect of mathematics, an aspect that is neglected in the ordinary math curriculum of secondary education in the Netherlands. That goal was pursued by giving the students ample opportunity to conduct investigations on their own and in a way that they would be responsible for judging the correctness of their arguments in making mathematical deductions. In that way proving is imbedded in a larger structure and becomes a tool for the students to convince each other. During the courses we searched for the right way for teachers to coach these investigations, that is: to find a balance between mere concentration on guiding the process of the students' investigations and active intervention in the learning process of proving. In this article we illustrate with two examples – the first from the '95–'96 course and the second from the '96–'97 course – in what way we adjusted our coaching. Our results are explorative, but our approach appears promising and we are convinced that investigations as learning environment for proving would also be a valuable part for the regular secondary math education in the Netherlands. This revised version was published online in July 2006 with corrections to the Cover Date.     

COLLEGE TEACHING IN LEGAL EDUCATION: Teaching Method,Students' Time-on-Task,and Achievement

Based on the theory of teaching as assistedperformance (Tharp and Gallimore, 1988), the UtrechtUniversity Law School has implemented college teachingfocusing on a particular combination of various methods of assisting students' performance. In additionto the conventional case method in legal education, theproblem method was adopted as the primary method ofinstruction. This method focused on how students learn to solve legal problems by actuallyfinding, framing, and analyzing issues themselves.Contingency management was used to motivate students tocomplete problem-solving assignments and attend theseminars in which training was provided for legalproblem-solving skills. Analyses of variance wereperformed to examine the relation between the problemand the case method, and students' time-on-task andachievement. The results were favorable to the problemmethod: Compared to the case method, students spent moretime on learning activities and performed better,controlling for the analysis for their previousachievement, time-on-task in previous terms, and motivationfor the particular course. The differences inachievement between both teaching methods were mainlyrelated to the difference in the time students spent on the particular course.     

Evaluating assessment quality in competence-based education: A qualitative comparison of two frameworks

Because learning and instruction are increasingly competence-based, the call for assessment methods to adequately determine competence is growing. Using just one single assessment method is not sufficient to determine competence acquisition. This article argues for Competence Assessment Programmes (CAPs), consisting of a combination of different assessment methods, including both traditional and new forms of assessment. To develop and evaluate CAPs, criteria to determine their quality are needed. Just as CAPs are combinations of traditional and new forms of assessment, criteria used to evaluate CAP quality should be derived from both psychometrics and edumetrics. A framework of 10 quality criteria for CAPs is presented, which is then compared to Messick's framework of construct validity. Results show that the 10-criterion framework partly overlaps with Messick's, but adds some important new criteria, which get a more prominent place in quality control issues in competence-based education.     

The LAAS: A Computerized Scoring System for Small- and Large-Scale Developmental Assessments

The evaluation of developmental interventions has been hampered by a lack of practical, reliable, and objective developmental assessment systems. This article describes the construction of a domain-general computerized developmental assessment system for texts: the Lexical Abstraction Assessment System (LAAS). The LAAS provides assessments of the order of hierarchical complexity of oral and written texts, employing scoring rules developed with predictive discriminant analysis. The LAAS is made possible by a feature of conceptual structure we call hierarchical order of abstraction, which produces systematic quantifiable changes in lexical composition with development. The LAAS produces scores that agree with human ratings of hierarchical complexity more than 80% of the time within one-third of a complexity order across 6 complexity orders (18 levels), spanning the portion of the lifespan from about 4 years of age through adulthood. This corresponds to a Kendall's tau of .93.     

Promoting versatility in mentor teachers’ use of supervisory skills

Mentor teachers need a versatile supervisory skills repertoire. Besides taking the prevalent role of daily advisor and instructor, mentor teachers should also be able to stimulate reflection in student teachers. Video recordings of 60 mentoring dialogues were analysed, both before and after a mentor teacher training aimed at developing the encourager role. Mentor teachers’ repertoires of supervisory skills were found to consist of an average of seven supervisory skills. After training, a shift was observed in the frequencies and duration with which supervisory skills were used. Although considerable inter-individual variability existed between mentor teachers, training positively affected the use of supervisory skills for stimulating reflection in student teachers.     

Development and evaluation of a Chinese version of the Questionnaire on Teacher Interaction (QTI)

Teacher–student interpersonal relationships play an important role in education. The Questionnaire on Teacher Interaction (QTI) was designed to measure students’ interpersonal perceptions of their teachers. There are two Chinese versions of the QTI for student use, and that inherited the weaknesses of the previous English versions, such as items that focus on class behaviour, instead of teacher behaviour, or that include conditionals and negations. The present study aimed to develop an improved Chinese version of the QTI which is conceptually parallel with the original QTI and with the use of optimal item wording. The process contained several steps, including expert panels and student and teacher interviews. New items were also created based on the Chinese secondary classroom context, rather than only using translations of English items. The final version of the Chinese version of the QTI presented in this paper was evaluated with a sample of 2000 students from 4 secondary schools in mainland China who rated a total of 80 teachers. The resulting version of the Chinese QTI had adequate validity and reliability, and it distinguished clearly between teachers. The predictive validity was supported by the relation between the students’ perceptions of their teachers and their academic emotions in class. Although further improvement of the instrument is recommended, it can be used to study interpersonal teacher behaviour in China and to help to improve Chinese teachers’ teaching practices.     

Assessors’ perceptions of their judgement processes: Successful strategies and threats underlying valid assessment of student teachers

As current procedures for teacher assessment are often based on non-standardized, qualitative information derived from multiple sources, the overall validity of the assessment depends heavily on the judgement processes of the assessors. Because it is of great importance for assessors to be aware of their own judgement processes and of the possible threats to validity in these processes, investigating assessors’ perceptions is of vital significance. In the present study, the perceptions of 22 assessors who judged a student teacher pair-wise using a specific assessment procedure were explored using semi-structured interviews. A qualitative analysis of the individual assessors’ perceptions with regard to the essential judgement processes of consideration of evidence and combination of evidence to attain an overall judgement resulted in an overview of successful strategies and threats underlying a valid assessment process. General implications for ensuring the validity of the assessment process and the preparation of assessors are discussed.     

Teachers’ practical knowledge about classroom management in multicultural classrooms

Creating a positive working atmosphere in the classroom is the first concern of many student and beginning teachers in secondary education. Teaching in multicultural classrooms provides additional challenges for these teachers. This study identified shared practical knowledge about classroom management strategies of teachers who were successful in creating a positive working atmosphere in their multicultural classrooms. Twelve teachers were selected who were regarded as successful classroom managers in Dutch multicultural classes by their principals and students. Video-stimulated interviews were used to elicit data about the practical knowledge of these teachers. The teachers were aware of the importance of providing clear rules and correcting student behaviour whenever necessary, but they also wanted to reduce potential negative influences of corrections on the classroom atmosphere. They aimed at developing positive teacher–student relationships and adjusted their teaching methods anticipating students' responses. Most teachers seemed reluctant to refer to the cultural and ethnic background of their students.