Comparison tasks and intentional errors – towards mathematical flexibility by looking for different solution strategies
Thesis event information
Date and time of the thesis defence
Place of the thesis defence
IT115
Topic of the dissertation
Comparison tasks and intentional errors – towards mathematical flexibility by looking for different solution strategies
Doctoral candidate
Master of Science Riikka Palkki
Faculty and unit
University of Oulu Graduate School, Faculty of Science, Reseach Unit of Mathematical Sciences
Subject of study
Mathematics
Opponent
Professor Minna Hannula-Sormunen, University of Turku
Custos
Professor Peter Hästö, University of Turku
Comparison tasks and intentional errors – towards mathematical flexibility by looking for different solution strategies
In this dissertation I discuss mathematical flexibility and the learning of mathematics using
comparison tasks and intentional errors. The term “flexibility” is used to denote the ability to
produce different strategies for tackling a given task and choose the most appropriate. I use the term “comparison task” to mean an example where the solutions to a task by two imaginary pupils’ are presented and examined by using given questions. The task may include some typical, intentional error. By using comparison tasks, it is possible, according to previous research, to develop mathematical skills and flexibility. Flexibility is important in mathematics for solving problems of different types, as well as for developing thinking. Furthermore, the use of intentional errors has produced good learning results in quantitative research. However, from a qualitative perspective, comparison tasks and especially intentional errors are still under-researched.
My dissertation includes four articles and the summary. The research material consists of 266
students’ answers to an equation solving test; the recorded discussions of 25 class teachers, subject teachers and teacher trainees; 23 mathematics teachers’ answers to a questionnaire and classroom discussions (4 teachers, 74 pupils). The research methods are quantitative (regression analysis and cross tabulation) and qualitative (phenomenography, content analysis and case study).
The flexibility of the Finnish students who participated in the research was not very high.
According to the teachers’ conceptions and beliefs, comparison tasks and intentional errors could be used to help improve for example the ability to view problems from different perspectives, analytic skills and learning to live with errors. A teacher’s role could develop into more that of a coach. These tasks seemed to be useful in the classroom. On the other hand, teachers were concerned about whether pupils possess the skills or interested to use these methods or if they may confuse an erroneous solution with the right one. On the whole, teachers’ openness to these task types suggested that flexibility could be brought to the school classroom, but teachers’ concerns about, for example, students’ pre-knowledge or possible misconceptions should be considered in teacher training and future research.
comparison tasks and intentional errors. The term “flexibility” is used to denote the ability to
produce different strategies for tackling a given task and choose the most appropriate. I use the term “comparison task” to mean an example where the solutions to a task by two imaginary pupils’ are presented and examined by using given questions. The task may include some typical, intentional error. By using comparison tasks, it is possible, according to previous research, to develop mathematical skills and flexibility. Flexibility is important in mathematics for solving problems of different types, as well as for developing thinking. Furthermore, the use of intentional errors has produced good learning results in quantitative research. However, from a qualitative perspective, comparison tasks and especially intentional errors are still under-researched.
My dissertation includes four articles and the summary. The research material consists of 266
students’ answers to an equation solving test; the recorded discussions of 25 class teachers, subject teachers and teacher trainees; 23 mathematics teachers’ answers to a questionnaire and classroom discussions (4 teachers, 74 pupils). The research methods are quantitative (regression analysis and cross tabulation) and qualitative (phenomenography, content analysis and case study).
The flexibility of the Finnish students who participated in the research was not very high.
According to the teachers’ conceptions and beliefs, comparison tasks and intentional errors could be used to help improve for example the ability to view problems from different perspectives, analytic skills and learning to live with errors. A teacher’s role could develop into more that of a coach. These tasks seemed to be useful in the classroom. On the other hand, teachers were concerned about whether pupils possess the skills or interested to use these methods or if they may confuse an erroneous solution with the right one. On the whole, teachers’ openness to these task types suggested that flexibility could be brought to the school classroom, but teachers’ concerns about, for example, students’ pre-knowledge or possible misconceptions should be considered in teacher training and future research.
Last updated: 23.1.2024