24 SES 12, Cognitive Development and Mathematics Learning
Proportional reasoning and related concepts such as ratio and proportion are essential for and have an essential influence on the learning of other topics in mathematics and science, and understanding many context and situations in daily life (Cramer & Post, 1993). Despite being such important in the educational system and everyday life, proportional reasoning and the concepts of ratio and proportion are difficult for students to acquire (Tourniaire & Pulos, 1985). Resnick and Singer (1993) claimed that learning of the ratio and proportion concepts are challenging for students, and they constitute “one of the stumbling blocks of the middle school curriculum” (p. 107). Indeed, it is argued that the topics of ratio, proportion and proportional reasoning are among the subjects that are found to be the most difficult and problematic to learn and teach at all school levels (Cramer, Post, & Currier, 1993; Lamon, 2007: Post, Harel, Behr, & Lesh, 1991; Simon & Blume, 1994).
National Council of Teachers of Mathematics [NCTM] stresses that “the ability to reason proportionally develops in students throughout grades 5-8” (1989, p. 82). Hence, it is reasonable to consider middle school years as critical for the development of proportional reasoning. On the other hand, a serious amount of time is allocated for the development of proportional reasoning in most of the seventh grade mathematics curricula implemented worldwide. Having this and the aforementioned considerations in mind, there seems to be a need for improving proportional instruction in seventh grade classrooms. As a first step of this improvement, it is essential to determine the possible problems and shortcomings of the existing instruction of proportional reasoning in seventh grade. Previous studies have shown that some of the problems are related to inadequate content and pedagogical content knowledge of teachers (Harel & Behr, 1995: Hines & McMahon, 2005) and a superficial and procedural way of teaching (Lesh, Post, & Behr, 1988). However, it would be interesting and valuable to learn teachers’ experiences and perceptions regarding the difficulties encountered in the teaching and learning of proportional reasoning, and there was no study focusing on this aspect in the available literature. Therefore, in this study the purpose was to investigate the experiences and perceptions of mathematics teachers on the difficulties in the teaching and learning of ratio and proportion in seventh grade.
Since the purpose of this study was to describe and interpret teachers’ experiences and their perceptions based on their experiences, a phenomenological study was utilized (Bogdan & Biklen, 1992). To this purpose, two mathematics teachers were interviewed face to face in a time period of approximately half an hour. The two teachers were selected based on their year of experience in teaching proportional reasoning topics in seventh grade and volunteerism. Both teachers were female and had at least 10 years in teaching 7th grade mathematics in public schools. Interview questions were prepared beforehand and checked with a professor who is experienced in qualitative analysis. Interview questions included two parts as background questions and questions about teaching and learning of proportional reasoning, and ratio and proportion concepts. The questions were focused on the objectives in the curriculum, planning of the teachers, textbooks, extra resources, students’ knowledge and challenges in teaching these concepts. Interviews were audiotaped, and reflections during the interview and after the interview were written down by the researcher. The interview data were transcribed, and qualitative data analysis was conducted. In this data analysis process some procedures were followed. First, the transcripts were studied repeatedly having the questions of what these data possibly meant and how these meanings could contribute to the development of emerging categories and themes. The transcripts were also studied horizontally in which the segments are grouped into the themes until no new themes were possible to emerge (Marshall & Rossman, 2011).
The results of the study indicated that there were several aspects of difficulties in the teaching and learning of ratio and proportion in 7th grade such as curriculum related difficulties, book related difficulties, teacher related difficulties, student related difficulties and difficulties related to the nature of mathematics and the topic of ratio and proportion. Even though these five aspects were separated into distinct themes, an emphasis on the procedural knowledge and algorithms such as cross multiplication was present in most of the themes. To be more specific, the results demonstrated that not only students focused on the procedures and algorithms but also the teachers put an emphasis on the procedures and algorithms for ratio and proportion instruction rather than conceptual understanding. Furthermore, they indicated that procedures and algorithms were highly emphasized in all official textbooks and exercise books and curriculum rather than conceptions. Lastly, the results revealed that the information and the problems related to the ratio and proportion were not organized from easy to more complex in curriculum, books and instruction. These themes were further discussed, and some suggestions were provided.
Bogdan, R. C, & Biklen, S. K. (1992). Qualitative research for education: An introduction to theory and methods. Boston: Allyn & Bacon. Cramer, K., & Post, T. (1993). Making connections: A case for proportionality. Arithmetic Teacher, 60 (6), 342-346. Cramer, K., Post, T., & Currier, S. (1993). Learning and teaching ratio and proportion: Research implications. In D. T. Owens (Ed.), Research Ideas for the Classroom, (pp. 159-178). New York, NY: Macmillan. Harel, G., & Behr, M. (1995). Teachers' solutions for multiplicative problems. Hiroshima Journal of Mathematics Education, 3, 31-51. Hines, E., & McMahon, M. T. (2005). Interpreting middle school students' proportional reasoning strategies: Observations from preservice teachers. School Science and Mathematics, 105(2), 88-105. Lamon, S. (2007). Rational numbers and proportional reasoning. In F. K. Lester Jr. (Ed.), Second handbook of research on mathematics teaching and learning (pp. 629-667). Charlotte, NC: Information Age Publishing. Lesh, R., Post, T. & Behr, M. (1988). Proportional reasoning. In J. Hiebert & M. Behr (Eds.), Number concepts and operations in the middle grades (pp. 93–118). Reston, VA: Lawrence Erlbaum & National Council of Teachers of Mathematics. Marshall, C. & Rossman, G. B. (2011). Designing qualitative research (5th ed.). Thousand Oaks, CA: Sage. National Council of Teachers of Mathematics [NCTM] (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: The Council. Post, T. R., Harel, G., Behr, M. J., & Lesh, R. (1991). Intermediate teachers' knowledge of rational number concepts. In E. Fennema, T. P. Carpenter & S. J. Lamon (Eds.), Integrating research on teaching and learning mathematics (pp. 177-198). Albany, NY: State University of New York Press. Resnick, L., & Singer, J. (1993). Protoquantitative origins of ratio reasoning. In T. Carpenter, E. Fennema, & T. Romberg (Eds.), Rational numbers: an integration of research (pp. 107-130). Hillsdale, New Jersey: Lawrence Erlbaum Associates. Simon, M. A., & Blume, G. W. (1994). Mathematical modeling as a component of understanding ratio-as-measure: A study of prospective elementary teachers. Journal of Mathematical Behavior, 13, 183-197. Tourniaire, F., & Pulos, S. (1985). Proportional reasoning: A review of the literature. Educational Studies in Mathematics, 16(2), 181-204.
00. Central Events (Keynotes, EERA-Panel, EERJ Round Table, Invited Sessions)
Network 1. Continuing Professional Development: Learning for Individuals, Leaders, and Organisations
Network 2. Vocational Education and Training (VETNET)
Network 3. Curriculum Innovation
Network 4. Inclusive Education
Network 5. Children and Youth at Risk and Urban Education
Network 6. Open Learning: Media, Environments and Cultures
Network 7. Social Justice and Intercultural Education
Network 8. Research on Health Education
Network 9. Assessment, Evaluation, Testing and Measurement
Network 10. Teacher Education Research
Network 11. Educational Effectiveness and Quality Assurance
Network 12. LISnet - Library and Information Science Network
Network 13. Philosophy of Education
Network 14. Communities, Families and Schooling in Educational Research
Network 15. Research Partnerships in Education
Network 16. ICT in Education and Training
Network 17. Histories of Education
Network 18. Research in Sport Pedagogy
Network 19. Ethnography
Network 20. Research in Innovative Intercultural Learning Environments
Network 22. Research in Higher Education
Network 23. Policy Studies and Politics of Education
Network 24. Mathematics Education Research
Network 25. Research on Children's Rights in Education
Network 26. Educational Leadership
Network 27. Didactics – Learning and Teaching
The programme is updated regularly (each day in the morning)
- Search for keywords and phrases in "Text Search"
- Restrict in which part of the abstracts to search in "Where to search"
- Search for authors and in the respective field.
- For planning your conference attendance you may want to use the conference app, which will be issued some weeks before the conference
- If you are a session chair, best look up your chairing duties in the conference system (Conftool) or the app.