INTRODUCTION
The importance of teachers’ content knowledge for teaching is acknowledged as a factor of teacher quality in mathematics and science. Research with regard to both subjects suggests that teachers’ knowledge of the discipline, in particular knowledge of the discipline in relation to teaching, emerges as a key factor in student learning. Recognising this important link between what teachers and their students know about mathematics and science, we seek insights into the knowledge and understanding of their subject areas possessed by teachers and prospective teachers. The topic of ratio, which is relevant to teaching and learning in both subjects, and which has been shown to cause problems for learners, is a natural focus for the project.
FIRST PHASE OF THE STUDY, 2011-2012
At the 2011 annual conference of ATEE in Riga, the Science and Mathematics Education RDC initiated a multiple case study of prospective teachers’ content knowledge of ratio for teaching mathematics and science. The main research questions specified initially were: (a) What are prospective teachers’ representations of ratio? and (b) How are these representations similar and different between primary, secondary mathematics, and secondary science prospective teachers?
Four members of the group worked together to develop an instrument in which students were asked about the meanings they ascribed to ratio and the representations they associated with it. For the first round of data collection, the focus was on the first research question, which was refined as follows:
(a) What meanings do prospective teachers at primary and secondary levels in [specific institutions in the participants’ home countries] Ireland, Portugal and the USA give to the term ‘ratio’?
(b) What multiple representations do these prospective teachers associate with the term ‘ratio’?
(c) Do the prospective teachers’ descriptive meanings and representations indicate different levels of understanding for teaching ratio?
The importance of teachers’ content knowledge for teaching is acknowledged as a factor of teacher quality in mathematics and science. Research with regard to both subjects suggests that teachers’ knowledge of the discipline, in particular knowledge of the discipline in relation to teaching, emerges as a key factor in student learning. Recognising this important link between what teachers and their students know about mathematics and science, we seek insights into the knowledge and understanding of their subject areas possessed by teachers and prospective teachers. The topic of ratio, which is relevant to teaching and learning in both subjects, and which has been shown to cause problems for learners, is a natural focus for the project.
FIRST PHASE OF THE STUDY, 2011-2012
At the 2011 annual conference of ATEE in Riga, the Science and Mathematics Education RDC initiated a multiple case study of prospective teachers’ content knowledge of ratio for teaching mathematics and science. The main research questions specified initially were: (a) What are prospective teachers’ representations of ratio? and (b) How are these representations similar and different between primary, secondary mathematics, and secondary science prospective teachers?
Four members of the group worked together to develop an instrument in which students were asked about the meanings they ascribed to ratio and the representations they associated with it. For the first round of data collection, the focus was on the first research question, which was refined as follows:
(a) What meanings do prospective teachers at primary and secondary levels in [specific institutions in the participants’ home countries] Ireland, Portugal and the USA give to the term ‘ratio’?
(b) What multiple representations do these prospective teachers associate with the term ‘ratio’?
(c) Do the prospective teachers’ descriptive meanings and representations indicate different levels of understanding for teaching ratio?
Data from 158 students were collected and analysed.
Some RDC participants at the presentation
A grounded theory approach was used to examine the meanings and representations offered; the conjecture was made that participants who offered meanings reflecting two variables, and who provided many, varied, and relevant representations, possessed relational understanding of ratio. These findings were reported at the 2012 conference. (Unfortunately, the paper submitted to the Conference Proceedings was corrupted when turned into PDF format; for the correct version, click here.) Additional work done by one of the authors, Elsa Price, was also presented at the conference, and her paper is included in the Conference Proceedings.
SECOND PHASE OF THE STUDY, 2012 - 2015
The 2012 conference provided an opportunity to plan for further data collection, with the possibility of involving different groups, such as in-career as well as prospective teachers. Other members of the RDC expressed interest in joining the project.
During the year 2012-2013, three of the four original participants collected further data independently in their own countries (in two cases collaborating with national colleagues). They presented their papers at the conference in Norway in Summer 2013; two papers appear in the 2013 Conference Proceedings while the third has been published in the journal BOLEMA. Further work, broadening the scope of the project, was presented at the conferences in subsequent years, and findings are reported in the 2014 Conference Proceedings and the 2015 Conference Proceedings.
During the year 2012-2013, three of the four original participants collected further data independently in their own countries (in two cases collaborating with national colleagues). They presented their papers at the conference in Norway in Summer 2013; two papers appear in the 2013 Conference Proceedings while the third has been published in the journal BOLEMA. Further work, broadening the scope of the project, was presented at the conferences in subsequent years, and findings are reported in the 2014 Conference Proceedings and the 2015 Conference Proceedings.
ONGOING WORK, 2016 -
A related contribution on the ratio concept - not using the Ratio Project instrument - was introduced at the 2016 conference, further extending the scope of the project. This and other papers appear in the 2016 Conference Proceedings and the 2017 Conference Proceedings.
Additional contributions are welcome. For those who wish to use or adapt the Ratio Project instrument, a suggested version in English (aimed at pre-service teachers) is available here. It is not copyright, but acknowledgment of the ATEE project and the original authors (Berenson et al., 2013) would be welcomed! Additionally, the possibility of adapting the instrument to address other topics of interest for mathematics and science education has been considered, and was realised in 2018 with the initiation of the Functions Project. Any other work involving such adaptation is also of interest to the RDC.
A related contribution on the ratio concept - not using the Ratio Project instrument - was introduced at the 2016 conference, further extending the scope of the project. This and other papers appear in the 2016 Conference Proceedings and the 2017 Conference Proceedings.
Additional contributions are welcome. For those who wish to use or adapt the Ratio Project instrument, a suggested version in English (aimed at pre-service teachers) is available here. It is not copyright, but acknowledgment of the ATEE project and the original authors (Berenson et al., 2013) would be welcomed! Additionally, the possibility of adapting the instrument to address other topics of interest for mathematics and science education has been considered, and was realised in 2018 with the initiation of the Functions Project. Any other work involving such adaptation is also of interest to the RDC.
