Source: Journal of Mathematics Teacher Education, volume 25 issue 3, pages 293–320
(Reviewed by the Portal Team)
This study focuses on problem posing within primary classrooms; these are contexts where teachers are not mathematics subject specialists and need to be responsive to pedagogical considerations of many curricular areas.
Rather than generate their own problems, it is common for primary teachers to depend heavily on textbook problems or make cosmetic changes to textbook problems (Silver, Mamona-Downs, Leung & Kenny, 1996; Koichu, Harel & Manaster, 2013).
Consequently, within this study the authors consider problem posing to refer both to the generation of new problems and to the modification of existing textbook problems to form new problems.
This study explores the impact of a letter writing exchange on the problem posing skills of prospective primary teachers.
The research questions are:
(a)In what ways does engagement in a letter writing initiative influence the characteristics of mathematical problems posed and understandings of a good mathematical problem?
(b)What are the challenges experienced by prospective primary teachers when posing good mathematical problems?
Methodology
This research study is informed by design research (Cobb, Confrey, diSessa, Lehrer, & Schauble, 2003; Kelly, Lesh, & Baek, 2008) and involves the collaboration of researchers, prospective and practicing teachers to tackle a real teaching and learning problem, i.e. posing good mathematical problems for use in elementary classrooms.
This partnership with classroom teachers and children aligns the research with Barab and Squire’s (2004, p. 2) definition of design-based research as having the ‘intent of producing new theories, artefacts and practices that account for and potentially impact learning and teaching in naturalistic settings’.
The study is interventionist and focuses on designing an intervention—in this case, the letter writing initiative.
It incorporates cycles of analysis, design and development, evaluation and revision (in this case, to mathematics problems) and consequently is considered iterative.
It requires involvement of practitioners and places emphasis on understanding and improving the intervention thus is considered process oriented.
The study is utility oriented in that it attributes success or value based on the practicality of the intervention for users in real contexts, i.e. the design of worthwhile mathematical problems for use by primary children. Findings from this study will be used to inform and evaluate the intervention, and subsequent prototypes, thereby contributing to theory building.
Participants
Participants were 28 prospective primary teachers [PTs] enrolled in year 3 of a 4-year (eight semester) undergraduate initial teacher education programme.
They had completed five compulsory mathematics education courses.
At the time of this study, participants were in their sixth semester and registered in a mathematics education elective.
Participants had completed three school placements in primary classrooms from grades 1–6.
The letter writing initiative
The letter writing initiative was embedded in a 12-week mathematics education elective, consisting of 3 contact hours per week, and co-taught by the two authors of this paper.
The purpose of the elective was to develop the problem posing skills of participants. Problem posing skills were developed by having participants pose mathematics problems, which were delivered to primary school ‘penpals’ as part of the letter writing initiative.
The accompanying letters served as an avenue to gain access to children’s experiences when engaging with the problems, acted as a communication tool between PTs and their primary school penpals, and informed the design and modification of subsequent problems.
Feedback and reflections on problem development
On a weekly basis (weeks 4–10), PTs critiqued their posed problems and analysed their penpals’ responses; this promoted reflection upon problem characteristics and modifications to their problems.
Structured feedback was provided on four occasions (weeks 4, 6, 7, 8) from instructors and peers.
In week 4, problems were discussed and modified within small groups and then in the whole class forum.
In week 6, each PT met with and received feedback on problems from one of the instructors.
Feedback focused on the problem features, the penpal’s success in responding to the problem, and considerations for future problem posing (e.g. identifying changes in problem features).
Alongside these activities, instructors provided regular structured feedback in the form of brief focused written messages.
Data sources
The data sources were closely coordinated with the semester activities.
Analysis of the pre- and post-intervention questionnaires, in weeks 1 and 12, revealed the impact of the initiative on understandings and design of problems and thus informed research question 1.
The pre-problem appraisal and reflection on penpal response completed from weeks 4-10 revealed the challenges of posing good mathematical problems and were used to address research question 2.
Pre‑problem appraisal and reflection on pupils’ response
Each PT designed five unique problems (weeks 5–7, 9–10) for each of their penpals.
Each problem had an associated ‘pre-problem appraisal’ and ‘reflection on penpal response’.
The appraisal and reflection activities were designed according to recommendations arising in the literature (Crespo & Sinclair 2008) and provided insights into PTs’ reasoning and understandings related to the selected problem.
Findings and discussions
A welcome finding of this study was the influence of previous mathematics education instruction, as part of the initial teacher education programme, on PTs’ initial understandings of problem posing.
Whereas studies report that initial problems posed are limited to arithmetic, one step problems requiring recall of memorised facts and implementation of routinized procedures (Leavy & Hourigan, 2020, Chapman, 2012), the initial problems posed required more than one step to solve.
They were also contextually rich, realistic and incorporated some of the cultural artefacts and real-life connections recommended by Bonotto (2013) and Lee (2012).
Participants communicated awareness of the importance of problems being appropriately challenging, involving mathematical reasoning, having no immediately clear strategy and more than one correct solution; although 80% of participants did not succeed in incorporating these characteristics into the initial problems.
This suggests the presence of inert theoretical understandings regarding the characteristics of good problems but limited ability to incorporate these features into posed problems.
The letter writing initiative changed participants’ understandings of good problems and also significantly changed the types of problems posed at the conclusion of the study.
They had greater appreciation for problems being appropriately challenging, having no immediately obvious solution or strategy, having multiple steps and multiple solutions.
A unique finding was participants’ use of multiple part problems as a way to meet two emerging goals: supporting students in experiencing success and increasing cognitive demand.
They launched multiple part problems by placing one step procedural questions at the beginning; this ensured that their penpal experienced success in the initial stages of problem solving.
This was followed by more complex questions that increased the challenge and cognitive demand of the problem.
Nonetheless posing mathematically worthwhile problems remained a challenge and participants required considerable support, opportunities for reflection, and peer and tutor feedback to improve problems.
The study also lends support to Chapman’s (2012) findings that the construction of multiple solution problems is most challenging.
Although there was an appreciable increase in problems that had more than one correct solution, and a strong appreciation for the importance of these problem types, the majority of problems posed at the conclusion of the study had only one correct solution.
There was also a demonstrable increase in cognitive demand with just over half categorized as ‘procedures with connections’, although 40% remained at the level of low cognitive demand with only 9% categorised as ‘doing mathematics’.
Although the authors agree to some extent with Tichá and Hospesová’s (2013) finding that prospective primary teachers failed to engage in deep analysis of problems, they contend that the competing demands of writing non-traditional problems and incorporating desirable problem features impacted their ability to attend to cognitive demand.
References
Barab, S., & Squire, K. (2004). Design-Based Research: Putting a Stake in the Ground. The Journal of the Learning Sciences, 13(1), 1–14.
Bonotto, C. (2013). Artifacts as sources for problem-posing activities. Educational Studies in Mathematics, 83(1), 37–55.
Chapman, O. (2012). Prospective elementary school teachers’ ways of making sense of mathematical problem posing. PNA, 6(4), 135–146.
Cobb, P., Confrey, J., diSessa, A., Lehrer, R., & Schauble, L. (2003). Design experiments in education research. Educational Researcher, 32(1), 9–13.
Crespo, S., & Sinclair, N. (2008). What makes a problem mathematically interesting? Inviting prospective teachers to pose better problems. Journal of Mathematics Teacher Education, 11(5), 395–415.
Kelly, A. E., Lesh, R. A., & Baek, J. Y. (Eds.). (2008). Handbook of design research methods in education: Innovations in science, technology, engineering, and mathematics learning and teaching. Routledge.
Koichu, B., Harel, G., & Manaster, A. (2013). Ways of thinking associated with mathematics teachers’ problem posing in the context of division of fractions. Instructional Science, 41(4), 681–698.
Leavy, A. M., & Hourigan, M. (2020). Posing mathematically worthwhile problems: developing the problem-posing skills of prospective teachers. Journal of Mathematics Teacher Education, 23(4), 341–361.
Lee, J. E. (2012). Prospective elementary teachers’ perceptions of real-life connections reflected in posing and evaluating story problems. Journal of Mathematics Teacher Education, 15, 429–452.
Silver, E. A., Mamona-Downs, J., Leung, S., & Kenny, P. A. (1996). Posing mathematical problems: An exploratory study. Journal for Research in Mathematics Education, 27(3), 293–309.
Tichá, M., & Hošpesová, A. (2013). Developing teachers’ subject didactic competence through problem posing. Educational Studies in Mathematics, 83(1), 133–143.