24. Mathematics Education Research
Paper
A Problem-Posing Intervention to Enhance Secondary Students' Mathematical Problem-Posing Competence, Problem-Solving Competence, and Creative Thinking
Ling Zhang1,2, Andreas Stylianides2, Gabriel Stylianides3
1Southwest University, China; 2University of Cambridge, United Kingdom; 3University of Oxford, United Kingdom
Presenting Author: Stylianides, Andreas;
Stylianides, Gabriel
Motivation and research question
Mathematical problem posing, the process of interpreting concrete or abstract situations and formulating them as meaningful mathematical problems (Stoyanova & Ellerton, 1996), is a form of authentic mathematical inquiry and creation recognised as important for students’ learning by educators and curriculum frameworks internationally (e.g., Chinese Ministry of Education, 2022; National Council of Teachers of Mathematics [NCTM], 2000). Further to being important in its own right, problem posing has been associated with improved competence in mathematical creative thinking, a key transferrable skill for life and work, and mathematical problem solving, which is problem posing’s twin activity central to virtually all mathematics curricula internationally (Shriki, 2013; Wang et al., 2022).
Recognising problem posing’s importance, researchers designed and implemented problem-posing interventions, albeit with mixed results. In a systematic review of 39 problem-posing intervention studies and a meta-analysis of 26 of them (Zhang et al., under review a&b), we synthesised key intervention components and measured their relative or combined effect on students’ problem-posing competence. Thus, we gained insights into what works best, for whom, and under what conditions. Yet, those promising components were not all integrated into the same intervention, nor was the impact of such an intervention explored on all of the following: problem posing, problem solving, and creative thinking.
Based on best knowledge in the literature about problem posing interventions (Zhang et al., under review a&b), we designed a new problem posing intervention aiming to enhance secondary students’ mathematical problem-posing and problem-solving competences and creative thinking, incorporating the components with the most evidence of impact in the literature. In this paper, we report findings about the effectiveness of the intervention to achieve its intended learning outcomes, by addressing the following research question:
To what extent does the developed problem-posing intervention, implemented in secondary school classrooms, enhance students’ mathematical problem-posing competence, problem-solving competence, and creative thinking?
The problem-posing intervention
We developed the problem-posing intervention using our findings from a systematic review and a meta-analysis of interventions published between 1990 and 2021 that aimed at fostering participants’ mathematical problem-posing competence (Zhang et al., under review a&b). We identified three categories of intervention components from the review (ibid): activity-based practice that engaged participants in experiencing problem posing (e.g., overview of what problem posing is–WPP, discussion of what “good” problems are–WGP), method-based assistance that helped participants pose problems (e.g., use of strategies involved in problem posing–SPP, use of problem posing examples–PPE), and environment-based support that guided interaction among participants and the teacher (e.g., interactive learning environment–ILE). The results of our meta-analysis showed that the problem-posing interventions had a significant and positive impact on participants’ mathematical problem-posing competence (g=0.72, p<.001). Particularly, the effect sizes of interventions that incorporated method-based assistance or environment-based support were on average 84% or 83% higher than those of interventions without such kinds of intervention components, respectively.
Based on these findings, our designed intervention, in the form of annotated lesson plans for delivery by the teachers, incorporated all three categories of intervention components, including the following five specific components that we found to be particularly promising: WPP, WGP, SPP, PPE, and ILE. The intervention duration was 220 minutes and is aimed for 13-to-15-year-olds who tend not to be occupied by high-stakes assessments. Also, these students tend to be at a critical juncture in their schooling when the intervention can better equip them for further mathematical studies. Finally, the intervention is not meant to be treated as extracurricular due to its intended impact on the recognised, key learning goals of mathematical problem solving and creativity.
Methodology, Methods, Research Instruments or Sources Used
Participants
We implemented the intervention in two secondary, mixed-attainment classes in China with a total of 81 students (13 to 15 years of age). Both classes were taught by the same mathematics teacher who worked closely with the first author to understand and enact the intervention, following the annotated lesson plans we had provided. Over a two-week period, the teacher implemented five structured intervention lessons, each corresponding to one of the five distinct components identified in the literature and in the following sequence: WPP, WGP, SPP, PPE, and ILE. The intervention took a total instructional time of 220 mins, as intended.
Instruments
To measure mathematical problem-posing and mathematical problem-solving, we used the QUASAR cognitive assessment instrument (QCAI) (Parke et al., 2003). This included a set of mathematical open-ended problem-solving and corresponding problem-posing tasks designed for secondary school students of similar age to assess the effectiveness of instructional programs. QCAI tasks have undergone extensive scrutiny to ensure their quality and validity. Two forms of QCAI as pre-and post-tests, including the QCAI-problem posing and QCAI-problem solving test, were sequentially implemented in two class periods of approximately 40 minutes each.
To measure mathematical creative thinking, we used the Multiple Solution Tasks (MSTs) developed by Leikin (2009). The MSTs, a well-established instrument, has been used in a range of comprehensive studies with school students. The MSTs were completed by the students within 40 minutes.
The mean difficulty levels of the pre- and post-tests were found to be comparable through the use of Rasch model analysis.
Data analysis
To address the research question, we compared students’ performance in the pre- and post-tests using quantitative methods. In more detail, these methods included observed-score equating analysis, paired-sample t-test, and Ne McNemar-tests to evaluate students’ changes in performance in terms of mathematical problem-posing, problem-solving, and creative thinking. We also collected qualitative data documenting the implementation of the intervention and the discussions between the researcher and the teacher prior and after the lessons, but reports of analyses of these data is beyond the scope of this paper.
Conclusions, Expected Outcomes or Findings
The intervention was found to have a positive impact on students’ problem-posing competence (d=0.58), problem-solving competence (d=1.61), and creative thinking (d=0.65), indicating medium to large effects. These findings are encouraging as there is a scarcity of interventions of short duration with a broad-based impact on academically important, higher-order skills, such as those targeted by our intervention, which can prepare students not only for advanced mathematical studies but also for life and work (Stylianides & Stylianides, 2013). The findings also serve as a critique of several mathematics curricula internationally, including the English, which make no reference to mathematical problem-posing. Given that problem posing’s twin activity is central to virtually all mathematics curricula internationally, including the English, our findings make a case for the merits of a concerted problem-posing-and-solving curricular coverage.
The fact that our intervention was developed based on the findings of our systematic review and meta-analysis of prior problem-posing interventions (Zhang et al., under review a&b), which allowed us to see what works best, for whom, and under what conditions, possibly explains the positive intervention outcomes. Yet, we need to be cautious about the relatively small sample (81 students, 2 classes, 1 teacher) and the possible role played by the cultural context where the intervention was implemented (the Chinese). In the next stage of our research program, we plan to conduct pre-trial development and early evaluation of our intervention in England (with minor adaptations to account for the new cultural context), working with a larger number of schools (10) and teachers (20) as part of a 1-day professional development training program. Through the preliminary evaluation of the intervention’s feasibility and efficacy with Year 10 students in England, who are of equivalent age to the Chinese student participants, we will aim to pave the ground for a future randomised control trial.
ReferencesChinese Ministry of Education. (2022). Mathematics Curriculum Standard of compulsory education. Beijing, China: People’s Education Press.
Leikin, R. (2009). Exploring mathematical creativity using multiple solution tasks. In R. Leikin, A. Berman, & B. Koichu (Eds.), Creativity in mathematics and the education of gifted students (pp. 129-145). Sense Publisher.
National Council of Teachers of Mathematics (NCTM). (2000) Principles and standards for school mathematics. Reston, VA: Author.
Parke, C. S., Lane, S., Silver, E. A., & Magone, M. E. (2003). Using assessment to improve middle-grades mathematics teaching & learning: suggested activities using QUASAR tasks, scoring criteria, and students’ work. Reston, VA: NCTM.
Shriki, A. (2013). A model for assessing the development of students’ creativity in the context of problem posing. Creative Education, 4(7), 430.
Stoyanova, E., & Ellerton, N. F. (1996). A framework for research into students’ problem posing in school mathematics. In P. Clarkson (Ed.), Technology in Mathematics Education (pp. 518-525). Melbourne: Mathematics Education Research Group of Australasia.
Stylianides, A. J., & Stylianides, G. J. (2013). Seeking research-grounded solutions to problems of practice: Classroom-based interventions in mathematics education. ZDM – The International Journal on Mathematics Education, 45(3), 333-341.
Wang, M., Walkinton, C., & Rouse, A. (2022). A meta-analysis on the effects of problem-posing in mathematics education on performance and dispositions. Investigations in Mathematics Learning, 14(4), 265–287.
Zhang, L., Stylianides, G. J., & Stylianides, A. J. (under review a). Enhancing mathematical problem posing competence: A meta-analysis of intervention studies. International Journal of STEM Education.
Zhang, L., & Stylianides, A. J., & Stylianides, G. J. (under review b). Approaches to supporting and measuring mathematical problem posing: A systematic review of interventions in mathematics education. International Journal of Science and Mathematics Education.
24. Mathematics Education Research
Paper
Exploring the Dynamics of Problem Posing and Solving Skills of Pre-Service Primary School Teachers
Elif Tuğçe Karaca1, Vuslat Şeker2
1KIRIKKALE UNIVERSITY, Turkiye; 2HU University of Applied Sciences Utrecht, Netherlands
Presenting Author: Karaca, Elif Tuğçe
The pedagogical landscape of elementary mathematics education is significantly influenced by the type and quality of problems presented in the classroom. Traditional methodologies, which often emphasize rote learning and procedural mastery, fall short of fostering critical thinking and inquiry, essential components for cultivating mathematical proficiency (Henningsen & Stein, 1997). Recognizing this, the literature advocates a paradigm shift toward integrating problem solving and reasoning as fundamental aspects of mathematics education, thereby enriching students' learning experiences and enhancing their conceptual understanding (e.g., Miranda & Mamede, 2022; National Council of Teachers of Mathematics (NCTM), 2000; Van de Walle et al., 2010).
Problem solving, as described by the NCTM (2000), should not be an isolated segment of the curriculum but an integral part of mathematics learning, integrated into the core of education. NCTM (2000) further notes that problem solving highlights mathematical engagement. In addition, the cognitive and metacognitive dimensions of problem solving underscore the importance of engaging with problems in ways that go beyond mere computation. Jonassen (2000) articulates that the significance of a problem derives from its potential to contribute to “societal, cultural, or intellectual” domains, which requires a solver's engagement in mental representation and manipulation of the problem space (p. 65). This perspective is complemented by Lester and Kehle's definition, which emphasizes problem solving as an active engagement process “using prior knowledge and experience” (cf. Santos-Trigo, 2007, p. 525).
Problem-posing, similar to problem-solving, is an integral part of this pedagogical development. It is recognized as a sophisticated mathematical activity that promotes creativity, flexibility, and deeper understanding (Silver, 1994). It is defined as the ability to formulate, reformulate, and explore problems based on existing mathematical situations or concepts. It could be described as “one of the highest forms of mathematical knowing and a sure path to gain status in the world of mathematics” (Crespo, 2015, p. 494). NCTM (2000) also points out that students need “to create engaging problems by drawing inspiration from various scenarios encompassing mathematical and non-mathematical contexts” (p. 258). In light of these considerations, this study explores the interplay between problem solving and problem posing in the context of mathematics education for preservice elementary teachers. Specifically, it seeks to examine the nature and quality of problems posed by preservice teachers, the challenges they face in the problem-posing process, and how their problem-solving skills influence their ability to generate meaningful mathematical problems. Through a comprehensive analysis of pre-service teachers' engagement with problem solving and problem posing, this research aims to contribute to how to support the pre-service teachers' skills and the interplay between them to create better learning environments for their students by improving their teaching strategies.
Methodology, Methods, Research Instruments or Sources Used
This research will employ a qualitative research design, focusing on understanding and interpreting the nature of pre-service primary school teachers' problem-posing abilities and exploring the challenges they face during the process. The study will also investigate the relationship between prospective teachers' problem-solving and problem-posing abilities in specific mathematics content areas, aiming to examine the nature and quality of problems they pose, the difficulties encountered in problem-posing, and how their problem-solving skills influence their problem-posing capabilities.Qualitative data will be gathered from a convenience sample of 28 primary school pre-service teachers enrolled in a mathematics teaching course during the Spring 2024 semester. This course, a required part of the undergraduate primary school teacher education program at a public university in Turkey, includes weekly three-hour lectures over twelve weeks, focusing on problem-solving processes and integrated problem-posing activities within topics such as early algebra, numbers, and operations. Each weekly session will feature problem-solving and problem-posing tasks based on relevant literature. Data collection will use a two-part instrument: the first part will be a paper-pencil test for each week's content, starting with a problem-solving task followed by a problem-posing task. Students will solve the given problem, then create and solve their own posed problems, identifying any issues in their problem formulation. The second part will involve in-depth think-aloud protocols with a subset of participants to understand their cognitive processes during problem-solving and posing, including their strategies and awareness of problem-posing challenges.
Data from the paper-pencil tests and think-aloud protocols will be analyzed qualitatively. The paper-pencil tasks will undergo content analysis using thematic coding procedures based on established frameworks (e.g., Problem-Solving Task Rubric and Problem Posing Task Rubric by Rosli et al., 2015). The think-aloud protocols will be transcribed and analyzed to gain insights into participants’ thought processes during problem-posing and solving, and a comparative analysis will be conducted to explore the nature of their problem-solving and posing abilities and their effectiveness in formulating and solving problems.
Conclusions, Expected Outcomes or FindingsBuilding on the exploratory studies by Grundmeier (2015), Hospesova & Ticha (2015), and insights from Rosli et al. (2015), this research aims to deepen our understanding of the problem-solving and posing skills of pre-service primary school teachers. Grundmeier (2015) observed that practice enhances problem-posing efficiency and creativity among prospective elementary and middle school teachers. Hospesova and Ticha (2015) identified significant knowledge gaps and challenges in problem-posing, despite teachers acknowledging its importance in mathematics education. Complementing these findings, Rosli et al. (2015) revealed middle school preservice teachers’ proficiency in solving more straightforward arithmetic tasks. However, they had difficulties in abstract generalization and algebraic interpretation. Notably, these teachers could formulate fundamental yet meaningful problems. The results suggested the integral role of problem-solving in facilitating effective problem-posing.
Aligned with these studies, the current research is expected to uncover similar findings within pre-service primary school teachers' problem-solving and posing competencies. This research will explore the nature and quality of problems posed, the challenges encountered in the problem-posing process, and the interrelation between problem-solving prowess and problem-posing skills. Employing comprehensive data collection and analysis methods inspired by Rosli et al. (2015) and others, this research aims to offer new insights and validate existing findings.
ReferencesCrespo, S. (2015). A collection of problem-posing experiences for prospective mathematics teachers that make a difference. In Ed.Mix & Battista (Eds.), Mathematical problem posing: From research to effective practice (493-511).USA: Springer.
Grundmeier, T. A. (2015). Developing the problem-posing abilities of prospective elementary and middle school teachers. In Ed.Mix & Battista (Eds.), Mathematical problem posing: From research to effective practice (411-431).USA: Springer.
Henningsen, M., & Stein, M. K. (1997). Mathematical tasks and student cognition: Classroom-based factors that support and inhibit high-level mathematical thinking and reasoning. Journal for Research in Mathematics Education, 28(5), 524-549.
Hošpesová, A., & Tichá, M. (2015). Problem posing in primary school teacher training. In Ed.Mix & Battista (Eds.), Mathematical problem posing: From research to effective practice,(433-447).USA: Springer.
Jonassen, D. H. (2000). Toward a design theory of problem solving. Educational Technology Research and Development, 48(4), 63–85. http://doi.org/10.1007/BF02300500
Miranda, P., & Mamede, E. (2022). Appealing to creativity through solving and posing problems in mathematics class. Acta Scientiae. Revista de Ensino de Ciências e Matemática, 24(4), 109-146.
National Council of Teachers of Mathematics (NCTM). (2000) Principles and standards for school mathematics. Author.
Rosli, R. Capraro, M. M., Goldsby, D., Gonzalez, E. G., Onwuegbuzie, A. J., & Capraro, R. B. (2015).Middle-grade preservice teachers’ mathematical problem solving and problem posing. In Ed.Mix & Battista (Eds.), Mathematical problem posing: From research to effective practice,(333-354).USA: Springer.
Santos-Trigo, M. (2007). Mathematical problem solving: An evolving research and practice domain. ZDM - International Journal on Mathematics Education, 39(5-6), 523–536. http://doi.org/10.1007/s11858-007-0057-9
Silver, E. A. (1994). On the teaching and learning of mathematical problem posing. Journal for Research in Mathematics Education, 25(1), 25-43.
Van De Walle, J. A., Karp, K. S. & Bay-Williams, J. M. (2010). Elementary and middle school mathematics: Teaching developmentally (7th ed). Allyn and Bacon/Pearson Education.
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