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.

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.

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.

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