Although teaching mathematics through problem-solving has been advocated and promoted internationally for decades, this vision has failed to be consistently implemented in classrooms (United States: R. K. Anderson et al., 2018; United States: Berry et al., 2023; Chile, Canada, Israel: Felmer et al., 2019; Finland: Kaitera & Harmoinen, 2022; Indonesia: Yayuk et al., 2020). Therefore, while this research was conducted in Australia, it has broader relevance.

Thus, Stigler and Hiebert (2009) advise that teaching patterns observed in videotapes of Year Eight mathematics instruction in 1995 “were similar to those described by American researchers for the past 100 years” and suggest that “the concept of teaching as a cultural activity could help explain the stability of teaching patterns over time” (p.33). The issue then, as Liljedahl (2020) states it is that “student difficulty with mathematics has been a pervasive and systematic problem since the advent of public education - not because students can’t learn mathematics, but because, by and large, students can’t learn by being told how to do it” (p.172). It is therefore imperative to further explore problem-solving practice in the classroom.

The research which this paper reports explored the practices of two primary classroom teachers as they planned and taught mathematics through problem-solving. It sought to address the following research questions:

1. What rationales are given by the primary teachers for teaching mathematics through problem-solving?

2a. How do the primary teachers plan to teach mathematics through problem-solving and to what extent are the plans enacted?

2b. How is mathematics taught through problem-solving in the primary classroom?

3. What outcomes are perceived as flowing from student mathematics learning?

A qualitative research approach was adopted that was located within a constructivist worldview (Bernard, 2017) with a case study approach being seen as the most appropriate method that would enable a rich investigation of the teachers’ and students’ experiences (Creswell & Poth, 2018). Importantly, it allowed a positive and strengths-based analysis of teacher pedagogical practices. How the teachers implemented problem-solving with their students and navigated the research practice gap could then be explored.

Self-determination theory (Deci & Ryan, 1985) focusing on autonomy, competency, and relatedness, Polya’s (1945) problem-solving steps, Takahashi’s (2021) lesson structure, and Stacey’s (2018) approaches to teaching mathematics through problem-solving were used to assist in analysing and interpreting the collected data.

Problem-solving discussions have long focused on a balance of teaching mathematics for, about, and through problem-solving. Siemon and Booker (1990) suggest that teaching for problem-solving concerns knowledge, skills, and strategies, for future problem-solving situations. Whereas teaching about problem-solving aligns with “the means to access, monitor and direct what is known and what can be done” (Siemon & Booker, 1990, p.26). Moreover, teaching through problem-solving provides a context for further learning (Siemon & Booker, 1990). It is also worth noting that teaching for, about, and through problem-solving have different foci and that each are crucial in a mathematics classroom.

Findings revealed that the teachers implemented problem-solving to develop skills and strategies seen as essential for high school and beyond while also acknowledging the positive impact on student confidence and engagement. Additionally, the study found that student choice, cooperative learning, maintaining cognitive challenge, and task design resulted in teachers focus on teaching mathematics for and about problem-solving. The study also examined student outcomes perceived as flowing from mathematics learning as enjoyment, engagement, and confidence. This research concludes that teaching mathematics for and about problem-solving occurred frequently in both mathematics classrooms, however due to adopted lesson structures and a strong focus on mathematical content over mathematical processes teaching mathematics through problem-solving opportunities were limited.

Teachers and students from Stage Two (ages 9-10) and Stage Three (ages 10-12), together with two executive teachers responsible for student learning and curriculum development, participated in the study. Data were collected via teacher, executive teacher, and student interviews, informal teacher conversations, and classroom observations. Over two school terms, 26 classroom lesson observations were conducted in mathematics and other subjects. Documentation was also collected and included teaching and learning plans, a school scope and sequence, and student work samples and exit slips. The elements of the effective pedagogy in mathematics (Anthony & Walshaw, 2009), the framework for engagement with mathematics (Attard, 2014), behavioural indicators of cognitive engagement in the mathematics classroom (Helme & Clarke, 2001), and the Australian Mathematics K-10 Syllabus and National Numeracy Learning Progressions (NESA, 2019) assisted as data collecting tools. The LessonNote App (2019) was also used to assist data collection. The App is designed to document the flow of a lesson and its impact on student learning. This App was decided upon after extensive familiarisation of its features and was considered an appropriate tool for this study as it was able to capture the sequence of a lesson. All data sources were initially read and then reread to apply codes. Deductive coding was applied, with codes originating from the data sources, the researcher’s analytical reflection, or related to self-determination theory (Deci & Ryan, 1985), Polya’s (1945) problem-solving steps, Takahashi’s (2021) lesson structure, or Stacey’s (2018) four approaches to teaching mathematics through problem-solving. The contributions of Polya, Takahashi, and Stacey offer opportunities to apply self-determination theory to problem-solving and explore how it is supported in the classroom. Analytical coding stems from the researcher's interpretation and reflection on meaning (Richards, 2005). Stake (1995) describes qualitative data analysis as the process of giving meaning to both initial impressions and final compilations. In this study, data sources were revisited to revise coding as data was continuously collected. This ongoing regrouping of data was necessary to reflect previously collected data. A series of Excel documents was used to assist with collating, coding, and analysing the data. The data were coded and further organised into categories and themes, reflecting both data derived and framework derived themes. Ten themes were identified from the data, which informed the two considerations of lesson structure and process outcomes.

The research outcomes enhance understanding of the role of problem-solving in teaching mathematics. Most importantly, they point to the relative weight that the teachers gave to teaching for, about, and through problem-solving with the latter being much less in evidence. The findings point to two considerations that teachers need to contemplate when teaching mathematics through problem-solving. The first consideration emphasises the importance of lesson structure in mathematics. Predominantly using traditional structures may lead to a teacher-centric approach focused on content. Flexibility in lesson structure, particularly regarding the timing of explicit teaching, allows instruction to occur as needed, enabling support to be gradually introduced. Lesson structures that promote independent learning offer more opportunities for students to engage with mathematical processes. The second consideration emphasises the need to focus more on process outcomes, addressing the gap between the intended and implemented curriculum, as these outcomes are often not a priority in the classroom. Moreover, an additional focus on process outcomes, contributes to more of a student-centric approach in the classroom. Piggott (2011) advocates for students observing teachers as they struggle with a task and describing what they do as problem-solvers, as a way of modelling the problem-solving process, therefore further addressing process outcomes. A view of independent practice as independent learning may contribute to the lack of opportunities to engage with process outcomes. This study identifies three teacher implications that support student autonomy, competence, and relatedness, aligning with the two considerations. These implications are: (1) the importance of independent learning, (2) focusing on process outcomes to bridge the gap between the intended and implemented curriculum, and (3) planning and teaching to facilitate student-centred learning.

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