Introduction

Problem-solving, involving solution of unfamiliar (Kilpatrick et al., 2001) and cognitively demanding problems (McCormick, 2016), is important in mathematics education (e.g. Liljedahl et al., 2016) and a twenty-first century skill that teachers should help students develop (English & Sriraman, 2009). Hence, there is motivation for primary school teachers to consider teaching approaches to support problem-solving.

The novel nature and open-endedness of problem-solving, teacher difficulties in choosing appropriate problems, knowing appropriate times to intervene to support problem-solving and dealing with student struggle were issues reported by year 1 teachers in three Australian schools (Cheeseman, 2018). In a survey of 108 Australian upper primary teachers, McCormick (2016) found that teachers’ focus on teaching the mathematics for problems removed the problem-solving nature of many tasks. During problem-solving lessons, the extent and nature of teacher assistance is an important consideration, as well as problem choice, as they impact the problem-solving nature of tasks and the potential for students to engage with problems. The study reported in this article investigated two Australian primary teachers’ implementation of lessons based on a problem-solving intervention.

Problem-Solving in the Curriculum

Education is state-based in Australia, although based on the Australian curriculum (ACARA, n.d.). The Victorian mathematics curriculum had four proficiencies (problem-solving, understanding, fluency and reasoning), developed across three content strands (number and algebra; measurement and geometry; statistics and probability) at the time of this study (VCAA, n.d.). Problem-solving was identified as “the ability of students to make choices, interpret, formulate, model and investigate problem situations, select and use technological functions and communicate solutions effectively” (VCAA, n.d., para. 4), aligned with Schoenfeld (1992) who suggested students should use higher-order thinking skills and select solution methods when problem-solving. The Victorian curriculum provided an imperative for teachers to develop students’ problem-solving strategies noting that “capable problem-solvers [i.e., students] solve unfamiliar and meaningful problems, create their own investigations, use prior knowledge and strategies to aide in their solving of problems and check the viability of their solutions” (VCAA, n.d., para 4). Importantly, students should choose and use strategies to solve unfamiliar problems. Although problem-solving was an expectation in the Australian curriculum, Stacey et al. (2015) reviewed mathematics pedagogies and resources in Australian school education and found problem-solving was often overlooked due to perceived lack of teaching resources and time. Cheeseman (2018) and McCormick (2016) also noted time constraints and accessibility to resources as barriers to problem-solving.

Problem-Solving Process and Structure of Lessons

Problem-solving interventions require two considerations: the structure and features to support development of problem-solving strategies. Mason et al. (1985) provided a three-phase problem-solving process (i.e. entry, attack and review) incorporating problem-solving strategies, aligned with Pólya (1945). The entry phase involves understanding the problem and deciding on appropriate strategies; this may include trialling cases (specialise) to decide on a strategy (Mason et al., 1985). In the attack phase, numbers may be varied systematically to search for a pattern and generalising is introduced. The review phase involves checking for both accuracy and transferability of strategies to other problems. Mason et al. noted ineffective problem-solving strategies can cause a student to be “stuck”, resulting in distilling (i.e. articulating difficulty) or mulling (i.e. abandon problem momentarily to think and hopefully arrive at a successful strategy). Inherent in distilling and mulling is that finding a strategy is the remit of the student, rather than provided by the teacher. Given that students may gain insight into a problem while solving (Liljedahl, 2004), teachers should allow students time to engage with problems to develop strategies for solving.

Sullivan et al. (2016) reported a successful intervention with four phases (“posing the problem”; “differentiating the problem” (i.e. individual work before teacher provides a prompt to increase/decrease problem difficulty); “reviewing student activity on the problem”, including sharing strategies; “consolidating the learning” (i.e. solving-related problems)). Teachers (n = 34) solved problems before classroom use to better anticipate students’ responses. They were provided with enabling prompts (e.g. Could pictures help?) to support “differentiating the problem” as the ability to provide suitable prompts to individuals was identified as important. Provision of suitable prompts requires teachers to respond at a student’s point-of-need, based on their thinking. NCTM (2014) also noted that teachers should “elicit and use evidence of student thinking” (p. 10). Twenty-six teachers in Sullivan et al. (2016) believed the intervention structure (outlined above) was successful, noting the importance of phase two (i.e. students attempt problem, prior to teacher use of prompts) and phase four (i.e. students explain thinking and identify efficient problem-solving strategies). All but five of the teachers would plan future lessons using the intervention, highlighting its efficacy.

In a study with year 1 and 2 students (n = 75), Russo and Hopkins (2017) found lesson structure did not impact students’ problem-solving outcomes. Some students preferred problem-first (i.e. challenging problem, class discussion of strategies, worksheets and teacher-led summary), while others preferred teach-first (i.e. discussion of mathematical content, worksheets, challenging problem and teacher-led summary). Given positive responses to consistent features reported for both teachers and students, there is potential in having consistent features in problem-solving lessons, supporting development of a problem-solving intervention to be implemented across a range of lessons.

Effective Problem-Solving Interventions

Features that contribute to successful problem-solving include time for teachers to help students understand language in a problem and teacher anticipation of students’ responses (Sullivan et al., 2016); time for student struggle before teachers provide enabling/extending prompts (Cheeseman et al., 2017; Sullivan et al., 2016); and time for students to collaborate and share knowledge (Ingram et al., 2016). NCTM (2014) noted teachers should include tasks “that promote mathematical reasoning and problem solving and allow multiple entry points and varied solution strategies” (p. 10) and provide “students, individually and collectively, with opportunities and supports to engage with productive struggle” (p. 10). This call for student struggle supported inclusion of time where students are not provided with strategies by either the teacher or peers; hence, there should be time for individual struggle. Ingram et al. (2020) noted the importance of students exploring problems without assistance initially, even though some students may struggle. Problem-solving interventions that included time to struggle with problems improved students’ problem-solving skills (Roche et al., 2013). Sullivan et al. (2014) found that primary students (n = 758) preferred to struggle before being given solutions, with 88% preferring problems of equal or greater difficulty to the problems in their study, highlighting students’ desire for challenging problems. Teachers in Russo et al. (2022) generally noted the importance of providing prompts after students had time to “grapple” with problems; struggle time was valued to enable students to understand problems and attempt solutions prior to teacher support through prompts.

Ingram et al. (2016) reported strategies that contributed to student persistence in problem-solving, following a survey of ten teachers who trialled strategies that encouraged students (aged 9–11) to persist with challenging problems. Strategies included encouraging collaboration; providing access to materials; encouraging students to reflect on their ability to persist; and posing probing questions to direct student thinking; as well as implementing extending prompts (i.e. encourage finding more responses; looking for patterns in results) and enabling prompts (i.e. support students to understand and attempt problems). Russo et al. (2022) also identified that teacher provision of enabling prompts (i.e. related to problem, but requiring fewer steps or of lower complexity) and extending prompts could cater for a range of students. Cheeseman et al. (2017) reported year 3–6 teachers (n = 37) who found extending and enabling prompts supported students to understand and solve problems and extend thinking. Clear expectations of students (e.g. recording thinking, independent or group work) are beneficial, also time for teachers to pay attention to students’ explanations and teacher questions to understand students’ thinking (Roche et al., 2013). The discussion above highlights a range of considerations in developing a problem-solving intervention for mathematics. Following is a problem-solving intervention and a report on its implementation in two classes.

The research question is: How did teachers implement the problem-solving intervention and what were the implications for teaching and learning of problem-solving?

Method

The literature review identified ten features to include in a problem-solving intervention (first column, Table 1). The first-named researcher developed the intervention, with two versions, based on these ten features and planned four lessons using the intervention (Table 2).

Table 1 Problem-solving intervention features and rationale (adapted from Stewart (2020))
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Table 2 Overview of lessons: intervention-IP (Coin lesson) and intervention-WCP (Counter lesson, Cookie lesson and Over-and-Over lesson) (adapted from Stewart (2020))
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Teachers in one coeducational primary school in Victoria, Australia, were invited to participate by the first-named researcher, who taught at the school; hence, convenience sampling was used. Prior to this, ethics was approved by the University of Melbourne and permission approved from the Department of Education and Training Victoria and school principal. Two teachers, Beth and Rachel (pseudonyms), volunteered and provided information about the study to their classes (composite classes, year 1/2 students, aged 6–8 years). Seven students from Beth’s class and eight students from Rachel’s class volunteered to participate, with permission from parents/guardians. Each teacher implemented two problem-solving lessons with student data collected from participants.

Problem-Solving Intervention Development

Two versions of the intervention were created: intervention-WCP (whole class prompts included, teachers anticipate students’ responses) and intervention-IP (whole class prompts excluded, teachers not expected to anticipate students’ responses); this enabled the impact of whole class prompts and teacher anticipation of students’ responses to be investigated. Table 1 summarises intervention features.

Problem-Solving Lesson Plans Based on Intervention

Four lessons were planned with four problems. Each problem:

  • Supported use of a range of problem-solving strategies.

  • Catered for a range of abilities as students can look for patterns or rules.

  • Required finding a number of cases for the answer.

The suggested problems (Table 2) were anticipated to be challenging for the students as they required interpretation of a worded problem, determination of a strategy and provision of an answer that fully addressed all components of the problem. Fülöp (2021) defines a problem as “a challenge for which the participant has a goal, but does not have direct access to a method or an algorithm that gives the solution” (p. 1312). For the Coin problem, the students had not yet studied money, so they utilised understanding from outside the classroom; the Cookie problem was open-ended, so the students needed to produce an answer to address the requirements of the problem. Students had to select and use strategies to solve these non-routine problems; the intention was that the teacher would provide prompts only, rather than suggest or model approaches for solving. Teachers were provided with one lesson based on intervention-WCP (i.e. Coin) and three on intervention-IP (i.e. Counter, Cookie and Over-and-Over). Both teachers taught the Coin problem and one lesson based on intervention-IP (Rachel Cookie; Beth Over-and-Over). Table 2 shows the lessons taught and some assumptions behind problems.

Figure 1 shows pages 1 and 2 of intervention-WCP populated with “How many ways can you make 50c using coins”? Fig. 2 shows pages 1 and 2 of intervention-IP populated with “Everyone got the same number of cookies. How many cookies may there have been”? Problem-specific additions from researcher are in different fonts (e.g. title of task in Fig. 1). Boxes indicate features in Table 1.

Fig. 1
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Problem-solving intervention-WCP (whole class prompts), page 1 and page 2 populated with Coin problem (Stewart, 2020)

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Fig. 2
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Problem-solving intervention-IP (individual prompts), page 1 and page 2 populated with Cookie problem (Stewart, 2020)

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For each lesson, the following were added to the intervention:

  • Challenging problem

  • Individual prompts

  • General enabling prompts (e.g. Could you use a table?)

  • General extending prompts (e.g. Are there more answers? How do you know?)

  • Problem-specific enabling prompts (e.g. Could drawing an array help?)

  • Problem-specific extending prompts (e.g. How does changing the number of people or the amount of cookies they receive change your answer?).

  • Whole class prompts specific to the challenging problem (only for intervention-WCP)

Draft lessons were discussed, and teachers invited to make changes; neither teacher suggested changes to the intervention structure.

Teacher Choices—Lessons Based on Intervention

Teachers were invited to add prompts to lessons; none was added to the Coin problem. Each teacher added one prompt to their second lesson (Rachel, “Could you use pictures of cookies?” (RL2); Beth, “Could you skip count?” (BL2)). Rachel altered the Cookie problem wording to “Everyone (in our class) got the same number of cookies. How many cookies may there have been?”, potentially changing the problem by indicating the number of students is the number in the class. Despite this, Rachel implemented the problem as intended by directing students to choose the number of students for the problem.

For the Coin problem, teachers selected from two student recording sheets, one with visuals (i.e. showing coins) and one without. Both teachers chose visuals, suggesting this scaffolding was perceived to be helpful. Enabling/extending prompts could be provided to students verbally or through individual prompts printed on paper strips; Rachel chose verbal prompts only, while Beth used both types.

Methodology

The study was a qualitative case study of two teachers’ implementation of a problem-solving intervention; Yin (2015) noted case study enabled in-depth understanding of the complexity of a phenomenon in its real-life context. Corbett-Whittier and Hamilton (2013) support use of case studies in educational settings to allow for collection of context-specific and meaningful data, suggesting educational contexts are too complex for widespread generalisations due to diversity of settings, individuals and pedagogies.

This study used a case study approach to gain rich data about the two teachers’ implementation of the problem-solving intervention. Teachers’ implementation of the problem-solving intervention was the phenomenon of interest with a focus on their perspectives and reflections. A range of research instruments were used to gain multiple perspectives on the intervention implementation.

Lesson observation notes about teachers and students’ responses to each lesson were recorded and students’ work samples were collected and photographed by the first-named researcher at the end of each lesson. Two semi-structured interviews were conducted with each teacher, before and after the lessons. Interview 1 focussed on teachers’ views on (and previous experience with) problem-solving and introduced the Coin problem. Possible student solutions were discussed to assist teachers in anticipating students’ responses (i.e. feature of intervention-WCP). Interview 2 focussed on each teacher’s reflections on their two lessons. Photos of students’ work samples were available for teachers to refer to when discussing students’ responses to the lessons.

Data sets (i.e. interview transcripts, student work samples, lesson observation notes) were examined using content analysis. Weber (1990) noted content analysis as systematically and objectively making valid inferences by examining texts that represent social communication. To implement content analysis, coding was used. Data sets were read three times each to familiarise the researcher with their content, followed by a combination of deductive and inductive coding. Features of the intervention identified in the problem-solving literature (Table 1) guided the deductive analysis to ensure key features of the intervention were the focus of the analysis. Inductive coding was used to reduce bias and ensure that new insights that arose from the data were identified. Data sets were revisited multiple times to check that findings correctly reflected the data.

Deductive Coding

Data sets were coded based on words, phrases and portions of the data related to one or more of the problem-solving intervention features in Table 3. Codes were combined and findings were drawn from these combined codes. Literature was revisited multiple times throughout this process.

Table 3 Features of problem-solving interventions identified from literature
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Table 4 shows comments from Rachel relating to struggle time. Several codes were combined related to “Teaching strategies Rachel uses to reduce the amount of struggle for students”. For example, Rachel noted she would “model an example” (R2040) if she was to teach the lesson again, which inherently suggested a reduction in struggle time (Feature 3) as students would be provided with a successful problem-solving strategy by the teacher. This comment provides an example where the teacher did not use the term struggle time, but this could be inferred. Additional codes include those where Rachel reported prompting students (e.g. by identifying links with prior knowledge) where the intervention suggested students work without teacher support.

Table 4 Sample comments relating to struggle: Rachel (interviews)
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Inductive Coding

Inductive coding was based on Krippendoff’s (2004) six components of content analysis where meaning is developed from data through the process of considering each component and revisiting components as the researcher reviews and rereads the text; there is an expectation that components be reviewed considering new learning from data. In Table 5, each separate component is described, including the role of the researcher. Following the deductive and inductive coding, five themes were identified.

Table 5 Krippendorff’s components of content analysis including researcher’s procedure
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Results and Discussion: Problem-Solving

Teacher deviations from the intervention mainly centred on students’ abilities to understand the requirements of a problem, develop problem-solving strategies and develop appreciation of efficiency of approaches through class discussion of strategies. Five themes were identified related to problem-solving: struggle, teachers anticipate student responses, prompts, student share time and problem-solving practice: teachers.

Theme One: Struggle

Time for students to struggle with a problem was an intervention feature included in each lesson plan, both during the problem introduction, where teachers were asked not to model approaches for solving, and for the first 15 min of independent student work. In interviews, both teachers identified inclusion of struggle time as problematic for some students. The following sections explain teachers’ views on struggle and provide insight into students’ responses to struggle.

Prior to teaching, neither teacher identified that struggle would support students’ problem-solving. Despite having discussed the intervention with the researcher, thus being aware that struggle was an intervention feature, Rachel did not mention struggle in interview one. Beth discussed student struggle as a negative consequence of lessons that were too fast paced and focussed on students with good understanding of mathematics, noting “… I think that’s the problem in maths, that a lot of teachers just move onto the next [i.e., mathematical topic, problem, or lesson] and the next. And this kid, you can see… Losing their confidence. Losing their enjoyment…” (B1072–B1074). Student struggle was believed to contribute to loss of student confidence and enjoyment in mathematics, so something to be minimised. Beth identified two teaching strategies to reduce her students’ struggle, both of which scaffolded students:

  • Providing students with a random problem (e.g. 2 + 2) that they can confidently answer when they have difficulty with a problem

  • Modelling an example to provide a strategy for solving the problem.

Both can be good teaching strategies; however, the point at which they are implemented will impact the extent to which students have a chance to struggle and the problem-solving nature of a problem.

Although the intervention purposefully included struggle in the initial stages of problem-solving, both teachers deviated to reduce the struggle experienced by students. Reducing the amount of struggle in lessons was consistent with interview responses, where the benefits of struggle for supporting students’ problem-solving were not identified. Although Rachel reduced the suggested struggle time, she did include some struggle time and the observed benefits of this in students’ work are discussed in the next section. Table 6 summarises foci of the intervention, teacher deviations and the impact of deviations.

Table 6 Summary of the deviations made by the two teachers in all lessons based on the intervention (adapted from Stewart (2020))
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Struggle Promoted a Range of Problem-Solving Strategies

Beth modelled successful problem-solving strategies (number sentences; look for a pattern) in both lessons, contrasting with Rachel’s approach which incorporated some struggle time and students had to find and use their own strategies. Teacher modelling of strategies and excluding specific struggle time resulted in fewer problem-solving strategies used by Beth’s students; most used the strategy demonstrated. Table 7 shows that for the Coin problem (RL1/BL1) Beth’s students used the two strategies demonstrated (i.e. 2 and 3), while Rachel’s students used one more strategy (i.e. 1, 2 and 3). The largest number of strategies (6) was used by Rachel’s students for the Cookie problem (RL2). Rachel prompted some students early in this lesson, noting that she “definitely deviated [i.e., from lesson] and sort of pushed, nudged them [i.e., the students] in the right direction” as she observed students were not attempting the problem. Although struggle time was reduced through providing a prompt, Rachel did not model a strategy and included some dedicated struggle time. A greater range of strategies were demonstrated in lessons without teacher modelling and where class struggle time was included, so these intervention features supported students in choosing and using strategies.

Table 7 Frequency of problem-solving strategies in lessons (adapted from Stewart (2020))
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Impact of Reducing Struggle

Neither Beth nor Rachel included the recommended struggle time (Tables 4 and 6), prompting students earlier than suggested. When introducing the Coin problem (BL1), Beth modelled the strategy of using a number sentence (Fig. 3). However, the examples may have detracted from consideration of the real-life context constrained by Australian currency (5c, 10c, 20c, 50c); the first number sentence is impossible. In this case, teacher modelling, intending to reduce struggle, may have hindered students’ problem-solving as it focussed on a numerical problem of adding two numbers to give 50, rather than consider the real-life constraints of adding coins to give 50c.

Fig. 3
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Beth’s number sentence visual for introduction to Coin problem (BL1) (Stewart, 2020)

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Beth identified the impact of some of her choices to reduce struggle. In introducing Over-and-Over (BL2), Beth focussed on the words “same” and “over” to reach “16”; some students focussed on the number 16 and added a range of numbers, rather than the same number, to equal 16 (e.g. student 2 in Fig. 4).

Fig. 4
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Incorrect solutions as numbers not repeated: student 2 (BL2)

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“We talked about the key words being the “number”, “sixteen”. We talked about repeated… the “same number” … really reinforced at the start of the lesson and some students … They’ve been focussed on one element of that… like the “sixteen” being the important thing but the “repeated” being not so important” (B2099).

Not providing time for students to struggle with the problem prior to class discussion resulted in some students not understanding the problem. This might be counterintuitive for teachers who view struggle as negative, rather than supporting students to develop understanding of a problem, to support problem-solving.

Although Rachel included struggle time at the start of lessons, she provided enabling prompts earlier than suggested in both lessons (Tables 4 and 6). Some students’ negative response to struggle at the start of lessons impacted her decision to reduce struggle (“I could see frustration from a lot of kids especially in the first ten minutes”, R2042). Individual struggle was also reduced through students observing successful strategies used by peers; Rachel noted “when they saw another kid doing a strategy… they had that light bulb [moment] where they went “Oh, they’re doing ‘groups of’ so I might draw groups or I’ll do an array….”” (R2010). Rachel credited this for some students’ success in problem-solving. It is not only the teacher who determines struggle time, particularly when students work in groups. An intervention feature was that students have individual struggle time, prior to discussion of strategies with their teacher or peers. If the intention of problem-solving is purely finding a correct answer, then observing and applying a successful strategy would result in success. However, given the imperative in the curriculum for developing “capable problem-solvers” and the importance of the problem-solving process (e.g. Pólya, 1945), students must be able to choose and use strategies, as well as find a correct answer.

Understanding the goal of struggle time did not always result in inclusion of struggle time in teaching. Rachel acknowledged students were “meant to be in a state of frustration and confusion” (R2042) during struggle time, however suggested that reduced struggle time could foster students’ perseverance with problems and maintain engagement, thus avoiding students’ “I don’t understand it so I’m not gonna do it attitude” (R2042). This suggested that unless students could solve problems relatively quickly, they would disengage with mathematics. There was inherent struggle in the Coin problem due to the need for students to consider calculations with money, a topic yet to be taught. Rachel noted student 3 crossed out her working and did not persevere “…she completely gave up and refused almost to participate because she didn’t understand…” (R2064–R2066). Although Rachel attributed student 3’s inability to solve the problem to lack of understanding of money, the student recorded three correct responses, namely, 5 10’s is 50, 20c + 20c + 10c is 50c and 20c + 10c + 10c + 10 = 50c (Fig. 5), providing evidence of use of a successful strategy to partially solve the problem.

Fig. 5
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Three correct solutions for Coin problem (*)—(RL1) (adapted from Stewart (2020))

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Rachel suggested her extension students, with good mathematical understanding, did not respond positively to struggle as “they couldn’t figure out how to approach this [i.e., the problem]” (R2004). These students, who may be used to solving problems quickly, may be perturbed by having to struggle and not finding a correct answer immediately. In contrast, students who find mathematics difficult were expected to be inclined to persist with problems as they were “…used to maybe failing or not doing as well” (R2002), so anticipating struggle in mathematics. Due to perceived student resilience issues during the Coin problem, Rachel indicated that if she taught the lesson again she would support students when introducing the problem by “model(ling) an example” (R2040) or having a “discuss(ion) with the kids” (R2040). In this case, reducing struggle is aligned with maintaining students’ perseverance with problems. Although both teachers wanted to reduce struggle for students with good understanding and those who found mathematics difficult, a potential benefit of providing the opportunity to struggle might be that students are more willing to struggle as it becomes a normal part of problem-solving.

Overall, struggle time was not recognised as contributing to students’ problem-solving skills by Beth or Rachel, but instead negatively impacting students’ resilience and confidence in mathematics; hence, both teachers deviated to reduce the amount of struggle their students experienced. For teachers used to scaffolding students’ learning, the notion of struggle may be counterintuitive, as it is related to inability to solve a problem rather than thinking time to engage with a problem and consider strategies. Cheeseman (2018) also found many year 1 teachers did not see the benefit of students struggling with problems. One conclusion could be that teachers do not want students in the early years of primary school to struggle in mathematics, to remain positive about learning mathematics. Although our study was limited to two teachers and their classes, the increased range of problem-solving strategies used by the class where struggle was included suggests that struggle time could be an important consideration for teachers to support students’ problem-solving. If problem-solving includes having opportunities to select solution methods and use higher-order thinking (Schoenfeld, 1992), particularly in the entry phase where problems are introduced and students decide on strategies for solving (Mason, et al., 1985), then struggle time is important. Our findings suggest that some teachers may find dedicated struggle time difficult to implement as they perceive it to be negative and to result in less student confidence and resilience in mathematics.

Theme 2: Teachers Anticipate Student Responses

Two different implementations of the intervention (i.e. teachers either did or did not anticipate students’ responses prior to teaching) provided insight into whether a teacher’s anticipation of students’ responses supported teachers in promoting effective problem-solving by students.

In the first lesson (RL1 and BL1, intervention-WCP), each teacher anticipated students’ responses to the Coin problem prior to teaching and there were few misunderstandings identified in students’ work (Table 8), highlighting the efficacy of this approach. This builds on Sullivan et al. (2016) who found that year 3 and 4 teachers were better prepared to support students if they anticipated students’ possible responses before a problem was implemented. The current study, with year 1 and 2 students, suggests this approach might also be beneficial for supporting problem-solving of younger children.

Table 8 Problem misunderstandings identified in student work samples for all lessons based on the intervention (adapted from Stewart (2020))
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In the second lessons (Over-and-Over, BL2; Cookie, RL2), teachers did not anticipate students’ responses prior to teaching (intervention-IP). Many of Beth’s students produced some incorrect responses (Table 8), suggesting there were misunderstandings about the goal of the problem. The teacher introduction, where Beth modelled a strategy (Table 5 and 6), may have contributed, as students largely replicated her approach rather than decide on a strategy to solve. Beth did not discuss the assumption behind the problem (i.e. same number needed to be added to make 16); the need for this discussion may have been apparent if students’ responses had been anticipated. Rachel’s students did not demonstrate any misunderstanding of the problem in RL2 which may be due to the nature of Cookie (i.e. open-ended and simple wording). It seemed that for straightforward problems the students could understand the problem and work towards a solution; however, where the problem was more complex (e.g. BL2), then possibly having the teacher anticipate students’ answers might have highlighted potential student difficulties and helped the teacher to orchestrate the lesson to support students’ problem-solving.

Theme 3: Prompts

The range of prompts enabled teacher choice, with teachers including some provided in the lessons and additional prompts added in-the-moment while teaching (Table 9). Rachel used more prompts than Beth across both lessons (14, 22; 10, 9). Teacher use of a range of prompts suggested they were useful in the problem-solving lessons. Neither teacher used all provided prompts, indicating that teachers made decisions about appropriateness. Table 10 shows the Coin prompts, including those provided in the lesson but not used by either teacher. Neither teacher used general enabling prompts to foster an approach for solving (e.g. “Could you make a list?”). However, they utilised general enabling prompts that encouraged students to read and understand the problem (e.g. Can you think of an easier question like this?).

Table 9 Summary table of general and problem-specific prompts identified in all lesson observation data
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Table 10 Prompts in Coin problem lessons (adapted from Stewart (2020))
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Both teachers provided a rationale for use of a range of prompts, with decisions focussed on scaffolding students. Rachel identified that specific and general prompts enabled her to scaffold students’ solution of problems, rather than model one given approach for solving or direct students to a particular strategy; choice of problem-solving strategy remained with the students. Beth noted an enabling prompt (“What if the number you reached was 6?”) to scaffold a student having difficulty with Over-and-Over, reducing the size of the number (i.e. 6 cf 16) to create a simpler, but related problem. This supported the student “to realise that she could do more for the sixteen” (B2083) and that additional solutions existed for the original problem. Prompts provided suggested phrases to support problem-solving with Rachel noting they “were fantastic because there were some kids that I didn’t want to lead directly but I didn’t know exactly how to tell them where to go without giving them an answer” (R2012). In this case, prompts provided Rachel with pedagogical support by providing phrases that scaffolded students’ problem-solving. Cheeseman et al. (2017) noted the importance of extending and enabling prompts for supporting students’ problem-solving and our study highlights that prompts can also support teaching of problem-solving, by providing teachers with phrases to prompt students in the problem-solving process. Teachers’ repertoires of enabling and extending prompts for problem-solving were enhanced through the intervention.

Rachel kept prompts “general” to minimise the scaffolding provided to students, so they had to determine solution strategies, contrasting with the approach of modelling strategies. Rachel’s use of “general” was different to the categorisation of general prompts in the intervention; for Rachel, “general” prompts referred to questions that promoted student reflection and she noted “I tried to keep it really general. I tried to keep a lot of my conversation in questions, so it was pushing them to do the thinking” (R2050). Rachel identified affordances of prompts, particularly the benefit for students in determining strategies, for example, “Could you change the number of cookies each person gets?” (R2044) was identified as effective for encouraging consideration of the full extent of the Cookie problem, rather than assume one cookie per person (i.e. approach used by most of her students initially).

Recognising affordances of prompts will assist teachers in choosing appropriate prompts to scaffold or extend students at different stages of the problem-solving process. A challenge for teachers is identifying when specific prompts are most effective, particularly when presented with several enabling and extending prompts and selecting prompts in-the-moment in response to students’ discussion, work or questions in class. Effectiveness of prompts could relate to helping students understand a problem, choose a strategy and reflect on a solution or the capacity to extend students through an additional challenge. Both teachers identified the purposes of prompts and reflected on their use of prompts, which suggested they were also assessing the effectiveness of prompts to achieve goals, such as scaffolding students.

Timing of prompts can impact the extent of scaffolding students receive at each stage of a problem-solving lesson, or the extent to which students are extended beyond the scope of the original problem (Sullivan et al., 2016). Timing is inextricably linked to selection of prompts, as to have a well-timed prompt a teacher needs to select an appropriate prompt that helps students move forward with their problem-solving; it can also relate to a decision not to use a prompt at a particular point in a lesson. Both teachers provided some prompts earlier than suggested, deviating from the intervention to scaffold students rather than allow students the struggle time suggested. Two possible reasons for this could be that the planned struggle time was too long, and teachers noted it was not productive, or else teachers wanted students to solve problems relatively quickly and did not see benefit in students spending time struggling.

Enabling prompts support students who either have misunderstood a problem or are not making progress. Strategic use of enabling prompts was demonstrated when Rachel was observed prompting students who were unclear about the Cookie problem to reread the question. There were also instances where selection of prompts did not support students who misunderstood a problem. Beth prompted students to draw pictures for Over-and-Over, which can be a useful strategy for many problems, but not helpful for supporting understanding of this problem; the given prompts in the intervention targeted the goal of the problem (e.g. Have you added the same number over and over to reach 16?). Timing of extending prompts was also important, as providing such a prompt too early can result in students focussing on the extension problem, without answering the original problem. Figure 6 shows an example where Beth gave an extending prompt (i.e. to increase the total) to student 6, encouraging consideration of an extension problem before completing the original problem. Prior to this, the student had found six (out of the 13) correct responses (i.e. numbers 1–6). Once the new total (i.e. $1) was provided, the student did not complete the original problem. Some students prompted in ways that did not appear to target their misconceptions or guide them to fully answer problems; in the case of student 6, the prompt did not address the incorrect thinking behind responses 7 and 8. Selection and timing of prompts here impacted the extent to which the student engaged successfully in the problem-solving lesson.

Fig. 6
figure 6

Six correct (1–6) and two incorrect (8, 9) responses for original total and three correct responses for new total: student 6 (BL1)

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Theme 4: Student Share Time

The intervention included 10 min share time at the end of each lesson to foster both sharing of problem-solving strategies and discussion of the effectiveness and efficacy of these strategies. Neither teacher included this full 10 min of share time; however, Rachel facilitated an additional share time in the middle of the second lesson. Neither teacher asked students who used effective strategies to share their problem-solving strategies in the share time sections of their lessons; thus, the opportunity for students to learn about effective strategies from peers was missed.

Both teachers commented on the need for more time for discussion and reflection on problem-solving strategies at the end of their lessons. Beth noted the need for more share time (“I wish we’d done it a little bit more”, B2054) and Rachel noted her “end of lesson sharing was a bit too short” (R2082) and that “next time I’d probably give myself more time” (R2082). This highlights a tension in teaching between allowing students time to explore strategies and allocating sufficient time for class discussion of the efficacy of different strategies. After Rachel’s additional share time in the middle of RL2 (see Table 4 and 6), her students were observed being encouraged to apply strategies observed from peers (“If you’re stuck please stay on the floor [i.e., for further group discussion]. If not go back to your table and find another way. Think about the strategies people shared and use one that might work for you.”). Rachel reflected on the success of encouraging students to apply strategies observed during mid-lesson share time and suggested that “the kids that maybe didn’t have as many strategies or many ways to approach it could sort of steal ideas from people who were sharing” (R2080), so the perceived benefit was that students could learn successful strategies from their peers. This deviation from the intervention appeared to have a positive impact on Rachel’s students’ problem-solving in the lesson.

Neither teacher reported on why they did not choose specific students to share their problem-solving strategies during share time. To ensure that share time encourages students to reflect upon their choice of problem-solving strategies, having a range of students share their approaches and discuss their effectiveness is imperative. It is unknown whether teachers did not see benefits in choosing specific students or whether this was due to perceived time constraints.

Although both teachers reported that share time at the end of their lessons was too short, their reflections highlighted that they would increase share time in future problem-solving lessons. Ingram et al. (2016) highlighted the role of student sharing and collaboration in successful problem-solving and both teachers recognised that they needed to allow more time for sharing. The intervention made share time explicit; this may bring the need for inclusion of share time to the attention of teachers.

Theme 5: Problem-Solving Practice: Teachers

Both Rachel and Beth indicated that involvement in the study had positively impacted their teaching of problem-solving. The specific impact Beth mentioned was that she would encourage students to use more manipulatives and pictures in her problem-solving lessons in the future to make mathematics more concrete for her students. “When you’ve got the manipulatives or when you’ve got the students drawing it, they can see it. It’s something concrete it’s not just… a blur of numbers” (B2341).

Rachel identified positive aspects of the intervention, including students being flexible in their approaches to solving, noting one student who “modelled her thinking in… three different ways and was able to articulate it … fantastic” (R2070). She also appreciated the extending and enabling prompts which helped her to guide her students’ thinking. Rachel noted that she would implement lessons based on the intervention in the future; one such lesson, focussed on multiplication, had already been planned. For this multiplication lesson, Rachel planned to reduce the amount of struggle suggested in the intervention by giving students a problem with different totals depending on whether students’ mathematical understanding was categorised as low (total of 10), middle (total of 30) or high (total of 50 or 100). Not expecting all students to grapple with the more difficult total (i.e. 50 or 100) may impact the potential for students to develop problem-solving skills.

Teachers’ comments suggested that the intervention had a positive impact on their teaching of problem-solving, in particular the availability of prompts. Including specific features in the problem-solving intervention and naming them could foster teacher recognition that these features are important considerations.

Conclusion

Both teachers were able to implement the intervention, including enabling and extending prompts in their lessons. The teachers gave few comments about whether the intervention improved their students’ problem-solving abilities, but they both noted challenges experienced by students, predominantly related to struggle. Despite this, the study showed that there is potential for students to demonstrate a wider range of problem-solving strategies when provided with struggle time, compared to the situation where they are shown an approach for solving by the teacher or another student. Although Sullivan et al. (2014) found that most students in their study preferred to struggle with problems and solve difficult problems, the teachers in our study wanted to reduce students’ struggle. Their rationale for reducing struggle was mainly focussed on maintaining students’ confidence and resilience in tackling challenging problems. This tension between students’ expectation for challenging problems (and implicit expectation of struggle) and teachers’ desire to reduce struggle may result in problems being set that are not challenging enough for all students. McCormick (2016) noted the need for cognitively demanding problems, but if teachers view students’ struggle as negative then approaches to reduce struggle, for example by modelling a strategy, may reduce the cognitive demand of problems.

Struggle time was an integral feature of the intervention and teacher deviation to reduce struggle can impact the efficacy of the intervention. Struggle was intended to be positive, to promote development of persistence with problem-solving and to support individuals to consider strategies that might be used to solve problems where the answer is not evident immediately. Struggle time was not anticipated to be reduced; hence, for future implementation of the intervention, it would be beneficial to include an explicit rationale for struggle time to encourage teachers not to reduce this.

The intervention included numerous enabling and extending prompts, but teachers made selections as to which ones they included. Having a range of prompts enabled teacher choice to target the scaffolding for individuals and to extend students. However, it may be beneficial to provide a limited number of general enabling or extending prompts so that students can develop familiarity with these prompts over time and teachers could encourage students to ask themselves these prompts, thus facilitating problem-solving.

The fact that the teachers did not always implement the intervention as intended in the lessons could be viewed in a positive light. It implies that the two teachers were using their professional judgement about effective problem-solving practices during the lessons. Roche et al. (2013) identified the importance of struggle time for improving students’ problem-solving. Sharing in the middle of problem-solving lessons may help resolve the tension between allowing student struggle and providing support when teaching problem-solving.

Both teachers reflected on their teaching of problem-solving lessons through the experience of trialling lessons based on the intervention and planned to implement aspects of the intervention in the future. The teachers perceived that the intervention had an impact on their teaching of problem-solving, with share time in lessons identified as an aspect to increase in future problem-solving lessons. Given that Stacey et al. (2015) identified a perceived lack of problem-solving resources to support Australian teachers with problem-solving, the teachers’ perception of the benefits of the intervention on their problem-solving practices provides promise for use of the intervention to support the teaching of problem-solving. Explicit statements about features of the intervention which should be included and those where teacher choice is expected, for example in selection of appropriate prompts, could support teachers in the implementation of problem-solving lessons.