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))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))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

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

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

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

“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

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

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))

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 dataTable 10 Prompts in Coin problem lessons (adapted from Stewart (2020))

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

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

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.