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## 6 Tips for Teaching Math Problem-Solving Skills

Solving word problems is tougher than computing with numbers, but elementary teachers can guide students to do the deep thinking involved.

A growing concern with students is the ability to problem-solve, especially with complex, multistep problems. Data shows that students struggle more when solving word problems than they do with computation , and so problem-solving should be considered separately from computation. Why?

Consider this. When we’re on the way to a new destination and we plug in our location to a map on our phone, it tells us what lane to be in and takes us around any detours or collisions, sometimes even buzzing our watch to remind us to turn. When I experience this as a driver, I don’t have to do the thinking. I can think about what I’m going to cook for dinner, not paying much attention to my surroundings other than to follow those directions. If I were to be asked to go there again, I wouldn’t be able to remember, and I would again seek help.

If we can switch to giving students strategies that require them to think instead of giving them too much support throughout the journey to the answer, we may be able to give them the ability to learn the skills to read a map and have several ways to get there.

Here are six ways we can start letting students do this thinking so that they can go through rigorous problem-solving again and again, paving their own way to the solution.

When we can remind students that they already have many comprehension skills and strategies they can easily use in math problem-solving, it can ease the anxiety surrounding the math problem. For example, providing them with strategies to practice, such as visualizing, acting out the problem with math tools like counters or base 10 blocks, drawing a quick sketch of the problem, retelling the story in their own words, etc., can really help them to utilize the skills they already have to make the task less daunting.

We can break these skills into specific short lessons so students have a bank of strategies to try on their own. Here's an example of an anchor chart that they can use for visualizing . Breaking up comprehension into specific skills can increase student independence and help teachers to be much more targeted in their problem-solving instruction. This allows students to build confidence and break down the barriers between reading and math to see they already have so many strengths that are transferable to all problems.

## 2. Avoid boxing students into choosing a specific operation

It can be so tempting to tell students to look for certain words that might mean a certain operation. This might even be thoroughly successful in kindergarten and first grade, but just like when our map tells us where to go, that limits students from becoming deep thinkers. It also expires once they get into the upper grades, where those words could be in a problem multiple times, creating more confusion when students are trying to follow a rule that may not exist in every problem.

We can encourage a variety of ways to solve problems instead of choosing the operation first. In first grade, a problem might say, “Joceline has 13 stuffed animals and Jordan has 17. How many more does Jordan have?” Some students might choose to subtract, but a lot of students might just count to find the amount in between. If we tell them that “how many more” means to subtract, we’re taking the thinking out of the problem altogether, allowing them to go on autopilot without truly solving the problem or using their comprehension skills to visualize it.

## 3. Revisit ‘representation’

The word “representation” can be misleading. It seems like something to do after the process of solving. When students think they have to go straight to solving, they may not realize that they need a step in between to be able to support their understanding of what’s actually happening in the problem first.

Using an anchor chart like one of these ( lower grade , upper grade ) can help students to choose a representation that most closely matches what they’re visualizing in their mind. Once they sketch it out, it can give them a clearer picture of different ways they could solve the problem.

Think about this problem: “Varush went on a trip with his family to his grandmother’s house. It was 710 miles away. On the way there, three people took turns driving. His mom drove 214 miles. His dad drove 358 miles. His older sister drove the rest. How many miles did his sister drive?”

If we were to show this student the anchor chart, they would probably choose a number line or a strip diagram to help them understand what’s happening.

If we tell students they must always draw base 10 blocks in a place value chart, that doesn’t necessarily match the concept of this problem. When we ask students to match our way of thinking, we rob them of critical thinking practice and sometimes confuse them in the process.

## 4. Give time to process

Sometimes as educators, we can feel rushed to get to everyone and everything that’s required. When solving a complex problem, students need time to just sit with a problem and wrestle with it, maybe even leaving it and coming back to it after a period of time.

This might mean we need to give them fewer problems but go deeper with those problems we give them. We can also speed up processing time when we allow for collaboration and talk time with peers on problem-solving tasks.

## 5. Ask questions that let Students do the thinking

Questions or prompts during problem-solving should be very open-ended to promote thinking. Telling a student to reread the problem or to think about what tools or resources would help them solve it is a way to get them to try something new but not take over their thinking.

These skills are also transferable across content, and students will be reminded, “Good readers and mathematicians reread.”

## 6. Spiral concepts so students frequently use problem-solving skills

When students don’t have to switch gears in between concepts, they’re not truly using deep problem-solving skills. They already kind of know what operation it might be or that it’s something they have at the forefront of their mind from recent learning. Being intentional within their learning stations and assessments about having a variety of rigorous problem-solving skills will refine their critical thinking abilities while building more and more resilience throughout the school year as they retain content learning in the process.

Problem-solving skills are so abstract, and it can be tough to pinpoint exactly what students need. Sometimes we have to go slow to go fast. Slowing down and helping students have tools when they get stuck and enabling them to be critical thinkers will prepare them for life and allow them multiple ways to get to their own destination.

## Center for Teaching

Teaching problem solving.

Print Version

## Tips and Techniques

Expert vs. novice problem solvers, communicate.

• Have students  identify specific problems, difficulties, or confusions . Don’t waste time working through problems that students already understand.
• If students are unable to articulate their concerns, determine where they are having trouble by  asking them to identify the specific concepts or principles associated with the problem.
• In a one-on-one tutoring session, ask the student to  work his/her problem out loud . This slows down the thinking process, making it more accurate and allowing you to access understanding.
• When working with larger groups you can ask students to provide a written “two-column solution.” Have students write up their solution to a problem by putting all their calculations in one column and all of their reasoning (in complete sentences) in the other column. This helps them to think critically about their own problem solving and helps you to more easily identify where they may be having problems. Two-Column Solution (Math) Two-Column Solution (Physics)

## Encourage Independence

• Model the problem solving process rather than just giving students the answer. As you work through the problem, consider how a novice might struggle with the concepts and make your thinking clear
• Have students work through problems on their own. Ask directing questions or give helpful suggestions, but  provide only minimal assistance and only when needed to overcome obstacles.
• Don’t fear  group work ! Students can frequently help each other, and talking about a problem helps them think more critically about the steps needed to solve the problem. Additionally, group work helps students realize that problems often have multiple solution strategies, some that might be more effective than others

## Be sensitive

• Frequently, when working problems, students are unsure of themselves. This lack of confidence may hamper their learning. It is important to recognize this when students come to us for help, and to give each student some feeling of mastery. Do this by providing  positive reinforcement to let students know when they have mastered a new concept or skill.

## Encourage Thoroughness and Patience

• Try to communicate that  the process is more important than the answer so that the student learns that it is OK to not have an instant solution. This is learned through your acceptance of his/her pace of doing things, through your refusal to let anxiety pressure you into giving the right answer, and through your example of problem solving through a step-by step process.

Experts (teachers) in a particular field are often so fluent in solving problems from that field that they can find it difficult to articulate the problem solving principles and strategies they use to novices (students) in their field because these principles and strategies are second nature to the expert. To teach students problem solving skills,  a teacher should be aware of principles and strategies of good problem solving in his or her discipline .

The mathematician George Polya captured the problem solving principles and strategies he used in his discipline in the book  How to Solve It: A New Aspect of Mathematical Method (Princeton University Press, 1957). The book includes  a summary of Polya’s problem solving heuristic as well as advice on the teaching of problem solving.

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## Teaching problem solving: Let students get ‘stuck’ and ‘unstuck’

Subscribe to the center for universal education bulletin, kate mills and km kate mills literacy interventionist - red bank primary school helyn kim helyn kim former brookings expert @helyn_kim.

October 31, 2017

This is the second in a six-part  blog series  on  teaching 21st century skills , including  problem solving ,  metacognition , critical thinking , and collaboration , in classrooms.

In the real world, students encounter problems that are complex, not well defined, and lack a clear solution and approach. They need to be able to identify and apply different strategies to solve these problems. However, problem solving skills do not necessarily develop naturally; they need to be explicitly taught in a way that can be transferred across multiple settings and contexts.

Here’s what Kate Mills, who taught 4 th grade for 10 years at Knollwood School in New Jersey and is now a Literacy Interventionist at Red Bank Primary School, has to say about creating a classroom culture of problem solvers:

Helping my students grow to be people who will be successful outside of the classroom is equally as important as teaching the curriculum. From the first day of school, I intentionally choose language and activities that help to create a classroom culture of problem solvers. I want to produce students who are able to think about achieving a particular goal and manage their mental processes . This is known as metacognition , and research shows that metacognitive skills help students become better problem solvers.

I begin by “normalizing trouble” in the classroom. Peter H. Johnston teaches the importance of normalizing struggle , of naming it, acknowledging it, and calling it what it is: a sign that we’re growing. The goal is for the students to accept challenge and failure as a chance to grow and do better.

I look for every chance to share problems and highlight how the students— not the teachers— worked through those problems. There is, of course, coaching along the way. For example, a science class that is arguing over whose turn it is to build a vehicle will most likely need a teacher to help them find a way to the balance the work in an equitable way. Afterwards, I make it a point to turn it back to the class and say, “Do you see how you …” By naming what it is they did to solve the problem , students can be more independent and productive as they apply and adapt their thinking when engaging in future complex tasks.

After a few weeks, most of the class understands that the teachers aren’t there to solve problems for the students, but to support them in solving the problems themselves. With that important part of our classroom culture established, we can move to focusing on the strategies that students might need.

Here’s one way I do this in the classroom:

I show the broken escalator video to the class. Since my students are fourth graders, they think it’s hilarious and immediately start exclaiming, “Just get off! Walk!”

When the video is over, I say, “Many of us, probably all of us, are like the man in the video yelling for help when we get stuck. When we get stuck, we stop and immediately say ‘Help!’ instead of embracing the challenge and trying new ways to work through it.” I often introduce this lesson during math class, but it can apply to any area of our lives, and I can refer to the experience and conversation we had during any part of our day.

Research shows that just because students know the strategies does not mean they will engage in the appropriate strategies. Therefore, I try to provide opportunities where students can explicitly practice learning how, when, and why to use which strategies effectively  so that they can become self-directed learners.

For example, I give students a math problem that will make many of them feel “stuck”. I will say, “Your job is to get yourselves stuck—or to allow yourselves to get stuck on this problem—and then work through it, being mindful of how you’re getting yourselves unstuck.” As students work, I check-in to help them name their process: “How did you get yourself unstuck?” or “What was your first step? What are you doing now? What might you try next?” As students talk about their process, I’ll add to a list of strategies that students are using and, if they are struggling, help students name a specific process. For instance, if a student says he wrote the information from the math problem down and points to a chart, I will say: “Oh that’s interesting. You pulled the important information from the problem out and organized it into a chart.” In this way, I am giving him the language to match what he did, so that he now has a strategy he could use in other times of struggle.

The charts grow with us over time and are something that we refer to when students are stuck or struggling. They become a resource for students and a way for them to talk about their process when they are reflecting on and monitoring what did or did not work.

For me, as a teacher, it is important that I create a classroom environment in which students are problem solvers. This helps tie struggles to strategies so that the students will not only see value in working harder but in working smarter by trying new and different strategies and revising their process. In doing so, they will more successful the next time around.

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## Jabberwocky

Problem-solving is the ability to identify and solve problems by applying appropriate skills systematically.

Problem-solving is a process—an ongoing activity in which we take what we know to discover what we don't know. It involves overcoming obstacles by generating hypo-theses, testing those predictions, and arriving at satisfactory solutions.

Problem-solving involves three basic functions:

Seeking information

Generating new knowledge

Making decisions

Problem-solving is, and should be, a very real part of the curriculum. It presupposes that students can take on some of the responsibility for their own learning and can take personal action to solve problems, resolve conflicts, discuss alternatives, and focus on thinking as a vital element of the curriculum. It provides students with opportunities to use their newly acquired knowledge in meaningful, real-life activities and assists them in working at higher levels of thinking (see Levels of Questions ).

Here is a five-stage model that most students can easily memorize and put into action and which has direct applications to many areas of the curriculum as well as everyday life:

## Expert Opinion

Here are some techniques that will help students understand the nature of a problem and the conditions that surround it:

• List all related relevant facts.
• Make a list of all the given information.
• Restate the problem in their own words.
• List the conditions that surround a problem.
• Describe related known problems.

## It's Elementary

For younger students, illustrations are helpful in organizing data, manipulating information, and outlining the limits of a problem and its possible solution(s). Students can use drawings to help them look at a problem from many different perspectives.

Understand the problem. It's important that students understand the nature of a problem and its related goals. Encourage students to frame a problem in their own words.

Describe any barriers. Students need to be aware of any barriers or constraints that may be preventing them from achieving their goal. In short, what is creating the problem? Encouraging students to verbalize these impediments is always an important step.

Identify various solutions. After the nature and parameters of a problem are understood, students will need to select one or more appropriate strategies to help resolve the problem. Students need to understand that they have many strategies available to them and that no single strategy will work for all problems. Here are some problem-solving possibilities:

Create visual images. Many problem-solvers find it useful to create “mind pictures” of a problem and its potential solutions prior to working on the problem. Mental imaging allows the problem-solvers to map out many dimensions of a problem and “see” it clearly.

Guesstimate. Give students opportunities to engage in some trial-and-error approaches to problem-solving. It should be understood, however, that this is not a singular approach to problem-solving but rather an attempt to gather some preliminary data.

Create a table. A table is an orderly arrangement of data. When students have opportunities to design and create tables of information, they begin to understand that they can group and organize most data relative to a problem.

Use manipulatives. By moving objects around on a table or desk, students can develop patterns and organize elements of a problem into recognizable and visually satisfying components.

Work backward. It's frequently helpful for students to take the data presented at the end of a problem and use a series of computations to arrive at the data presented at the beginning of the problem.

Look for a pattern. Looking for patterns is an important problem-solving strategy because many problems are similar and fall into predictable patterns. A pattern, by definition, is a regular, systematic repetition and may be numerical, visual, or behavioral.

Create a systematic list. Recording information in list form is a process used quite frequently to map out a plan of attack for defining and solving problems. Encourage students to record their ideas in lists to determine regularities, patterns, or similarities between problem elements.

Try out a solution. When working through a strategy or combination of strategies, it will be important for students to …

Keep accurate and up-to-date records of their thoughts, proceedings, and procedures. Recording the data collected, the predictions made, and the strategies used is an important part of the problem solving process.

Try to work through a selected strategy or combination of strategies until it becomes evident that it's not working, it needs to be modified, or it is yielding inappropriate data. As students become more proficient problem-solvers, they should feel comfortable rejecting potential strategies at any time during their quest for solutions.

Monitor with great care the steps undertaken as part of a solution. Although it might be a natural tendency for students to “rush” through a strategy to arrive at a quick answer, encourage them to carefully assess and monitor their progress.

Feel comfortable putting a problem aside for a period of time and tackling it at a later time. For example, scientists rarely come up with a solution the first time they approach a problem. Students should also feel comfortable letting a problem rest for a while and returning to it later.

Evaluate the results. It's vitally important that students have multiple opportunities to assess their own problem-solving skills and the solutions they generate from using those skills. Frequently, students are overly dependent upon teachers to evaluate their performance in the classroom. The process of self-assessment is not easy, however. It involves risk-taking, self-assurance, and a certain level of independence. But it can be effectively promoted by asking students questions such as “How do you feel about your progress so far?” “Are you satisfied with the results you obtained?” and “Why do you believe this is an appropriate response to the problem?”

## TeacherVision Editorial Staff

The TeacherVision editorial team is comprised of teachers, experts, and content professionals dedicated to bringing you the most accurate and relevant information in the teaching space.

## ScholarWorks at University of Montana

Home > The Mathematics Enthusiast > Vol. 10 > No. 1 (2013)

## Article Title

Problem Solving in the Primary School (K-2)

Richard Lesh Lyn English Chanda Riggs Serife Sevis

This article focuses on problem solving activities in a first grade classroom in a typical small community and school in Indiana. But, the teacher and the activities in this class were not at all typical of what goes on in most comparable classrooms; and, the issues that will be addressed are relevant and important for students from kindergarten through college. Can children really solve problems that involve concepts (or skills) that they have not yet been taught? Can children really create important mathematical concepts on their own – without a lot of guidance from teachers? What is the relationship between problem solving abilities and the mastery of skills that are widely regarded as being “prerequisites” to such tasks? Can primary school children (whose toolkits of skills are limited) engage productively in authentic simulations of “real life” problem solving situations? Can three-person teams of primary school children really work together collaboratively, and remain intensely engaged, on problem solving activities that require more than an hour to complete? Are the kinds of learning and problem solving experiences that are recommended (for example) in the USA’s Common Core State Curriculum Standards really representative of the kind that even young children encounter beyond school in the 21st century? … This article offers an existence proof showing why our answers to these questions are: Yes. Yes. Yes. Yes. Yes. Yes. And: No. … Even though the evidence we present is only intended to demonstrate what’s possible, not what’s likely to occur under any circumstances, there is no reason to expect that the things that our children accomplished could not be accomplished by average ability children in other schools and classrooms.

## Recommended Citation

Lesh, Richard; English, Lyn; Riggs, Chanda; and Sevis, Serife (2013) "Problem Solving in the Primary School (K-2)," The Mathematics Enthusiast : Vol. 10 : No. 1 , Article 4. DOI: https://doi.org/10.54870/1551-3440.1259 Available at: https://scholarworks.umt.edu/tme/vol10/iss1/4

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## Learning to Teach Mathematics Through Problem Solving

• Open Access
• Published: 21 April 2022
• volume  57 ,  pages 407–423 ( 2022 )

• Judy Bailey   ORCID: orcid.org/0000-0001-9610-9083 1

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While there has been much research focused on beginning teachers; and mathematical problem solving in the classroom, little is known about beginning primary teachers’ learning to teach mathematics through problem solving. This longitudinal study examined what supported beginning teachers to start and sustain teaching mathematics through problem solving in their first 2 years of teaching. Findings show ‘sustaining’ required a combination of three factors: (i) participation in professional development centred on problem solving (ii) attending subject-specific complementary professional development initiatives alongside colleagues from their school; and (iii) an in-school colleague who also teaches mathematics through problem solving. If only one factor is present, in this study attending the professional development focussed on problem solving, the result was little movement towards a problem solving based pedagogy. Recommendations for supporting beginning teachers to embed problem solving are included.

## Working on a manuscript?

Introduction.

For many years curriculum documents worldwide have positioned mathematics as a problem solving endeavour (e.g., see Australian Curriculum, Assessment and Reporting Authority, 2018 ; Ministry of Education, 2007 ). There is evidence however that even with this prolonged emphasis, problem solving has not become a significant presence in many classrooms (Felmer et al., 2019 ). Research has reported on a multitude of potential barriers, even for experienced teachers (Clarke et al., 2007 ; Holton, 2009 ). At the same time it is widely recognised that beginning teachers encounter many challenges as they start their careers, and that these challenges are particularly compelling when seeking to implement ambitious methods of teaching, such as problem solving (Wood et al., 2012 ).

Problem solving has been central to mathematics knowledge construction from the beginning of human history (Felmer et al., 2019 ). Teaching and learning mathematics through problem solving supports learners’ development of deep and conceptual understandings (Inoue et al., 2019 ), and is regarded as an effective way of catering for diversity (Hunter et al., 2018 ). While the importance and challenge of mathematical problem solving in school classrooms is not questioned, the promotion and enabling of problem solving is a contentious endeavour (English & Gainsburg, 2016 ). One debate centres on whether to teach mathematics through problem solving or to teach problem solving in and of itself. Recent scholarship (and this research) leans towards teaching mathematics through problem solving as a means for students to learn mathematics and come to appreciate what it means to do mathematics (Schoenfeld, 2013 ).

Problem solving has been defined in a multitude of ways over the years. Of central importance to problem solving as it is explored in this research study is Schoenfeld’s ( 1985 ) proposition that, “if one has ready access to a solution schema for a mathematical task, that task is an exercise and not a problem” (p. 74). A more recent definition of what constitutes a mathematical problem from Mamona-Downs and Mamona ( 2013 ) also emphasises the centrality of the learner not knowing how to proceed, highlighting that problems cannot be solved by procedural effort alone. These are important distinctions because traditional school texts and programmes often position problems and problem solving as an ‘add-on’ providing a practice opportunity for a previously taught, specific procedure. Given the range of learners in any education setting an important point to also consider is that what constitutes a problem for some students may not be a problem for others (Schoenfeld, 2013 ).

A research focus exploring what supports beginning teachers’ learning about teaching mathematics through problem solving is particularly relevant at this time given calls for an increased curricular focus on mathematical practices such as problem solving (Grootenboer et al., 2021 ) and recent recommendations from an expert advisory panel on the English-medium Mathematics and Statistics curriculum in Aotearoa (Royal Society Te Apārangi, 2021 ). The ninth recommendation from this report advocates for the provision of sustained professional learning in mathematics and statistics for all teachers of Years 0–8. With regard to beginning primary teachers, the recommendation goes further suggesting that ‘mathematics and statistics professional learning’ (p. 36) be considered as compulsory in the first 2 years of teaching. This research explores what the nature of that professional learning might involve, with a focus on problem solving.

## Scoping the Context for Learning and Sustaining Problem Solving

The literature reviewed for this study draws from two key fields: the nature of support and professional development effective for beginning teachers; and specialised supports helping teachers to employ problem solving pedagogies.

## Beginning Teachers, Support and Professional Development

A teacher’s early years in the profession are regarded as critical in terms of constructing a professional practice (Feiman-Nemser, 2003 ) and avoiding high attrition (Karlberg & Bezzina, 2020 ). Research has established that beginning teachers need professional development opportunities geared specifically to their needs (Fantilli & McDougall, 2009 ) and their contexts (Gaikhorst, et al., 2017 ). Providing appropriate support is not an uncontentious matter with calls for institutions to come together and collaborate to provide adequate and ongoing support (Karlberg & Bezzina, 2020 ). The proposal is that support is needed from both within and beyond the beginning teacher’s school; and begins with effective pre-service teacher preparation (Keese et al., 2022 ).

Within schools where beginning teachers regard the support they receive positively, collaboration, encouragement and ‘involved colleagues’ are considered as vital; with the guidance of a 'buddy’ identified as some of the most valuable in-school support activities (Gaikhorst et al., 2014 ). Cameron et al.’s ( 2007 ) research in Aotearoa reports beginning teachers joining collaborative work cultures had greater opportunities to talk about teaching with their colleagues, share planning and resources, examine students’ work, and benefit from the collective expertise of team members.

Opportunities to participate in networks beyond the beginning teacher’s school have also been identified as being important for teacher induction (Akiri & Dori, 2021 ; Cameron et al., 2007 ). Fantilli & McDougall ( 2009 ) in their Canadian study found beginning teachers reported a need for many support and professional development opportunities including subject-specific (e.g., mathematics) workshops prior to and throughout the year. Akiri and Dori ( 2021 ) also refer to the need for specialised support from subject-specific mentors. This echoes the findings of Wood et al. ( 2012 ) who advocate that given the complexity of learning to teach mathematics, induction support specific to mathematics, and rich opportunities to learn are not only desirable but essential.

Akiri and Dori ( 2021 ) describe three levels of mentoring support for beginning teachers including individual mentoring, group mentoring and mentoring networks with all three facilitating substantive professional growth. Of relevance to this paper are individual and group mentoring. Individual mentoring involves pairing an experienced teacher with a beginning teacher, so that a beginning teacher’s learning is supported. Group mentoring involves a group of teachers working with one or more mentors, with participants receiving guidance from their mentor(s) (Akiri & Dori, 2021 ). Findings from Akiri and Dori suggest that of the varying forms of mentoring, individual mentoring contributes the most for beginning teachers’ professional learning.

## Teachers Learning to Teach Mathematics Through Problem Solving

Learning to teach mathematics through problem solving begins in pre-service teacher education. It has been shown that providing pre-service teachers with opportunities to engage in problem solving as learners can be productive (Bailey, 2015 ). Opportunities to practise content-specific instructional strategies such as problem solving during student teaching has also been positively associated with first-year teachers’ enactment of problem solving (Youngs et al., 2022 ).

The move from pre-service teacher education to the classroom can be fraught for beginning teachers (Feiman-Nemser, 2003 ), and all the more so for beginning teachers attempting to teach mathematics through problem solving (Wood et al., 2012 ). In a recent study (Darragh & Radovic, 2019 ) it has been shown that an individual willingness to change to a problem-based pedagogy may not be enough to sustain a change in practice in the long term, particularly if there is a contradiction with the context and ‘norms’ (e.g., school curriculum) within which a teacher is working. Cady et al. ( 2006 ) explored the beliefs and practices of two teachers from pre-service teacher education through to becoming experienced teachers. One teacher who initially adopted reform practices reverted to traditional beliefs about the learning and teaching of mathematics. In contrast, the other teacher implemented new practices only after understanding these and gaining teaching experience. Participation in mathematically focused professional development and involvement in resource development were thought to favourably influence the second teacher.

Lesson structures have been found to support teachers learning to teach mathematics through problem solving. Sullivan et al. ( 2016 ) explored the use of a structure comprising four phases: launching, exploring, summarising and consolidating. Teachers in Australia and Aotearoa have reported the structure as productive and feasible (Ingram et al., 2019 ; Sullivan et al., 2016 ). Teaching using challenging tasks (such as in problem solving) and a structure have been shown to accommodate student diversity, a pressing concern for many teachers. Student diversity has often been managed by grouping students according to perceived levels of capability (called ability grouping). Research identifies this practice as problematic, excluding and marginalising disadvantaged groups of students (e.g., see Anthony & Hunter, 2017 ). The lesson structure explored by Sullivan et al. ( 2016 ) caters for diversity by deliberately differentiating tasks, providing enabling and extending prompts. Extending prompts are offered to students who finish an original task quickly and ideally elicit abstraction and generalisation. Enabling prompts involve reducing the number of steps, simplifying the numbers, and/or varying forms of representation for students who cannot initially proceed, with the explicit intention that students then return to the original task.

Recognising the established challenges teachers encounter when learning about teaching mathematics through problem solving, and the paucity of recent research focussing on beginning teachers learning about teaching mathematics in this way, this paper draws on data from a 2 year longitudinal study. The study was guided by the research question:

What supports beginning teachers’ implementation of a problem solving pedagogy for the teaching and learning of mathematics?

## Research Methodology and Methods

Data were gathered from three beginning primary teachers who had completed a 1 year graduate diploma programme in primary teacher education the previous year. The beginning teachers had undertaken a course in mathematics education (taught by the author for half of the course) as part of the graduate diploma. An invitation to be involved in the research was sent to the graduate diploma cohort at the end of the programme. Three beginning teachers indicated their interest and remained involved for the 2 year research period. The teachers had all secured their first teaching positions, and were teaching at different year levels at three different schools. Julia (pseudonyms have been used for all names) was teaching year 0–2 (5–6 years) at a small rural school; Charlotte, year 5–6 (9–10 years) at a large urban city school; and Reine, year 7–8 (11–12 years), at another small rural school. All three beginning teachers taught at their respective schools, teaching the same year levels in both years of the study. Ethical approval was sought and given by the author’s university ethics committee. Informed consent was gained from the teachers, school principals and involved parents and children.

Participatory action research was selected as the approach in the study because of its emphasis on the participation and collaboration of all those involved (Townsend, 2013 ). Congruent with the principles of action research, activities and procedures were negotiated throughout both years in a responsive and emergent way. The author acted as a co-participant with the teachers, aiming to improve practice, to challenge and reorient thinking, and transform contexts for children’s learning (Locke et al., 2013 ). The author’s role included facilitating the research-based problem solving workshops (see below), contributing her experience as a mathematics educator and researcher. The beginning teachers were involved in making sense of their own practice related to their particular sites and context.

The first step in the research process was a focus group discussion before the beginning teachers commenced their first year of teaching. This discussion included reflecting on their learning about problem solving during the mathematics education course; and envisaging what would be helpful to support implementation. It was agreed that a series of workshops would be useful. Two were subsequently held in the first year of the study, each for three hours, at the end of terms one and two. Four workshops were held during the second year, one during each term. At the end of the first year the author suggested a small number of experienced teachers who teach mathematics through problem solving join the workshops for the second year. The presence of these teachers was envisaged to support the beginning teachers’ learning. The beginning teachers agreed, and an invitation was extended to four teachers from other schools whom the author knew (e.g., through professional subject associations). The focus of the research remained the same, namely exploring what supported beginning teachers to implement a problem solving pedagogy.

Each workshop began with sharing and oral reflections about recent problem solving experiences, including successes and challenges. Key workshop tasks included developing a shared understanding of what constitutes problem solving, participating in solving mathematical problems (modelled using a lesson structure (Sullivan et al., 2016 ), and learning techniques such as asking questions. A time for reflective writing was provided at the end of each workshop to record what had been learned and an opportunity to set goals.

During the first focus group discussion it was also decided the author would visit and observe the beginning teachers teaching a problem solving lesson (or two) in term three or four of each year. A semi-structured interview between the author and each beginning teacher took place following each observed lesson. The beginning teachers also had an opportunity to ask questions as they reflected on the lesson, and feedback was given as requested. A second focus group discussion was held at the end of the first year (an approximate midpoint in the research), and a final focus group discussion was held at the end of the second year.

All focus group discussions, problem solving workshops, observations and interviews were audio-recorded and transcribed. Field notes of workshops (recorded by the author), reflections from the beginning teachers (written at the end of each workshop), and lesson observation notes (recorded by the author) were also gathered. The final data collected included occasional emails between each beginning teacher and the author.

## Data Analysis

The analysis reported in this paper drew on all data sets, primarily using inductive thematic analysis (Braun & Clarke, 2006 ). The research question guided the key question for analysis, namely: What supports beginning teachers’ implementation of a problem solving pedagogy for the teaching and learning of mathematics? Alongside this question, consideration was also given to the challenges beginning teachers encountered as they implemented a problem solving pedagogy. Data familiarisation was developed through reading and re-reading the whole body of data. This process informed data analysis and the content for each subsequent workshop and focus group discussions. Colour-coding and naming of themes included comparing and contrasting data from each beginning teacher and throughout the 2-year period. As a theme was constructed (Braun & Clarke, 2006 ) subsequent data was checked to ascertain whether the theme remained valid and/or whether it changed during the 2 years. Three key themes emerged revealing what supported the beginning teachers’ developing problem solving pedagogy, and these constitute the focus for this paper.

Mindful of the time pressures beginning teachers experience in their early years, the author undertook responsibility for data analysis. The author’s understanding of the unfolding ‘story’ of each beginning teacher’s experiences and the emerging themes were shared with the beginning teachers, usually at the beginning of a workshop, focus group discussion or observation. Through this process the author’s understandings were checked and clarified. This iterative process of member checking (Lincoln & Guba, 1985 ) began at a mid-point during the first year, once a significant body of data had been gathered. At a later point in the analysis and writing, the beginning teachers also had an opportunity to read, check and/or amend quotes chosen to exemplify their thinking and experiences.

## Findings and Discussion

In this section the three beginning teachers’ experiences at the start of the 2 year research timeframe is briefly described, followed by the first theme centred on the use of a lesson structure including prompts for differentiation. The second and third themes are presented together, starting with a brief outline of each beginning teacher’s ‘story’ providing the context within which the themes emerged. Sharing the ‘story’ of each beginning teacher and including their ‘voice’ through quotes acknowledges them and their experiences as central to this research.

The beginning teachers’ pre-service teacher education set the scene for learning about teaching mathematics through problem solving. A detailed list brainstormed during the first focus group discussion suggested a developing understanding from their shared pre-service mathematics education course. In their first few weeks of teaching, all three beginning teachers implemented a few problems. It transpired however this inclusion of problem solving occurred only while children were being assessed and grouped. Following this, all three followed a traditional format of skill-based (with a focus on number) mathematics, taught using ability groups. The beginning teachers’ trajectories then varied with Julia and Reine both eventually adopting a pedagogy primarily based on problem solving, while Charlotte employed a traditional skill-based mathematics using a combination of whole class and small group teaching.

## A Lesson Structure that Caters for Diversity Supports Early Efforts

Data show that developing familiarity with a lesson structure including prompts for differentiation supported the beginning teachers’ early efforts with a problem solving pedagogy. This addressed a key issue that emerged during the first workshop. During the workshop while a ‘list’ of ideas for teaching a problem solving lesson was co-constructed, considerable concern was expressed about catering for a range of learners when introducing and working with a problem. For example, Charlotte queried, “ Well, what happens when you are trying to do something more complicated, and we’re (referring to children) sitting here going, ‘I’ve no idea what you're talking about” ? Reine suggested keeping some children with the teacher, thinking he would say, “ If you’re unsure of any part stay behind” . He was unsure however about how he would then maintain the integrity of the problem.

It was in light of this discussion that a lesson structure with differentiated prompts (Sullivan et al., 2016 ) was introduced, experienced and reflected on during the second workshop. While the co-constructed list developed during the first workshop had included many components of Sullivan’s lesson structure, (e.g., a consideration of ‘extensions’) there had been no mention of ‘enabling prompts’. Now, with the inclusion of both enabling and extending prompts, the beginning teachers’ discussion revealed them starting to more fully envisage the possibilities of using a problem solving approach, and being able to cater for all children. Reine commented that, “… you can give the entire class a problem, you've just got to have a plan, [and] your enabling and extension prompts” . Charlotte was also now considering and valuing the possibility of having a whole class work on the same problem. She said, “I think … it’s important and it’s useful for your whole class to be working on the same thing. And … have enablers and extenders to make sure that everyone feels successful” . Julia also referred to the planning prompts. She thought it would be key to “plan it well so that we’ve got enabling and extending prompts” .

## Successful Problem Solving Lessons

Following the second workshop all three beginning teachers were observed teaching a lesson using the structure. These lessons delighted the beginning teachers, with them noting prolonged engagement of children, the children’s learning and being able to cater for all learners. Reine commented on how excited and engaged the children were, saying they were, “ just so enthusiastic about it ”. In Charlotte’s words, “ it really worked ”, and Julia enthusiastically pondered this could be “ the only way you teach maths !”.

During the focus group discussion at the end of the first year, all three reflected on the value of the lesson structure. Reine called it a ‘framework’ commenting,

I like the framework. So from start to finish, how you go through that whole lesson. So how you set it up and then you go through the phases… I like the prompts that we went through…. knowing where you could go, if they’re like, ‘What do I do?’ And then if they get it too easy then ‘Where can you go?’ So you've got all these little avenues.

Charlotte also valued the lesson structure for the breadth of learning that could occur, explaining,

… it really helped, and really worked. So I found that useful for me and my class ‘cause they really understood. And I think also making sure that you know all the ins and outs of a problem. So where could they go? What do you need to know? What do they need to know?

While the beginning teachers’ pre-service teacher education and the subsequent research process, including the use of the lesson structure, supported the beginning teachers’ early efforts teaching mathematics through problem solving, two key factors further enabled two of the beginning teachers (Julia and Reine) to sustain a problem solving pedagogy. These were:

Being involved in complementary mathematics professional development alongside members of their respective school staff (a form of group mentoring); and

Having a colleague in the same school teaching mathematics through problem solving (a form of individual mentoring).

Charlotte did not have these opportunities and she indicated this limited her implementation. Data for these findings for each teacher are presented below.

## Complementary Professional Development and Problem Solving Colleague in Same School

Julia began to significantly implement problem solving from the second term in the first year. This coincided with her attending a 2-day workshop (with staff from her school) that focused on the use of problem solving to support children who are not achieving at expected levels (see ALiM: Accelerated Learning in Maths—Ministry of Education, 2022 ). She explained, “ … I did the PD with (colleague’s name), which was really helpful. And we did lots of talking about rich learning tasks and problem solving tasks…. And what it means ”. Following this, Julia reported using rich tasks and problem solving in her mathematics teaching in a regular (at least weekly) and ongoing way.

During the observation in term three of the first year Julia again referred to the impact of having a colleague also teaching mathematics through problem solving. When asked what she believed had supported her to become a teacher who teaches mathematics in this way she firstly identified her involvement in the research project, and then spoke about her colleague. She said, “ I’m really lucky one of our other teachers is doing the ALiM project… So we’re kind of bouncing off each other a little bit with resources and activities, and things like that. So that’s been really good ”.

At the beginning of the second year, Julia reiterated this point again. On this occasion she said having a colleague teaching mathematics through problem solving, “ made a huge difference for me last year ”, explaining the value included having someone to talk with on a daily basis. Mid-way through the second year Julia repeated her opinion about the value of frequent contact with a practising problem solving colleague. Whereas her initial comments spoke of the impact in terms of being “ a little bit ”, later references recount these as ‘ huge ’ and ‘ enabling ’. She described:

a huge effect… it enabled me. Cause I mean these workshops are really helpful. But when it’s only once a term, having [colleague] there just enabled me to kind of bounce ideas off. And if I did a lesson that didn’t work very well, we could talk about why that was, and actually talk about what the learning was instead…. . It was being able to reflect together, but also share ideas. It was amazing.

Julia’s comments raise two points. It is likely that participating in the ALiM professional development (which could be conceived as a form of group mentoring) consolidated the learning she first encountered during pre-service teacher education and later extended through her involvement in the research. Having a colleague (in essence, an individual mentor) within the same school teaching mathematics through problem solving appears to be another factor that supported Julia to implement problem solving in a more sustained way. Julia’s comments allude to a number of reasons for this, including: (i) the more frequent discussion opportunities with a colleague who understands what it means for children to learn mathematics through problem solving; (ii) being able to share and plan suitable activities and resources; and (iii) as a means for reflection, particularly when challenges were encountered.

Reine’s mathematics programme throughout the first year was based on ability groups and could be described as traditional. He occasionally used some mathematical problems as ‘extension activities’ for ‘higher level’ children, or as ‘fillers’. In the second year, Reine moved to working with mixed ability groups (where students work together in small groups with varying levels of perceived capability) and initially implemented problem solving approximately once a fortnight. In thinking back to these lessons he commented, “ We weren’t really unpacking one problem properly, it was just lots of busy stuff ”. A significant shift occurred in Reine’s practice to teaching mathematics primarily by problem solving towards the last half of the second year. He explained, “ I really ramped up towards terms three and four, where it’s more picking one problem across the whole maths class but being really, really conscious of that problem. Low entry, high ceiling, and doing more of it too ”.

Reine attributed this change to a number of factors. In response to a question about what he considered led to the change he explained,

… having this, talking about this stuff, trialling it and then with our PD at school with the research into ability grouping... We’ve got a lot of PD saying why it can be harmful to group on ability, and that’s been that last little kick I needed, I think. And with other teachers trialling this as well. Our senior teacher has flipped her whole maths program and just does problem solving.

Like Julia, Reine firstly referred to his involvement in the research project including having opportunities to try problems in his class and discuss his experiences within the research group. He then told of a colleague teaching at his school leading school-wide professional development focussed on the pitfalls of ability grouping in mathematics (e.g., see Clarke, 2021 ) and instead using problem solving tasks. He also referred to having another teacher also teaching mathematics through problem solving. It is interesting to consider that having positive experiences in pre-service teacher education, the positive and encouraging support of colleagues (Reine’s principal and co-teacher in both years), regular participation in ongoing professional development (the problem solving workshops), and having a highly successful one-off problem solving teaching experience (the first year observation) were not enough for Reine to meaningfully sustain problem solving in his first year of teaching.

As for Julia, pivotal factors leading to a sustaining of problem solving teaching practice in the second year included complementary mathematics professional development (a form of group mentoring) and at least one other teacher (acting as an individual mentor) in the same school teaching mathematics through problem solving. It could be argued that pre-service teacher education and the problem solving workshops ‘paved the way’ for Julia and Reine to make a change. However, for both, the complementary professional development and presence of a colleague also teaching through problem solving were pivotal. It is also interesting to note that three of the four experienced teachers in the larger research group taught at the same level as Reine (see Table 1 below) yet he did not relate this to the significant change in his practice observed towards the end of the second year.

Charlotte’s mathematics programme during the first year was also traditional, teaching skill-based mathematics using ability groups. At the beginning of the second year Charlotte moved to teaching her class as a whole group, using flexible grouping as needed (children are grouped together in response to learning needs with regard to a specific idea at a point in time, rather than perceived notions of ability). She reported that she occasionally taught a lesson using problem solving in the first year, and approximately once or twice a term in the second year. Charlotte did not have opportunities for professional development in mathematics nor did she have a colleague in the same school teaching mathematics through problem solving. Pondering this, Charlotte said,

It would have been helpful if I had someone else in my school doing the same thing. I just thought about when you were saying the other lady was doing it [referring to Julia’s colleague]. You know, someone that you can just kind of back-and-forth like. I find with Science, I usually plan with this other lady, and we share ideas and plan together. We come up with some really cool stuff whereas I don’t really have the same thing for this.

Based on her experiences with teaching science it is clear Charlotte recognised the value of working alongside a colleague. In this, her view aligns with what Julia and Reine experienced.

Table 1 provides a summary of the variables for each beginning teacher, and whether a sustained implementation of teaching mathematics through problem solving occurred.

The table shows two variables common to Julia and Reine, the beginning teachers who began and sustained problem solving. They both participated in complementary professional development with colleagues from their school, and the presence of a colleague, also at their school, teaching mathematics through problem solving. Given that Julia was able to implement problem solving in the absence of a ‘research workshop colleague’ teaching at the same year level, and Reine’s lack of comment about the potential impact of this, suggests that this was not a key factor enabling a sustained implementation of problem solving.

Attributing the changes in Julia and Reine’s teaching practice primarily to their involvement in complementary professional development attended by members of their school staff, and the presence of at least one other teacher teaching mathematics through problem solving in their school, is further supported by a consideration of the timing of the changes. The data shows that while Julia could be considered an ‘early adopter’, Reine changed his practice reasonably late in the 2 year period. Julia’s early adoption of teaching mathematics through problem solving coincided with her involvement, early in the 2 years, in the professional development and opportunity to work alongside a problem solving practising colleague. Reine encountered these similar conditions towards the end of the 2 years and it is notable that this was the point at which he changed his practice. That problem solving did not become embedded or frequent within Charlotte’s mathematics programme tends to support the argument.

Understanding what supports primary teachers to teach mathematics through problem solving at the beginning of their careers is important because all students, including those taught by beginning teachers, need opportunities to develop high-level thinking, reasoning, and problem solving skills. It is also important in light of recent calls for mathematics curricula to include more emphasis on mathematical practices (such as problem solving) (e.g., see Grootenboer et al., 2021 ); and the Royal Society Te Apārangi report ( 2021 ). Findings from this research suggest that learning about problem solving during pre-service teacher education is enough for beginning teachers to trial teaching mathematics in this way. Early efforts were supported by gaining experience with a lesson structure that specifically attends to diversity. The lesson structure prompted the beginning teachers to anticipate different children’s responses, and consider how they would respond to these. An increased confidence and sense of security to trial teaching mathematics through problem solving was enabled, based on their more in-depth preparation. Beginning teachers finding the lesson structure useful extends the findings of Sullivan et al. ( 2016 ) in Australia and Ingram et al. ( 2019 ) in Aotearoa to include less experienced teachers.

In order for teaching mathematics through problem solving to be sustained however, a combination of three factors, subsequent to pre-service teacher education, was needed: (i) active participation in problem solving workshops (in this context provided by the research-based problem solving workshops); (ii) attending complementary professional development initiatives alongside colleagues from their school (a form of group mentoring); and (iii) the presence of an in-school colleague who also teaches mathematics through problem solving (a form of individual mentoring). It seems possible these three factors acted synergistically resulting in Julia and Reine being able to sustain implementation. If only one factor is present, in this study attending the problem solving workshops, and despite a genuine interest in using a problem based pedagogy, the result was limited movement towards this way of teaching.

Akiri and Dori ( 2021 ) have reported that individual mentoring contributes the most to beginning teachers’ professional growth. In a manner consistent with these findings, an in-school colleague (who in essence was acting as an individual mentor) played a critical role in supporting Reine and Julia. However, while Akiri and Dori, amongst others (e.g., Cameron et al., 2007 ; Karlberg & Bezzina, 2020 ), have identified the value of supportive, approachable colleagues, for both Julia and Reine it was important that their colleague was supportive and approachable, and actively engaged in teaching mathematics through problem solving. Having supportive and approachable colleagues, as Reine experienced in his first year, on their own were not enough to support a sustained problem solving pedagogy.

## Implications for Productive Professional Learning and Development

This study sought to explore the conditions that supported problem solving for beginning teachers, each in their unique context and from their perspective. The research did not examine how the teaching of mathematics through problem solving affected children’s learning. However, multiple sets of data were collected and analysed over a 2-year period. While it is neither possible nor appropriate to make claims as to generalisability some suggestions for productive beginning teacher professional learning and development are offered.

Given the first years of teaching constitute a particular and critical phase of teacher learning (Karlberg & Bezzina, 2020 ) and the findings from this research, it is imperative that well-funded, subject-focussed support occurs throughout a beginning teacher’s first 2 years of teaching. This is consistent with the ninth recommendation in the Royal Society Te Apārangi report ( 2021 ) suggesting compulsory professional learning during the induction period (2 years in Aotearoa New Zealand). Participation in subject-specific professional development has been recognised to favourably influence new teachers’ efforts to adopt reform practices such as problem solving (Cady et al., 2006 ).

Findings from this study suggest professional development opportunities that complement each other support beginning teacher learning. In the first instance complementarity needs to be with what beginning teachers have learned during their pre-service teacher education. In this study, the research-based problem solving workshops served this role. Complementarity between varying forms of professional development also appears to be important. Furthermore, as indicated by Julia and Reine’s experiences, subsequent professional development need not be on exactly the same topic. Rather, it can be complementary in the sense that there is an underlying congruence in philosophy and/or focus on a particular issue. For example, it emerged in the problem solving workshops, that being able to cater for diversity was a central concern for the beginning teachers. Attending to this issue within the problem solving workshops via the introduction of a lesson structure that enabled differentiation, was congruent with the nature of the professional development in the two schools: ALiM in Julia’s school, and mixed ability grouping and teaching mathematics through problem solving in Reine’s school. All three of these settings were focussed on positively responding to diversity in learning needs.

The presence of a colleague within the same school teaching mathematics through problem solving also appears to be pivotal. This is consistent with Darragh and Radovic ( 2019 ) who have shown the significant impact a teacher’s school context has on their potential to sustain problem based pedagogies in mathematics. Given that problem solving is not prevalent in many primary classrooms, it would seem clear that colleagues who have yet to learn about teaching mathematics through problem solving, particularly those that have a role supporting beginning teachers, will also require access to professional development opportunities. It seems possible that beginning and experienced teachers learning together is a potential pathway forward. Finding such pathways will be critical if mathematical problem solving is to be consistently implemented in primary classrooms.

Finally, these implications together with calls for institutions to collaborate to provide adequate and ongoing support for new teachers (Karlberg & Bezzina, 2020 ) suggest there is a need for pre-service teacher educators, professional development providers and the Teaching Council of Aotearoa New Zealand to work together to support beginning teachers’ starting and sustaining teaching mathematics through problem solving pedagogies.

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Bailey, J. Learning to Teach Mathematics Through Problem Solving. NZ J Educ Stud 57 , 407–423 (2022). https://doi.org/10.1007/s40841-022-00249-0

Accepted : 04 April 2022

Published : 21 April 2022

Issue Date : December 2022

DOI : https://doi.org/10.1007/s40841-022-00249-0

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Primary resources with your students in mind

## Problem Solving in Mathematics for primary students

Teaching your primary students problem solving in mathematics can be maddening at times, but it’s a lifelong skill that is so important. In this case, I will be talking about how to implement problem solving in your classroom specifically in math. There are so many strategies on how to solve a math problem, and it’s important for students to think about how they want to tackle it, and the steps they used to get there. It’s also great for students to hear how others came up with their solution too.

A few years ago, I was actually filmed teaching these problem-solving steps as a model for the school district I was teaching in. Kindergarten problem solving takes a lot of practice, but once you and your class get the steps down, it’s like clockwork, and it’s really beneficial!

## Here 6 steps to follow when conducting math problem solving:

Each student gets their own sheet of paper with the word problem printed at the top. Have the students sit on the carpet, and read the problem several times. I point out the numbers and circle them. I don’t want to give away what method to do or if the question is asking for addition or subtraction. So I keep it pretty basic when I introduce the problem.

Here is a sample of a problem the students would do. This is the paper they would have in front of them with space to work under the problem.

## 2. Student Work:

Students work independently for about 3-5 minutes at their tables. At the tables, I provide buckets full of manipulatives. Inside are Unifix cubes, 10-frames, number lines, number bonds, and their pencils and crayons. Students are able to choose whatever method works for them. They are not to work with their partner at this time. They are to think for themselves on which strategy they would like to use. I also created an anchor chart that can be a good reminder for them.

I have this anchor chart posted in front of the room so the students can refer back to this when picking a strategy.

## 3. Partner share:

This part is tricky, but since we always practice this engagement strategy, it’s not that difficult for them to discuss with their partner how they solved the problem. I had to model it a few times pretending like I was a student. Basically, they are to show their partner how they solved it, and I give them the sentence starter, “I solved the problem by _____.” This should be roughly 1-2 minutes.

One student discusses the strategy they used. The other student listens. Switch.

## 4. Class discussion:

I have the students bring their paper and pencil, and I usually pick one manipulative to bring to the carpet. For example, I say bring 10 cubes with you. Then I call on about 6 students to give me their answer. That’s it. No explanation yet. I write down the correct answers and incorrect answers. Then I go back and call on the students so they can tell the class how they solved the problem. I usually illustrate or model what they are saying on my whiteboard or doc camera. For example: If the student says, “I drew a picture of 5 balloons.” I ask them, “why did you draw 5?” They can explain that it says 5 in the sentence. So then I go ahead and draw 5 balloons on the board. Then the student continues to explain, “then I crossed out 2 balloons.” I again ask why. I proceed to cross out 2 balloons. Then I ask what did you do after that? They say, “I got 3 balloons.” I ask the class if that makes sense. I love it when they go further and write it into a number sentence. 5-2=3. Then I move on to the next kid. I make sure to pick kids who solved the problem differently this way the class can see multiple ways to solve a problem. I make sure that the students who provided the incorrect answer explain their way. Then we realize that their way didn’t make sense and I cross out their answer (in a super positive, nice try, you were on the right track sort of way!).

## 5. Class Solution:

After we have gone through a few different strategies from step 4, we pick ONE way to solve it and have everyone try it together with their own manipulatives, paper, and pencil. Most of the time, I just pick it. I have everyone count out 5 cubes and then take 2 away. We then all write the number sentence on our paper.

## 6. Class Summary:

This is teacher-led and can be done on your whiteboard or on chart paper. I pretty much sequence our steps in order using First, Next, Last. For example: “First, we counted out 5 cubes because Jack had 5 balloons. Then, we took 2 away because he gave them to his friend. Last, we counted how many cubes we had left.” We know that Jack has 3 balloons left so we wrote the number sentence 5-2=3. Then after we come up those steps, we all read it together, and that’s problem-solving! This process should take about 30-40 minutes, but don’t be surprised if it takes you up to 45 minutes to even an hour the first few times!

Our class summary – keep it short and simple!

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## Preschool Problem-Solving

Discover five ways parents can help preschoolers develop problem-solving abilities..

Three-year-old Sarah tries to display the leaves she has collected on a sheet of paper, but they keep falling off. She remembers seeing her teacher use the glue in a plastic bottle to stick a picture onto the paper. Fascinated with exploring new materials, Sarah decides to try to solve her problem by using the glue. Sarah squeezes streams of glue into a pile on her paper and then pushes the leaves on top. Like most three-year-olds, she's solving a problem through trial and error, relying primarily on her senses rather than reasoning. So it may take several experiments before she understands that the leaves won't stick readily to the big pile of glue.

Focused but Frustrated Threes enjoy using their imagination to solve problems as they arise. Wanting a construction worker's hard hat for his dramatic play, Max enthusiastically decides to use an upsidedown plastic bowl. Delighted, he then repeatedly demonstrates how to use the pretend supervisor's walkie-talkie he creatively made from a juice box. At this age, children can sometimes become frustrated in their problem-solving attempts because they can see only one possible solution — which may not be workable. For example, when Tommy's jacket zipper is stuck, he keeps pulling it up, convinced that this is the only available approach.

If at First ... Adventuresome four-year-olds frequently charge ahead in their quest to solve problems. While they may need some help in focusing on the actual problem, they are more patient than three-year-olds and can try out different solutions.

For example, several four-year-olds struggle to get their wagon out of the mud on the playground. First they try pushing it. Then they attempt to pull it. When these methods fail to budge the wagon, they decide to take the heavy rocks out and then try again. Typical of this age, the children then boast about how strong and what good thinkers they are!

Team Efforts Using their larger vocabularies, four-year-olds are ready to negotiate with one another. Their developing language skills help them as they work together and engage in group decision-making. With practice, they learn to choose from among several different solutions. For instance, a few of the children decide to build a house. They gather a variety of materials — colored foil, corrugated cardboard, twigs, dandelions, tree bark — and then work together to decide which ones to use. They discuss their predictions about which of the materials might work and how best to use them.

What You Can Do Preschoolers learn best when they're given frequent opportunities to solve problems that are meaningful to them — those that arise in their day-to-day life.

• Provide opportunities for hands-on investgations. Offer children interesting items to explore, such as magnets, found objects, and broken (but safe) appliances. Rotate your materials to keep them fresh and thought-provoking.
• Foster creative- and critical-thinking skills by inviting children to use items in new and diverse ways. Strings of colored beads, for example, can become reins for a racehorse, hair for a doll, links for measuring, or tools to press into clay to make designs.
• Encourage children's suggestions and solutions. Promote brainstorming by asking openended questions: "What can you do with a...?" "How many ways can you...?" Listen carefully to children's ideas.
• Allow children to find their own solutions. Offer help when they become frustrated, but don't solve their problems for them.
• Use literature as a springboard. Share books that show how characters solve problems, such as King of the Playground by Phyllis Naylor and Caps for Sale by Esphyr Slobodkina.

## ORIGINAL RESEARCH article

Promoting Innovative Learning-oriented Feedback Practice: Engagement, Challenges, and Solutions

## Visual Analysis on Commognitive Conflict in Collaborative Problem Solving in Classrooms

• 1 Jing Hengyi School of Education, Hangzhou Normal University, China

The final, formatted version of the article will be published soon.

In recent years, international education research has adopted a new paradigm of communication-focused inquiry and taken a social turn. The study used multi-camera video recordings of 32 groups of peers made up of 64 Year 7 students for observation, recording, and coding. It also used open-ended contextualized mathematics problems created by an Australian partner. Thus, we investigate the types and expressions of commognitive conflict in collaborative problem solving, as well as visualize and diagnose the process of high- and low-quality peer commognitive conflict. Finally, it was discovered that there is a need to encourage students to focus and resolve commognitive conflicts and make timely feedback; visual studies of commognitive conflict can empower AI-assisted teaching as well as the intelligent diagnosis and visual analysis of collaborative problem solving provide innovative solutions for teaching feedback.

Keywords: commognitive conflict, collaborative problem solving, cognitive diagnosis, visualization, learning-oriented feedback

Received: 04 May 2023; Accepted: 25 Sep 2023.

Copyright: © 2023 Lu, Zhang and Li. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY) . The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

* Correspondence: PhD. Yangjie Li, Jing Hengyi School of Education, Hangzhou Normal University, Hangzhou, 311121, Jiangsu Province, China

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