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The study, which is derived from a larger study, compares grades 10 – 12 mathematics learners’ non-routine problem solving. An exploratory study was conducted on a convenience sample drawn from three high performing high schools located in Tshwane North District, Gauteng province of South Africa.Learners wrote a non-routine problem solving test. Findings revealed that the 11th grade learners obtained the highest mean score while that of the 10th grade learners was the lowest. High school learners’ level of strategy use on solving non-routine problems improved significantly as they progress from grade 10 to higher grades. No significant difference was discovered as learners progress from grade 11 to 12.

Creative thinking transforms existing information, either from long-term memory or external sources, into new representations and innovative ideas. Creative thinking is an activity that processes received information to produce new representations and innovative ideas. Developing this skill is essential for students; however, recent research has yet to classify their creative thinking in solving non-routine problems. This study created a new classification based on three aspects of creative thinking in solving mathematical problems. The research used a qualitative descriptive approach, employing 72 students from Indonesia as subjects in a test involving the use of the think-aloud approach in problem-solving. Data collection consisted of tests, observations, and interviews. There were three stages of data analysis: data reduction, presentation and conclusions, and verification. This study's results grouped students' creative thinking processes in solving non-routine problems into three classifications: direct, flexible, and elaborative. The three classifications differ in the strategies or steps used to solve problems. The similarities of the three classifications include using similar aspects of creative thinking, namely flexibility, fluency, and elaboration in solving non-routine problems, even though some of these aspects have yet to be realised optimally due to students' creative thinking in solving mathematical problems, especially non-routine problems, still needing improvement. Study offers a comprehensive study of creative thinking for mathematical problem-solving. Creative thinking is part of mathematical thinking, which is a crucial component of 21st century mathematics skills. Future educational challenges are very complex. For this reason, this study is here to present a classification of mathematical thinking for mathematical problem-solving. This research has never been conducted before, so the study is truly suitable to fill the gap and provide high novelty value. Hopefully, readers will be academically entertained by reading this study and gain new insights.

Understanding a problem is as important as solving it to understand the meaning of mathematics. It is a widespread view that problem-solving skills help individuals to overcome the problems easily in their daily lives. An individual with problem-solving skills is a self-confident, creative and independent thinker. It can be explicable that the societies formed by these individuals can easily solve the problems. In this vein, this study is aimed to determine the relationship between the acquisition of problem-solving skills that play an important role in mathematics alongside two dimensions of classroom management which are time management and plan-program management. It is thought that there is a close relationship between these two dimensions; problem-solving skills. The types of problems are divided into two; routine and non-routine. The aim is to get opinions about how these problems will be taught by mathematics teachers, at what level they will be used in class and what methods they will use for them. The study was conducted by doing a content analysis of qualitative research methods. The sample group which is randomly determined from five provinces of North Cyprus is composed of mathematics teachers who teach at the 9th grade level. The data was collected through semi-structured interview forms. Content analysis technique was used in the evaluation of the obtained data. As a result, it was found out that the mathematics teachers who work in the Ministry of National Education and teach at the 9th grade, have been given importance to problem-solving skills such as, problem-solving duration and problem-solving methods in their classes. In addition, the teachers pointed out that the time given is not sufficient at schools and they could not give enough importance to plan-program activities. They also pointed out that although the annual and monthly plans specified by the Ministry, the desired variety of questions could not be provided by different methods.

In the Netherlands, mathematics textbooks are a decisive influence on the enacted curriculum. About a decade ago, Dutch primary school mathematics textbooks provided hardly any opportunities to learn problem solving. In this study we investigated whether this provision has changed. In order to do so, we carried out a textbook analysis in which we established to what degree current textbooks provide non-routine problem-solving tasks for which students do not immediately have a particular solution strategy at their disposal. We also analyzed to what degree textbooks provide ‘gray-area’ tasks, which are not really non-routine problems, but are also not straightforwardly solvable. In addition, we inventoried other ways in which present textbooks facilitate the opportunity to learn problem solving. Finally, we researched how inclusive these textbooks are with respect to offering opportunities to learn problem solving for students with varying mathematical abilities. The results of our study show that the opportunities that the currently most widely used Dutch textbooks offer to learn problem solving are very limited, and these opportunities are mainly offered in materials meant for more able students. In this regard, Dutch mainstream textbooks have not changed compared to the situation a decade ago. A textbook that is the Dutch edition of a Singapore mathematics textbook stands out in offering the highest number of problem-solving tasks, and in offering these in the materials meant for all students. However, in the ways this textbook facilitates the opportunity to learn problem solving, sometimes a tension occurs concerning the creative character of genuine problem solving.

Mathematical problem solving remains a struggle for many students today (Celebioglu et al., 2010; Englard, 2010; Gavaz et al., 2021; N. H. Lee et al., 2014; Yeo, 2011). Though intervention studies have reported measured impact on improving students’ problem-solving abilities (Gavaz et al., 2021; Gidalevich & Kramarski,  2019; N. H. Lee et al., 2014), few studies examine how learners would approach a non-routine problem without learning problem-solving strategies in school (Celebioglu et al., 2010; Saygılı, 2017; Yazgan, 2015; Yeo, 2011). The latter would complement instruction related to strategies by making known the entry point of young problem solvers. In this study, 10 Grade Two students in a neighborhood primary school in Singapore were individually engaged in solving a non-routine problem with access to manipulatives. Based on their mathematics instructional history in school, the students have not been introduced to the relevant problem-solving strategies related to the task. The study examines how the students approach and solve the problem. Findings revealed that all the ten students were able to solve the problem with eight of them using manipulatives to help them. The study recommends that prior to classroom teaching, teachers can invite students to share their problem-solving methods and demonstrate how formal notations can capture these mathematically. Appropriate manipulatives should also be made accessible during problem-solving lessons.

This study aims to determine the profile of the argumentation ability of undergraduate chemical education students. The research was conducted by giving 3 questions based on non-routine problems. In these questions, questions are given that demand statements of claims, data, warrants, backings, qualifiers, and rebuttals. The answers of 47 students were analyzed using descriptive qualitative analysis techniques. The results of the analysis show that the students' argumentation skills are included in the type of argumentation without rebuttal. Based on the rubric of the argumentation level without rebuttal, it shows that most of the students' abilities are at level 2, namely being able to make claims, but the data, warrants, and backing used are quite good in supporting the claims. The results of the distributed questionnaires showed that most of the students had not mastered argumentation. Experiencing the rigors of thinking in working on argumentative questions makes students motivated to study further. These findings will be the basis for the need to train students' argumentation skills with various methods so as to improve students' argumentation skills.

This study aims to determine the effect of an instruction dealing with non-routine problem solving on fifth graders' strategy flexibility and success in problem-solving. For this aim, a quasi-experimental pre-test- post-test design without a control group was designed. The sampling method of the research is convenience sampling. There were 65 fifth graders (11–12 years of age) who came from two different classes of a public middle school located in Istanbul/Turkey. The instruction carried out by the first researcher in the students' classrooms lasted ten weeks (20 lesson hours). Pre-test and post-test consisted of eight non-routine problems which can be solved by using guess and check, make a systematic list, work backward, look for a pattern, simplify the problem, and make a drawing strategies. The results showed that instruction that focuses on non-routine problem solving could improve students' strategy flexibility in this area. Besides, non-routine problem-solving instruction was associated with a significant positive improvement in students' problem-solving achievement. Based on these results, some educational implications and suggestions for future studies were discussed.

How might teachers transform routine tasks into non-routine ones? The purpose of this article is to address this question. The article opens with a discussion of why non-routine problems require creative and original thought. Specifically, I discuss how non-routine problems require students to confront uncertainty and how uncertainty can serve as a catalyst for creative thought and action. Next, I discuss how the logic of routine tasks can impede original and creative thought. I then introduce the concept of lesson unplanning and explain how it can be used to convert routine tasks into non-routine problems. I also discuss how non-routine problems can range from more modest in-classroom assignments to more ambitious efforts. The paper closes with a brief discussion of directions for future research and practice.

Metacognitive skills play an important role in solving mathematical problems. However, there is a lack of empirical studies on the role of metacognitive skills in solving mathematical problems, particularly non-routine ones. Therefore, this study was undertaken to identify students' metacognitive skills and the impact of such skills on non-routine mathematical problem solving. By using a quantitative method, a total of 304 students in Johor Bahru district were involved in the study. A Self-Monitoring Questionnaire (SMQ) and a mathematical test were used in data collection. Data were analysed using descriptive and inferential statistics such as frequency, percentage, mean, the Mann-Whitney U test, and the Kruskal-Wallis H test. Results showed that the level of the students' performance in solving non-routine mathematical problems was very low. There was also a significant difference in the metacognitive skills among students with different performance levels in solving non-routine mathematical problems, and we concluded that these metacognitive skills should be emphasised in this process.

The analysis of teacher–student interaction when jointly solving routine problems in the primary education mathematics classroom has revealed that there is scarce reasoning and little participation on students’ part. To analyze whether this fact is due to the routine nature of the problems, a sample of teachers who solved, together with their students, a routine problem involving three questions with different cognitive difficulty levels (task 1) was analyzed, describing on which part of the problem-solving process (selection of information or reasoning) they focused their interaction. Results showed that they barely focused the interaction on reasoning, and participation of students was scarce, regardless of the cognitive difficulty of the question to be answered. To check whether these results could be due to the routine nature of the problem, a nonroutine problem (task 2) was solved by the same sample of teachers and students. The results revealed an increase in both reasoning and participation of students in processes that required complex reasoning. This being so, the main conclusion of the present study is that including nonroutine problem solving in the primary education classroom as a challenging task is a reasonable way to increase students’ ability to use their own reasoning to solve problems, and to promote greater teacher–student collaboration. These two aspects are relevant for students to become creative, critical, and reflective citizens.