In scaling up the use of heuristic trees to facilitate students’ mathematical problem solving, we developed a design course and design protocol for heuristic trees. However, designing heuristic trees is a challenging task. The study reported in this paper aims to collate an inventory of teachers’ difficulties in designing heuristic trees. We analyzed a sample of heuristic trees teachers designed after participating in the course. Through open coding we arrived at a list of mistakes in the designs, and showed how these relate to the design principles for heuristic trees. We see support for the conclusion that most of the design principles are not straightforward to implement. In particular, teachers need to address the general techniques and concepts in the problem, and provide support for students in a way that needs no further intervention by the teachers.

This study is an exploratory study of primary school teachers conducting in utilizing Graphic Organizer as a tool to teach problem solving in a professional development setting. The importance of problem solving among children has been highlighted in the Brunei educational reform, particularly in mathematics. One of the inhibiting factors in teaching problem solving is the low level of comprehension and transformation skills needed to solve mathematics word problems. Graphic Organizer is an instructional strategy used to help students to compartmentalize the necessary information to solve word problems. A group of mathematics teachers was introduced to embedding Graphic Organizer as a tool to address issues in problem solving in a professional development workshop. They implemented the strategy in their respective classes and found ways to apply and assess students’ problem-solving strategies. Their reflections on the challenges and affordances Graphic Organizer posed in teaching problem solving are also discussed.

My Ph.D.’s research (Favier, 2022) aims at characterizing the processes used by students when they solve problems in the ordinary context of the classroom, i.e., when the problem-solving session is led by the teacher. The chosen problems may require students to make trials and errors. I consider two levels of characterization: the outer structure (Lehmann et al., 2015) of problem solving processes in terms of timing and organizing of processes and the inner structure (Ibid) considering heuristics. In this paper, I focus on the outer structure of the processes. Embedded cameras installed on the students’ heads were used to collect audio-visual data as close as possible to the students’ work. The recorded work of 33 groups of two or three students (20 groups at the primary school and 13 at the secondary school level) for a total of 79 students are coded independently by a research assistant and by us using the framework for the analysis of videotaped problem-solving sessions by Schoenfeld (1985). It consists of cutting the students’ work into macroscopic chunks called episodes. The analysis of these empirical data leads us to discuss and enrich the descriptive model of problem-solving processes proposed by Rott et al (2021) with an additional dimension that allows us to take into account the interactions between students and teacher. The use of this enriched model allows us to identify three problem solvers’ profiles in terms of the process implemented. Some of these processes are linear like Pólya (1957) proposed in his model. Some others are cyclics like in Schoenfeld’s model (1985). The large majority are not supported by these two very well-known models which shows the contingency of the student’s work and, therefore, the limit of these two models (in particular Pólya’s model) used to teach problem solving.

In this work, we establish a heuristic strategy, the purpose of which is enhancing the posing of new problems in the school context. The strategy is supported by a cognitive framework consisting of six stages: Selecting, Classifying, Associating, Searching, Verbalizing, and Transforming. The first five actions make up an essentially creative process, while the last stage is present within the nucleus of the previous ones. This provides the process with a high level of complexity. Compactly, we call the strategy SCASV+T. We reflect on the heuristic nature of the strategy, as well as the didactic actions that are required for its implementation. We also describe a didactic situation in elementary geometry, where the posing of new problems based on one already solved is discussed. The analysis is carried out with students who are studying a Bachelor’s degree in Mathematics Education, who know the strategy and try to put it into practice collectively. Analysis and discussion are led by a professor, who provides suggestions and demonstrates the importance of each action in the development of heuristic reflection.

In this paper, the problems and problem-posing tasks in six series of secondary school mathematics textbooks were analysed, the distribution of the number, location and types of problems from the perspective of historical comparison, as well as the types of problem-posing tasks and the distribution of problem-posing tasks across content areas were studied. It was found that although the number of problem-posing tasks has increased, the percentage is still quite small with a maximum of 0.4%, and the distribution of problem-posing tasks across content areas is uneven. It was found that a large number of problems had been included in the content text section since 1990s. The distribution of these problems across grade levels and content areas are well balanced, indicating that problem-guided learning has become a new feature of the textbooks. From the perspective of types, these problems provide rich mathematical learning opportunities for students to acquire knowledge (“knowing” and “understanding”) and to go through the thinking process (“to abstract and generalize” “to explore and discover” “to reflect and summarize”). However, the distribution of each type of problems across different grades and content areas are both uneven.

This study reports a small-scale international comparative study investigating rural elementary students’ mathematical thinking on division, through analyzing the similarities and differences between division problems posed by elementary students in Inner Mongolia in China and Montana in the United States (U.S.). Bruner’s (1985) paradigmatic and narrative modes of thought served as an analytic framework in this study. The primary data source for this study was students’ responses to the open-ended prompt, “Write two different types of division problems.” Each student’s responses were coded according to the perspectives of paradigmatic and narrative modes of thought. The structures and contexts of posed problems and students’ characterization of different division problems were examined. Our findings show that most students in both countries posed problems involving partitive (i.e., group size unknown) and equal groups division. No students in either country posed array/area problems. Of the ten common structures for division problems, students in China created problems aligned with six structures while the students from the United States used only two structures. An examination of the contexts used in each problem revealed that different types of food were the most common context used by students in both countries, although with unique cultural contexts. None of the students in either group situated their story problems in a rural context.

This is an exploratory study of 18 grade 9 students working on two problem-posing tasks involving the quadratic function. There are a variety of problem-posing strategies used by students, including the use of associated sub-topics, using the quadratic formula as a guide, working backwards, and adopting a trial-and-error approach. The free-posing task seems to help students to bring out more variety of sub-topics that they can connect, perhaps reflecting some confidence for such type of task. This is less so in the semi-structured task. It also appears that the number of sub-topics used is not dependent on student achievement type. Some implications for teaching and for teachers are also discussed. Specifically in the context of differentiated instruction in a classroom, problem posing activities can be one strategy to engage students. The findings of this exploratory study have the potential to add to the body of local knowledge about how problemposing instructions can be engendered in the classroom to bring about deeper classroom engagement in mathematics.