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Abstract Using data from a problem-posing project, this study analyzed the characteristics of middle school students’ responses to problem-posing prompts that did not match our assumptions and expectations to better understand student thinking. The study found that the characteristics of middle school students’ unexpected responses were distributed across three different problem-posing processes: 1) orientation responses related to different interpretations of the problem-posing prompt or situation accounted for the majority; 2) connection responses related to making connections among pieces of information accounted for the second most common type; and 3) generation responses related to generation of problems only accounted for a very small proportion. Additionally, it was found that the problem-posing prompts influenced the distribution of types of unexpected responses. These findings contribute to our understanding of problem-posing processes and have implications for the design of problem-posing tasks. Most importantly, this analysis reveals that even though these responses are unexpected, students’ responses make sense to them and our objective should be to make sense of their responses. 

Problem posing engages students in generating new problems based on given situations (including mathematical expressions or diagrams) or changing (i.e., reformulating) existing problems. Problem posing has been at the forefront of discussion over the past few decades. One of the important topics studied is the process of problem posing as experienced by students and teachers. This paper focuses on problem-posing processes and models thereof. We first provide an overview of previous research and then present the results of a scoping review regarding recent research on problem-posing processes. This review covers 75 papers published between 2017 and 2022 in top mathematics education research journals. We found that some of the prior research directly attempted to examine problem-posing processes, whereas others examined task variables related to problem-posing processes. We conclude this paper by proposing a model for problem-posing processes that encompasses four phases: orientation, connection, generation, and reflection. We also provide descriptions of the four phases of the model. The paper ends with suggestions for future research related to problem-posing processes in general and the problem-posing model proposed in particular. 

Ha habido un mayor énfasis en la integración del planteamiento de problemas en el currículo y la instrucción, con la promesa de proporcionar potencialmente más oportunidades y de mayor calidad para que los estudian- tes aprendan matemáticas a medida que participan en actividades en las que plantean problemas. Este artículo tiene como objetivo proporcionar una sínte- sis de lo que dice la investigación sobre la enseñanza de las matemáticas a través de la formulación de problemas. En particular, aborda las siguientes preguntas: (1) ¿Cómo es la enseñanza de las matemáticas a través de la for- mulación de problemas? (2) ¿Qué es el planteamiento de problemas? (3) ¿Qué es una tarea sobre planteamiento de problemas? (4) ¿Cómo deben los maestros manejar los problemas planteados por los estudiantes en la instrucción en el aula? (5) ¿Cómo se puede apoyar a los maestros para que aprendan a enseñar a través del planteamiento de problemas? (6) ¿Cuál es el efecto de la instrucción del Aprendizaje Basado en el Planteamiento de Problemas (ABPP) en maestros y estudiantes? A lo largo de las secciones, se plantean varias preguntas rela- cionadas sin respuesta y, el documento termina con un modelo de instrucción ABPP propuesto. Con el fin de que, las ideas presentadas en este documento puedan servir como un trampolín para alentar a más académicos a participar en la investigación de problemas, para que podamos brindar más oportunida- des y que los estudiantes aprendan matemáticas a través de la formulación de problemas 

It is important to understand teachers’ views about problem-posing (PP) tasks and the prompts that are used in such tasks to engage students in posing problems. In this study, we explored 15 middle school mathematics teachers’ views about PP prompts. We found that the teachers’ views were motivated by their curricular reasoning around engaging and challenging their students and addressed five main prompt characteristics: openness, promoting critical thinking, providing scaffolding, more or less intimidating, and allowing for differentiation. The teachers’ reasoning suggested they attended to how PP can create opportunities for sensemaking, deepen students’ learning of mathematics, and foster students’ identities as creative doers of mathematics. How- ever, they did not address connecting students’ life experiences to mathematics, another key goal of teaching mathematics through PP. The findings have implications for curriculum developers and researchers regarding the design of PP tasks and the implementation of such tasks in the classroom, and they suggest several directions for future research. 

This study concerns the cognitive process of mathematical problem posing, conceptualized in three stages: understanding the task, constructing the problem, and expressing the problem. We used the eye tracker and think-aloud methods to deeply explore students’ behavior in these three stages of problem posing, especially focusing on investigating the influence of task situation format and mathematical maturity on students’ thinking. The study was conducted using a 2×2 mixed design: task situation format (with or without specific numerical information)×subject category (master’s students or sixth graders). Regarding the task situation format, students’ performance on tasks with numbers was found to be significantly better than that on tasks without numbers, which was reflected in the metrics of how well they understood the task and the complexity and clarity of the posed problems. In particular, students spent more fixation duration on understanding and process- ing the information in tasks without numbers; they had a longer fixation duration on parts involving presenting uncertain numerical information; in addition, the task situation format with or without numbers had an effect on students’ selection and processing of information related to the numbers, elements, and relationships rather than information regarding the context presented in the task. Regarding the subject category, we found that mathematical maturity did not predict the quantity of problems posed on either type of task. There was no significant main group difference found in the eye-movement metrics. 

This study used three pairs of problem-posing tasks to examine the impact of different prompts on students’ problem posing. Two kinds of prompts were involved. The first asked students to pose 2–3 different mathematical problems without specifying other requirements for the problems, whereas the second kind of prompt did specify additional requirements. A total of 2124 students’ responses were analyzed to examine the impact of the prompts along multiple dimensions. In response to problem-posing prompts with more specific requirements, students tended to engage in more in-depth mathematical thinking and posed much more linguistically and semantically complex problems with more relationships or steps required to solve them. The findings from this study not only contribute to our understanding of problem-posing processes but also have direct implications for teaching mathematics through problem posing.