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Learning to teach through mathematical problem posing: Theoretical considerations, methodology, and directions for future research
Abstract
Teachers are at the heart of implementing any educational innovation or improvement. One critical need is to investigate how teachers learn to use problem posing to teach mathematics in the classroom. This article conceptualizes issues about mathematical problem posing (MPP) and about learning to teach through problem posing. It highlights significant recent findings on using problem posing for teachers’ learning. It discusses methodological issues about problem-posing research, as well as the future directions of research on learning to teach through problem posing. The international perspective provides a broad view of ways to integrate MPP in classrooms in general and the use of problem posing for teacher professional development in particular.
... Utifrån en genomgång av forskning om problemlösning framhåller de forskning med fokus på elevers lärande av matematikinnehåll i samband med problemlösning som extra viktigt. Speciellt saknas studier i form av interventioner där problemlösning och problemformulering designas och implementeras i aktuella verksamheter (Cai & Hwang, 2020;English & Sriraman, 2010;Palmér & van Bommel, 2020;Singer m.fl., 2013). Studier likt den som presenteras här ämnar bidra till att fylla detta tomrum. ...
... Eftersom problemformulering möjliggör en annan inblick i elevernas matematiska förståelse än vad problemlösning gör (Cai & Hwang, 2020) erbjuder undervisningskontexten i den här presenterade studien en rik kontext att analysera. ...
... Sammantaget ger studiens teoretiska val (Marton, 2014) inte bara inblick i hur elever urskiljer, differentierar och slutligen sammanför nödvändiga aspekter av ett lärandeobjekt division. Studien tillför även viktig kunskap om problemlösning och problemformulering generellt och specifikt mot skolans tidigare år och hur studier innehållande problemlösning och problemformulering kan designas och implementeras i aktuella verksamheter (Cai & Hwang, 2020;English & Sriraman, 2010), något som till stor del saknas i såväl nationell som internationell forskning (Palmér & van Bommel, 2020;Singer m.fl., 2013). ...
I denna studie arbetade förskoleklasselever med division genom problemlösning och problemformulering. Data kommer från en undervisningsaktivitet uppdelad på två tillfällen. Aktiviteten planerades i samarbete mellan förskoleklasslärare och forskare och genomfördes i 11 förskoleklasser med 205 elever. Vid problemlösning urskilde eleverna relationen mellan helhet och delar, storleken på varje del, dela som division samt kontinuerlig och diskret mängd som aspekter av division. Vid problemformulering återkom dessa aspekter samt tillkom aspekten att täljaren kan vara ett rationellt tal. Utöver dessa aspekter av division formulerade eleverna till exempel uppgifter med en liknande kontext (kakor) men med ett annat matematikinnehåll (till exempel subtraktion). Då det finns få studier om problemlösning och problemformulering med yngre elever bidrar denna studie med kunskap av värde för både (förskoleklass)lärare och forskare. Division in preschool class through problem solving and problem posing In this study, preschool-class students worked with problem solving and problem posing on division. Data comes from an activity divided into two sessions. The activity was planned in collaboration between preschool-class teachers and researchers and carried out in 11 preschool classes with 205 students. While solving problems, students distinguished the relationship between the parts and whole, the size of each part, dividing as division and continuous and discrete quantities as aspects of division. While posing problems, these aspects reappeared as well as the aspect that the numerator can be a rational number. Apart from problems on division, the students posed problems with a similar context (cookies) but a different mathematical content (e.g., subtraction). As there are few studies on problem solving and problem posing with younger students, this study contributes with knowledge of value to both (preschool class) teachers and researchers.
... Within the context of the development of algebraic thinking, more studies are focusing on the invention of algebraic problems (e.g. Cai & Hwang, 2020;Cañadas et al., 2018;Fernández-Millán & Molina, 2017) than on the invention of patterns (e.g. Rivera & Rossi-Becker, 2016). ...
... We highlighted the importance of inventing patterns as a resource to observe how students understand mathematics. From the teachers' perspective, the invention is a way to evaluate students' conception of a particular topic (Cai & Hwang, 2020;Fernández-Millán & Molina, 2017), and it allows students' skills to apply mathematical knowledge (Cañadas et al., 2018). Characterising the type of patterns invented could provide information for teaching patterns by attending to both the representations that have been closest to them and the structures with which they identify pattern creation. ...
... As mentioned above, in research on algebraic development, there are more studies on the invention of algebraic problems related to an equation (e.g. Cai & Hwang, 2020;Cañadas et al., 2018;Fernández-Millán & Molina, 2017). One of the few studies focusing on the invention of patterns was conducted by Rivera and Rossi-Becker (2016). ...
... In recent years, many researchers have turned their attention to problem posing, exploring various aspects of this activity, such as the nature of this task or its role and implementation in math classrooms. This interest is reflected in publications of special issues journals (Cai & Hwang, 2020;Cai & Leikin, 2020;Singer et al., 2013) and books (e.g., Felmer et al., 2016;Singer et al., 2015). Those publications conducted so far on this topic point out its importance and the need to conduct further investigation and future lines of research. ...
... The literature contains different ways of interpreting the expression Problem Posing (in lowercase from now on), depending on the perspective from which it is implemented in the classroom or analyzed in the research. Cai and Hwang (2020), building on the definition forwarded by , they proposed the following: ...
... In that same sense, Tichá and Hošpesová (2009) indicate that problem posing contributes to the development of mathematics knowledge during the pre-service education of primary school teachers. Although little is known about how teachers integrate problem posing into mathematics teaching (Cai & Hwang, 2020), research has shown that only if pre-service or in-service teachers gain experience by developing problem-posing activities, they will be able to incorporate it into their practice and promote it among their students (Singer et al., 2013). It is therefore necessary to develop specific educational programs that give teachers the knowledge to effectively use problem posing in their classrooms (Cai & Hwang, 2020). ...
This research presents a study on the problems posed by pre-service primary school teachers by focusing on the problem-posing tasks situation as the research variable. The investigation was carried out with 205 students of a bachelor's degree in Primary Education Teacher in Spain. They were asked to pose problems with fractions based on two given initial situations: numerical and contextualized. For each problem, we analyze its plausibility, the meanings of fractions, the mathematical structure, and the reasonability of the context. Results indicate that mostly posed problems use part-whole or operator meaning of fractions, as well as the additive or multipli-cative structure. There are no differences between the plausibility and reasonability of the problems based on the initial situation, although it has shown better results when the given situation is contextualized. In addition, in contextualized situations, teachers show greater ability in formulating problems with a wide variety of structures and meanings of fractions.
... Despite being foregrounded as a fundamental tool for mathematics education in classic works (Freudenthal, 1973), problem posing has only relatively recently attracted the attention of researchers (Cai & Hwang, 2020;Silver, 1994). This attention has been seen across educational stages, including primary (English, 1997), secondary (Koichu, 2020), and graduate (Rosli et al., 2015) levels, but to date little has been done in the sphere of early childhood education (Palmér & van Bommel, 2020). ...
... There are numerous definitions of problem posing (Baumanns & Rott, 2022). We consider the posing of mathematical problems by pupils themselves either based on a given problem situation or generated by modifying existing problems (Cai & Hwang, 2020;Silver, 1994). Our focus henceforth will be on problems based on a given problem situation. ...
... According to Cai et al. (2022), the problem situation can be purely mathematical or real. The data provided can be qualitative or quantitative and can include mathematical expressions and diagrams (Cai & Hwang, 2020). In addition, a problem-posing task can be structured or unstructured (Baumanns & Rott, 2020, 2022 according to the level of openness made available to the problem poser. ...
This research focused on understanding the variables inherent in the design and implementation of a mathematical problem-posing task. We developed a single case study of a problem-posing lesson by an Early Childhood Education teacher in a classroom with 4-to 5-year-old children who were unfamiliar with such activities. The results of this study show the potential of considering five variables serving as critical points that pose dilemmas linked to the design and implementation of problem-posing tasks. We found that the task changed from its original design during implementation, implying that the choices the teacher made about the variables were not static and were strongly linked to the purpose of the problem-posing task as well as to the contextual characteristics of the early childhood classroom. This study provides a potentially useful framework for analyzing the design and implementation of problem-posing tasks as a dynamic process.
... Although recent, research on problem posing has different objectives and, consequently, can be characterized methodologically in three ways: as a construct, variable and intervention (Cai & Hwang, 2020). ...
... In this case, it is assumed that the construct is well defined, being used as a variable in the analysis of a larger phenomenon, or of its relations with other aspects. Problem posing can be used, for example, to analyze relationships with creativity or problem solving (Cai & Hwang, 2020). ...
... As an intervention, the research aims to analyze how problem posing is used to promote better learning outcomes and improve creativity, among others objectives, making it possible to help teachers use problem posing in their classes (Cai & Hwang, 2020). ...
Problem posing has been considered a cognitively demanding activity, with potential for the development and evaluation of critical and creative thinking. In the classroom it is considered one of the authentic forms of mathematical investigation in which students have the possibility to associate their experiences, interests and knowledge in the creation of problems. Contributing to this research strand, this paper aims to discuss the potential of problem posing for the development of creative thinking in mathematics classes. In particular, this qualitative study aims to analyze a problem posing activity carried out with a class of first grade elementary school children. The results indicate that problem posing has significant potential for the development of creativity, allowing children to assign meaning and critically analyze data, relating their experiences, knowledge, and interests.
... Apesar de recente, a pesquisa sobre proposição de problemas tem diferentes objetivos e, por consequência, pode ser caracterizada metodologicamente de três maneiras: como constructo, variável e intervenção (Cai & Hwang, 2020). ...
... Assumese, neste caso, que o constructo está bem definido, sendo utilizado como variável em análises de um fenômeno maior, ou de suas relações com outros aspectos. A proposição de problemas pode ser utilizada, por exemplo, para analisar as relações com a criatividade ou com a resolução de problemas (Cai & Hwang, 2020). ...
... Como intervenção, a pesquisa visa analisar como a proposição de problemas é utilizada para promover melhores resultados de aprendizagem e melhorar a criatividade, dentre outros objetivos, possibilitando auxiliar professores a utilizar a proposição de problemas em suas aulas (Cai & Hwang, 2020). ...
A proposição de problemas tem sido considerada uma atividade cognitivamente exigente, com potencial para o desenvolvimento e a avaliação do pensamento crítico e criativo. Em sala de aula é considerada umas das formas autênticas de investigação matemática em que os estudantes têm a possibilidade de associar suas vivências, interesses e conhecimentos na criação dos problemas. Contribuindo com essa vertente de pesquisa, este artigo tem como objetivo discutir potencialidades da proposição de problemas para o desenvolvimento do pensamento criativo nas aulas de Matemática. Em particular, este estudo, de natureza qualitativa,se propõe a analisar uma atividade de proposição de problemas realizada junto a uma turma de crianças do primeiro ano do Ensino Fundamental. Os resultados indicam que a proposição de problemas tem expressivo potencial para o desenvolvimento da criatividade, além de possibilitar que as crianças atribuam significado e analisem criticamente os dados, relacionando suas experiências, seus conhecimentos e interesses.
... There is no agreement on how mathematical problem posing is defined, though it is generally used to refer to "the process by which, on the basis of mathematical experience, students construct personal interpretations of concrete situations and formulate them as meaningful mathematical problems" (Stoyanova & Ellerton, 1996, p. 519). As a complex notion, mathematical problem posing has been described in different ways: as a logical process (Cai & Hwang, 2020;Cai & Rott, 2024;Stoyanova & Ellerton, 1996); as a product-oriented phenomenon (Silver, 1994); as a role-centered accomplishment shaped by the norms of particular communities (Klinshtern et al., 2015;Kontorovich, 2020); and a cognitive activity, a research or instructional tool, or a learning goal (Cai & Leikin, 2020;Liljedahl & Cai, 2021). ...
... Higher grade level participants, who are more accustomed to the conventional teacher-led instruction and are relatively successful in learning mathematics in this way of teaching, are more likely to possess low motivation in posing problems (Silver, 1994) and thus benefit less from the interventions. Furthermore, Cai and Hwang (2020) delved into the nuances of problem posing from a pedagogical standpoint, highlighting the difference between students and teachers. For teachers, problem posing extends beyond merely generating problems based on given problem situations or modifying existing problems, which are the areas students are solely focusing on. ...
Mathematical problem posing, generally defined as the process of interpreting given situations and formulating meaningful mathematical problems, is academically important, and thus several interventions have been used to enhance this competence among students and teachers. Yet little is known about the interventions’ various components and their relative or combined effectiveness. In this meta-analysis of 26 intervention studies in mathematics, we identified nine intervention components and found that the interventions had a medium, positive, and significant mean weighted effect size. A stepwise meta-regression analysis revealed that intervention efficacy varied by moderators relevant to the research design, sample characteristics, and intervention characteristics. The findings obtained from this meta-analysis are expected to serve as a foundation for future efforts to design and implement (more) effective interventions to improve mathematical problem posing competence.
... Señalan Singer et al. (2013) que para que el profesorado genere espacios dedicados a la formulación de problemas, previamente debe haber experimentado esta actividad con el fin de reconocer las características relevantes a tener en cuenta. Cai y Hwang (2020) indicaron que son necesarias más investigaciones acerca de cómo el profesorado formula problemas y cómo lo integra en la enseñanza de las matemáticas. En España, tenemos varios ejemplos de investigaciones en esta línea (García-Alonso et al., 2022;Sosa-Martín et al., enviado;Torregrosa et al., 2021). ...
En este trabajo se estudia la formulación de problemas de fracciones por parte de estudiantes del Grado en Maestro en Educación Primaria. El objetivo es analizar los errores y la ausencia de información de los enunciados planteados en función de la situación inicial de la que parten. Se solicita a 205 futuros docentes plantear problemas a partir de dos situaciones iniciales diferentes: datos numéricos (1/4 y 3/8) y un contexto (un viaje escolar). Se analizaron 274 problemas extraídos de una muestra de problemas de fracciones procedentes de un estudio más amplio, donde se utilizan distintos significados de fracción (Sosa-Martín et al., enviado). Los resultados muestran dificultades asociadas al significado de la unidad de referencia de la fracción en los casos parte-todo que indican la necesidad de incidir en ello a lo largo de la formación inicial de los docentes.
... El desarrollo de habilidades para la resolución de problemas se ha tornado central (Ellerton et al., 2015;Lester y Cai, 2016). Aunque la formulación de problemas también ha sido reconocida como esencial en el avance de las matemáticas (Polya, 1985), la investigación sobre formulación de problemas es relativamente reciente en el ámbito de la Educación Matemática (Aktaş, 2022;Cai y Hwang, 2020). Hasta los años 80 y 90 no se resalta su importancia (Kilpatrick, 1987;Silver, 1994) ni encontramos las primeras investigaciones (Brown y Walter, 2005;Ellerton, 1986;Silver y Cai, 1996;Stoyanova y Ellerton, 1996). ...
Esta investigación se centró en la gestión de aula que una maestra realiza durante una tarea de formulación de problemas con alumnado de Educación Infantil (4-5 años). Mediante un estudio de caso único, nos enfocamos en identificar las demandas matemáticas que la maestra establece para favorecer el avance en la formulación de un problema, la actividad del alumnado que las detona y las relaciones entre ambas. Los resultados indican que la mayoría de las demandas matemáticas están orientadas a favorecer el desarrollo del contexto del problema y revelan que las dificultades que las detonan están ligadas al significado no matemático de problema que prevalece en el alumnado.
This research focused on the classroom management that a teacher performs during a problem formulation task with students in Early Childhood Education (4-5 years old). By means of a single case study, we focused on identifying the mathematical demands that the teacher establishes to favor progress in the formulation of a problem, the students' activity that triggers them and the relationships between both. The results indicate that most of the mathematical demands are oriented to favor the development of the context of the problem and reveal that the difficulties that trigger them are linked to the non-mathematical meaning of problem that prevails in the students.
... All experts agree that teachers are at the heart of education and learning (Hanifah et al., 2022;Cai & Hwang, 2020;Eliza et al., 2022). Teachers play a very important role in improving the level of education. ...
The purpose of this study is to explore the implementation of self-reflection in Microteaching practice to improve the teaching quality of prospective Arabic language teachers. This study also analyzes some of the problems in implementing self-reflection and its solutions. The research was conducted at the Arabic Language Education Study Program, Faculty of Tarbiyah and Keguruan, Sunan Ampel State Islamic University Surabaya, Indonesia. The research method used was descriptive-analytical method. The data collection techniques were participant observer, in-depth interview, and document review. The data analysis technique used the interactive analysis model of Miles, Huberman, and Saldana. The results showed that Microteaching is a teaching practice for prospective Arabic teachers and one form of teaching practice is the implementation of Self-Reflection. Self reflection is carried out at the end of each teaching exercise orally and in writing in the form of a reflective journal. In the implementation of Self-Reflection, there are several problems, including: they are embarrassed to reveal their weaknesses in teaching practice and have difficulty determining solutions to their weaknesses. The solution to these problems is that they are asked to conduct oral and written self-reflection and discuss with peers to determine an action plan in order to improve the quality of their teaching practice.
... On the other hand, teachers must be able to formulate and provide challenging problems to their students (Cai & Hwang, 2020). Educational strategies to improve mathematical problem-solving skills must also provide opportunities for students to learn mathematics while solving problems (Yapatang & Polyiem, 2022). ...
This study aims to improve the mathematical problem-solving ability of students at Vocational High School (VHS) through implementing Project Based Learning (PjBL). The research method used was a quasi-experiment with a matching-only pretest-posttest control group design, with a sample of 64 students divided into two classes, namely experimental and control classes, each consisting of 32 students. The sampling technique used was purposive sampling, where the sample was selected based on specific criteria relevant to the research objectives. Data were collected using tests of mathematical problem-solving ability administered before (pre-test) and after (post-test) the treatment. Data were analyzed using an independent sample t-test and a two-way ANOVA test with univariate general linear model (GLM) analysis. The results showed a difference between the experimental and control groups before the treatment. However, after the treatment, the experimental group had higher mathematical problem-solving ability than the control group, including the improvement and interaction between PjBL implementation and the Early Mathematics Skills (EMS) category.
... While historically research has focused on students as problem solvers (Schoenfeld, 2010), in the last decade there has been increasing emphasis on understanding the role of teachers in the problem solving process within the context of school mathematics. Thus, studies have investigated how teachers interact with curriculum materials (Ahl et al., 2015), how problem solving may be used as a pedagogical approach for teaching mathematics (Cai & Hwang, 2019), what kinds of challenges teachers encounter when integrating problem solving into their instruction (Cheeseman, 2018) and shifts in teachers' conceptions of problem posing after participating in professional development activities (Cai et al., 2020;Saadati & Felmer, 2021). ...
The decisions that teachers make in transforming the curriculum into specific lesson plans determine the real enactment or otherwise of curricular ideals. These decisions are shaped by the resources available and by each teacher’s goals and orientations. This exploratory study employs Schoenfeld’s decision-making model to examine how resources, goals and orientations influence lesson planning for mathematics problem solving, for different profiles of primary teachers in Chile. To this purpose, a survey was conducted among 40 teachers of varying degrees of ability and experience: some were beginning teachers, others were experienced but had no further training in teaching problem solving and a third group was composed of experienced teachers with specific training in this question. Interviews with two teachers from each profile revealed important differences between the three groups. Beginning teachers relied more heavily on official resources such as the official curriculum and standard textbooks, aligning themselves with school requirements. Experienced teachers with problem solving training demonstrated a strong inclination towards teaching through a problem solving approach. While beginning teachers acknowledged the importance of promoting problem solving strategies, they did not usually adapt problems to the mathematical content or to the age/competence of their students. Interestingly, all three groups under-utilised sections of curricular resources that emphasise the present curricular focus on problem solving. Finally, the study found that experience alone is not enough to develop a problem solving approach and that focused professional development programmes are needed to equip teachers with the necessary skills. In addition, a diagnostic teaching approach should be incorporated into initial teacher training.
... 809). By problem posing in mathematics education, I refer to several related types of activity that entail or support teachers and students formulating (or reformulating) and expressing a problem or task based on a particular context (which I refer to as the problem situation; Cai & Hwang, 2020;Silver 1994). Research has clearly shown that both students and teachers are capable of posing mathematical problems based on given situations (Cai et al., 2015). ...
Thius document includes a collection of the position documents presented at the International Workshop of Israel Science foundation. https://curiosity-isf.edu.haifa.ac.il/
... Evidence from research suggests that using problem posing in mathematics instruction could improve the quality of teaching mathematics (cf. Cai & Hwang, 2020). Teaching through problem posing could further teachers' mathematical goals for their classes, in addition to improving students' mathematical communication skills (National Council of Teachers of Mathematics, 1991;. ...
This qualitative case study used a Qualtrics questionnaire to elicit experts' views regarding the role of problem posing and effective instructional approaches related to the use of problem posing in the teaching of mathematics content courses for preservice elementary teachers in the United States. A thematic analysis of responses to the questionnaire provided by six of the experts who participated in the study suggest that problem posing could serve as a tool for assessing mathematical understanding, or as a tool for promoting critical thinking, among other things in mathematics education. Additionally, most of the experts noted that getting started with problem posing tasks is particularly challenging for many students. Modeling problem posing and creating classroom environments/cultures that promote problem prosing are some of the teaching approaches identified by most of the experts as effective in supporting students' learning about problem posing. Furthermore, nearly all the expects expressed disappointment at the paucity of problem posing opportunities provided by textbooks for mathematics content courses for preservice elementary teachers. Implications for instruction are included.
... In the case of the former, problem posing can be used as an instructional task or problems can be posed for the students to solve. From the perspective of the student, they may be asked to pose problems with predefined conditions or to redesign existing problems (Cai, Hwang 2020). This article will tackle different classroom situations which present both the teacher's and the students' perspectives. ...
Problem posing and problem solving serve as a crucial element of classroom instruction in early mathematics education, and have long been a topic of study of many practitioners and researchers. Used as a powerful tool for differentiation, they affect the ways in which the practice of mathematics is perceived by students and also help teachers gauge children’s understanding of concepts. In the student-centered approach, problem posing and problem solving can successfully engage students in creative educational situations. This research has been part of a broader design research project, the aim of which was to investigate the relationship between a classroom environment that allows for dialogue and young children’s propensity to design and solve their own tasks. Methodology included taking field notes and photographs, followed by reflection sessions. Turning young mathematicians into independent inquirers helped them gain authentic ownership of their knowledge. Additionally, it aided in the development of the young children’s competencies in effective engagement in problem posing activities. The toolbox of instructional techniques for problem posing in the classroom evolved, transforming mathematical classrooms into inquiry polygons for all learners.
... Note the assumptions underpinning this statement: the researchers do not highlight a non-satisfactory understanding of the phenomenon in general, but the cognitive and affective perspectives on it. In a similar fashion, editors of recent special issues indicate that a significant portion of their studies attend to cognitive processes (e.g., Cai & Hwang, 2020;Cai & Leikin, 2020). ...
... A formulação de problemas, associada à resolução, pode favorecer habilidades de resolução de problemas, viabilizar a compreensão e o desenvolvimento de processos cognitivos complexos (BROWN;WALTER, 2005;CIFARELLI;CAI et al., 2015;SILVER, 1994;, além de colocar o estudante como centro do processo de ensino-aprendizagem e dar mais visibilidade sobre o pensamento matemático do estudante HWANG, 2020). ...
A formulação de problemas, quando associada à resolução, apresenta potencial para melhorar a aprendizagem da matemática no contexto escolar. Nesse contexto, objetivou-se descrever os movimentos do trabalho pedagógico de pesquisas que abordaram a formulação de problemas por estudantes do ensino fundamental e médio e, a depender dos resultados, estabelecer um movimento alternativo em relação a essa estratégia de uso de problemas. O estudo teórico e qualitativo teve objetivo exploratório-descritivo e procedimento bibliográfico. Os resultados possibilitaram identificar a abordagem utilizada e descrever os movimentos em relação à formulação de problemas e, especificamente, apontaram que o referencial teórico/metodológico pode ter levado à concentração do trabalho pedagógico na resolução de problemas, sendo necessário estabelecer um movimento alternativo para orientar a constituição de sequências didáticas constituídas pela formulação e resolução de problemas: o movimento bidirecional cíclico. Considera-se, portanto, que esse movimento apresente potencial para fundamentar uma metodologia de ensino-aprendizagem da matemática por meio do uso de problemas.
... Observa-se que esta concepção de Resolução de Problemas está totalmente em consonância com o movimento atual das pesquisas de Resolução de Problemas, que sugerem a Proposição de Problemas como uma forma de continuar avançando nas práticas e pesquisas em Resolução de Problemas (Jurado, 2016;Cai, Hwang, 2020). ...
É com grande entusiasmo que apresentamos este e-book, resultado das reflexões e investigações do Grupo de Trabalho de Educação Matemática – GT13. Em um cenário educacional em constante transformação, o GT13 dedicou-se à compreensão aprofundada de diversos aspectos cruciais no universo da Matemática. Ao longo desta jornada intelectual, exploramos com diligência o Ensino de Matemática, desbravando caminhos que visam aprimorar as práticas pedagógicas e proporcionar experiências de aprendizagem mais significativas. Esta obra é mais do que um registro de nossas descobertas; é um convite à reflexão e à colaboração contínua. A Matemática, longe de ser uma disciplina estática, é um terreno fértil para a exploração constante, repleto de desafios e descobertas. Que este e-book inspire educadores, pesquisadores e entusiastas da Matemática a se lançarem nesse universo dinâmico e enriquecedor.
... Pengajuan masalah memiliki keterkaitan dengan problem solving (pemecahan masalah). Pengajuan masalah yang efektif sangat penting untuk pengajaran matematika berkualitas tinggi (Cai & Hwang, 2020). ...
Creative thinking merupakan salah satu kemampuan yang perlu dikuasai siswa namun kondisi menunjukkan bahwa creative thinking belum dikuasai dengan baik. Penelitian ini akan memberikan penguatan literatur mengenai model pembelajaran yang dapat menguatkan creative thinking. Tujuan penelitian ini adalah mengetahui keefektivan pembelajaran pengajuan masalah terhadap penguatan kemampuan creative thinking siswa SMA. Penelitian kuantitatif dengan jenis eksperimen semu, rancangan dengan kelompok kontrol hanya pascaperlakuan. Populasi adalah siswa SMA Negeri 1 Suruh kelas XI MIPA. Melalui clustered random sampling terpilih kelas XI MIPA 1 (kelompok eksperimen) dan kelas XI MIPA 2 (kelompok kontrol). Data diambil melalui tes dan observasi, instrumen berupa soal tes, lembar observasi keterlaksanaan pembelajaran. Data pascaperlakuan dianalisis menggunakan uji normalitas, uji homogenitas, uji perbedaan rata-rata. Data hasil observasi dikonversi menjadi persentase lalu dikategorikan menjadi skala kualitatif berdasarkan tabel kategori keterlaksanaan pembelajaran. Berdasar analisis data diketahui hasil uji perbedaan rata-rata postest kelompok eksperimen dan postest kelompok kontrol yaitu nilai sig.2 tailed 0,00 < 0,05 sehingga H0 ditolak yang bermakna terdapat perbedaan rata-rata kemampuan creative thinking kelompok eksperimen dan kelompok kontrol. Mean difference kelompok eksperimen dan kelompok kontrol bernilai positif yaitu 12,60 berarti rata-rata kemampuan creative thinking kelompok eksperimen lebih tinggi dari rata-rata kemampuan creative thinking kelompok kontrol. Pembelajaran pengajuan masalah terlaksana “sangat baik”. Kesimpulan penelitian ini adalah pembelajaran pengajuan masalah efektif terhadap penguatan kemampuan creative thinking siswa kelas XI MIPA SMA Negeri 1 Suruh.
... A number of studies have found that difficulties that are often exhibited by pre-service elementary mathematics teachers when working with various mathematical ideas/concepts, including posing problems in the context of fractions, are widespread (cf. Cai & Hwang, 2020;Isik & Kar, 2012;Lee & Lee, 2022;Osana & Royea, 2011;Xie & Masingila, 2017). Other studies have reported that students' performance in mathematical tasks is often related to the opportunities they have had to learn, such as via course textbooks, about the content covered in these tasks (cf. ...
This qualitative document analysis study reports on opportunities to learn provided by textbooks for mathematics content courses for pre-service elementary teachers. Specifically, the study examined opportunities to engage in problem posing and quantitative reasoning, in the context of fractions, provided by six textbooks commonly used in the teaching of content knowledge courses for pre-service elementary mathematics teachers in the United States. Among other things, this examination revealed two key findings. First, all the textbooks provide numerous opportunities that could potentially trigger students’ quantitative reasoning. This is commendable, and a positive response to growing calls to increase such opportunities in mathematics education. Second, there is a paucity of opportunities to engage in problem posing in all the textbooks. Given the numerous benefits of using problem posing in the teaching and learning of mathematics, we recommend that authors of textbooks for mathematics content courses for pre-service elementary teachers provide more opportunities to engage in problem posing in their textbooks. We argue that the act of problem posing requires significant quantitative reasoning, and thus problem posing provides the opportunity for students to develop such reasoning. Implications for instruction and directions for future research are discussed.
... Crespo (2015) considers the teacher and the students as creators of mathematics problems themselves and suggests a framework for the problem-posing activity of teachers in the classroom and sets four possible screenplays: (1) posing problems to students, (2) posing problems with students, (3) posing a mathematical problem of personal interest, and (4) posing socially relevant mathematical problems. Problem-posing can be a tool for the teacher to use when planning lessons; it can also be a means of mathematical instruction, i.e., the teacher creates a situation in which they use student problem-posing to implement the lesson's task order (Cai & Hwang, 2020). In all classroom implementations, the teacher must have the ability to pose problems and must be able to anticipate the problems that the students will raise (Mason, 2015). ...
The ability to create problems is included in the Hungarian National Core Curriculum as a mathematical skill to be developed in students, which is one of the reasons why problem-posing is a fundamental pedagogical skill of mathematics teachers. Therefore, it is necessary to include the development of this competence in the training program for prospective mathematics teachers. The study results were obtained during the “Mathematics Competitions” course for prospective mathematics teachers, where one of the objectives is to develop problem-posing skills. The students involved have already completed problem-solving courses, so they are already familiar with the essential aspects of problem-posing. In particular, they have used successive reformulations of problems during the problem-solving process. However, they have not systematically addressed the practice of problem-posing based on an initial problem. The author compares the problem-posing products of two groups: novices and experts. Novices had only problem-solving practice, whereas the expert group received training in the “what if not” method of problem-posing. By adapting the existing evaluation categories in the literature, the author created a system of categories suitable for describing the corpus and providing adequate information for evaluating problem-posing products. The conclusion is that competent problem-posing is a learnable activity. After dedicated training, prospective teachers with similar mathematical backgrounds and problem-solving experience posed problems more competently. The difference can be explained by the intervention resulting in better problem perception by the expert group than by the novices. The comparative analysis’s pedagogical implication is the need for focused training for problem posing in teacher education. Additionally, the author says that the skills needed to pose a problem go beyond those needed to solve a problem.
The aim of this paper is to describe and analyze how a group of prospective teachers create problems to develop proportional reasoning either freely or from a given situation across different contexts, and the difficulties they encounter. Additionally, it identifies their beliefs about what constitutes a good problem and assesses whether these beliefs are reflected in their problem creation. This is a descriptive-qualitative study that utilizes theoretical and methodological tools from the Onto-semiotic Approach in the content analysis of participants' responses. The results indicate that the prospective teachers' beliefs about what makes a good problem do not always manifest in their practice. The prospective teachers faced challenges in inventing problems that meet the established didactic-mathematical purpose, related to insufficient didactic-mathematical knowledge of proportional reasoning, achieving better outcomes in the arithmetic context and in free creation.
El propósito de esta investigación fue profundizar en la caracterización del pensamiento matemático de los docentes de primaria mediante la creación de problemas de álgebra temprana. Para lograrlo, se empleó la metodología investigación acción en donde siguiendo un ciclo continuo de planificación, acción, observación y reflexión, se diseñaron e implementaron talleres de formación que incluyeron diversas actividades: el Taller I de contextualización y el Taller II de creación de problemas que involucraban la identificación de patrones, Además, se realizó una entrevista final a los docentes participantes para conocer sus percepciones sobre el proceso desarrollado en la investigación y las contribuciones a su labor docente. La implementación de estos talleres y los resultados obtenidos en la creación y resolución de problemas permitieron evidenciar las estrategias utilizadas por los docentes en el desarrollo de estos procesos. Como resultado, se lograron avances significativos en la identificación de los aspectos que caracterizan el pensamiento matemático de los docentes al crear problemas algebraicos, así como en las contribuciones para su práctica pedagógica.
The Direct Method in language teaching, which emphasizes teaching language through immersion and avoiding the use of the learner’s native language, has been widely used in educational settings for over a century. Its focus on oral communication, everyday vocabulary, and inductive grammar teaching has made it a popular approach. However, debates about its effectiveness compared to more modern teaching methods have persisted, particularly in terms of long-term language acquisition and proficiency. This literature review aims to evaluate the effectiveness of the Direct Method by analyzing studies conducted across various educational contexts. The review examines how this method impacts learners’ speaking, listening, and overall communicative competence. By synthesizing both qualitative and quantitative research, the study provides a comprehensive view of the strengths and limitations of the Direct Method. The findings indicate that while the Direct Method improves oral proficiency and listening skills in the short term, it may not be as effective for developing reading and writing skills. Studies show that the method fosters greater student engagement and motivation but may not adequately address the complexities of advanced grammar and academic language. Furthermore, the method’s effectiveness depends heavily on the teacher’s skill in creating immersive language environments. In conclusion, the Direct Method offers valuable benefits for beginner and intermediate learners, especially in communicative aspects of language. However, it requires supplementation with other approaches to develop comprehensive language proficiency. Future research should explore ways to integrate the Direct Method with more advanced instructional techniques for a balanced approach to language teaching.
The present study aimed to describe teachers' instructional flows when implementing a mathematical problem-posing task using scriptwriting technique. With match-sticks, a growing pattern that increases by a constant unit was created and presented to the teachers as a problem-posing situation. We analyzed the instructional flows in 50 scripts, taking into account situations recognized in the problem-posing field as critical for integrating problem posing into mathematics classrooms. We determined three instructional flows in the scripts: pose and solve cycle-based, observation-based, and problem-solving based, the first being the most common. We presented a new problem-posing instructional model and discussed its potential benefits for student learning.
Resumo A proposição de problemas tem ganhado destaque em documentos curriculares e pesquisas atuais, em que muitas questões estão ainda em aberto, e não se manifestam clara e coerentemente nas práticas escolares. Este artigo tem como objetivo identificar e estabelecer entendimentos para as expressões formulação, elaboração, criação e proposição de problemas com vistas a subsidiar o contexto de pesquisas e práticas brasileiras, no âmbito da Educação Matemática. Para tanto, apresenta-se uma revisão da literatura retomando aspectos históricos e constituindo um referencial teórico acerca das implicações no contexto da aprendizagem matemática e da educação integral dos estudantes. Então, a partir dos resultados obtidos de um estudo de revisão sistemática realizado, e avançando na constituição de um corpo teórico, apresenta-se os sentidos atribuídos a essas expressões, em diversos estudos brasileiros e internacionais, para, então, serem apresentadas as definições por nós assumidas: entendemos que a criação de problemas envolve os processos de formulação e elaboração de problemas e está inserida na proposição de problemas, que avança para apresentar o problema criado para um potencial resolvedor. A relevância dessas definições explicitadas abrange tanto contextos de práticas educativas que contemplem a associação com a resolução de problemas, quanto a pesquisa em Educação Matemática, com condicionamentos e reflexos sobre a coleta e análise de dados, com vistas ao aprofundamento das compreensões acerca de aspectos específicos ligados à proposição de problemas a serem considerados em estudos futuros.
The problem posing has gained prominence in curriculum documents and current research, in which many questions are still open, and are not manifested clearly and coherently in school practices. This article aims to identify and establish understandings for the terms formulation, elaboration, creation, and problem posing to support the context of Brazilian research and practice in the field of Mathematics Education. To this end, a literature review is presented, resuming historical aspects and constituting a theoretical reference about the implications on the context of mathematics learning and students' comprehensive education. Then, based on the results obtained from a systematic study review and moving forward in the constitution of a theoretical body knowledge, we present the meanings attributed to these expressions in several Brazilian and international studies and then present the definitions we assume: we understand that problem creation involves the processes of problem formulation and elaboration and it is inserted in the problem posing which advances to present the created problem to a potential solver. The relevance of these definitions covers both contexts of educational practices that contemplate the association with problem solving and research in Mathematics Education, with conditions and reflections on the collection and analysis of data, in order to deepen the understanding of specific aspects related to problem posing to be considered in future studies.
This study examined characteristics of Chinese cultural group students’ mathematical problem posing and the impact of cultural contexts on students’ mathematical problem posing. Forty-four (N = 44; girls: 68.2%; boys: 31.8%) fifth-grade cultural group students from a province in Southwest China responded to a four-task-based questionnaire and posed mathematical problems based on given problem situations, two with a cultural context and two without a cultural context. Characteristics of mathematical problem posing were analyzed through the total numbers, appropriateness, difficultly, and flexibility levels of the problems posed by the students. Results show that Chinese cultural group students were capable of posing a considerable number of mathematical problems; the medium and large effect sizes show that cultural contexts impact students’ problem-posing performance. Practical implications of the study and possible further studies are discussed.
Mathematical problem posing was advocated as one major area needing systematic investigation. It was also recommended that school mathematics learning should involve experientially real problem situations and using handheld devices to bridge across formal and informal contexts for mathematical learning. This research study explored how 11 middle school students generated mathematical problems in a real-life setting with the help of a mobile device. The multiple case study gathered observations, participants’ generated artifacts, and individual interviews and reported a number of themes related to the general process, barriers or challenges participants encountered, and the role of the handheld device.
Problem posing become one of the essential skills to promote in class. Meanwhile, there is still limited research on electronic teaching materials that integrated local wisdom to enhance student problem posing. The aim of this research is to investigated the perception of teacher about problem posing and the need to include local wisdom in electronic module. This research is descriptive research. The teachers who have implemented and have not implemented the teaching material to foster problem posing in Lombok Island, Indonesia become the subject of this study. The data was collected by questionnaire and interview. The questionnaire contains subject’s perception about problem posing and their need to integrate local wisdom in e-module. The data was analyzed by statistic descriptive and qualitative method. The majority of teachers believe that students need to enhance their problem posing skill. Furthermore, they require educational resources based on wisdom, such as the Rinjani Geopark, to strengthen that skill. The findings of this study recommended that the development of an e-module to promote problem posing could be the next area of investigation..
In this study, 176 Turkish seventh-grade students were asked to pose three problems for each of the swimming and purchasing tasks to examine how the solvability and complexity of the problems changed depending on the order of responses. The purchasing task required to pose a problem to derive a specific answer before posing new problems, whereas the swimming task required to pose problems based on existing data. The findings indicated that students performed poorly in terms of posing solvable problems in both tasks. The majority of solvable responses in the purchasing task were at the high complexity level, but most in the swimming task were at the low complexity level. No evidence of a relationship between the order of the responses and any of the analysis units (solvability and complexity) was found. Furthermore, when comparing the groups that posed and did not pose solvable problems for the specific answer in the purchasing task, the former group had a stronger tendency to maintain the complexity points in their other responses compared to their first response.
O objetivo deste trabalho é analisar uma atividade de exploração de problemas desenvolvida em uma experiência de estágio de docência, no contexto de formação inicial de professores de matemática. A atividade abordou noções de análise combinatória ao mesmo tempo que contemplou um debate sobre gênero e sexualidade, a partir de um problema inicial que questionava de quantas maneiras diferentes dois personagens de uma situação poderiam se vestir para ir a um determinado lugar. A atividade foi desenvolvida em dois encontros com alunos de uma licenciatura em matemática e os registros escritos dos alunos foram coletados para análise, além de parte das discussões acontecidas em sala de aula. Essa pesquisa é de abordagem qualitativa. A discussão sobre qual tipo de roupa cada pessoa poderia vestir surgiu naturalmente no início da atividade. Muitos alunos indicaram nos registros escritos roupas de mulher e roupas de homem para contar as possibilidades, embora tenham considerado as mesmas possibilidades para os dois personagens. Os alunos, de uma maneira geral, apresentaram dificuldades na proposição de problemas a partir da situação inicial e não souberam identificar os conceitos matemáticos que poderiam ser trabalhados nos problemas propostos por eles. Esta pesquisa indicou a dificuldade dos alunos de licenciatura em matemática em lidar com a resolução, exploração e proposição de problemas e a necessidade de desenvolver debates sobre gênero e sexualidade na formação de professores de matemática. PALAVRAS-CHAVE: Gênero, Exploração de Problemas, Análise Combinatória, Sexualidade.
The study of the multiple applications of mathematics in different spheres of economic, cultural, environmental and social life can serve to understand the need to use it for the good of society, with actions that promote a more harmonious relationship between man and the environment. In this sense, the ability to formulate problems is receiving increasing attention, being identified as one of the fundamental axes of the teaching of mathematics. Studies on this subject show this skill as a qualitatively superior stage of problem-solving processes because it contributes to the creative, logical and reflective reasoning of students. The objective of the article is to reveal the procedures for obtaining the actions and operations of the ability to formulate mathematical problems in pre-university students, first by analogy with the operational system of text construction and its enrichment through the Experimental Theoretical Mode. The systematization of the obtained model contributes to transform the traditional teaching by one that assumes, in the first place, the development of skills and preparation for life in a society that constantly conceives innovations.
The study focuses on the problem posing of three primary school teachers of the last grade of primary school, which was based on the textbook of the first grade of secondary school. The problem posing took place during a professional development program aimed to support the transition in mathematics between the two educational levels. We explore the characteristics of the modified task in relation to its presentation, content, solution, as well as the teachers' views in relation to their modifications. The results have shown that the modifications in the provided task were related to the teachers' varying views about teaching and learning for facilitating the transition.
Öğretmen adaylarının kurulan problemleri nasıl değerlendirdiği; öğretmenlik mesleğini anlamalarına ve içselleştirmelerine katkı sağlayacaktır. Bu bağlamda yapılan bu çalışmada matematik öğretmen adaylarının matematik problemlerini değerlendirme yaklaşımlarının incelenmesi amaçlanmıştır. Araştırmada nitel araştırma yöntemlerinden biri olan durum çalışması kullanılmıştır. Araştırma Türkiye’nin bir ilindeki bir devlet üniversitesinde ilköğretim matematik öğretmenliği dördüncü sınıfta öğrenim gören 20 öğrenci ile gerçekleştirilmiştir. Yapılan bu çalışmanın veri toplama sürecinde öğrenci yanıtlarını içeren form öğretmen adaylarına dağıtılmış ve öğretmen adaylarının öğrencilerinin problem kurma etkinliklerine verdikleri yanıtları değerlendirmeleri istenmiştir. Katılımcılara alanyazındaki değerlendirme kriterleri konusunda herhangi bir bilgi verilmeyip öğrenci yanıtlarını değerlendirmede serbest bırakılmıştır. Araştırmada elde edilen verilerin analizinde içerik analizden yararlanılmıştır. Katılımcılardan elde edilen bulgular öğrencilerin problem kurma etkinliklerine verdikleri yanıtları değerlendirirken 6 ana kriterden yararlandıkları görülmüştür. Bu ana kriterler; problem mi?, problem kurma durumuna uygunluk, çözülebilirlik, bağlamsallık, dil kullanımı ve karmaşıklıktır. Problem kurma durumuna uygunluk ve çözülebilirlik kriterleri tüm öğretmen adayları tarafından kullanılan kriterler iken diğer kriterlerin öğretmen adayları tarafından kullanılma sıklıkları farklılık göstermektedir. Bu sonuçlar doğrultusunda öğretmen adaylarının zihinlerinde bir değerlendirme şeması olmadığı düşünülmüştür. Bu nedenle de öğretmen adaylarına kurulan matematik problemlerini değerlendirmeye yönelik eğitimler verilmesi önerilmiştir.
The invention of problems is a fundamental competence that enhances the didactic-mathematical knowledge of mathematics teachers and therefore should be an objective in teacher training plans. In this paper, we revise different proposals for categorizing problem-creation activities and propose a theoretical model for problem posing that, based on the assumptions of the Onto-Semiotic Approach, considers both the elements that characterize a problem and a categorization of different types of problem-posing tasks. In addition, the model proposes a description of the mathematical processes that occur during the sequence of actions carried out when a new problem is created. The model is illustrated by its application to analyze the practices developed by pre-service teachers in three problem-posing tasks aimed at specific didactic-mathematical purposes (mobilizing certain mathematical knowledge or reasoning, contributing to achieving learning goals, or addressing students’ difficulties). We conclude discussing the potential of our model to analyze the mathematical processes involved in problem creation from the perspective of teacher education.
The regulatory phases of cognition during problem posing are being presented through a case study of a grade 9 student, Tan, engaged in a geometric problem-posing task. The present study follows Schoenfeld’s notion of regulation of cognition as it is used in his episode-based framework for analysis of problem-solving protocols. The study paints the different phases in the regulation of cognition during problem posing, namely, property noticing, problem construction, checking solution, and looking back. The looking back phase is not strongly exhibited. Discussion of these phase descriptors in classroom problem-posing instructions are also made.
In this chapter, we articulate a perspective on problem posing research grounded in our own research in modeling. We begin by identifying a shared commitment between researchers in modeling and problem posing: to honor and amplify students’ (and teachers’) mathematical agency. We believe this shared commitment can serve as the basis for a shared agenda of modeling and problem posing, going forward. However, we next identify a potential issue, which is rooted in a tendency to assume a tight coupling between problem types and individual topics in the mathematics curriculum. To illuminate the nature and significance of this issue, we consider problem solving and posing through the lens of Donald Schön (1983) study of epistemological crises across professions, and we consider how Schön’s dilemma of “rigor or relevance” applies to mathematics education. We urge a shift toward activities that highlight problematic situations and that ask students to engage with such situations by interpreting them mathematically. We argue that to achieve relevance, mathematics education needs to engage with a broader range of problem situations, to utilize a wider array of representational tools, and to embrace a more encompassing view of mathematical problem posing, problem solving, and modeling, as interpretive human activities.
Problem posing is an important aspect of mathematics education. It has received increasing attention in the literature on curricular and pedagogical innovation. Some investigations on problem posing have looked at how students formulate mathematical problems, while others looked at how teachers pose them. In this chapter, this issue was discussed from two different perspectives both referring to how mathematics teachers deal with problem posing. Study 1 analyzed the characteristics of problems they posed, by asking them to formulate eight mathematical word problems whose resolution would involve multiplication and/or division. Study 2 analyzed the practice of teachers who use problem posing to teach mathematics to their students. In this study, participants answered in writing the following questions: What do your students learn by formulating math problems? What do they need to know to be able to formulate math problems? What are the main difficulties they face in formulating math problems? What are the main difficulties you encounter when working with the formulation of math problems in the classroom? The data obtained in the first study allowed to know the characteristics of mathematical word problems posed by teachers, especially in relation to the types of problems they formulate. On the other hand, data in the second study allow to know their opinions about their students and their teaching practice regarding problem posing in the classroom. The results of both studies were discussed in an articulated way, emphasizing the formulation of mathematical problems as part of the didacticknowledge that needs to be included in teacher education programs.
Problem solving is a distinctive human activity that shapes and influences what individuals do when facing social, professional, or academic situations. Reflecting on what the process of identifying, formulating, and solving problems involves becomes a relevant task to understand how people develop resources, strategies, and ways of reasoning to solve problems in different domains. How are mathematical problems formulated? And what does the process of approaching and solving problems involve? How do students develop problem-solving competencies? How do teachers and students’ use of digital tools shape the ways they reason and solve mathematical problems? These questions have inspired mathematicians and mathematics educators to investigate what the process of formulating and solving mathematical problems entails and ways for students to understand mathematical concepts and to solve problems. In this chapter, some seminal conceptual frameworks are reviewed to shed light on principles and tenets to support and frame learning scenarios that foster students’ problem-solving competencies. Further, the consistent and systematic use of digital technologies becomes relevant for learners to enhance their ways of reasoning to work on mathematical tasks and to engage in and extend mathematical discussions beyond classrooms. Thus, digital tools provide a set of affordances for students to dynamically model tasks and rely on heuristic strategies such as orderly dragging objects, quantifying parameters, tracing loci, using sliders, etc. to work and solve the problems.
This study reports a project on teaching word problems in early algebra for primary and lower secondary students. The word problems approach consisted of a set of connected activities in which problem solving, argumentation and problem posing were infused. The project aims to promote in students, within the usual classroom activities, inquiry attitude in problem solving, and awareness of the emerging mathematical objects. It engages them in argumentation activities to justify mathematical properties and in modelling processes. Results of one of the experiments of the project in a class of 5th grade students are reported. Within this study, we aim to: (a) orient the students to make argumentations of a solving process in general terms, (b) facilitate them to explore the corresponding numerical expressions for different series of numerical data, and (c) engage them in solving new questions that arise during their activities. The viability of the project is discussed in light of the results of the experimentation. The chapter is concluded with suggestions for the research and reflections on the teacher’s role in such a problem solving approach.
In our first editorial (Cai et al., 2017), we highlighted the long-standing, critical issue of improving the impact of educational research on practice. We took a broad view of impact, defining it as research having an effect on how students learn mathematics by informing how practitioners, policymakers, other researchers, and the public think about what mathematics education is and what it should be. As we begin to dig more deeply into the issue of impact, it would be useful to be more precise about what impact means in this context. In this editorial, we focus our attention on defining and elaborating exactly what we mean by “the impact of educational research on students' learning.”
This paper reports on 2 studies that examine how mathematical problem posing is integrated in Chinese and US elementary mathematics textbooks. Study 1 involved a historical analysis of the problem-posing (PP) tasks in 3 editions of the most widely used elementary mathematics textbook series published by People’s Education Press in China over 3 decades. Study 2 compared the PP tasks in Chinese and US elementary mathematics textbooks. This allows for the examination of PP tasks from an international comparative perspective, which provides one point of view about the kinds of learning opportunities that are available to students in China and the USA. We found evidence that the inclusion of PP tasks in the Chinese textbook series reflected, to some degree, changes in the curricular frameworks in China. However, the distribution of PP tasks across grade levels and content areas, as well as the variety of types of PP tasks included, suggest a need for greater intentionality in the design and placement of PP tasks in both the Chinese and US textbook series. Findings from the 2 studies reported in this paper not only contribute to our understanding about the inclusion of PP tasks in curriculum both historically and internationally, but also suggest a great need to systematically integrate PP activities into curriculum and instruction. The fact that both Chinese and US curriculum standards have heavily emphasized PP in school mathematics, despite there being only a small proportion of PP activities in both Chinese and US elementary mathematics curricula, suggests the existence of challenges that are delaying the implementation of reform ideas such as problem posing in school mathematics.
In this chapter, the authors note that during the past 30 years there have been significant advances in our understanding of the affective, cognitive, and metacognitive aspects of problem solving in mathematics and there also has been considerable research on teaching mathematical problem solving in classrooms. However, the authors point out that there remain far more questions than answers about this complex form of activity. The chapter is organized around six questions: (1) Should problem solving be taught as a separate topic in the mathematics curriculum or should it be integrated throughout the curriculum? (2) Doesn’t teaching mathematics through problem require more time than more traditional approaches? (3) What kinds of instructional activities should be used in teaching through problems? (4) How can teachers orchestrate pedagogically sound, problem solving in the classroom? (5) How can productive beliefs toward mathematical problem solving be nurtured? (6) Will students sacrifice basic skills if they are taught mathematics through problem solving?
This chapter synthesizes the current state of knowledge in problemposing research and suggests questions and directions for future study. We discuss ten questions representing rich areas for problem-posing research: (a) Why is problem posing important in school mathematics? (b) Are teachers and students capable of posing important mathematical problems? (c) Can students and teachers be effectively trained to pose high-quality problems? (d) What do we know about the cognitive processes of problem posing? (e) How are problem- posing skills related to problem-solving skills? (f) Is it feasible to use problem posing as a measure of creativity and mathematical learning outcomes? (g) How are problem-posing activities included in mathematics curricula? (h) What does a classroom look like when students engage in problem-posing activities? (i) How can technology be used in problem-posing activities? (j) What do we know about the impact of engaging in problem-posing activities on student outcomes?.
The present study aimed to analyse the potential difficulties in the problems posed by pre-service teachers about first degree equations with one unknown and equation pairs with two unknowns. It was carried out with 20 pre-service teachers studying in the Department of Elementary Mathematics Educations at a university in Eastern Turkey. The problem Posing Test (PPT) including five items concerning the equation types was used as the data collection instrument. Furthermore, semi-structured interviews were made with each preservice teacher. It was found that the pre-service teachers had difficulties in seven categories of problem posing. These difficulties were centered on incorrect translation of mathematical notations into problem statements, unrealistic values assigned to unknowns, and posing problems by changing the equation structure. Moreover, the pre-service teachers were found to have greater difficulty in posing problems about equations pairs.
“Success stories,” i.e., cases in which mathematical problems posed in a controlled setting are perceived by the problem posers or other individuals as interesting, cognitively demanding, or surprising, are essential for understanding the nature of problem posing. This paper analyzes two success stories that occurred with individuals of different mathematical backgrounds and experience in the context of a problem-posing task known from past research as the Billiard Task. The analysis focuses on understanding the ways the participants develop their initial ideas into problems they evaluate as interesting ones. Three common traits were inferred from the participants' problem-posing actions, despite individual differences. First, the participants relied on particular sets of prototypical problems, but strived to make new problems not too similar to the prototypes. Second, exploration and problem solving were involved in posing the most interesting problems. Third, the participants' problem posing involved similar stages: warming-up, searching for an interesting mathematical phenomenon, hiding the problem-posing process in the problem's formulation, and reviewing. The paper concludes with remarks about possible implications of the findings for research and practice.
In this paper, I comment on the set of papers in this special issue on mathematical problem posing. I offer some observations about the papers in relation to several key issues, and I suggest some productive directions for continued research inquiry on mathematical problem posing.
In this study, we used problem posing as a measure of the effect of middle-school curriculum on students' learning in high school. Students who had used a standards-based curriculum in middle school performed equally well or better in high school than students who had used more traditional curricula. The findings from this study not only show evidence of strengths one might expect of students who used the standards-based reform curriculum but also bolster the feasibility and validity of problem posing as a measure of curriculum effect on student learning. In addition, the findings of this study demonstrate the usefulness of employing a qualitative rubric to assess different characteristics of students' responses to the posing tasks. Instructional and methodological implications of this study, as well as future directions for research, are discussed.
A Test of Arithmetic Problem Posing was developed by the authors to examine the arithmetic problem-posing behaviours of sixty-three
prospective elementary school teachers. Results of analysis were then used to examine task format (i.e., the presence or absence
of specific numerical information) on subjects’ problem posing and the relationship between subjects’ problem posing and their
mathematics knowledge and verbal creativity. The major findings were that the test effectively evaluated arithmetic problem
posing, and that most subjects were able to pose solvable and complex problems. In addition, problem-posing performance was
better when the task contained specific numerical information than when it did not, and that problem-posing performance was
significantly related to mathematical knowledge but not to verbal creativity.
In this article, I model how a problem-posing framework can be used to enhance our abilities to systematically generate mathematical problems by modifying the attributes of a given problem. The problem-posing model calls for the application of the following fundamental mathematical processes: proving, reversing, specializing, generalizing, and extending. The given problem turned out to be a rich source of interesting, worthwhile mathematical problems appropriate for secondary mathematics teachers and high school students. As a student and teacher of mathematics, I was intrigued about the origin of mathematical problems, especially nontrivial problems whose solutions are not obtained by a formula or algorithm, problems that somehow extend the frontiers of our personal mathematical knowledge. When, as a student, I solved nonroutine textbook problems, I thought, not only of devising a plan or a solution, but also of how the textbook authors and mathematicians generated mathematical problems. Their origin remained an enigma to me until later when, as a teacher, my curiosity led me to read The Art of Problem Posing (Brown & Walter, 1990). This book provided insights into the origin of mathematical problems and motivated me to examine more closely the relationships among related problems. In their book, Brown and Walter propose the "What-if" strategy as a generic means to modify a given problem to create additional related problems. Because I am a geometry lover, I first applied Brown and Walter's "What-if" problem-posing strategy to geometric problems. As a result, I developed a problem-posing framework that has guided my students and me to pose mathematical problems systematically. This problem-posing framework calls for the application of the following prototypical problem-posing strategies: proof problems, converse problems, special problems, general problems, and extended problems. I often use the Geometer's Sketchpad (GSP) (Jackiw, 2001) to verify the reasonability of the resulting conjectures. The main purpose of this article is twofold. First, I model how the framework can be used to generate nonroutine mathematical problems from a given problem and, as a consequence, to discover mathematical patterns and relationships. Second, I discuss some of the difficulties that my students, prospective secondary mathematics teachers, experience when generating mathematical problems. The approach described below reflects the approach that I have followed in class. Because the focus of this article is on problem posing, proofs for most of the resulting theorems are not provided.
This study explored the mathematical problem posing and problem solving of 181 U.S. and 223 Chinese sixth-grade students.
It is part of a continuing effort to examine U.S. and Chinese students’ performance by conducting a cognitive analysis of
student responses to mathematical problem-posing and problem-solving tasks. The findings of this study provide further evidence
that, while Chinese students outperform U.S. students on computational tasks, there are many similarities and differences
between U.S. and Chinese students in performing relatively novel tasks. Moreover, the findings of this study suggest that
a direct link between mathematical problem posing and problem solving found in earlier studies for U.S. students is true for
Chinese students as well.
Learning to pose mathematical tasks is one of the challenges of learning to teach mathematics. How and when preservice teachers
may learn this essential practice,however, is not at all clear. This paper reports on a study that examined the changes in
the problem posing strategies of a group of elementary preservice teachers as they posed problems to pupils. It reports that
their later problem posing practices significantly differed from their earlier ones. Rather than posing traditional single
steps and computational problems, these preservice teachers ventured into posing problems that had multiple approaches and
solutions, were open-ended and exploratory, and were cognitively more complex. Their problem posing style also changed. Rather
than making adaptations that made students' work easier or narrowed the mathematical scope of the problem, their adaptations
became less leading and less focused on avoiding pupils' errors. Posing problems to an authentic audience, engaging in collaborative
posing, and having access and opportunities to explore new kinds of problems are highlighted as important factors in promoting
and supporting the reported changes.
The mathematical problems generated by 509 middle school students, who were given a brief written “story-problem” description and asked to pose questions that could be answered using the information, were examined for solvability, linguistic and mathematical complexity, and relationships within the sets of posed problems. It was found that students generated a large number of solvable mathematical problems, many of which were syntactically and semantically complex, and that nearly half the students generated sets of related problems. Subjects also solved eight fairly complex problems, and the relationship between their problem-solving performance and their problem posing was examined to reveal that “good” problem solvers generated more mathematical problems and more complex problems than “poor” problem solvers did. The multiple-step data analysis scheme developed and used herein should be useful to teachers and other researchers interested in evaluating students' posing of arithmetic story problems.
In this study, 53 middle school teachers and 28 prospective secondary school teachers worked either individually or in pairs to pose mathematical problems associated with a reasonably complex task setting, before and during or after attempting to solve a problem within that task setting. Written responses were examined to determine the kinds of problems posed in this task setting, to make inferences about cognitive processes used to generate the problems, and to examine differences between problems posed prior to solving the problem and those posed during or after solving. Although some responses were ill-posed or poorly stated problems, subjects generated a large number of reasonable problems during both problem-posing phases, thereby suggesting that these teachers and prospective teachers had some personal capacity for mathematical problem posing. Subjects posed problems using both affirming and negating processes; that is, not only by generating goal statements while keeping problem constraints fixed but also by manipulating the task's implicit assumptions and initial conditions. A sizable portion of the posed problems were produced in clusters of related problems, thereby suggesting systematic problem generation. Subjects posed more problems before problem solving than during or after problem solving, and they tended to shift the focus of their posing between posing phases based at least in part on the intervening problem-solving experience. Moreover, the posed problems were not always ones that subjects could solve, nor were they always problems with “nice” mathematical solutions.
This study investigated the problem-posing abilities of third-grade children who displayed different profiles of achievement in number sense and novel problem solving. The study addressed (a) whether children recognize formal symbolism as representing a range of problem situations, (b) whether children generate a broader range of problem types for informal number situations, (c) how children from different achievement profiles respond to problem-posing activities in formal and informal contexts, and (d) whether children's participation in a problem-posing program leads to greater diversity in problems posed. Among the findings were children's difficulties in posing a range of problems in formal contexts, in contrast to informal contexts. Children from different achievement profiles displayed different response patterns, reflected in the balance of structural and operational complexity shown in their problems.
This one-year study involved designing and implementing a problem-posing program for fifth-grade children. A framework developed for the study encompassed three main components: (a) children's recognition and utilisation of problem structures, (b) their perceptions of, and preferences for, different problem types, and (c) their development of diverse mathematical thinking. One of the aims of the study was to investigate the extent to which children's number sense and novel problem-solving skills govern their problem-posing abilities in routine and nonroutine situations. To this end, children who displayed different patterns of achievement in these two domains were selected to participate in the 10-week activity program. Problem-posing interviews with each child were conducted prior to, and after the program, with the progress of individual children tracked during the course of the program. Overall, the children who participated in the program appeared to show substantial developments in each of the program components, in contrast to those who did not participate.
Problem posing (not only in lesson planning but also directly in teaching whenever needed) is one of the attributes of a teacher’s subject didactic competence. In this paper, problem posing in teacher education is understood as an educational and a diagnostic tool. The results of the study were gained in pre-service primary school teacher education. Students were asked to pose problems containing some given data (namely fractions). The subsequent analysis of the problems posed by the students revealed shortcomings in their conceptual understanding of fractions. Classroom-based joint reflection became the means of reeducation.
The paper introduces an exploratory framework for handling the complexity of students’ mathematical problem posing in small groups. The framework integrates four facets known from past research: task organization, students’ knowledge base, problem-posing heuristics and schemes, and group dynamics and interactions. In addition, it contains a new facet, individual considerations of aptness, which accounts for the posers’ comprehensions of implicit requirements of a problem-posing task and reflects their assumptions about the relative importance of these requirements. The framework is first argued theoretically. The framework at work is illustrated by its application to a situation, in which two groups of high-school students with similar background were given the same problem-posing task, but acted very differently. The novelty and usefulness of the framework is attributed to its three main features: it supports fine-grained analysis of directly observed problem-posing processes, it has a confluence nature, it attempts to account for hidden mechanisms involved in students’ decision making while posing problems.
This study examined US and Chinese 6th grade students’ generalization skills in solving pattern-based problems, their generative thinking in problem posing, and the relationships between students’ performance on problem solving and problem posing tasks. Across the problem solving tasks, Chinese students had higher success rates than US students. The disparities appear to be related to students’ use of differing strategies. Chinese students tend to choose abstract strategies and symbolic representations while US students favor concrete strategies and drawing representations. If the analysis is limited to those students who used concrete strategies, the success rates between the two samples become almost identical. With regard to problem posing, the US and Chinese samples both produce problems of various types, though the types occur in differing sequences. Finally, this study revealed differential relationships between problem posing and problem solving for US and Chinese students. There was a much stronger link between problem solving and problem posing for the Chinese sample than there was for the US sample.
Our work is inspired by the book Imagining Numbers (particularly the square root of minus fifteen), by Harvard University
mathematics professor Barry Mazur (Imagining numbers (particularly the square root of minus fifteen), Farrar, Straus and Giroux, New York, 2003). The work of Mazur led us to question whether the features and steps of Mazur’s re-enactment of the imaginative work of
mathematicians could be appropriated pedagogically in a middle-school setting. Our research objectives were to develop the
framework of teaching mathematics as a way of imagining and to explore the pedagogical implications of the framework by engaging
in an application of it in middle school setting. Findings from our application of the model suggest that the framework presents
a novel and important approach to developing mathematical understanding. The model demonstrates in particular the importance
of shared visualizations and problem-posing in learning mathematics, as well as imagination as a cognitive space for learning.
This study identifies the kinds of problems teachers pose when they are asked to (a) generate problems from given information and (b) create new problems from ones given to them. To investigate teachers’ problem posting, preservice and inservice teachers completed background questionnaires and four problem-posing instruments. Based on previous research, a classification scheme was developed and used to categorize the problem statements. The findings indicate that the teachers--both preservice and in-service--struggled when asked to generate their own problems from a given set of information. Teachers had more success when they posed related problems from problems given to them. Teachers posed a greater number of responses for problem generation than reformulation. However, the reformulation responses were more often considered mathematical problems (versus exercises or nonproblems) than the problem generation responses. Teacher background and experience factors also play a role in problem posing. Implications for future research directions are discussed.
Children as arithmetic book authors
- van den Brink
Problem formulating: Where do good problems come from?
- Kilpatrick
A framework for research into students’ problem posing in school mathematics
- Stoyanova