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In this article, we argue that folding back is successful when the learners engage in exploratory talk. To support our argument, we sourced data from a Grade 10 mathematics classroom of 54 learners who participated in a four-week teaching experiment conducted by the second author. We mainly focused on talks in two groups of learners to address the silence of literature on folding back that alludes to the kind of talk that learners engage in. Data were captured through video recording of learners' interactions as they worked on the tasks in different sessions. We present these data as transcribed extracts of talks that the learners held and synthesise them into stories through Polkinghorne's narrative mode of data analysis, also using a process that Kim referred to as narrative smoothing. Pirie and Kieren's conception of folding back and Wegerif and Mercer's three ways of talking and thinking among learners were used as a heuristic device for synthesising the stories. The narratives illustrate that exploratory talk promotes folding back, where learners build on each other's ideas to develop geometry understanding. Therefore, the significance of this article is that for classrooms that wish to promote growth in understanding through folding back, the type of talk that should be normative is exploratory talk.

Researchers have long debated the meaning of mathematical understanding and ways to achieve mathematical understanding. This study investigated experienced Chinese mathematics teachers’ views about mathematical understanding. It was found that these mathematics teachers embrace the view that understanding is a web of connections, which is a result of continuous connection making. However, in contrast to the popular view which separates understanding into conceptual and procedural, Chinese teachers prefer to view understanding in terms of concepts and procedures. They place more stress on the process of concept development, which is viewed as a source of students’ failures in transfer. To achieve mathematical understanding, the Chinese teachers emphasize strategies such as reinventing a concept, verbalizing a concept, and using examples and comparisons for analogical reasoning. These findings draw on the perspective of classroom practitioners to inform the long-debated issue of the meaning of mathematical understanding and ways to achieve mathematical understanding.

The understanding process of primary education students was analyzed when they solve tasks related to the concept of ratio. The study was based on the theoretical framework of Pirie and Kieren (1994). The methodology was qualitative with the case study method. The study was carried out in three stages: planning, development and analysis, using the field observation technique. Data collection was carried out through a task and an interview. The data were analyzed based on theoretical articulation. The results revealed that students lack the prior knowledge necessary to understand the concept of ratio. In conclusion, it can be noted that students do not present logical arguments to formalize the concept, and their understanding process is reduced to memorization or the use of mathematical strategies without understanding the relationship between the task and the mathematical concept.

Mathematical understanding continues to be one of the major goals of mathematics education. However, what is meant by “mathematical understanding” is underspecified. How can we operationalize the idea of mathematical understanding in research? In this article, I propose particular specifications of the terms mathematical concept and mathematical conception so that they may serve as useful constructs for mathematics education research. I discuss the theoretical basis of the constructs, and I examine the usefulness of these constructs in research and instruction, challenges involved in their use, and ideas derived from our experience using them in research projects. Finally, I provide several examples of articulated mathematical concepts.

Mathematicians suggest that some proofs are valued for their explanatory value. This has led to a philosophical debate about the distinction between explanatory and non-explanatory proofs. In this paper, we explore whether contrasting views about the explanatory value of proof are possible and how to understand these diverging assessments. By considering an epistemic and contextual conception of explanation, we can make sense of disagreements about explanatoriness in mathematics by identifying differences in the background knowledge, skill corpus, or epistemic aims of mathematicians or mathematical communities. We focus on the relation between explanation, epistemic aims and diverging explanatory assessments by looking at cases from mathematical practice.

This study explores relationships among prospective secondary teachers' skills of attending to relevant mathematics elements in students' answers, interpreting students' mathematical understanding, and proposing instructional actions. Thirty prospective secondary mathematics teachers analyzed three high school students' answers to three problems of derivatives of a function at a given point and proposed instructional actions to help them progress in their understanding. Findings indicate that the more prospective teachers identified links between the mathematical elements and characteristics of students' understanding, the more suitable the instructional activities were. Furthermore, our results suggest practical implications for teacher education programs since the type of task presented in this research may help to foster prospective secondary teachers' noticing of students mathematical understanding.

The changes in high school mathematics books in recent years have led to some changes in the concept of calculus, and majority of learners used to work hard on memorizing rather than understanding. In the early years after entering university, they will be challenged by the same concepts and their application in other sciences. Review of the previous studies have shown that no research instrument has been developed to address such challenges till now. Then, the researchers of the study developed and standardized an instrument. The population consisted of all first-year undergraduate students of engineering and basic sciences fields from of University in Tehran. Using purposeful sampling technique, 162 male and female students were selected. All of them selected had passed the course of calculus 1. The researchers considered some factors and developed an instrument and finally validated it based on those components. Moreover, since there were no available tools to measure the derivative’s relational and instrumental understanding, some questions regarding the concept of derivative definition were designed and added to the end of the questionnaire. Finally, after the investigations conducted through exploratory factor analysis, a research instrument was developed and introduced to measure the understanding of the concept of derivative and its instrumental and relational understanding. Los cambios en los libros de matemáticas de la escuela secundaria en los últimos años han llevado a algunos cambios en el concepto de cálculo, y la mayoría de los estudiantes solían trabajar duro para memorizar en lugar de comprender. En los primeros años después de ingresar a la universidad, serán desafiados por los mismos conceptos y su aplicación en otras ciencias. La revisión de los estudios anteriores ha demostrado que hasta ahora no se ha desarrollado ningún instrumento de investigación para abordar tales desafíos. Luego, los investigadores del estudio desarrollaron y estandarizaron un instrumento. La población estaba compuesta por todos los estudiantes de primer año de pregrado de ingeniería y ciencias básicas de la Universidad de Teherán. Usando una técnica de muestreo intencional, se seleccionaron 162 estudiantes varones y mujeres. Todos los seleccionados habían pasado el curso de cálculo 1. Los investigadores consideraron algunos factores y desarrollaron un instrumento y finalmente lo validaron con base en esos componentes. Además, dado que no había herramientas disponibles para medir la comprensión relacional e instrumental de la derivada, se diseñaron y agregaron algunas preguntas sobre el concepto de definición de derivada al final del cuestionario. Finalmente, después de las investigaciones realizadas a través del análisis factorial exploratorio, se desarrolló e introdujo un instrumento de investigación para medir la comprensión del concepto de derivada y su comprensión instrumental y relacional.

The frame concept from linguistics, cognitive science and artificial intelligence is a theoretical tool to model how explicitly given information is combined with expectations deriving from background knowledge. In this paper, we show how the frame concept can be fruitfully applied to analyze the notion of mathematical understanding. Our analysis additionally integrates insights from the hermeneutic tradition of philosophy as well as Schmid’s ideal genetic model of narrative constitution. We illustrate the practical applicability of our theoretical analysis through a case study on extremal proofs. Based on this case study, we compare our analysis of proof understanding to Avigad’s ability-based analysis of proof understanding.

This study investigates how eighth-grade students conceptualize the Pythagorean theorem through an instructional design grounded in realistic mathematics education (RME), analyzed using the APOS (action, process, object, schema) theoretical framework. Employing a design-based research methodology, a classroom activity was developed in which students used square tiles and right triangles to explore and ultimately discover the relationship now known as the Pythagorean theorem. The study was conducted with an entire eighth-grade classroom, and data-including group work observations and student interviews- were analyzed to trace students’ transitions across APOS stages. Three focal student cases were examined in depth to illustrate diverse developmental trajectories. The results indicate that while all students began with physical manipulation and informal reasoning, two students progressed to object-level understanding. The findings provide insights into how RME-based tasks can foster conceptual development and highlight areas for refining instructional design to better scaffold students’ transitions across cognitive stages.

This study aims to find a pattern of correcting mathematical understanding errors in solving mathematical problems for junior high school students. The method used in this research is qualitative. Data collection using triangulation technique: observation, test results, and interviews. Interviews were conducted on three subjects for one month at MTs. The instrument used is a mathematical problem-solving test. This study found patterns of improvement in mathematical problems in geometry, namely the area and perimeter of rectangles and lines. This study found that the pattern of mathematical understanding errors can be corrected by applying patterns such as reminding the material of rectangular shapes, parallelograms, algebraic forms, and units of measure, writing down everything that is known in the problem, and making an example of a variable if it cannot be written directly. The pattern of improving mathematical understanding can solve the mathematics problems of junior high school students. Furthermore, this pattern of mathematical understanding errors can be applied by teachers to improve math problems and embed geometric concepts in mathematics learning.