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This paper aims to broaden the scope of the construct concept image by addressing the mathematically oriented notion of "solution of an algebraic equation." The notion solution of an algebraic equation is not mathematical per se, but it is most prevalent in mathematical communications and in mathematics education. The paper describes five episodes of lesson segments that were collected in a secondary school mathematics teaching certificate program. The specific tasks, the episodes, and the related class discussions, involving the notion of solution of algebraic equation, are described. Several issues, arising from the various concept images of solution of equations, including miscommunication, ambiguity about correctness, ranking correctness, and minimizing errors, are discussed. Along an analysis of these data, 4 suggestions are offered for improving the communication around the notion solution of an algebraic equation: promote awareness, offer instructional guidelines, suggest a vocabulary, and be sensitive.

In this article, we examine how 10th graders make use of different intertwined representations when solving algebraic word problems. It is structured in a quantitative phase, with 61 students and a subsequent follow-up case study with nine pairs of students. We found that drawings and tables, used as auxiliary representations within variants of the algebraic solving method, were useful in the construction of situation and/or problem models. However, this did not entail a higher rate of correct solutions, because students made errors in the subsequent conversion to equations due to misconceptions about the notion of equation. Teaching implications are discussed.

To improve students' mathematics achievement, their errors should be treated as an opportunity to stimulate conceptual and procedural understanding. Using a teaching scenario, this study investigated 40 high school teachers' analyses of and responses to a student error(s) in solving a quadratic equation by using the factoring method. The teachers' responses were analyzed quantitatively and qualitatively. Analyses results revealed a gap between identified errors and addressed ones, as well as a gap between interpreting a knowledge weakness and explaining a knowledge focus. While the errors came from conceptual aspects of the factoring method, most of the participants offered conceptual explanations addressed in a teacher-centered way. The results also revealed that student errors and mathematical knowledge were two foci of the teachers' responses to students. The broader implications of the study for the international community are discussed in accordance with the findings.

In linear programming, many students find it difficult to translate a verbal description of a problem into a valid mathematical model. To better understand this, we examine the existing characteristics of college engineering students' errors across linear programming (LP) problems. We examined textbooks to identify the types of problems typically found in introductory linear programming courses. We then developed a comprehensive set of tasks and analyzed students' work to create a taxonomy of the errors and issues that students exhibited. From our findings, we define four categories of identified error types: (1) decision variable errors, (2) variable relationship errors, (3) notation errors, and (4) form errors. This study contributes to the research by investigating students' work in an area of undergraduate mathematics that has not been heavily explored before. Findings suggest specific areas of focus for future work in helping students develop their understanding of linear programming models and mathematical modeling in word problems in general.

Teacher empathy has proven to be highly relevant for the educational process and for educational outcomes, while integrating it into the teacher practicing process is challenging. This study suggests using PSTs' (pre-service students) incorrect responses as a pedagogical tool for improving empathy awareness among them so that PSTs can feel empathy and not merely learn about empathy. Incorrect responses to conservation tasks, recorded in the process of interviewing 47 PSTs who major in science education, are used as a first step research. A possible approach to the arousal of empathy during the interview was unexpectedly revealed to learn about the PSTs' incorrect responses to conservation tasks. Confronting PSTs with their incorrect responses resulted in feelings of embarrassment, confusion, and even loss of self-confidence. Such feelings may help in evoking an attitude of empathy toward student errors, increasing PSTs' awareness to tacit knowledge and to students' tacit feelings and integrating bottom-up vs. top-down processes during PSTs' reflective practice; and might be regulated effectively to better instruction.

When translating word problems into equations, a common error consists of using the same letter to refer to two different quantities. This error, called multiple referents for the unknown, can be connected to the existence of deictic or indexical expressions within the problem statement. This paper aims to analyse the influence of indexical expressions on the incidence of the multiple referents for the unknown error. Results from a quantitative study with 117 Spanish secondary students show a significantly higher number of such type of error when the problem statement contains indexical expressions. The analysis of students' performances in subsequent interviews indicates that this error may be due to the fragmented reading of statements.

The present study uses the intuitive rules theory as a framework to examine whether some of the difficulties in dealing with errors and uncertainties observed among students in the university physics laboratory can stem from their use of intuitive rules. The study also examines the relationship between the use of intuitive rules and laboratory experience level. Based on interviews with 20 students and a research sample, which included 52 students, having different laboratory experiences, our findings show that students used the intuitive rules in the tasks they were provided. Moreover, students with less laboratory experience used the intuitive rules more frequently than students with more laboratory experience. Understanding the influence of intuitive rules on students' performance, when dealing with experimental data, may be a great help to educators in designing better learning environments to address related science and engineering practices.

In this article, we present the results of a research study that explores secondary students' capacity to perform translations of algebraic statements between the verbal and symbolic representation systems through the lens of errors. We classify and compare the errors made by 2 groups of students: 1 at the beginning of their studies in school algebra and another 1 completing their studies on algebra in compulsory education. This comparison allows us to detect errors which require specific attention in instruction due to its persistence and to identify errors that disappear as students advance in their study of algebra. The results and conclusions have a pedagogic value to inform instruction and also lead to backed conjectures and research questions to push forward research on student's translation capacity and students' knowledge of algebraic symbolism.

Recent research on the phenomenon of improper proportional reasoning focused on students' understanding of elementary functions and their external representations. So far, the role of basic function properties in students' concept images of functions remained unclear. We add to this research line by investigating how accurate students are in connecting functions to their corresponding properties and how this accuracy depends on function types and representations. A large group of 10th graders evaluated for different function types, represented in either a graphical, a formulaic, or a tabular mode, the correctness of statements about their general properties and behavior. Results show that students succeeded rather well in making the right connections between properties and functions. Errors depended not only on the type of function for which the properties were evaluated but also on the kind of representation in which the function was presented. These results highlight the importance of function properties in students' concept images of functions and suggest positive effects of making these properties explicit to students.

This study examines how third-graders' conceptions about the origin of species are affected by formal instruction and whether children can learn not only about evolution but also about natural selection. We interviewed the same group of third-grade children (8-9 years old) twice, before and after following a curriculum about these topics. Creationist answers were very rare at both pre- and post-tests. The number of evolutionist answers increased significantly after instruction. However, children learned about evolution in a piecemeal way; they gave evolutionist answers for only some of the animals they were asked about and made several errors when putting classes of vertebrates on a time line. In addition, only a minority of children learned something about evolutionary mechanisms. They said that new-born animals can be different from the parents because of mutations but did not mention differential survival. These results challenge the views, according to which, cognitive biases (i.e. essentialism, teleological and intentional thinking) make children unreceptive to evolution.

The purpose of the present study was to portray students' misconceptions and errors while solving conceptually and procedurally oriented tasks involving length, area, and volume measurement. The data were collected from 445 sixth grade students attending public primary schools in Ankara, Türkiye via a test composed of 16 constructed-response format tasks. The findings revealed a wide range of misconceptions and errors such as "believing that all rulers are 30 cm long," "confusing area formula with perimeter formula," "believing a box has more than one surface area," "using the volume formula for surface area," "believing that ruler must be longer than the object measured," etc. These misconceptions and errors could be considered as the evidences indicating the sixth graders' lack of comprehension of the fundamental concepts of spatial measurement and their relationships and the procedures and formulas used for measuring length, area, and volume. The possible causes of such misconceptions and overcoming ways were also discussed.

This study focuses on teacher learning of student thinking through grading homework, assessing and analyzing misconceptions. The data were collected from 10 teachers at fifth-eighth grade levels in the USA. The results show that assessing and analyzing misconceptions from grading homework is an important approach to acquiring knowledge of students' thinking. By engaging in the inquiry process of the 4 steps of identifying errors, analyzing reasons for the errors, designing approaches for correction, and taking action for correction, the teachers made obvious progress in their knowledge of students' thinking, understood the difficulties and challenges their students had in learning mathematics, and enhanced their pedagogical content knowledge.

The high-stakes Lebanese Brevet examinations are undertaken by almost all school students at the end of year 9 and include papers in Chemistry and Physics. This research presents an analysis of the 2007 examinations in these two science subjects using official statistics and response patterns arising from samples of candidate scripts. The examinations themselves appeared to be a significant source of student shortcomings, and recommendations are put forward to address their deficiencies and exploit their potential feedback functions.