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This study explored the unique integration of mathematics tutors in teaching high school mathematics in a Virtual mathematics school (VMS). The tutors were three excelling STEM students who did not have any formal preparation for teaching before their work in the VMS. The goal of the study presented in this paper was to design a model of proficiency for these tutors. The model was designed using multiple case study methodology by tracking the learning through teaching of the tutors. Activity theory (Leontiev, 1978) framed the design of the study and led to a "goal-action" model of the tutors' proficiency in the VMS. The goals part of the model is rooted in the construct of students' mathematical potential (Leikin, 2021), and the action part was unraveled using the construct of a Teaching Triad (Jaworski in "For the Learning of Mathematics," 12(1), 8-14, 1992). In this paper, we illustrate how our model can be used to analyze changes in the tutors' proficiency, and how it can be applied to descriptive, explanatory, and analytical power.

Argumentative problem solving in mathematics classrooms is a crucial practice that supports important student learning goals via collaborative deliberation and consensus building, but also places substantial cognitive and affective demands on both students and teacher. In this in-depth qualitative study, we considered how students' emotions during argumentative discourse relate to their learning of real-life functional situations, an area of mathematics highlighted in the literature as foundational yet difficult. Observations, written reflections, and interviews of six Year 9 (14-15-year-old) students' experience of argument construction, critique, and consensus building were analyzed for insights into emotions experienced during small-group argumentative problem solving, and how various emotions might support or hinder students' mathematics learning. The emotion of frustration was experienced frequently and was found to facilitate persistence and the correction of misconceptions, but also to hinder the learning in contributing to ongoing tension and eventual resignation. Peer critique was found to stimulate negative emotions but positive learning processes, such as seeking other ways to explain one's reasoning, and re-evaluating one's own incorrect solutions.

This study explores secondary mathematics teachers' attention to the three components of Jaworski's (1992) "teaching triad" (i.e., "mathematical challenge," "learning management," and "sensitivity to students") as they planned a lesson revolving around a rich mathematics problem and assessed pre-designed student solutions for that problem. Written work was gathered from 17 cohorts. Qualitative analysis generated categories. Quantitative analysis revealed that some components of the teaching triad were attended to in both activities, some were not attended to in either activity, and some were attended to in one activity but not in the other. Findings are interpreted in light of theory.

The aim of this study is to explore Israeli high school graduates' mathematical explanations for the spread of the coronavirus, given that the mathematics required to do so was part of their school curriculum. An online questionnaire consisting of two sections provided a variety of potential framings for explaining the phenomenon. The first section invited the participants to explain the spread of the coronavirus in terms of their school majors in general, with no specific reference to mathematics. The second section asked explicitly to explain the mathematical context underlying the phenomenon. In this section, the participants were asked to discuss the Prime Minister's speech given in the media a few weeks earlier, in which he described the spread of the coronavirus as a geometric series. Data analysis of 87 participants' responses to the questionnaire revealed 11 different mathematical ideas used to explain the spread of the coronavirus. These ideas included are as follows: doubling, sequence, exponential growth, using powers, tree diagram, recursion, fast-growing rate with covariation, probability, parabola and quadratic function, acceleration, and factorial. It was also found that the second section of the questionnaire elicited a wider range of mathematical ideas than the first one. We suggest possible explanations for the emergence of the mathematical ideas, which seem to reflect the graduates' intuitive knowledge, influenced not only by their mathematics track level but also by their chosen high school majors. Possible implications are discussed.

Formative assessment practices for secondary mathematics have been advocated as valuable for students, but difficult for teachers to learn. There have been calls in the literature to increase the emphasis on formative assessment in mathematics teacher preparation courses. This study explored the use of peer-assessment strategies for helping pre-service secondary mathematics teachers (PSTs) cultivate formative assessment principles and practices for assessing school students. Twenty-seven PSTs participated in a peer-assessment cycle comprised of: sourcing a rich mathematics task; constructing an assessment rubric for it; and collecting and analysing a selection of secondary student responses to the task. Each PST then provided written and verbal feedback to a peer on his/her rubric and student solution assessments. We draw on theoretical conceptions of Teacher Assessment Literacy in Practice to characterize the PSTs' perceptions of their experience of formative assessment processes for learning to assess school students, in terms of cognitive and affective dimensions of their conceptions of assessment. The cohort evidenced a wide range of levels of confidence with the various aspects of formative assessment practices but on average less confidence in assessing school student task responses themselves than in assessing peer work. In addition to highlighting specific changes to different types of assessment knowledge, the PSTs also evidenced an awareness of shifts in their attitudes, in coming to view student task responses with more appreciation and humility.

Researching students' responses to tasks at different year levels and in varied curriculum contexts can provide insights that relate their understandings to prior learning experiences and teaching approaches. In this article, we discuss evidence of students from three curriculum contexts (English, Australian, and Israeli) (n = 350) ways of attending to the functional relationship between two variables at different levels of secondary school in their responses to two linear functions tasks. We found that the students from an English national curriculum context were more likely than the other cohorts to focus only on the dependent variable when presented with a table of values. With a figural pattern generalization task, no examples of invalid proportional reasoning were found among student responses from the Israeli curriculum context, unlike those in the English and Australian curriculum contexts. A high percentage of the Year seven-eighths responses from the Israeli curriculum context evidenced symbolic generalization whereas several responses from the Australian curriculum indicated explicit descriptive, but not symbolic, generalization. Twenty teachers also participated in the study. Analysis of national curriculum content and task examples from each context, along with the teachers' expectations of their students' responses to the tasks and described teaching approaches for linear functions, led to our exploration of possible reasons for differences and similarities found among the responses. We suggest possible implications for teaching and learning linear functions and avenues for further research.

A foundational component of developing algebraic thinking for meaningful calculus learning is the idea of "function" that focuses on the relationship between varying quantities. Students have demonstrated widespread difficulties in learning calculus, particularly interpreting and modeling dynamic events, when they have a poor understanding of relationships between variables. Yet, there are differing views on how to develop students' functional thinking over time. In the Australian curriculum context, linear relationships are introduced to lower secondary students with content that reflects a hybrid of traditional and reform algebra pedagogy. This article discusses an investigation into Australian secondary students' understanding of linear functional relationships from Years 7 to 12 (approximately 12 to 18 years old; n = 215) in their approaches to three tasks (finding rate of change, pattern generalisation and interpretation of gradient) involving four different representations (table, geometric growing pattern, equation and graph). From the findings, it appears that these students' knowledge of linear functions remains context-specific rather than becoming connected over time.

Our research focuses on ways teachers envision argumentation in their secondary mathematics classrooms. Semi-structured interviews were conducted with 31 secondary school mathematics teachers in Israel. They were asked to envisage a hypothetical classroom dialogue focusing on mathematical tasks which would engage their students in some argumentative activity. Analysis of the teachers’ responses identified categories related to both structural and dialogic aspects associated with three dimensions of argumentation: (1) task characteristics, (2) teaching strategies, and (3) students’ characteristics. From a crossanalysis of the teachers’ statements certain categories appeared more frequently than others. We interpret our results from a theoretical and practical perspective and make recommendations for areas of future research. [ABSTRACT FROM AUTHOR]

Whole-class discussions in mathematics are envisioned as spaces for the sharing of ideas and making connections among them. We pursue how Palestinian/Arab Israeli teachers consider making use of multiple solutions to a task: which three they indicate that they would select for a whole-class discussion and in what sequence. Under the hypothetical assumption that each solution has been authored by boys and girls (with below-average and with above-average grades), how do teachers consider distributing opportunities to present the designated solutions? Participants most commonly selected a direct model, an error, or an inductive approach supported by a geometric representation, typically in that sequence (or error, direct model, inductive). Participants tended to indicate that they would invite a girl with below-average grades to present the direct model; a student with above-average grades to present the error; and a boy with above-average grades to present the inductive/geometric solution. Our analysis extends to participants' explanations of their choices and illuminates their intentions: making use of existing status arrangements, leveraging existing hierarchies to benefit others in the class, or modifying the indicated student's status by making space for them to participate on the public floor. Our analysis highlights implications of these patterns, especially in terms of classroom opportunities to learn. [ABSTRACT FROM AUTHOR]

This paper arises from a study of how concepts related to understanding functions develop for students across the years of secondary/high school, using small samples from two different curricula systems: England and Israel. We used a survey consisting of function tasks developed in collaboration with teachers from both curriculum systems. We report on 120 higher achieving students, 10 from each of English and Israeli, 12-18 years old. Iterative and comparative analysis identified similarities and differences in students' responses and we conjecture links between curriculum, enactment, task design, and students' responses. Towards the end of school, students from both curriculum backgrounds performed similarly on most tasks but approached these by different routes, such as intuitive or formal and with different understandings, including correspondence and covariational approaches to functions.

This study aims to explore the conceptualisations of function that some students express when they are responding to fictitious students' statements about functions. We also asked them what is meant by "function" and many voluntarily used examples in their responses. The task was developed in collaboration with teachers from two curriculum systems, England and Israel. It was given to 10 high-achieving English students from each of the years 10-13 and to 10 high-achieving Israeli students from comparable years (total of 80 students). Data analysis included identifying students' dominant ideas for functions as expressed in their responses, and analysing the types of examples that students used to explain their responses. Differences found between the samples from the countries led to conjectures about the influence of curriculum and teaching, and in particular, about the role of word, in this case "function", in concept image development. Whereas most students showed that they had a meaning for the word, those students whose relevant experience of earlier concepts had been organised around the word "function" generally showed stronger understanding of function as object.

This study examines expressions of reasoning by some higher achieving 11 to 18 year-old English students responding to a survey consisting of function tasks developed in collaboration with their teachers. We report on 70 students, 10 from each of English years 7-13. Iterative and comparative analysis identified capabilities and difficulties of students and suggested conjectures concerning links between the affordances of the tasks, the curriculum, and students' responses. The paper focuses on five of the survey tasks and highlights connections between informal and formal expressions of reasoning about variables in learning. We introduce the notion of "schooled" expressions of reasoning, neither formal nor informal, to emphasise the role of the formatting tools introduced in school that shape future understanding and reasoning.

Identifying and expressing relations between quantities is a key aspect of understanding and using functions. We are aiming to understand the development of functions understanding throughout school years in Israel. A survey instrument was developed with teachers and given to 20 high and average achieving students from each of years 7-11 and to 10 high achieving students from year 12, a total of 110 students. Our analytical approach is to identify qualitatively what students appeared to do and whether their approaches led to complete solutions. We look for progress in understanding variables and relations between them, and we found that there does not appear to be a strong link between curriculum and informal understandings of variables and covariation, but there are other strengths.

This study examines how students' opportunities to engage in argumentative activity are shaped by the teacher, the class, and the mathematical topic. It compares the argumentative activity between two classes taught by the same teacher using the same textbook and across two beginning algebra topics--investigating algebraic expressions and equivalence of algebraic expressions. The study comprises two case studies in which each teacher taught two 7th grade classes. Analysis of classroom videotapes revealed that the opportunities to engage in argumentative activity with the topic investigating algebraic expressions were similar in each teacher's two classes. By contrast, substantial differences were found between one teacher's classes with regard to the opportunities to engage in argumentative activity with the topic equivalence of algebraic expressions. The discussion highlights how the interplay between the characteristics of the mathematical topic, the characteristics of the class, and the characteristics of the teacher contributed to the shaping of students' opportunities to engage in argumentative activity.

This study investigates students' ways of attending to linear sequential data in two tasks, and conjectures possible relationships between those ways and elements of the task design. Drawing on the substantial literature about such situations, we focus for this paper on linear rate of change, and on covariation and correspondence approaches to linear data. Data sources included a survey instrument of six tasks that was developed in collaboration with a group of teachers, and the tasks for this paper are two concerned with linear functions. The whole survey was given to 20 students from each of UK years 7-11 and 10 students from each year 12-13 (total of 120 students). Our analytical approach was to identify what all students appear to do, not how correct they were or what pre-determined methods they might use. Our analysis uses theories of dual-process and dynamic graded continuum to suggest conjectures about how students' capabilities in acting with sequential data depend to some extent on task features, as well as on curriculum and pedagogy.

This study examines the views of people involved in mathematics education regarding the commonly stated goal of using mathematics learning to develop deductive reasoning that is usable outside of mathematical contexts. The data source includes 21 individual semi-structured interviews. The findings of the study show that the interviewees ascribed different meanings to the above-stated goal. Moreover, none of them said that it is possible to develop formal logic-based reasoning useful outside of mathematics, but for different reasons. Three distinct views were identified: the intervention-argumentation view, the reservation-deductive view, and the spontaneity-systematic view. Each interviewee's view was interrelated with the interviewee's approach to deductive reasoning and its nature in mathematics and outside it.