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In this paper, we report on the outcomes of a survey of students across 12 higher education institutions in Ireland and the UK. Participants were asked to give their views on what tutors should do and not do when they are providing Mathematics Learning Support to students. We present the categories that emerged from an analysis of student responses, including tutor behaviour, teaching methods, content knowledge and management items. We discuss the implications that these responses have for tutor selection and training, how they relate to mathematical pedagogy and we consider the high expectations that students appear to have for tutors, who are providing this supplemental support. [ABSTRACT FROM AUTHOR]

This study examined the Core Mathematics assessment of the West African Senior School Certificate Examination (WASSCE) by the West African Examination Council (WAEC) in Ghana for Depth of Knowledge and Context Characteristics over a 20-year period. In all, 1245 assessment items were analysed. The results indicated that nearly 80% of the items assessed students’ ability to either recall basic facts or perform straightforward routine procedures. In terms of the context characteristics, the results showed that about 70% of the items was framed within abstract context, with about 15% each framed within semi-reality and real-life reference contexts, respectively. Overall, the results indicated that the nature of the WASSCE Core Mathematics has not changed much over the 20-year period in terms of depth of knowledge and context characteristics. Implications of the results for teaching and learning of school mathematics, as well as recommendations for the Ministry of Education and WAEC, are provided. [ABSTRACT FROM AUTHOR]

Given the vast experiences pre-service mathematics teachers (PSMTs) have with functions, one may assume that they have existing personal definitions and concept images. The purpose of this paper is to build an understanding of the variety of images that one might associate with a definition of function that includes 'for each input there is one output' and how these images are coordinated with the definition to determine whether an object is a function or not. Two cases are presented to describe PSMTs who were working from the same personal definition of function, yet used that definition differently within a task designed around the function concept. The task itself was novel in that it is presented in a non-algebraic context, namely a vending machine. As such, PSMTs were required to coordinate their personal definition and images within the novel context. Attending to PSMTs' concept images revealed that though they were working from the same personal definition, images they drew upon to determine whether or not an object is a function were varied. [ABSTRACT FROM AUTHOR]

This research investigates the influence that gender, single-sex and co-educational schooling can have on students’ mathematics education in second-level Irish classrooms. Although gender differences in mathematics education have been the subject of research for many years, recent results from PISA (Programme for International Student Assessment) show that there are still marked differences between the achievement and attitude of male and female students in Irish mathematics classrooms. This paper examines the influence of gender in more detail and also investigates the impact of single-sex or co-educational schooling. This is a follow on study which further analyses data collected by the authors when they designed a pedagogical framework and used this to develop, implement and evaluate a teaching intervention in four second-level Irish schools. The aim of this pedagogical framework was to promote student interest in the topic of algebra through effective teaching of the domain. This paper further analyses the quantitative data collected and investigates whether there were differences in students’ enjoyment and achievement scores based on their gender and whether they attended single-sex or co-educational schools. [ABSTRACT FROM PUBLISHER]

Proof and proving are important components of school mathematics and have multiple functions in mathematical practice. Among these functions of proof, this paper focuses on the discovery function that refers to invention of a new statement or conjecture by reflecting on or utilizing a constructed proof. Based on two cases in which eighth and ninth graders engaged in proofs and refutations, we demonstrate that facing a counterexample of a primitive statement can become a starting point of students’ activity for discovery, and that a proof of the primitive statement can function as a useful tool for inventing a new conjecture that holds for the counterexample. An implication for developing tasks by which students can experience this discovery function is mentioned. [ABSTRACT FROM PUBLISHER]

The availability of sophisticated computer programs such as Wolfram Alpha has made many problems found in the secondary mathematics curriculum somewhat obsolete for they can be easily solved by the software. Against this background, an interplay between the power of a modern tool of technology and educational constraints it presents is discussed. Using topics from algebra (equations) and elementary number theory (summation of powers of integers), the paper suggests ways of developing problems that are both technology-immune and technology-enabled in the sense that whereas software can facilitate problem solving, its direct application is not sufficient for finding an answer. Stemming from the author's work with secondary mathematics teacher candidates, this paper highlights the appropriate use of technology as support system for multiple ways of knowing and knowledge construction in the modern classroom. [ABSTRACT FROM AUTHOR]

A square of side length 2a, a> 0 is selected in the Euclidean plane. For each a> 0, two types of hexagons are constructed using the selected square. They are labelled type-1 hexagon and type-2 hexagon. For each a> 0, there are two type-1 hexagons and two type-2 hexagons. In relation to the selected square, they appear to be horizontal and vertical. The VKS ellipses circumscribe type-1 hexagons and type-2 hexagons. The construction of these two types of hexagons and the determination of the Cartesian equations of the circumscribing VKS type-1 and VKS type-2 ellipses form the main themes for this paper. The eccentricities of these VKS type-1 ellipses and VKS type-2 ellipses are also discussed. The author hopes that the construction of the hexagons and the ellipses will be useful to students in high schools and colleges. [ABSTRACT FROM PUBLISHER]

The political importance of mathematics in post-16 education is clear. Far less clear is how mathematics does and should relate to vocational education. Successive mathematics curricula (e.g. core skills, key skills) have been developed in England with vocational learners in mind. Meanwhile, general mathematics qualifications remain largely disconnected from vocational learning. Following a brief historical survey of mathematics within vocational education, the paper presents findings from a nested case study of student groups in three large Further Education colleges in England. The primary unit of analysis herein is student groups learning Functional Mathematics in two vocational areas: construction and hairdressing. We show how approaches to organising teaching, developing connected curricula and classroom pedagogy tend toisolateorintegratemathematics from/with the vocational experience.Integratedapproaches are shown to impact positively on student engagement and attitudes to learning mathematics. The paper concludes by discussing the potential impact of academic qualifications displacing vocationally relevant mathematics. [ABSTRACT FROM PUBLISHER]

In this paper, situated in the context of elementary and secondary school mathematics curriculum in India, I argue that centrally determined uniform curriculum would have very little to offer a large majority of students in a country that is diverse in many ways. [ABSTRACT FROM PUBLISHER]

This study investigates the potential of video analysis and a mathematical knowledge for teaching framework, the Knowledge Quartet (KQ), in mathematics teacher education programmes. It reports on the effectiveness of these tools in analysing and supporting secondary level pre-service mathematics teachers’ subject matter knowledge and pedagogical content knowledge. This paper describes how a videotaped lesson of one pre-service teacher, teaching a class of mature students, was analysed and makes comparisons between the teacher educators’ and the pre-service teacher's observations. Inter-rater reliability was investigated and a Kappa coefficient of .72 indicated substantial agreement between both coders. Findings are presented and implications of the use of video and the KQ for mathematics teacher education are drawn. [ABSTRACT FROM PUBLISHER]

In this paper, we give an account of the mathematics that took place in a high-school classroom in connection with two extremely simple models for the spread of an infectious disease within a population. We also provide some ideas as to how the model might be made somewhat more realistic, and briefly discuss the possibility for developing more sophisticated models. [ABSTRACT FROM AUTHOR]

In this paper, research on some problematic aspects high school students have in learning trigonometry is presented. It is based on making sense of mathematics through perception, operation and reason in the case of trigonometry. We analyzed students' understanding of trigonometric concepts in the frame of triangle and circle trigonometry contexts, as well as the transition between these two contexts. In the conclusion, we present some new problematic aspects we noticed. The research was carried out with two groups of high school students, one of them at the beginning of their trigonometry learning (17 years old) and the other at the end of their high school education (19 years old). The students were given a questionnaire similar to that of Chin and Tall, and we analyzed the students' response. In our research, we noticed that students have difficulties with properties of periodicity and the fact that trigonometric functions are not one-to-one. In addition, there is poor understanding of radian measure and a lack of its connection to the unit circle. [ABSTRACT FROM AUTHOR]

This case study investigates the impact of the integration of information and communications technology (ICT) in mathematics visualization skills and initial teacher education programmes. It reports on the influence GeoGebra dynamic software use has on promoting mathematical learning at secondary school and on its impact on teachers’ conceptions about teaching and learning mathematics. This paper describes how GeoGebra-based dynamic applets – designed and used in an exploratory manner – promote mathematical processes such as conjectures. It also refers to the changes prospective teachers experience regarding the relevance visual dynamic representations acquire in teaching mathematics. This study observes a shift in school routines when incorporating technology into the mathematics classroom. Visualization appears as a basic competence associated to key mathematical processes. Implications of an early integration of ICT in mathematics initial teacher training and its impact on developing technological pedagogical content knowledge (TPCK) are drawn. [ABSTRACT FROM PUBLISHER]

This paper describes an experience of same-age peer tutoring conducted with 19 eighth-grade mathematics students in a secondary school in Castellon de la Plana (Spain). Three constructs were analysed before and after launching the program: academic performance, mathematics self-concept and attitude of solidarity. Students’ perceptions of the method were also analysed. The quantitative data was gathered by means of a mathematics self-concept questionnaire, an attitude of solidarity questionnaire and the students' numerical ratings. A statistical analysis was performed using Student's t-test. The qualitative information was gathered by means of discussion groups and a field diary. This information was analysed using descriptive analysis and by categorizing the information. Results show statistically significant improvements in all the variables and the positive assessment of the experience and the interactions that took place between the students. [ABSTRACT FROM PUBLISHER]

Previous studies have produced several typologies of teacher questions in mathematics. Probing questions that ask students to explain are often included in the types of questions. However, only rare studies have created subtypes for probing questions or investigated how questioning differs depending on whether technology is used or not. The aims of this study are to elaborate on different ways of asking students to give explanations in inquiry-based mathematics teaching and to investigate whether questioning in GeoGebra lessons differs from questioning in other lessons. Data was collected by video recording 29 Finnish mathematics student teachers’ lessons in secondary and upper secondary schools. The lesson videos were coded for the student teachers’ probing questions. After this, categories for the types of probing questions were created, which is elaborated in this paper. It was found that the student teachers who used GeoGebra emphasized conceptual probing questions during the explore phase of a lesson slightly more than the other student teachers. [ABSTRACT FROM AUTHOR]

Participating in classroom mathematics conversations, particularly engaging with the opinions of others, can improve learners' mathematical comprehension. Teachers can use a variety of invitation and follow-up moves to encourage student engagement. The study aimed to explore teacher follow-up on learners' initial responses to teacher questions and building on learner ideas in Grade 9 mathematics classrooms. A qualitative approach and descriptive case study design were utilised. Data were collected from three schools in the same circuit in South Africa through classroom lesson observations, face-to-face teacher interviews and researchers' field notes. Three Grade 9 mathematics teachers, one from each school, were purposively selected, and the selected teachers' already established Grade 9 classes, with a maximum of 30 learners each (owing to the COVID-19 health protocol), were automatically involved. Cognitively guided instruction, a learner-centred approach that uses social constructivism as a research paradigm, guided this study. The exploration was supplemented by Brodie's conceptual framework on teacher follow-up moves. The findings revealed that the three teachers followed up on learner contributions and used different follow-up moves, namely insert, elicit, press, maintain and confirm. The study showed hybrid teacher moves. Although more teacher moves were skewed towards elicit and insert, which are closer to traditional practices, there is an indication of moving towards the press, maintain and confirm, which are reform-oriented moves. In addition to these moves identified by Brodie, acknowledging or praising the learner for responding was another move that emerged in this study. [ABSTRACT FROM AUTHOR]

An important and interesting area in the study of triangle geometry is the related issue of extrema problems and inequalities. These problems play a significant role in the mathematics study program in high school. In tasks such as these, the difficulty level is high when one does not know in advance what the expected answer is. When one knows what to prove, the difficulty level is lower and most of the effort is aimed at attaining a proof of the expected answer. This can be done using dynamic geometric software. The possibility of making frequent changes to the geometric objects and the ability of dragging objects, contributes to the process of deducing properties, checking hypotheses and generalizing. In this paper, eight investigative tasks in Euclidean geometry are presented together with the applets developed for carrying out the dynamic investigation. Some of the tasks are well known, while others are almost unknown and are worthy of presentation as enrichment for those interested in the subject. The tasks were given to preservice teachers of mathematics as part of an advanced course for integrating technological tools in the teaching of mathematics. [ABSTRACT FROM PUBLISHER]

Access to advanced study in mathematics, in general, and to calculus, in particular, depends in part on the conceptual architecture of these knowledge domains. In this paper, we outline an alternative conceptual architecture for elementary calculus. Our general strategy is to separate basic concepts from the particular advanced techniques used in their definition and exposition. We develop the beginning concepts of differential and integral calculus using only concepts and skills found in secondary algebra and geometry. It is our underlining objective to strengthen students' knowledge of these topics in an effort to prepare them for advanced mathematics study. The purpose of this reconstruction is not to alter the teaching of limit-based calculus but rather to affect students' learning and understanding of mathematics in general by introducing key concepts during secondary mathematics courses. This approach holds the promise of strengthening more students' understanding of limit-based calculus and enhancing their potential for success in post-secondary mathematics. [ABSTRACT FROM PUBLISHER]

Literature suggests using multimedia learning principles in the design of instructional material. However, these principles may not be sufficient for the design of learning objects for concept learning in mathematics. This paper reports on an experimental study that investigated the effects of an instructional approach, which includes two teaching techniques – (a) variation theory and (b) representations of subject matter – on the design of learning objects for secondary school algebra concept learning. The results of this study showed that the experimental group performed significantly better than the control group on algebra learning achievement. The results also showed that only the experimental design with the addition of the instructional approach resulted in higher-order mathematical thinking skills and improved procedural skills of the students. Further analysis reveals that concept learning was simplified when multimedia learning principles were applied and the information was presented by the instructional approach. [ABSTRACT FROM AUTHOR]

Collaborative problem solving, when students work in pairs or small groups on a curriculum-related task, has become an increasingly common feature of classroom education. This paper reports a study of a topic which has received relatively little attention: how teachers can most usefully intervene when students are working in a group, but have encountered some sort of problem. The data used comes from a large-scale interventional study of mathematics and science teaching in secondary schools in south-east England, in which interactions between teachers and students were recorded in their usual classrooms. We identify the typical problem situations which lead to teachers’ interventions, and describe the different ways teachers were observed to intervene. We examine the different types of intervention, and consider how effective they are in helping group work proceed in a productive manner. Finally, we offer some conclusions about the practical implications of these findings. [ABSTRACT FROM AUTHOR]

This paper investigates the validity of a national, standards-based teacher evaluation programme by examining the relationship between teachers’ evaluation results and their students’ learning progress. We used census achievement data that assessed the same cohort of students at the end of 8th and 10th grade. We applied multilevel modelling and calculated correlations between teachers’ value-added estimates and their results based on the evaluation system’s multiple measures of teaching quality, which include a portfolio of classroom teaching artefacts and a video-taped lesson, a peer interview, supervisor assessments and a self-assessment. Our study finds a medium-size positive relationship between value-added estimates in mathematics and teachers’ portfolio results, while relationships with the other instruments are weaker. Correlations are lower for Spanish, but showing a similar pattern across instruments. We also find significant yet weak relationships between value-added estimates and three of six emerging factors of teaching practice based on the evaluation’s portfolio data. [ABSTRACT FROM PUBLISHER]

This study integrates augmented reality (AR) technology into teaching activities to design a learning system that assists junior high-school students in learning solid geometry. The following issues are addressed: (1) the relationship between achievements in mathematics and performance in spatial perception; (2) whether system-assisted learning can improve the spatial perceptions of students; (3) whether students with high, average and low academic achievement learn effectively after taught with system assistance; (4) system usability; (5) system task load; and (6) the relationship among various factors. Study participants were 76 students from Tainan City, Taiwan. Qualitative and quantitative data are obtained using pre- and post-system-assisted learning paper–pencil tests, a system usability scale, National Aeronautics & Space Administration Task Load indeX, and observations and focus group interviews. Analysis results indicate that students' math scores and spatial perception during the test closely correspond to each other. Additionally, system-assisted can improve the spatial perceptions of student. Regarding group differences related to the significance of impact, students with average and low academic achievements exhibit small and medium levels of effectiveness, but are insignificant in high academic achievement students. As for system usability, students with low academic achievements demonstrate a positive attitude toward the intensive programs provided by the system. However, the correlation between students’ learning effectiveness and system usability/task load is low. [ABSTRACT FROM PUBLISHER]

This paper identifies the amount of variance in mathematics achievement in high- and low-achieving schools that can be explained by school-level factors, while controlling for student-level factors. The data were obtained from 2679 Iranian eighth graders who participated in the 2007 Trends in International Mathematics and Science Study. Of the total sample, 1422 and 1257 students were from high- and low-achieving schools, respectively. Two-level hierarchical linear modelling was applied. The results indicated that of the total variance in mathematics achievement, 27.95 and 6.70% were due to between-school differences in the high- and low-achieving schools, respectively. Controlling for the school-level factors, the better-performing students were those with a higher level of confidence in learning mathematics in both samples. After controlling for the student-level factors, inadequacies in school resources and school type yielded the strongest link to achievement in the high- and low-achieving schools, respectively. [ABSTRACT FROM PUBLISHER]

This paper reports a qualitative research that identifies Mexican high school students’ social representations of mathematics. For this purpose, the social representations of ‘mathematics’, ‘learning mathematics’ and ‘teaching mathematics’ were identified in a group of 50 students. Focus group interviews were carried out in order to obtain the data. The constant comparative style was the strategy used for the data analysis because it allowed the categories to emerge from the data. The students’ social representations are: (A) Mathematics is…(1) important for daily life, (2) important for careers and for life, (3) important because it is in everything that surrounds us, (4) a way to solve problems of daily life, (5) calculations and operations with numbers, (6) complex and difficult, (7) exact and (6) a subject that develops thinking skills; (B) To learn mathematics is…(1) to possess knowledge to solve problems, (2) to be able to solve everyday problems, (3) to be able to make calculations and operations, and (4) to think logically to be able to solve problems; and (C) To teach mathematics is…(1) to transmit knowledge, (2) to know to share it, (3) to transmit the reasoning ability, and (4) to show how to solve problems. [ABSTRACT FROM PUBLISHER]

This paper explored variation of strategy use in pattern generalization across different generalization types and across grade level. A test was designed to assess students’ strategy use in pattern generalization tasks. The test was given to a sample of 1232 students from grades 4 to 11 from five schools in Lebanon. The findings of this study showed that the frequency of student strategy use differed according to the generalization type and that the functional and recursive strategies were frequently used in each of the generalization tasks. The use of the recursive strategy decreased, whereas, the use of the functional strategy increased, as the demands of the task changed from constructing a step-by-step solution to finding a general formula. The findings also showed that the use of the recursive strategy increased across grades 4–8 and then it decreased across grades 8–11; whereas the use of the functional strategy increased across grades 4–11. [ABSTRACT FROM PUBLISHER]

In learning mathematics through English, one of the major challenges facing English as a Foreign Language (EFL) learners is understanding the language used to present word problems in mathematics texts. Without comprehending such language, learners are not able to carry out the targeted calculations no matter how familiar they are with the mathematical concepts behind. As many researchers have pointed out, distinctive combinations of certain vocabulary and grammatical structures are often found in the mathematical register. It is therefore important for teachers to discover, analyse and provide guidance on how learners can best tackle those linguistic challenges. This paper attempts to achieve this goal by first identifying and classifying, from word problem examples, the potential linguistic challenges at the word, phrase, clause and discourse levels. It then proceeds to suggest specific linguistic and multimodal strategies with good potential to help EFL learners overcome such challenges, with reference to the relevant literature and examples. It concludes by reiterating the importance of mathematics teachers' active engagement with the language used in word problems and suggesting possible collaboration between them and English language teachers in promoting EFL learners' awareness of the language of mathematics. [ABSTRACT FROM AUTHOR]

In this paper, we develop a multi-level comparative approach to analyse Trends in International Mathematics and Science Survey (TIMSS) and Programme of International Student Achievement (PISA) mathematics results for a country, Russia, where the two tests provide contradictory information about students’ relative performance. Russian students do relatively well on the TIMSS mathematics test but relatively poorly on the PISA. We compare the performance of Russian students with different levels of family academic resources over the past decade on these tests compared to students with similar family resources in Russia’s neighbours and to Russian students studying in Latvian and Estonian Russian-medium schools. These comparisons and interviews with educators in Latvia and Estonia help us understand why students in Russia may perform lower on the PISA and to draw education policy lessons for improving international test performance generally and Russian students’ PISA mathematics performance specifically. [ABSTRACT FROM PUBLISHER]

This paper examines the relationship of the policies and practices employed by 3 high school reform models – Early College High Schools, Redesigned High Schools, and High Schools That Work – with student success in college preparatory mathematics courses by the end of the 10th grade. Data on policies and practices collected through a survey of school principals in North Carolina are combined with administrative data on student course-taking and performance. The examined policies include course-taking requirements, rigorous instruction, academic support, personalization, and relevance. Results show that implementation of these policies varies across models and that higher levels of implementation of combinations of these policies are associated with improved outcomes. [ABSTRACT FROM PUBLISHER]

One of the best ways for STEM students to understand the methods and thought process of their discipline is through participation in an undergraduate research project. However, mathematics secondary education students at Hood College are historically less likely to self-select for these types of supplemental experiences than traditional mathematics majors. Hood's Noyce STEM Teacher Education Partnership (NSTEP) sought to address this by designing a shortened summer research experience specifically for secondary education students in mathematics. The Summer Research Experience in Mathematical Modeling (SREM2) was a 4-week summer research project that provided this opportunity at a small college. Here we describe the benefits of such an experience and present the design of our program and its outcomes. [ABSTRACT FROM AUTHOR]

This paper reports the outcomes of a ‘lesson study’ project conducted in a mathematics department with four serving teachers in a secondary school in England. Using Dudley’s lesson study framework and drawing on Hargreaves and Fullan’s notion of professional capital, the feasibility and value of collaborative lesson study as a vehicle for the development of teacher learning were explored. Planning and evaluation meetings as well as end-of-project interviews were analysed to investigate how teachers planned research lessons together and how these were evaluated. Despite time constraints, teachers who engaged in lesson study reported that the process improved understanding of their students; that collaboration helped them to develop less-teacher-centred approaches and created a stronger sense of teacher community. The project demonstrated that lesson study has potential as an alternative or complementary model of teachers’ learning, but it also throws up substantive organisational challenges if its use is to expand. [ABSTRACT FROM PUBLISHER]

In the context of a Capstone course for prospective secondary teachers (PSTs) of mathematics, this paper examines how a journal assignment was implemented and how it can help us understand PSTs’ views of mathematics and mathematics teaching. First, the course design and specifics of the assignment are described. Then, details about implementation of the assignment are included. Finally, affordances of incorporating this type of assignment are discussed with examples of PSTs’ responses. The identified benefits include PSTs: (a) connecting new mathematical ideas to their prior knowledge; (b) identifying what they do not know; and (c) articulating the importance of reasoning and proving in mathematics teaching and learning. These benefits support the notion that using a journal assignment in mathematics content courses designed for PSTs can help them develop a profound understanding of mathematics content and practices. [ABSTRACT FROM PUBLISHER]

This study aimed to adapt the Get Set2 quiz and design a diagnostic quiz for implementation under assumed knowledge admissions into university science degrees. The GetReady Quiz that was designed, comprised five modules benchmarked against the high school curriculum in Mathematics, Chemistry and Biology. The significance of this study stems from the development of an alternate approach to support students, with deficiencies in mathematical skills, who enter university STEM programmes with recommended foundational knowledge in mathematics and science. A mixed-mode research methodology was employed in which quiz design was coupled with implementation and evaluation with a pilot study. The GetReady Quiz design featured five discipline modules in Mathematics, Chemistry and Biology, with progression of question difficulty. Knowledge gaps and apparent difficulties for some students with conceptual understanding were evident. Students displayed a realization of their lack of knowledge by reporting that completion of the GetReady Quiz helped them identify areas for revision. The findings indicated that the GetReady quiz was appropriate for assumed knowledge self-diagnosis for commencing students. Approaches to diagnostic testing, such as the GetReady Quiz, should be recommended, if not compulsory, for university students who commence study with STEM assumed knowledge admission. [ABSTRACT FROM AUTHOR]

Teachers are implementing flipped instruction in an increasing number of mathematics classes but the research base is not yet well developed on this topic. Many studies of flipped instruction in mathematics have involved a small number of classes utilizing flipped instruction being compared to classes with non-flipped instruction, but this study design fails to account for variations in how teachers implement flipped instruction. There seems to be as much variation within flipped implementations and within non-flipped implementations as there are between the two categories. To aid systematic attention to variation, this article presents a framework for flipped mathematics lessons that identifies key features of the videos assigned as homework as well as features of the in-class time with students. The components of the framework allow for structurally similar implementations to be grouped together meaningfully whereas different implementations, though still under the banner of 'flipped instruction', can be distinguished from one another. The framework is illustrated with data from flipped mathematics classes at secondary and post-secondary levels. [ABSTRACT FROM AUTHOR]

Classroom dialogue is widely used in mathematics teaching and learning, and if managed strategically, it will have productive benefits for mathematics achievement. However, dialogic participants often lack awareness of how dialogue could be constructed, and few studies show the characteristics of dialogic patterns in different stages of education. Drawing on the data from the Chinese National Cloud Platform, this study therefore comparatively examined the dialogic patterns of mathematics lessons captured in primary, secondary and high schools in China, using 300 video-recorded mathematics lessons, with 100 lessons for each stage of education. Classroom dialogue was transcribed and systematically coded, after which a lag sequential pattern mining technique was used to examine the collective process of dialogic contributions. Findings indicated that there were both similarities and differences in terms of the dialogue pattern throughout the three stages. Dialogue concerning previously-learnt knowledge, subjective expressions and analysis appeared frequently in mathematics lessons in the three education stages, while speculative talk and querying were less often observed. There were commonalities between dialogic patterns captured in mathematics lessons in secondary and high schools, which were significantly different from those in primary schools. The variation was most obvious in dialogue showing high-level cognition, namely, analysis, coordination and speculation. Prominent sequences captured in secondary and high school lessons were able to involve dialogue at both low and high cognitive levels, which demonstrated the characteristics of exploratory talk. This knowledge could help create productive classroom dialogue, and benefit mathematics teaching and learning. Specific data mining methods were developed, drawing on computer science to achieve an understanding of dialogue patterns. A lag sequential mining technique was applied to capture the collective and constructive process of classroom dialogue. Similar dialogic patterns in terms of low-cognitive dialogues were recognized in primary, secondary and high schools. A skewed high-cognitive dialogues among three education stages was suggested as possible reasons which lead to transitional difficulties in mathematics learning. [ABSTRACT FROM AUTHOR]

Flipped classrooms serve as a new pedagogical approach for teaching and learning, which involves switching from the traditional teacher-centred academic agenda. Unlike the traditional classroom model, the flipped classroom enables students to understand the lesson better while learning at their own pace and enhancing their communication skills among peers. At the same time, lecturers are provided with an opportunity to extend the discussion time in class through a more interactive session between students. While research works regarding the success of implementing flipped classrooms have been widely undertaken in the literature, only a few of these works considered exposing such models to secondary students. Engaging secondary students in the flipped classroom and investigating its impact on students paves the way for preparing an academic environment for students' advanced courses in the future. In this work, a case study was conducted at Barstow High School (Barstow, California) to evaluate the implementation of flipped classrooms in Mathematics education from an interplay of factors such as gender, ethnicity, and students' attitude. The students showed positive usability of the model from the end-activity evaluation and further noted an improved communication channel inside the classroom, allowing them to pace their own learning. [ABSTRACT FROM AUTHOR]

Integrating digital technology in education is challenging. This study reports on three high school mathematics classes where teachers attempted to improve their teaching and student learning by using a digital tool. For analysis we use the Information System Artifact model Lee et al. (2015) which distinguishes between three integrated sub-artifacts, the technological, the informational and the social and the Structurational Practice Lens to educational technology Halperin (2017). Using interviews and observations we find the major obstacle for student learning is a less developed social artifact. Students have difficulties using the tool effectively when teachers do not work to develop shared practices in technology use. When teachers do not themselves use the tool actively, they do not fully understand how students can learn from it, neither can they help them in synthesizing teacher- and tool instructions. Students end up having "two masters" competing rather than integrated teacher instruction and technology assistance. [ABSTRACT FROM AUTHOR]

The purpose of this study was to analyze the prompts that preservice secondary mathematics teachers used for the acquisition of mathematics knowledge in dynamic geometry environment tasks. The participants, four preservice secondary mathematics teachers who were enrolled in a computer-supported mathematics education course, designed a dynamic geometry environment task based on a high school mathematics curriculum learning outcome. The main data sources were the participants' task documents and the transcripts of interviews that were conducted with the participants to examine the details of their tasks. The techno-pedagogic mathematics task design model was used to analyze the data. The results indicated that the focus of the designs was to help students realize the invariant properties of geometric figures that were embodied by robust construction steps in the tasks. The preservice teachers utilized several capabilities of the dynamic geometry environment (e.g. measuring, dragging, and changing the input box value) to help students discern these properties and expected students to make a generalization based on inductive reasoning. However, since students were directed to build robust constructions, the mathematical activities prompted by the preservice teachers were limited to observing, finding, generalizing, and explaining the previously constructed invariant features in the tasks. [ABSTRACT FROM AUTHOR]

The article describes an instrument consisting of 46 indicators organised under 10 dimensions to assess the presence and educational utility of historical resources in secondary education mathematics textbooks. The instrument was constructed on the basis of previous research on the didactic use of the history of mathematics, existing instruments on the analysis of mathematics textbooks, and mathematics teachers' assessments of the suitability of the indicators. The instrument can be used as a stand-alone tool or integrated into other procedures addressing textbook analysis more broadly. Its application is illustrated with a sample comparison of two secondary level mathematics textbooks. The instrument has proven to be a useful tool for comparing the historical resources in different publishers' textbooks, for it enables secondary school mathematics teachers to base their choice of resources on an appraisal of the indicators most relevant to classroom teaching. [ABSTRACT FROM AUTHOR]

Comparative studies on science curricula provide insights into educational standards worldwide. Accordingly, in this study, we analysed the intended curricula of mathematics and science subjects of the Czech Republic, Estonia, Poland and Slovenia by comparing their lower secondary (ISCED 2) National Curriculum Documents on mathematics, physics, chemistry, biology, geography and geology from the perspective of learning outcomes. By document analysis, we extracted obligatory learning outcomes from the national curricula, assessing their level of detail and structure. A team of seven coders also measured cognitive demands of the learning outcomes using the revised Bloom's taxonomy. Our results showed considerable differences in the number of learning outcomes across countries and in the structure of these outcomes across subjects, even within national documents. Cognitive demands determined by learning outcomes were similar across the countries, but metacognitive knowledge and cognitive processes of higher level (Evaluate and Create) were mostly overlooked. We also found a lack of learning outcomes explicitly requiring the use of ICT, experimental or field work or data analysis. Although these requirements are usually formulated in general sections of curriculum documents, we recommend explicitly incorporating them into these documents as individual learning outcomes. [ABSTRACT FROM AUTHOR]

We interviewed 40 students each in grades 7 and 11 to investigate their integer-related reasoning. In one task, the students were asked to write and interpret equations related to a story problem about borrowing money from a friend. All the students solved the story problem correctly. However, they reasoned about the problem in different ways. Many students represented the situation numerically without invoking negative numbers, whereas others wrote equations involving negative numbers. When asked to interpret equations involving negative numbers in relation to the story, students did so in two ways. Their responses reflect distinct perspectives concerning the relationship between arithmetic equations and borrowing/owing. We discuss these findings and their implications regarding the role of contexts in integer instruction. [ABSTRACT FROM AUTHOR]

Our work focuses on the transition from high school to university in the field of calculus. In France, recursive sequences are studied as one of the classical exercises in both institutions. Their studies use different theorems and notions, such as functions, convergence, monotonicity, induction, etc. The work expected at this transition requires the development of recognition and control activities on the part of the students. We propose a new task allowing the development of such activities from a dialectic between two paradigms of analysis using the calculator. We highlight that students at the end of high school have difficulties in studying these sequences, and they do not easily develop the activities of recognition and control independently. This could cause problems in understanding recursive sequences at the beginning of university. [ABSTRACT FROM AUTHOR]

Two common interest-enhancement approaches in mathematics curriculum design are illustrations and personalization of problems to students' interests. The objective of these experiments is to test a variety of illustrations and personalization approaches. In the illustrations experiment, students (n = 265) were randomly assigned to lessons with story problems containing decorative illustrations, contextual illustrations, diagrammatic illustrations, misleading illustrations, or no illustrations (only text [control condition]). Students' problem-solving performance and attitudes were not affected by illustration condition, but learning was better in the control compared with contextual illustrations. In the personalization experiment, students (n = 223) were randomly assigned to story problems that were either personalized based on: a survey of their interests, their choice of interest topics, a randomly assigned interest topic, or the original nonpersonalized story problem (control). The findings indicated there were benefits for choice personalization both for performance in the problem set as well as on a later learning assessment. [ABSTRACT FROM AUTHOR]

I examine experiences of former Australian schoolgirls in relation to mathematics during secondary school. This research scrutinises misunderstandings about success and impact on subject choice that can result in post-schooling trajectories that limit what girls can do in their lives beyond school. I examine ways affective relationality, as a sense of embodied belonging, may influence participation in subjects. I frame the discussion using the Baradian concept of intra-action, a co-production that engages an ethic of non-coincidence. For these participants, a reductive high-stakes testing environment and aspirations to become a master subject evoke a powerful not good enough assemblage. The responsibility to achieve enough success incites a soliciting of a particular self in affective regulation. The dread of not excelling in mathematics was often too much to endure thus participants chose to discontinue studying mathematics. They understood this as a sensible solution to prevent vulnerability, as not good enough. [ABSTRACT FROM AUTHOR]

In this article, we report the results of research that explores the intra-mathematical connections that high school students make when they solve Calculus tasks, in particular those involving the derivative and the integral. We consider mathematical connections as a cognitive process through which a person relates or associates two or more ideas, concepts, definitions, theorems, procedures, representations and meanings among themselves, with other disciplines or with real life. Task-based interviews were used to collect data and thematic analysis was used to analyze them. Through the analysis of the productions of the 25 participants, we identified 223 intra-mathematical connections. The data allowed us to establish a mathematical connections system which contributes to the understanding of higher concepts, in our case, the Fundamental Theorem of Calculus. We found mathematical connections of the types: different representations, procedural, features, reversibility and meaning as a connection. [ABSTRACT FROM AUTHOR]

Algebra has long been identified as an area of difficulty in the teaching and learning of mathematics. Evidence of this difficulty can be found in Irish secondary-level classrooms. Chief Examiner Reports have consistently identified algebra as an area of student weakness in State examinations. In light of poor student performance, and as part of a nationwide reform of secondary mathematics curricula, a functions-based approach to teaching algebra has been adopted in Irish schools. It was introduced in September 2011 in place of the transformational (rule and procedure)-based approach which was previously used. Through comparing the diagnostic test scores of incoming students in an Irish university in the years before and after the reform, this study finds that the reformed approach has coincided with a decline in students’ technical algebraic skills. However, interviews with practising mathematics teachers reveal that this decline is not a direct result of the functions-based approach, but rather of a mixture of approaches being implemented in classrooms. Such divergence of approaches can be linked to the common mismatch between the intended curriculum prescribed by policy-makers and the implemented curriculum that is actually carried out by teachers in their classrooms. [ABSTRACT FROM PUBLISHER]

The present study examined indirect effects of principal leadership on the mathematics achievement of 254,475 15-year-old students from 10,313 schools in 32 OECD economies. Results showed that the students could be divided into three categories (Disadvantaged,Average, andPrivileged) differing in levels of student SES and prior achievement, parental academic expectations, and access to school resources. Results also showed that principal leadership effects accounted for a greater proportion of between-school achievement variance forDisadvantagedvis-à-visPrivilegedorAveragestudents. In particular, instructional leadership had the largest positive effect onDisadvantagedvis-à-vis other students’ achievement via the mediating variables of teacher autonomy and morale. Distributed leadership negatively affected the achievement ofDisadvantagedbut not other students. The negative effects of principal goal-setting were the largest while those of principal problem-solving were the smallest forDisadvantagedstudents. The study contributes to the literature by examining contextual influences on the leadership–achievement relationship. [ABSTRACT FROM AUTHOR]

This study aimed to explore the cross-lagged association of achievements in mathematics and languages. While the effect of language on achievements in mathematics is well-documented, few studies have examined the reciprocal relationships among mathematics, the Chinese language and the English language in the same study. This study conducted a secondary analysis of longitudinal achievement data collected through the Territory-wide System Assessment (TSA) in Hong Kong. The sample comprised 48,547 third-grade, unbalanced bilingual students who were measured three times over six years: in 2007 (in Grade 3), 2010 (Grade 6) and 2013 (Grade 9). Multilevel cross-lagged analysis found prior achievement in a subject was the strongest predictor of achievement in that subject three years later. Furthermore, cross-subject bidirectional prediction was found among achievements in mathematics, Chinese language and English language for students from Grade 3 to Grade 6 and from Grade 6 to Grade 9. [ABSTRACT FROM PUBLISHER]

There are many things which can be made more useful and interesting through the application of creativity. Self-concept in mathematics and some school environmental factors such as resource adequacy, teachers’ support to the students, teachers’ classroom control, creative stimulation by the teachers, etc. were selected in the study. The sample of the study comprised 770 seventh grade students. Pearson correlation, multiple correlation, regression equation and multiple discriminant function analyses of variance were used to analyse the data. The result of the study showed that the relationship between mathematical creativity and each attitudinal and environmental characteristic was found to be positive and significant. Index of forecasting efficiency reveals that mathematical creativity may be best predicted by self-concept in mathematics. Environmental factors, resource adequacy and creative stimulation by the teachers’ are found to be the most important factors for predicting mathematical creativity, while social–intellectual involvement among students and educational administration of the schools are to be suppressive factors. The multiple correlation between mathematical creativity and attitudinal and school environmental characteristic suggests that the combined contribution of these variables plays a significant role in the development of mathematical creativity. Mahalanobis analysis indicates that self-concept in mathematics and total school environment were found to be contributing significantly to the development of mathematical creativity. [ABSTRACT FROM AUTHOR]

TheEffecting Principled Improvement in STEM Education[epiSTEMe] project undertook pedagogical research aimed at improving pupil engagement and learning in early secondary school physical science and mathematics. Using principles identified as effective in the research literature and drawing on a range of existing pedagogical resources, the project designed and trialled a classroom intervention, with associated professional development, in a form intended to be suited to implementation at scale. The most distinctive feature of theepiSTEMepedagogical approach is its inclusion of a component of dialogic teaching. Aimed at the first year of secondary education in English schools (covering ages 11–12), theepiSTEMeintervention consists of a short introductory module designed to prepare classes for this dialogic teaching component, and topic modules which employ theepiSTEMepedagogical approach to cover two curricular topics in each of science and mathematics. A field trial was conducted over the 2010/2011 school year in 25 volunteer schools, randomly assigned to intervention and control groups. Within the intervention group, observation of lessons indicated that the level of dialogic teaching was higher for one of the topic modules than others. Evaluation focused on the effectiveness of the topic modules, each trialled in more than 10 classes containing a total of over 300 pupils, and compared with a group of similar composition. Overall, at this first implementation, learning gains under theepiSTEMeintervention were no greater, although for individual topic modules the effects ranged from small negative to small positive. No difference was found between intervention and control groups either in the opinion of pupils about their classroom experience or in changes in their attitude towards subjects. [ABSTRACT FROM PUBLISHER]

We delineate some types of structured practice (modeling, requests, feedback, and space-making) that help students learn to pose appropriate questions and to initiate exploration of those questions. Developing skills requires practice, so we suggest ways to embed structured practice into existing class sessions. Including structured practice is nuanced, instructor-specific, and depends on context. Therefore, we give examples of implementation in an inquiry-based learning (IBL)-influenced calculus class, in a partially IBL mathematics for liberal arts class, and in a completely IBL class for gifted high school students. [ABSTRACT FROM PUBLISHER]