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Generalization is one of the core aspects of algebraic thinking, and research literature points to the importance of objectification when generalizing mathematical discourses. To increase the field's understanding of the objectification process in various algebra discourses, our study analyses Grade 6 teachers' use of words, symbols, routines and narratives when they introduce various algebraic concepts. Our results show that there exist instances of objectification in the teachers' discourses, but these are all implicit, which can make them difficult for students to notice. Thus, teachers need to be made more aware of the importance of making the objectification process explicit. Furthermore, our results show that instances of objectification are tightly connected to the use of formal algebraic symbols. This highlights the importance of the transition from concrete/iconic to formal/arbitrary representations. [ABSTRACT FROM AUTHOR]

This paper discusses several linear algebra activities designed to help enhance students' skills in collaborating, exploring mathematics, and linking together abstract and visual ways of approaching mathematics. Most of these activities are short, accessible, engaging, and easy to incorporate into any classroom. In addition, we discuss some questions instructors can ask themselves to design novel and engaging activities when constrained to teaching from a particular curriculum. [ABSTRACT FROM AUTHOR]

Considering the need to combine theory and practice in planning teaching situations for visually impaired students, this research adopted the assumptions of the Historical-Cultural Theory and the Activity Theory, and was based on the Teaching Guiding Activity. Learning triggering situations for algebraic knowledge were developed, and interventions carried out with a 7thgrade student and another from 8th grade, both visually impaired and attending the multifunctional resource room of a public school in the state network, were analyzed. The aim of this article is to recognize the appropriation of algebraic knowledge by visually impaired students from learning triggering situations. To this end, the data were organized into two categories: manifestations of conceptual nexuses and manifestations of thought and language. These two categories allow understanding the phenomenon of "appropriation of algebraic knowledge" from the situations presented to the students. At the end of the study, it is highlighted that the elaborated situations allowed symbolic and instrumental mediation, enabling the appropriation of the conceptual nexuses of algebra (variation, variation field, and fluency) and of some selected school contents (recognition of unknowns, dependency of variables, and operations with monomials and polynomials). It should be noted that the proposed triggering situations achieved these results through a teaching organization process that considered the conditions for accessibility of the students served. [ABSTRACT FROM AUTHOR]

Representations are key to mathematical activities and meaning-making processes as they are part of modeling, connecting, communicating, and understanding mathematical ideas and concepts. The current study sought to examine a group of novice algebra learners' interactions with different representations from an onto-semiotic approach. A case study method was employed to understand how different algebraic practices (abstracting, generalizing, justifying, and operating on symbols) and functional thinking types (recursive, covariational, and correspondence) were facilitated through working with multiple representations. Three 6th graders participated in the study by completing 12 algebra tasks and taking part in two interviews. The onto-semiotic approach guided the data analysis process that involved the identification of mathematical objects that emerged in the participating students' mathematical practices. Then, the configuration of objects and semiotic functions established by the students in the functional situations was examined to understand the role of representations in the students' development of algebraic thinking and practices. Findings showed that abstraction is an essential process for generalization. Thinking about far figures facilitated abstraction and generalization through helping students construct non-ostensive concrete/pictorial representations. Verbal representations interacted with all representations and preceded symbolic representations. Working with near figures promoted recursive and covariational thinking while examining the far figures usually resulted in correspondence thinking. Implications for the school curriculum are discussed in the paper. [ABSTRACT FROM AUTHOR]

Function is a multifaceted and essential concept in mathematics that includes various complex ideas such as mapping, relations between variables, algebra, input-output, and rule. Pre-service mathematics teachers' (PMTs) should have a rich understanding of these ideas for their future teaching practices. However, research showed that PMTs had a limited and superficial understanding of function as students do. Thus, a learning module was designed for improving PMTs' conceptualizations of function ideas as they relate to their personal concept definitions and examples of functions. A design-based research study was conducted with 17 senior PMTs. The study results suggest an improvement in the way PMTs define functions and identify a variety of function ideas in definitions and examples. Even if the mapping idea became dominant among all ideas in PMTs' personal concept definitions and examples, they could identify the implicit function ideas and develop an understanding of connections between ideas. We also provide suggestions for the second cycle of the intervention. [ABSTRACT FROM AUTHOR]

This paper discusses findings from an ongoing study investigating mental mechanisms involved in the conceptualisation of linear transformations from the perspective of Action (A), Process (P), Object (O), and Schema (S) (APOS) theory. Data reported in this paper came from 44 first-year linear algebra students' responses on a task regarding the range of a linear transformation. Our analysis revealed parallels between Levels/Stages of the range concept and the use of representations of matrix multiplications. More importantly, these representations appeared to have been the enablers of transitions from lower to higher APOS Stages for the range. Conversely, mental mechanisms employing other means showed little to no progressions, some, furthermore, revealed faulty knowledge structures. [ABSTRACT FROM AUTHOR]

The article presents the results of a study that aimed to understand how the design and implementation of a Professional Learning Task contributed to the identification of mathematical and didactic knowledge of mathematics teachers as participants in a professional development. It fits into a qualitative-interpretive perspective, in which data collection came from the planning audios, its design, audios and videos of the discussions in the subgroups and the plenary of the second meeting. The results show that, from the professional learning task planned and implemented, it was possible to raise the knowledge of teachers, in addition to showing the articulation between the components of the Professional Learning Tasks. [ABSTRACT FROM AUTHOR]

Prior research has shown that game‐based learning tools, such as DragonBox 12+, support algebraic understanding and that students' in‐game progress positively predicts their later performance. Using data from 253 seventh‐graders (12–13 years old) who played DragonBox as a part of technology intervention, we examined (a) the relations between students' progress within DragonBox and their algebraic knowledge and general mathematics achievement, (b) the moderating effects of students' prior performance on these relations and (c) the potential factors associated with students' in‐game progress. Among students with higher prior algebraic knowledge, higher in‐game progress was related to higher algebraic knowledge after the intervention. Higher in‐game progress was also associated with higher end‐of‐year mathematics achievement, and this association was stronger among students with lower prior mathematics achievement. Students' demographic characteristics, prior knowledge and prior achievement did not significantly predict in‐game progress beyond the number of intervention sessions students completed. These findings advance research on how, for whom and in what contexts game‐based interventions, such as DragonBox, support mathematical learning and have implications for practice using game‐based technologies to supplement instruction. Practitioner notesWhat is already known about this topicDragonBox 12+ may support students' understanding of algebra but the findings are mixed.Students who solve more problems within math games tend to show higher performance after gameplay.Students' engagement with mathematics is often related to their prior math performance.What this paper addsFor students with higher prior algebraic knowledge, solving more problems in DragonBox 12+ is related to higher algebraic performance after gameplay.Students who make more in‐game progress also have higher mathematics achievement, especially for students with lower prior achievement.Students who spend more time playing DragonBox 12+ make more in‐game progress; their demographic, prior knowledge and prior achievement are not related to in‐game progress.Implications for practice and/or policyDragonBox 12+ can be beneficial as a supplement to algebra instruction for students with some understanding of algebra.DragonBox 12+ can engage students with mathematics across achievement levels.Dedicating time and encouraging students to play DragonBox 12+ may help them make more in‐game progress, and in turn, support math learning. [ABSTRACT FROM AUTHOR]

The research reported in this article sought to develop a taxonomy of grade 11 students' levels of understanding in algebraic problem-solving tasks. The student sample was from high schools in the province of Ontario in Canada. Problems from four areas in algebra namely, variables, expressions, equations and word problems were chosen to be represented in the test paper. A six-level taxonomy was constructed by analyzing the structure of students' written responses and their subsequent interview transcripts. The first three levels of the taxonomy belong to the application of lower level thinking skills while the last two levels belong to the application of higher level thinking skills. The taxonomy serves two purposes. Teachers can use it for formulating objectives in classroom teaching, and they can also use it as an evaluation tool in constructing assessment items. [ABSTRACT FROM AUTHOR]

This paper introduces the basic knowledge of integral factors of first-order ordinary differential equations and Lie symmetry analysis. It then discusses the principle of constructing an integral factor of the first-order ordinary differential equation by the Lie symmetric method. Finally, it presents some examples to show the process of constructing the integral factor of the first-order ordinary differential equation and finding the general solutions. [ABSTRACT FROM AUTHOR]

This study investigated the methods that pre-service teachers employed while finding volume using integration and the reasons why they utilized these methods (problems); the accuracy of their solutions was also analyzed in this study. Case study design, a qualitative research method, was used for this study. 45 sophomore pre-service teachers studying for the Middle School Mathematics Education degree programme at an Education Faculty in Anatolia. It is used to mean 45-second class preservice teachers studying for the Middle School Mathematics Education degree programme at an Education Faculty in Anatolia. It is used to mean 45-second class preservice teachers studying for the Middle School Mathematics Education degree programme at an Education Faculty in Anatolia. Pre-service teachers were provided with a training programme that made use of GeoGebra software for four weeks as part of their mathematical analysis course. Pre-service teachers were provided with a written form containing five open-ended questions and given time to answer them. Data from the study was analyzed using the content analysis method. Results from these analyses showed that half of the pre-service teachers chose problems that requires the shell method to calculate volume and that the main determining factor was the ease of mathematical calculation provided by the shell method. Also, it was found out that the mistakes made by pre-service teachers originated from issues such as the inability to fully comprehend the subject at hand, errors in formulization and/or graphing, and problems taking the integral. [ABSTRACT FROM AUTHOR]

Modern mathematics education used a different approach when dealing with word problems in algebra because of how people perceive mathematics. The teacher can aid students in honing their problem-solving abilities to assist them build mathematical concepts that can be applied in real-world situations. This research is a case study involving junior high school (JHS) pupils in Ghana's Tema Metropolis. For the study, data collection and analysis used a qualitative methodology. Using a purposive sample technique, data from 12 participants-six JHS mathematics teachers and six students-were gathered. Data were gathered through observation, interviews, and document analysis. The primary finding is that JHS pupils at Tema Education Metropolis are unable to present their own issues as proof of conceptual knowledge of algebra word problems. [ABSTRACT FROM AUTHOR]

Schools can approach the task of sorting students to privileged learning opportunities in different ways, potentially creating distinct and durable educational inequality regimes. We test this idea by exploring variation in socioeconomic inequalities in advanced mathematics course-taking across California middle schools during a statewide algebra-for-all initiative. This case provides unique insight into local stratification processes since the state pressured schools to boost advanced course enrollments but provided little guidance about how to do so. We distinguish two critical organizational processes: the provision of different types of opportunities and the allocation of students to opportunities. The former, we argue, creates the potential for inequality; the latter determines what level of inequality is realized. Using panel data for all public middle schools in the state over a decade, we demonstrate a curvilinear association between opportunities and inequality, with disparities highest when opportunities are most differentiated. However, allocations at most schools were less unequal than would be expected under a test-based meritocratic allocation regime. Further, we find substantial school-level variation which is systematically related to organizational characteristics and consistent over time. These patterns provide evidence for local educational inequality regimes. [ABSTRACT FROM AUTHOR]

This paper describes the difficulties faced by a group of middle school students (13- to 15-year-olds) attempting to translate algebraic statements written in verbal language into symbolic language and vice versa. The data used were drawn from their replies to a written quiz and semi-structured interviews. In the former, students were confronted with a series of algebraic statements and asked to choose the sole translation, of four proposed for each, that was semantically congruent with the original. The results show that most of the errors detected were due to arithmetic issues, especially around the distinction between product and exponent or sum and product in connection with the notions of perimeter and area. As a rule, the error distribution by type varied depending on the type of task involved. [ABSTRACT FROM AUTHOR]

The typical competitive and results-driven approach to school mathematics has traditionally been conceived as devoid of play, joy, and positive affect. In this paper we address the following questions: What markers of positive affect are observed while students are doing early algebra? Specifically, how are students' markers of joy related to early algebraic practices? What are the characteristics of playful stances to learning early algebra that are observed when children express positive epistemic affect? We analyze three cases in which elementary school students engaged in tasks from an early algebra classroom teaching experiment. Drawing from two theoretical frameworks, epistemic affect and early algebra, we conducted microgenetic analyses of lesson transcripts to identify markers of joy and early algebraic practices. We conducted frequency analyses to determine their co-occurrence. Our results indicate that children expressed joy while engaging in early algebraic practices, evidence of positive epistemic affect. We describe the aspects of each of the cases we present in terms of prior literature on playful stances to learning to further bolster our claims about the relationship between joy and engagement with the early algebraic practices. We conclude that mathematical learning environments should include open opportunities for students to engage with mathematical content, with multiple entry points and ways to respond. We also conclude that early algebraic practices provide opportunities for playfully engagement and positive epistemic affect. • We use two theoretical frameworks—epistemic affect and algebraic practices—to study play and joy in mathematics. • The early algebraic practices of representing and reasoning were most frequently associated with play and joy. • Our results point to the importance of play in school-based learning. [ABSTRACT FROM AUTHOR]

In this note we describe the mathematics that emerges from the construction of an origami box. We first construct a simple origami box from two rectangular sheets and then discuss some of the mathematical questions that arise in the context of algebra, geometry and calculus. [ABSTRACT FROM AUTHOR]

We propose an algorithm that allows calculating the remainder and the quotient of division between polynomials over commutative coefficient rings, without polynomial long division. We use the previous results to determine the quadratic factors of polynomials over commutative coefficient rings and, in particular, to completely factorize in Z [ x ] any integral polynomial with degree less than 6. The arguments are suitable for building classroom/homework activities in basic algebra courses. [ABSTRACT FROM AUTHOR]

In the context of early algebra research and as part of a classroom teaching experiment (CTE), we investigated fourth grade (9- to 10-year-old) students' justifications of how they performed tasks involving the functional relationship y = 2x. We related their written justifications (part of the task) to the task characteristics, which included various semiotic systems (verbal, numerical and alphanumeric, among others) and the demand of different type of justifications. The role of classroom discussion in helping express the functional relationship orally in more sophisticated terms was also investigated. The findings showed that students' written justifications changed with the semiotic system involved in the task. Oral discussion helped students generalize in more sophisticated terms than in their written justifications, in which they omitted information or used less precise language. [ABSTRACT FROM AUTHOR]

This paper provides a narrative account describing how a first-year teacher's empowering care in her mathematics classes in an inner-city high school in Southern California contributed to better student learning and helped raise district-wide test scores. The purpose of the article, chronicled from a practitioner's point of view, examines how teacher beliefs that "all students are capable of learning," operationalised concretely in terms of empowering care, enabled students to achieve impressive academic performances as measured by the district standardised tests during the recent school year. Through applying the research methodologies of narrative inquiry and conversation analysis to interpret the data, this study focuses on how students from impoverished, urban neighbourhoods started to believe in themselves. As a result, these students demonstrated that they could improve their learning in Algebra I and consistently perform better on district tests, outperforming other classes at their school and other schools within the district. [ABSTRACT FROM AUTHOR]

Recent scholarship around teaching elementary mathematics supports the learning of early algebra with 5- to 12-year olds. However, in spite of the recognition of the affordances of early algebra, issues about how to introduce it remain open. Within this context, Davydov's work is often cited as a source of impressive demonstration of young learners' capacity for algebraic thinking. This work requires further exploration in order to yield a clearer picture of a very particular teaching approach, which focuses on early abstractions and symbolic language. We argue that in order to fully understand how Davydov's work contributes to current conversations and what Davydov was trying to do, we need to shed light on the context- and time-specific discourse of the 1960 Soviet educational reforms that made it possible for Davydov to develop his vision about algebraic thinking and to set in motion appropriate teaching approaches for young learners. In this paper, we look back to the Soviet debates that unfolded in Russia on the integration of early algebra in elementary school word-problem solving. Drawing on these debates and the results of Davydov's school experiments, we lay out the developmental axes of capacity building. This can be done by emphasizing ascent from the abstract to the concrete using a variety of representational modeling tools to support the emergence of algebraic thinking while targeting particular habits of mind within carefully designed learning activities. We conclude with some insights about current arithmetic-algebra debates, and how these could be enriched and deepened by Davydov's work, which yet remains open to future discussion and reflection. [ABSTRACT FROM AUTHOR]

This article presents a series of open-ended tasks exploring the development of algebraic expressions suitable for exploration with Year 7 students. The tasks use a familiar tool, the hundreds chart, to aid the transition between representational tools and abstract ways of working. The tasks outline a developmental progression of patterns and relationships found in the hundreds chart, providing students with the opportunity to generalise and use algebraic language and notation. [ABSTRACT FROM AUTHOR]

Students in secondary school often struggle with symbol sense, that is, the general ability to deal with symbols and to recognize the structure of algebraic formulas. Fostering symbol sense is an educational challenge. In graphing formulas by hand, defined as graphing using recognition and reasoning without technology, many aspects of symbol sense come to play. In a previous study, we showed how graphing formulas by hand could be learned. The aim of the study we present here is to explore the relationship between students' graphing abilities and their symbol sense abilities while solving non-routine algebra tasks. A symbol sense test was administered to a group of 114 grade 12 students. The test consisted of eight graphing tasks and twelve non-routine algebra tasks, which could be solved by graphing and reasoning. Six students were asked to think aloud during the test. The findings show a strong positive correlation between the scores on the graphing tasks and the scores on the algebra tasks and the symbol sense used while solving these tasks. The thinking-aloud protocols suggest that the students who scored high on the graphing tasks used similar aspects of symbol sense in both the graphing and algebra tasks, that is, using combinations of recognizing function families and key features, and qualitative reasoning. As an implication for teaching practice, learning to graph formulas by hand might be an approach to promote students' symbol sense. [ABSTRACT FROM AUTHOR]

The MAL-project (Multimodal Algebra Learning) develops a digital system for learning algebra. A key issue is its applicability in schools and textbooks. The paper reports about a partial study of the project which aims at uncovering connecting factors that allow adapting the system to praxeologies in schools and textbooks. In the paper, the study and preliminary results are presented. [ABSTRACT FROM AUTHOR]

This study aims to investigate the non-algebraic solution strategies that prospective mathematics teachers use to solve word problems and their thought process while they were solving the problems. A total of 76 prospective mathematics teachers were asked to solve two word problems without using algebraic solutions and, then, clinical interviews were conducted with a sub-sample of 20 participants. The solutions used by the participants were analysed according to the type and frequency of the strategy. The results show that prospective mathematics teachers did not use various non-algebraic solutions for word problems and that they used the guess and check strategy most often. According to the findings obtained from the interviews, it was found that the participants tended to rely on solutions that used algebraic equations related to the problem while producing solution strategies. Furthermore, the participants were found to have a limited understanding of how to interpret the algebraic equations within their context, the difference between algebraic and non-algebraic solutions, and problem-solving strategies, which prevented the participants from producing alternative solution strategies. [ABSTRACT FROM AUTHOR]

The objective of this paper is to present a characterization of algebraic competence articulated in levels and development indicators. This characterization was used to analize the mathematical practice of inservice teachers of mathematics. Through a qualitative and exploratory research design, the algebraic competence in the teachers on high school (with students between 16 and 19 years old) is investigated and described. The results suggest that teachers present inconsistencies when solving tasks that demand a level three of algebraic competence, so the articulation of training programs based on this proposal will contribute to the development of algebraic competences in teachers of mathematics. [ABSTRACT FROM AUTHOR]

Algebraic competence is a major determinant of college access and career prospects, and equal sign knowledge is taken to be foundational to algebra knowledge. However, few studies have documented a causal effect of early equal sign knowledge on later algebra skill. This study assessed whether second-grade students' equal sign knowledge prospectively predicts their fourth-grade algebra knowledge, when controlling for demographic and individual difference factors. Children (N = 177; Mage  = 7.61) were assessed on a battery of tests in Grade 2 and on algebraic knowledge in Grade 4. Second-grade equal sign knowledge was a powerful predictor of these algebraic skills. Findings are discussed in terms of the importance of foregrounding equal sign knowledge to promote effective pedagogy and educational equity. [ABSTRACT FROM AUTHOR]

This study reports the academic benefits of peer tutoring in algebra for middle school students. A total of 380 students enrolled in grades 7th and 8th participated in the study. Two peer tutoring sessions took place during each week (10 weeks). Interactions between peers lasted 20 to 25 minutes for each session. The typology of tutoring was fixed and same-age. A pretest posttest with control group design was used. Statistical significant improvements were reported in the academic achievement variable after the implementation of the peer tutoring program for 7th and 8th grade courses separately and altogether. Over 87% of the students in the experimental group improved their marks. The overall effect size for the experience was reported to be medium (Hedge's g = 0.48). The main conclusion of this study is that fixed and same-age peer tutoring in algebra may be very beneficial for middle school students. [ABSTRACT FROM AUTHOR]

We propose a generalization of the classical Remainder Theorem for polynomials over commutative coefficient rings that allows calculating the remainder without using the long division method. As a consequence we obtain an extension of the classical Factor Theorem that provides a general divisibility criterion for polynomials. The arguments can be used in basic algebra courses and are suitable for building classroom/homework activities for college and high school students. [ABSTRACT FROM AUTHOR]

An impediment to conducting high‐quality quantitative research studies in education is the paucity of valid measures of learning gains. Studies often seek to investigate students' deep, conceptual understanding yet many measures assess only surface, procedural understanding. One reason is that the development of validated measures of conceptual understanding is resource intensive and time consuming, and success is not guaranteed. We evaluated a novel and efficient technique, based on comparative judgement, for assessing conceptual understanding. We applied the technique to a randomised controlled trial in which students were taught simple algebra based on either the Grid Algebra or the MiGen software package. The participants were Year 5 students (N = 188) drawn from four primary schools who had not encountered algebra previously. An instrument from the literature (Concepts in Secondary Mathematics and Science: Algebra Scale) and a novel comparative judgement assessment were administered following the intervention. Students in the Grid Algebra condition outperformed those in the MiGen condition on both post‐test measures. The comparative judgement technique performed similarly to the standard instrument but was far more efficient to design and implement. The technique can, in principle, be quickly applied to any target concept of interest. We conclude that comparative judgement is a valid, reliable and practical tool that could help to increase both the quantity and quality of quantitative research in education. [ABSTRACT FROM AUTHOR]

This paper examines whether using From Here to There! (FH2T:E), a dynamic game-based mathematics learning technology relates to improved early algebraic understanding. We use student log files within FH2T to explore the possible benefits of student behaviors and gamification on learning gains. Using in app measures of student interactions (mouse clicks, resets, errors, problem solving steps, and completions), 19 variables were identified to summarize overall problem solving processes. An exploratory factor analysis identified five clear factors including engagement in problem solving, progress, strategic flexibility, strategic efficiency, and speed. Regression analyses reveal that after accounting for behavior within the app, playing the gamified version of the app contributed to higher learning gains than playing a nongamified version. Next, completing more problems within the game related to higher achievement on the post-test. Third, two significant interactions were found between progress and prior knowledge and engagement in problem solving and prior knowledge, where low performing students gained more when they completed more problems and engaged more with those problems. [ABSTRACT FROM AUTHOR]

Considering the importance of the knowledge that the graphical representation of functions of two variables provides for applications in engineering courses we present in this article a didactic situation that is part of the didactic sequence of a doctorate. The research was developed as a qualitative research of the type didactic engineering and had as objective to develop, in the students of second year of engineering, the visualization, according to Duval, researching, mainly, the articulations of the apprehensions in the graphic register in a computational algebraic system (CAS). We note that the articulation of the apprehensions in the graphic register allowed engineering students to conjecture properties of functions of two variables, when, for example, the student applies those notions to problems of optimization as is the case here presented. [ABSTRACT FROM AUTHOR]

This article presents some results of a research aimed to identified and discussed, through APOS Theory, the conceptions of Mathematics students from Northern Paraná University, in order to accomplish tasks concerning the concept of Ring. For that we conducted the data collection through five tasks applied to 11 students who were concluding the subject Algebraic Structures. From written records obtained, identified the conceptions (action, process, object, schema), of each student. The research showed that after taking classes of Algebraic Structures, five students showed they were still ah the starting process of the construction of the conception-action of ring, four students the conception-action, one student constructed a conception-object, one student the conception-process, and no student showed to have constructed the conception-schema. [ABSTRACT FROM AUTHOR]

This study examined effects of gamified comparison on sixth graders' performance of algebra word problem solving and attitude toward algebra learning. Seventy-two sixth graders were invited to participate in a four-week experimental instruction and assigned to three groups: gamified comparison, comparison, and control. The results showed (1) a significant effect on solving similar problems: the gamified comparison group performed significantly better than the comparison group and the control group respectively, and the comparison group performed significantly better than the control group; (2) a significant effect on solving transfer problems: the gamified comparison group gained significantly higher scores than the comparison group and the control group respectively, and the comparison group gained significantly higher scores than the control group; (3) a significant effect on students' learning attitude: while no significant differences found on students' confidence, the gamified comparison group made significantly more positive responses than the comparison group and the control group in terms of enjoyment, motivation, and perceived value. This study proposed a feasible combination of game rule and comparison strategy, as well as exploring implications for teachers' teaching design and students' gamified learning activities. [ABSTRACT FROM AUTHOR]

There is a compelling need to develop an algebra teacher self‐efficacy instrument (ATSEI) as algebra continues to be considered a gatekeeper course for postsecondary educational and career opportunities, which is seen as a crucial piece in closing the achievement gap. This paper reports on the development and validation of the ATSEI, an instrument that measures two domains, Efficacy To Do School Algebra (Knowledge Efficacy, KE‐A) and Efficacy to Teach Algebra (Personal Teaching Efficacy, PTE‐A) along six categories. Four of the categories represented content standards (variables, functions, patterns, and modeling) and two of the categories represented process standards (technology and concrete models, and multiple representations). Through conducting an exploratory factor analysis across two phases, the instrument was reduced and refined from an initial 118 items developed from a curriculum analysis to 36 items that reflected two significant categories, Functions and Technology. The ATSEI measure is validated for in‐service mathematics teachers and thus provides an instrument to examine need and impact in professional development venues. The specificity of the ATSEI allows those working with teachers to be better able to support them in the field and in return positively influence the learning outcomes of the students they teach. [ABSTRACT FROM AUTHOR]

This study investigates the effect of utilizing variation theory approach (VTA) on students' algebraic achievement and their motivation in learning algebra. The study used quasi-experimental nonequivalent control group research design. It involved 114 Form Two students in four intact classes (two classeswere froman urban school, another two classes from a rural school). The first group of students from each school learnt algebra in class which used the VTA, while the second group of students in each school learnt algebra through conventional teaching approach. Two-way analysis of covariance and two-way multivariate analysis of variance (MANOVA) were used to analyse the data collected. The result of this study indicated that the use of VTA has significant effect on both urban and rural students' algebraic achievement. There were evidences that VTA has significant effect on rural VTA students' overall motivation in its five subscales: attention, relevance, confidence, satisfaction and interest but it was not so for urban VTA students'motivation. This study provides further empirical evidence that utilization of variation theory as pedagogical guide can promote the teaching and learning of Form Two Algebra topics in urban and rural secondary school classrooms. [ABSTRACT FROM AUTHOR]

Research focusing on algebra from primary to early secondary school level has made several major advances over the past decades. Students’ difficulties have been identified and supportive teaching and learning environments have been set up (Cai & Knuth, 2011; Kieran, 2007; Radford, 2008, Mathematics Education Research Journal, 26, 257-277, 2014). The effectiveness of these environments relies on the teachers’ ability to pay careful attention to students’ thinking, which then guides their instructional decisions. This inevitably raises the crucial question of the teachers’ knowledge for managing these types of situations in the classroom. In this context, this paper focuses on the mathematical knowledge for teaching figural pattern activities of 100 teachers at the primary and early secondary school levels. The results show that many primary teachers lack the essential knowledge for teaching these types of activities: they do not have a clear idea of their goal, they do not consider non-standard algebraic generalisations to be correct and they generally seem unable to help students to improve their arithmetical generalisation. Secondary school teachers also seem unable to give adequate feedback to improve students’ arithmetical generalisation. Although they seem to recognise that these activities are intended to improve algebraic thinking, they do not have a clear perception of the role that primary school learning can play in the development of this algebraic thinking. [ABSTRACT FROM AUTHOR]

We present two empirical studies with 241 and 211 pre-service teachers that evaluate the explanatory power of word order matching and static comparison as models for the reversal error. We used tasks consisting of generating an algebraic equation representing a comparison given in a verbal statement. We introduce the types of magnitude involved in the statement as variables of analysis, something that was not previously tackled in previous works. Our results show that there are no statistical differences in the production of reversal errors depending on the information included in the name used to designate the variable, and that there are statistical differences depending on the syntactic configuration as well as the type of magnitude involved in the statement. The interpretation of these results indicates that both word order matching and static comparison have some potential as explanatory models for the reversal error, and that neither one of them, alone, is enough to completely explain the phenomenon. [ABSTRACT FROM AUTHOR]

The coupling of spin-3 gauge fields to three-dimensional Maxwell and AdS -Lorentz gravity theories is presented. After showing how the usual spin-3 extensions of the AdS and the Poincaré algebras in three dimensions can be obtained as expansions of sl ( 3 , R ) algebra, the procedure is generalized so as to define new higher-spin symmetries. Remarkably, the spin-3 extension of the Maxwell symmetry allows one to introduce a novel gravity model coupled to higher-spin topological matter with vanishing cosmological constant, which in turn corresponds to a flat limit of the AdS -Lorentz case. We extend our results to define two different families of higher-spin extensions of three-dimensional Einstein gravity. [ABSTRACT FROM AUTHOR]

Background/Context: In this article we explore equity issues related to school district decisionmaking about students' opportunities to learn algebra. We chose algebra because of the important role it plays in the U.S. as a gatekeeper to future academic success. Current research has not yet explored issues of equity in district-level decision-making. Purpose/Objective/Research Question/Focus of Study: We examine the extent to which district decision-makers for mathematics attend to aspects of equity when they make decisions about resources related to the teaching and learning of algebra. The research questions guiding this study were: How do district decision-makers for mathematics report considering issues of equity when making decisions about students' opportunities to learn Algebra I? How do district characteristics, particularly students' racial and ethnic diversity, affect the extent of equity considerations by mathematics decision-makers? Research Design: We surveyed a national probability sample of 993 district decision-makers for mathematics about criteria that they consider when they select and distribute resources and structure learning opportunities in algebra for students and teachers in their districts. These survey items were our attempt to identify district-level practices in relationship to an equity framework. In this study, we examine national patterns in criteria for decision-making about algebra resources and examine the relation of these criteria to district features using a structural equation model. Findings: Our findings suggest that fewer decision-makers considered equity-related criteria in their decision-making about algebra, while many tended to endorse equality-related items addressing considerations for all students, such as giving all students the same resources or attending to preparation for standardized testing. The vast majority of decision-makers reported considering real life contexts for algebra when making decisions about professional development (PD) and curriculum, while fewer considered the students' culture or culturally relevant teaching. Decision-makers in only about half of the districts reported considering structural aspects, such as tracking. Modeling of the survey responses indicates that decisionmakers in the most racially or ethnically and linguistically diverse districts have the greatest tendency to consider equity criteria in structuring students' opportunities to learn algebra. Conclusions/Recommendations: The extent to which district decision-makers for mathematics attend to aspects of equity is noteworthy because their decisions inform the selection and distribution of educational resources for learning algebra across districts. These findings raise important concerns with respect to how district decision-making mobilizes and shapes the resources available to teachers and students. Recommendations include supporting district decision-makers in a) expanding their conceptions of real life contexts to include students' culture, b) considering different framings of the problem of participation gaps, c) reconsidering ability grouping and understanding the negative consequences of tracking, and d) carefully examining the kinds of stated and unarticulated rules, rewards, and sanctions that get put into place to uncover how inequitable practices get perpetuated. [ABSTRACT FROM AUTHOR]

We perform an in-depth study of several different cancellation properties for modules. Among those we consider are (half) (weak) cancellation modules, restricted cancellation modules, and (half) join principal modules. We also investigate which commutative rings have every nonzero (finitely generated) ideal (respectively, module) satisfying some cancellation property. [ABSTRACT FROM AUTHOR]