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The main purpose of this study is to investigate mathematics teachers' mathematical thinking process while they are constructing the length of [cube root of 2] by paper folding. To carry out this aim, two teachers--who are PhD. students--were interviewed one by one. During the construction, it was possible to observe the consolidation process of Pythagorean and Thales Theorem. All interviews were videotaped and analyzed through descriptive methods, according to abstraction in context, characteristics of consolidation and mathematical habits of mind. It was found that both two teachers constructed the knowledge of Pythagorean and Thales Theorems before the study and also these knowledges were consolidated. In addition, it was determined that these two approaches (abstraction in context and mathematical habits of mind) were close and corroborated each other. Moreover, consolidation process corroborated mathematical habits of mind.

It is well documented in literature that rational number is an important area of understanding in mathematics. Therefore, it follows that teachers and students need to have an understanding of rational number and related concepts such as fraction multiplication and division. This practitioner reference paper examines models that are important to elementary and middle school teachers and students in the learning and understanding of fraction multiplication and division.

The purpose of this article is to describe a couple of challenging mathematical problems that involve paper folding. These problem solving tasks can be used to foster geometric and algebraic thinking among students. The context of paper folding makes some of the abstract mathematical ideas involved relatively concrete. When implemented appropriately these activities have the potential to address many of the mathematical proficiencies, as delineated by Australian Curriculum and Assessment Reporting Authority (ACARA, 2014). [This paper was first published in the "Australian Senior Mathematics Journal," in 2014 (EJ1093386).]

In this study I examine the issue of how to support students' coordination of computer algebra systems (CAS) and paper-and-pencil in solving mathematics problems. Together with a classroom teacher I designed and conducted a collaborative teaching experiment in a ninth-grade algebra classroom to understand how to support students' coordination of tools. I introduce a predict-act-reflect-reconcile activity structure for coordinating CAS and paper-and-pencil in solving problems involving linear expressions and equations and adopt this as an analytic lens to examine the role of curriculum materials on impacting students' learning. Findings suggest the activity structure of predict-act-reflect-reconcile is a potentially fruitful starting point for coordinating CAS and paper-and-pencil in classroom activity, but not sufficient on its own as a task design principle without focused classroom discussion and support during the enacted curriculum. This research points to the importance of teaching students how to communicate about their process of using multiple tools while solving problems, especially the mathematical thinking that students engage in while reconciling differences across tool-based representations.

This research addresses the issue of how to support students' representational fluency--the ability to create, move within, translate across, and derive meaning from external representations of mathematical ideas. The context of solving linear equations in a combined computer algebra system (CAS) and paper-and-pencil classroom environment is targeted as a rich and pressing context to study this issue. We report results of a collaborative teaching experiment in which we designed for and tested a functions approach to solving equations with ninth-grade algebra students, and link to results of semi-structured interviews with students before and after the experiment. Results of analyzing the five-week experiment include instructional supports for students' representational fluency in solving linear equations: (a) sequencing the use of graphs, tables, and CAS feedback prior to formal symbolic transpositions, (b) connecting solutions to equations across representations, and (c) encouraging understanding of equations as equivalence relations that are sometimes, always, or never true. While some students' change in sophistication of representational fluency helps substantiate the productive nature of these supports, other students' persistent struggles raise questions of how to address the diverse needs of learners in complex learning environments involving multiple tool-based representations.

Part of the research community that has followed the Early Algebra paradigm is currently delimiting the differences between arithmetic thinking and algebraic thinking. This trend could prevent new research approaches to the problem of learning algebra, hiding the importance of considering an arithmetico-algebraic thinking, a new approach which underpins the construction of a cognitive structure that links both types of thinking. This paper proposes a theoretical and practical framework for a learning approach that supports the construction of a cognitive structure which fosters arithmetico-algebraic thinking at the beginning of secondary school by means of cultural and technological activities relating to polygonal numbers.

This paper focuses on pattern generalisation as a way to introduce young students to early algebra. We build on research on patterning activities that feature, in their work with algebraic thinking, both looking for sameness recursively in a pattern (especially figural patterns, but also numerical ones) and conjecturing about function-based relationships that relate variables. We propose a new approach to pattern generalisation that seeks to help children (grades 2 and 3) work both recursively and functionally, and to see how these two modes are connected through the notion of variable. We argue that a crucial change must occur in order for young learners to develop a flexible algebraic discourse. We draw on Sfard's (2008) communication approach and on Châtelet's (2000) notion of the "virtual" in order to pursue this argument. We also root our analyses within a new materialist perspective that seeks to describe phenomena in terms of "material entanglement," which include, in our classroom research context, not just the children and the teacher, but also words, gestures, physical objects and arrangements, as well as numbers, operations and variables.

College algebra fulfills general education requirements at many colleges in the United States. The study reported here investigated differences in mathematics achievement between undergraduates in college algebra classes using one of two homework methods: "WeBWorK," an open-source system for web-based homework, or traditional paper-and-pencil homework. We assessed learning for 439 students in 19 college algebra classes at a large public university in the United States. Twelve classes used "WeBWorK" and seven had traditional paper-and-pencil homework. Analysis of covariance revealed no significant differences in algebra performance or achievement gain by homework group, ethnicity or gender when statistically controlling for previous mathematics achievement. Results support the conjecture that "WeBWorK" is at least as effective as traditionally graded paper-and-pencil homework for students learning college algebra in moderately sized lecture-based classes.

This study aims to investigate plane geometry problem-solving strategies of prospective mathematics teachers using dynamic geometry software (DGS) and paper-and-pencil (PPB) environments after receiving an instruction with GeoGebra (GGB). Four plane geometry problems were used in a multiple case study design to understand the solution strategies developed by 2 prospective teachers. The results revealed that although the participants mostly used algebraic solutions in the PPB environment, they preferred geometric solutions in the GGB environment even though algebraic solutions were still possible (the software did not preclude them). Furthermore, different proofing strategies were developed in each environment. This suggests that changing the environment may prompt students to seek for additional solutions, which, in turn, results in a deeper understanding of the problem. As such, using both environments simultaneously in solving the same problems appears to bring about important benefits.

This study was performed to determine preservice elementary mathematics teachers' solution performance of systems of linear equations using geometric representation, algebraic representation and transition from geometric representation to the algebraic representation and the predictive role of preservice elementary mathematics teachers' performance of different representation usage on their systems of linear equations self-efficacy perceptions. The research follows the prediction model, which is one of the relational screening models. The data were obtained by using the 'Systems of Linear Equations Self-Efficacy Test' developed by Kardes [Kardes, D. (2010). Matematik ögretmen adaylarinin lineer denklem sistemleri çözüm süreçlerinin öz-yeterlik algisi ve çoklu temsil baglaminda incelenmesi [An investigation of the processes of pre-services mathematics teachers? Solving systems of linear equations within the context of self-efficacy and multiple representations] [Master's thesis]. Marmara University. Turkish Higher Education Council thesis repository (#264144). https://tez.yok.gov.tr/UlusalTezMerkezi/] and the 'Systems of Linear Equations Performance Test' developed by the researchers. Because of the research, it was found out that preservice teachers' algebraic representation performances and their self-efficacy perceptions were at the maximum level when the number of equations and unknown variables was equal (m = n). Additionally, due to the multiple regression analysis, preservice teachers' performances on geometric representation, the transition from geometric representation to algebraic representation and systems of linear equations in R[superscript 3] were determined to be significant predictors of their self-efficacy perception. [This study was presented as a verbal announcement at the International Symposium on Active Learning-ISAL 2019 symposium.]

The paper describes the results of a study whose main aim was to find the interrelationship between pupils' school grades in Czech language (native), mathematics, and physics and pupils' cognitive predispositions to problem-solving in science and mathematics diagnosed by the Lawson Classroom Test of Scientific Reasoning and the Culture of Problem Solving test. A total of 180 pupils from the Czech Republic aged 14-15 took part in this study. The results show that pupils with better grades in the monitored subjects achieve better results in both tests. It also turns out that there are generally statistically insignificant differences between the results of pupils assessed by grades 1 or 2 and between those assessed by grades 3 or 4. Pupils' performance in the two tests might help to strengthen the objectivization of grading at school. They might also help identify the indicators important for developing problem-solving skills. The research specifically points to the need to develop algebraic thinking, the conception of infinity, spatial imagination, geometric imagination in the plane, proportional reasoning, and the ability to control variables.

Gaming the system is a persistent problem in Computer-Based Learning Platforms. While substantial progress has been made in identifying and understanding such behaviors, effective interventions remain scarce. This study uses a method of causal moderation known as Fully Latent Principal Stratification to explore the impact of two types of interventions--gamification and manipulation of assistance access on the learning outcomes of students who tend to game the system. The results indicate that gamification does not consistently mitigate these negative behaviors. One gamified condition had a consistently positive effect on learning regardless of students' propensity to game the system, whereas the other had a negative effect on gamers. However, delaying access to hints and feedback may have a positive effect on the learning outcomes of those gaming the system. This paper also illustrates the potential for integrating detection and causal methodologies within educational data mining to evaluate effective responses to detected behaviors.

Numbers and variables often follow the same principles of arithmetic operations, yet numbers can be computed to a value whereas variables cannot. We examined the effect of symbols--numbers versus variables--on middle school students' problem-solving behaviors in a dynamic algebra notation system by presenting problems in numbers (e.g., 3+5-3) or variables (e.g., x+y-x). We found that compared to problems presented in numbers, students attempted the problems more times and took more total steps when the problems were presented in variables. We did not find differences in pre-solving pause time or strategy efficiency on the two types of problems, indicating that students might notice problem structure in both types of problems. The results have implications for research on cognitive processes of symbols as well as the design of educational technologies. [This paper was presented at the Annual Meeting of the Cognitive Science Society (2021).]

In Victoria, Australia, senior secondary mathematics students are expected to use technology and thus need to make decisions about using pen-and-paper (P&P) or technology when solving mathematics problems. The predominant technology is a Computer Algebra System (CAS). This study investigated the beliefs about CAS held by twelve Year 11 students as they learnt to use CAS and whether these beliefs were stable over time. These students held a range of beliefs related to the usefulness of CAS, speed of CAS compared to P&P, whether CAS is proper mathematics, choice of CAS or P&P, ease of use, the correctness of answers and solving problems in Mathematical Methods (i.e. the mathematics subject studied). Beliefs are often described as being stable (e.g. McLeod, 1992), but some researchers stress stability needs to be determined empirically rather than being seen as a characteristic of beliefs (e.g. Liljedahl et al., 2012). For this sample of students, stability (rather than instability) is a feature of students' beliefs about CAS.

This paper is part of broader research being conducted in the area of algebraic thinking in primary education. Our general research objective was to identify and describe generalization of a 2nd grade student (aged 7-8). Specifically, we focused on the transition from arithmetic to algebraic generalization. The notion of structure and its continuity in the generalization process are important for this transition. We are presenting a case study with a semi-structured interview where we proposed a task of contextualized generalization involving the function y = x + 3. Special attention was given to the structures evidenced and the type of generalization expressed by the student in the process. We noted that the student identified the correct structure for the task during the interview and that he evidenced a factual type of algebraic generalization. Due to the student's identification of the appropriate structure and the application of it to other different particular cases, we have observed a transition from arithmetic thinking to algebraic thinking.

Literature has established that some learners encountered difficulties solving first order ordinary differential equations (ODEs). The use of error analysis in teaching ODEs is believed to make essential contribution towards calculus knowledge development. This paper therefore focuses on analyzing pre-service teachers' (PSTs) errors and misconceptions apropos of first order ODEs. The paper analyzed the nature of errors made in a test which was written by PSTs on the above topic. The test comprised various types of first order differential equations such as ODEs with separable variables, exact ODEs, ODEs that needed integrating factors, linear ODEs, and homogeneous ODEs. The purpose was to investigate the challenges faced by PSTs in various types of ODEs and the nature of misconceptions that they had in each particular type. This is a qualitative study that involved 63 PSTs who wrote a test on ODEs after being taught the topic for two weeks. The authors marked the work in order to ascertain the misconceptions and errors exhibited by the participants in the test. The PSTs' performance in the test was analyzed using the SOLO taxonomy and the Newman's theory mistake analysis. The study established that the topic was rather difficult for PSTs due to various reasons that included, among others, knowledge gaps in integration rules, algebraic computations and, in rare cases, differentiation, as well as misapplication of the rules of natural logarithms. This research therefore recommends that mathematics teacher educators ought to rather focus on the concept of integration and basic algebra before introducing the topic on ODEs to teachers on training.

The aim of this paper is to review some studies conducted with different learning areas in which the schemes of different participants emerge. Also it is about to show how mathematical proofs are handled in these studies by considering Harel and Sowder's classification of proof schemes with specific examples. As a result, it was seen that the examined studies we readdressed in the learning areas of Analysis, Geometry, Algebra, Linear Algebra, Elementary Number Theory, Probability, Combinatorics, and MIX. Students in early grades tend more towards external and empirical proof schemes. On the other hand, the characteristics of the proof schemes become more sophisticated as the participants' profiles change to pre-service teachers or as they become more specializing in mathematics. Some results are as follows: the academic achievement levels, genders, and grade levels of the participants in the studies examined in this paper have indicated that they have similar traces with the schemes they use. In addition, it has been determined that new perspectives such as examining Harel and Sowder's classification with new lenses, revealing the overlooked roles of some dynamics in proof, or improving the framework provide an important research area in terms of revealing students' potentials.

Motivated by the consideration of integrable systems in three spatial dimensions in Euclidean space with integrals quadratic in the momenta we classify three-dimensional Abelian subalgebras of quadratic elements in the universal enveloping algebra of the Euclidean algebra under the assumption that the Casimir invariant p â†' â‹... l â†' vanishes in the relevant representation. We show by means of an explicit example that in the presence of magnetic field, i.e. terms linear in the momenta in the Hamiltonian, this classification allows for pairs of commuting integrals whose leading order terms cannot be written in the famous classical form of Makarov et al [17]. We consider limits simplifying the structure of the magnetic field in this example and corresponding reductions of integrals, demonstrating that singularities in the integrals may arise, forcing structural changes of the leading order terms. [ABSTRACT FROM AUTHOR]

This paper addresses the recent body of research in algebra and algebraic thinking from 2018 to 2022. We reviewed 74 journal articles and identified four clusters of content areas: (a) literal symbols and symbolizing, (b) equivalence and the equal sign, (c) equations and systems, and (d) functions and graphing. We present the research on each of these content clusters, and we discuss insights on effective teaching practices and the social processes supporting algebraic reasoning. The research base shows that incorporating algebraic thinking into the elementary grades, emphasizing analytic and structural thinking processes, and emphasizing covariational reasoning supports students' meaningful learning of core algebraic ideas. We close with a discussion of the major theoretical contributions emerging from the past five years, offering suggestions for future research.

In this paper we review selected significant developments in the use of digital technology in the teaching and learning of mathematics over the last five years. We focus on a number of important topics in this field, including the evolvement of STEAM and critical making as well as the process of redefining learning spaces in the transformation of the mathematics classroom. We also address the increasing use of computer algebra systems and dynamic geometry packages; and the issue of student collaboration online, especially using learning environments and social media. We briefly touch on artificial intelligence systems, including hyper-personalisation of learning, multimodality and videos. We include a brief discussion on the impact of COVID-19 on mathematics education, and lastly on the more theoretical perspective of the epistemology of digital technology and the construct of humans-with-media. We conclude the discussion with some possible concerns and mentioning some possible new topics for research in the field.

The concept of function is critical in mathematics in general and abstract algebra in particular. We observe, however, that much of the research on functions in abstract algebra (1) reports widespread student difficulties, and (2) focuses on specific types of functions, including binary operation, homomorphism, and isomorphism. Direct, detailed examinations of the function concept itself--and such fundamental properties as well-definedness and everywhere-definedness--are scarce. To this end, in this paper we examine non-examples of function in abstract algebra by conducting a textbook analysis and semi-structured interviews with abstract algebra instructors. In doing so, we propose four key categories based upon the definitive function properties of well-definedness and everywhere-definedness. These categories identify specific characteristics of the kinds of non-examples of function that abstract algebra instruction should emphasize, enabling us to hypothesize how students might be able to develop a robust view of function and explain in greater detail the nature of the reported difficulties that students experience.

"Real Normed Algebras Revisited," the last paper of the late Gadi Moran, attempts to reconstruct the discovery of the complex numbers, the quaternions, and the octonions, as well as proofs of their properties, using only what was known to 19th-century mathematicians. In his research, Gadi had discovered simple and elegant proofs of the above-mentioned classical results using only basic properties of the geometry of Euclidean spaces and tools from high school geometry. His reconstructions underline an interesting connection between Euclidean geometry and these algebras, and avoid the advanced machinery used in previous derivations of these results. The goal of this article is to present Gadi's derivations in a way that is accessible to a wide audience of readers. [ABSTRACT FROM AUTHOR]

A study was conducted to explore the beliefs and practices of 49 New South Wales (NSW) primary school teachers regarding their beliefs and practices concerning the use of concrete materials in the learning and teaching of Number and Algebra. This paper reports on elements of the study regarding why and how teachers use concrete materials. Not only do teacher beliefs influence their classroom practice, Buehl and Beck (2014) propose that teacher practices may impact teacher beliefs suggesting they are interrelated. This paper sought to situate teacher beliefs and practices upon a conceptualisation of this interrelationship based on aspects of classroom practice involving concrete materials. The study employed survey methodology utilising a questionnaire and semi-structured interviews to gather data that were analysed to identify key themes regarding teachers' prevailing beliefs about the use of concrete materials. These themes included a belief in a social constructivist approach to learning and teaching mathematics, a cognitive dimension and engagement. On the basis of these findings an overview is suggested that positions teacher beliefs and practices relating to the use of concrete materials and how it might be used to shift inservice and preservice teachers' beliefs towards a more constructivist teaching approach.

This paper presents an analysis of fractions errors displayed by learners due to deficient mastery of prerequisite concepts. Fractions continue to pose a critical challenge for learners. Fractions can be a tricky concept for learners although they often use the concept of sharing in their daily lives. 30 purposefully sampled learners participated in this study. The research instrument consists of fractions test whose questions were selected from various Annual National Assessment examination papers containing in addition, subtraction multiplication and division of fractional operations. The different types of errors displayed by learners were then identified, coded and categorized. The analysis showed that the main sources for errors were lack of understanding of the basic concepts, learners' prior knowledge, misconceptions and misapplication of rules. This study recommends that teachers should help their learners to develop fractions conceptual understanding. Learners need to be explicitly taught that errors are opportunities for learning, and that they are springboard of inquiry.

In this paper, we make a case for the importance of teaching secondary school level algebra to students with learning disabilities (LD). Furthermore, we illustrate how they struggle and present best-practices on how they are best supported. We demonstrate effective ways of how teachers can show students with LD how to solve challenging algebra problems. In particular, we depict how educators can help learners with LD show their work on paper in ways that support their thinking processes as they engage with challenging algebra problems.

This paper explains teaching modern geometry for schools using software called Surfer and converting the algebraic surfaces into Batik motifs as one example of teaching mathematics with the integration of STEAM. Typical algebraic surfaces known in schools were, for example, cylinders and spheres which are identified as traditional geometry in this article. On the other hand, other geometrical objects that are not solely identified as traditional ones are called modern geometrical objects. The objects addressed here are governed by multiple algebraic surfaces and existed in the software called Surfer. The software is used for students to learn traditional geometry initially. After knowing how the software Surfer is doing, the software can be used for drawing more complex geometrical objects that are named to be modern geometry. This software is introduced for students in middle schools until students in the university such that technology and art are immersed in learning geometry. In this research, the activity involved creating a Batik piece which was a local textile in Indonesia based on algebraic surfaces created with Surfer. There were two methods of presenting the motif, these are after drawing the algebraic surfaces, students copied the surfaces into textiles using drawing Batik procedures. In the second method, students received some prototypes of algebraic surfaces and converted them into new patterns which were illustrated on a special paper for a copper stamp prototype later on. Using geometry as the routine topic which must be learned by any level of students in learning mathematics, the learning, innovation skills, and technology skills are direct educational values that are targeted by the activities with the approach presented in this article. The educational value is not only for the sake of mathematics and its content but also for integrating mathematics with other fields and exploring mathematical terms such that STEAM is introduced in this sense. The article here has addressed learning geometry by combining technology, art, and creating products to the related geometrical studied objects to develop learning skills, innovative skills, and technology skills that are educational values required skills in 21st-century education.

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.

[This paper is part of the Focused Collection on Instructional labs: Improving traditions and new directions.] This paper presents a historical analysis of the development, implementation, research, and assessment of an evidence-based introductory algebra-based physics course. The course is laboratory based and taught with an inquiry-based pedagogy. The course has remained unique and has been taught consistently for over 20 years. It is taught predominantly to health science majors and is completely laboratory based without a lecture and without a required text. Students learn by working through a guided inquiry manual in groups, doing experiments to explore the nature of the world around them. They develop both quantitative and qualitative models based on their experiments. The focus is as much on critical thinking, modeling, and thinking like a scientist, as on physics content. I discuss the course throughout the years, presented in the context of the field of physics education research historically and the state of our department locally. The results of multiple forms of assessment are presented chronologically. The assessments were designed to answer research questions about the students' conceptual understanding and thinking skills, the effects of class size expansion, and questions about learning assistants' pedagogical content knowledge. One finding was that the conceptual understanding of students in the inquiry-based course was significantly higher than students taught traditionally but not significantly different from students taught by other research-informed pedagogies. However, we found that students taught in the inquiry-based format showed more evidence of the application of thinking skills as assessed by our thinking skills rubric in free-response exam problems. We have demonstrated that it is possible to expand a laboratory-based, inquiry-based physics course from sections of 24 students to sections of 60 students with the help of learning assistants and we have developed and validated a set of questions that can be used to measure learning assistants' pedagogical content knowledge or for learning assistant training purposes.

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

This paper reports on a study designed to investigate preservice teachers' understanding of factorisation, a topic not explicitly taught within their teacher education programme, but one they will be required to teach when they graduate. We query if the knowledge they bring from secondary school, prepares them sufficiently to teach their future students for understanding. 83 preservice secondary school mathematics teachers' procedural and conceptual understanding of quadratic factorisation were assessed using Usiskin's Framework for understanding mathematics (2012) which identifies several dimensions of understanding. The study provides evidence that the preservice mathematics teachers have a strong procedural understanding, and while some conceptual understanding does exist, there was very limited conceptual understanding within most of the dimensions of the framework (Usiskin, 2012). We conclude the paper by considering how teacher educators can address the issues of preservice teacher knowledge and understanding of content not formally covered within their teacher education programmes.

Educational research has reported different benefits related to the use of practice tests. In the case of the teaching and learning of mathematics, evidence has been found that the use of practice tests is associated with an improved performance in standardized tests. However, it is less known about the emotions that students experience during such practice tests. This paper reports on a study on the use of practice test in mathematics instruction at the undergraduate level, which focuses on exploring students' emotions during a practice test for linear algebra. 78 students answered a questionnaire one day after having participated in an oral practice test on linear algebra. The results suggest that before the practice test nervousness was predominant among students, but this emotion decreases as the activity progresses.

The present study explored 285 11th-grade students' preconceptions, misconceptions, and errors in solving mathematics tasks by graphical method. A descriptive-explorative study design was adopted. Cluster sampling was used to select students from sampled secondary schools in eastern and central Uganda. Students' paper and pen solution sketches together with the task-based interviews were used to identify students' preconceptions, misconceptions, and errors in linear programming (LP). Students' responses were analyzed thematically and interpreted as students' learning gaps. The results indicated that students lacked proficiency in relating basic algebraic concepts and procedures to the mathematical language with which LP is conveyed. Generally, most students could not adequately use their previous knowledge and connect it to the learning and solving of LP tasks. Besides applying wrong mathematical algebraic concepts, students had difficulties interpreting and writing correct models (inequalities) from LP word problems. Misconceptions and errors were common and peculiar to and in individual student's solution sketches especially in applying the concepts of equations and inequalities to graphically solve and optimize LP tasks. Students held extremely weak concept images of graphing equations, and inequalities and their linkage to optimizing feasible regions. This research provides insight into the learning of mathematics word problems(LP) and recommends that mathematics educators should effectively apply students' preconceptions, misconceptions, and errors as opportunities for enhancing students' LP conceptual changes. For mathematical proficiency, suitable learning approaches, methods, and strategies should be adapted to address specific individual student's flawed conceptual and procedural knowledge and understanding. These approaches may guide educators in helping students to construct the connections between the old and the new knowledge.

This paper examines Black Algebra I educators' perceptions of their experiences in a professional learning community, "Algebra I Academy," designed specifically to equip Black Algebra I educators with liberatory pedagogical practices for their instruction with Black students taking Algebra I. Utilizing survey and focus group data from Black Algebra I educators who participated in the Algebra I Academy, findings from this study indicate that the Algebra I Academy had a positive effect on Black Algebra I educators by creating a sense of community with fellow Black Algebra I educators, providing a space for Black Algebra I educators to reimagine who they are as Black math educators, and by offering a collaborative space for Black Algebra I educators to learn and apply new, more liberatory, Algebra I pedagogical practices. More universally, this study highlights the need and value-add of educator learning communities that are racially affinity-based, professional learning spaces that are responsive to the historical and contemporary socio-racial contexts of educators' lives in and outside of schooling, and attentive to the social and emotional, as well as pedagogical, needs of educators.

An important topic of study in secondary mathematics is non-linear functions, including quadratic equations. In this study, findings from 25 Year 11 students indicated that difficulties with critical prerequisite concepts such as algebraic conventions impeded students' success in understanding and working with quadratics. Analysis of student errors identified misconceptions associated with the null factor law, and the nature of quadratic equations. This paper proposes that these findings are a result of limited timeframes nominated for learning quadratic topics outlined in the enacted curriculum. The implication of this is that the enactment of the Australian Curriculum: Mathematics F-10 did not support the development of conceptual understanding or procedural fluency with key mathematical concepts for these students. Without purposeful attention to prerequisite knowledge, and suitable time allocated to develop understanding and fluency, students' proficiency with topics such as quadratics is negatively influenced. A mastery approach to the hierarchically organised curriculum is supported by findings of this study.

QR decomposition is widely used for solving the least squares problem. However, existing materials about it may be too abstract for non-mathematicians, especially STEM students, and/or require serious background in linear algebra. The paper describes theoretical background and examples of GNU Octave compatible MATLAB scripts that give relatively simple but complete explanations about how to use QR decomposition for the least squares problem solution. Only basic knowledge of linear algebra and calculus are required. Both Givens rotations and Householder reflections usage for the linear least squares problem were considered. It was shown that the algorithm based on Givens rotations is even easier to program than explicit formation of the normal equations with subsequent usage of Gaussian elimination.

Numerous studies affirm the importance of students' attitudes for mathematical education. This study uses mathematical autobiographical essays to examine the mathematical experiences and attitudes of undergraduate students enrolled in mathematics classes at a Midwestern University in the United States. The essays of 41 students (20 men, 21 women) out of 170 enrolled in College Algebra, Precalculus, and Calculus were analyzed using the categories of (1) confidence in mathematics ability, (2) value of mathematics, and (3) liking of mathematics. This paper provides context and theoretical depth to student reports of their attitudes in these categories. Statements at the intersections of both confidence and liking, and value and liking provide additional insights. First, students who perceive mathematics as valuable are more likely to like it. Second, the vast majority of students talk about confidence and liking together, and with a positive correlation. Further, these students tend to talk about liking or not liking mathematics because of their perceived ability in mathematics. Our study suggests the importance of proactive teaching strategies to improve students' perceptions of the value, liking of, and their confidence in, mathematics, and also suggests self-efficacy as a potential theoretical basis for this work.

Recent research has highlighted the role of functional relationships in introducing elementary school students to algebraic thinking. This functional approach is here considered to study essential components of algebraic thinking such as generalization and its representation, as well as the strategies used by students and their connection with generalization. This paper jointly describes the strategies and representations of generalization used by a group of 33 sixth-year elementary school students, with no former algebraic training, in two generalization tasks involving a functional relationship. The strategies applied by the students differed depending on whether they were working on specific or general cases. To answer questions on near specific cases they resorted to counting or additive operational strategies. As higher values or indeterminate quantities were considered, the strategies diversified. The correspondence strategy was the most used and the common approach when students generalized. Students were able to generalize verbally as well as symbolically and varied their strategies flexibly when changing from specific to general cases, showing a clear preference for a functional approach in the latter.

This paper reports on a novel corequisite design and implementation for College Algebra at Florida International University. The corequisite course uses online, just-in-time, prerequisite assignments delivered on an open-educational platform. Students get help from near-peer learning assistants inside a math emporium environment. The course remains three credits after the redesign and now supports all students who would otherwise be taking Intermediate Algebra as a prerequisite course.

This paper describes Alkaline, a size-reduced version of Kyber, which has recently been announced as a prototype NIST standard for post-quantum public-key cryptography. While not as simple as RSA, I believe that Alkaline can be used in an undergraduate classroom to effectively teach the techniques and principles behind Kyber and post-quantum cryptography in general. Classroom experiences with individual concepts used in Alkaline support this belief. In addition to cryptography, linear algebra and abstract algebra classes would be good candidates for the use of Alkaline. A few exercises suitable for use in these classes are included.

The Miller-Rabin test is a useful probabilistic method for finding large primes. In this paper, we explain the method in detail and give three variations on this test. These variations were originally developed as student projects to supplement a course in error correcting codes and cryptography.

Backward transfer is defined as the influence that new learning has on individuals' prior ways of reasoning. In this article, we report on an exploratory study that examined the influences that quadratic functions instruction in real classrooms had on students' prior ways of reasoning about linear functions. Two algebra classes and their teachers at two comprehensive high schools served as the participants. Both schools drew from low-socioeconomic urban populations. The study involved paper-and-pencil assessments about linear functions that were administered before and after a four- to five-week instructional unit on quadratic functions. The teachers were instructed to teach the quadratic functions unit using their regular approach. Qualitative analysis revealed three kinds of backward transfer influences and each influence was related to a shift in how the students reasoned about functions in terms of an action or process view of functions. Additionally, features of the instruction in each class provided plausible explanations for the similarities and differences in backward transfer effects across the two classrooms. These results offer insights into backward transfer, the relationship between prior knowledge and new learning, aspects of reasoning about linear functions, and instructional approaches to teaching functions.

In this paper, we examine the relationships between teachers' subject matter preparation and experience in teaching and their performance on an instrument measuring mathematical knowledge for teaching Algebra 1. We administered the same instrument to two different samples of teachers-high school practicing teachers and community college faculty--who teach the same algebra content in different levels of institutions, and we compared the performance of the two different samples and the relationships between the measured knowledge and their educational and teaching background across the samples. The comparison suggested that the community college faculty possess a higher level of mathematical knowledge for teaching Algebra 1 than high school teachers. The subsequent analyses using the Multiple Indicator Multiple Causes (MIMIC) models based on our hypothesis on the factors contributing to the differences in the knowledge between the two teacher samples suggest that experience teaching advanced algebra courses has positive effects on the mathematical knowledge for teaching Algebra 1 in both groups. Highlighting the positive effect of algebra-based teaching experience on test performance, we discuss the implications of the impact of subject specific experience in teaching on teachers' mathematical content knowledge for teaching.

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.

Reflection on the theme of the paper makes one to remember Sir Snow's 1959 lecture on The Two Cultures in which Snow had advanced the thesis that the sciences and the humanities had become split into two cultures and had argued that this division had become a major handicap to solving the world's problems. Mathematics education itself has always been negatively be influenced by this rift. For example, in educational context it is frequently claimed that "mathematics is a language since it provides both a conveyance for and a substantiation of our thoughts. It is that aspect of mathematics that explains the key role it plays in modern science" (EFFROS 1998, p.132). But mathematics is no mere language, as everyone will find out if he tries to order a new door to his apartment with words alone and without measurements. The paper will argue historically and in semiotic terms proposing a methodology based on the conception of a complementarity of meaning and reference of knowledge representations.

The overwhelming presence of a procedural meaning of equality and equations reported in previous research has led to a call for suitable pedagogical interventions to nurture a relational meaning of these concepts. This paper is a response to that call. Drawing on the theory of objectification, the first part deals with the configuration of a Grade 3 (8-9-year-old students) teaching-learning activity that seeks to create the classroom conditions for the formation of the mathematical operations and operation-based rules that underpin the algebraic simplification of linear equations. Instead of using problems involving abstract open arithmetic sentences or alphanumeric equations (e.g., 5+__ = 16; 2n+3=11), the teaching-learning activity resorts to story-problems. Two visual semiotic systems serving to model and solve the story-problems were devised. The story-problems were framed in narratives that allowed the teacher and the students to infuse equations, their equating parts, and the mathematical operations with contextual meanings. The first part of the paper includes the theoretical assumptions about the teaching-learning activity and its configuration, and a rationale behind the devisal of the semiotic systems. The second part presents a Vygotskian multimodal genetic analysis of the teaching-learning activity; that is, an analysis that shows the formation of concepts "in motion," in the process of their genesis. The genetic analysis sheds some light on the way students, in their work with the teacher, encountered and refined the cultural-historical algebraic meanings of the equal sign and equations, and the concepts required in solving equations.

This paper builds on our previous research and investigates how students' fractional competence and reasoning can provide clear evidence of non-symbolic algebraic thinking and its progressive transition towards fully generalised algebraic thinking. In a large-scale study, 470 primary students completed a written paper and pencil test. This included three reverse fraction tasks which required students to find an unknown whole when presented with a quantity representing a fraction of that whole. Seventeen students from one participating primary school undertook a semi-structured interview which included reverse fraction tasks, similar to those on the written test, but with progressive levels of abstraction, starting with particular instances and becoming more generalised. Two important products of the study are the Classification Framework for Reverse Fraction Tasks and the Emerging Algebraic Reasoning Framework. The interview results highlight two critical transition points for the emergence of students' algebraic reasoning. The first is the ability to transition from additive strategies to multiplicative strategies to solve reverse fraction problems. Students reliant on diagrams and additive strategies struggled to solve more generalised tasks that required multiplicative rather than additive strategies. The second transition is the shift from multiplicative thinking to algebraic reasoning where students could generalise their multiplicative knowledge to deal with any quantity represented in a reverse fraction task.

Early algebraic thinking is the reasoning engaged in by 5- to 12-year-olds as they build meaning for the objects and ways of thinking to be encountered within the later study of secondary school algebra. Ever since the 1990s when interest in developing algebraic thinking in the earlier grades began to emerge, there has been a steady growth in the research devoted to exploring ways of fostering this thinking. While in its early days this research had to grapple with the question of what kinds of algebraic thinking might be feasible for the younger student, the evolution of the field over the past 30 years has led to an ever-increasing range of activity that is truly multi-dimensional. In this survey paper, I have framed the multi-dimensionality of early algebraic thinking according to three overarching types, namely, that of analytic thinking, structural thinking, and functional thinking, with generalizing being the scarlet thread that runs through all three. The first part of the paper looks back to the history of the notion of early algebra and the initial research efforts aimed at characterizing early algebraic thinking. The second part delineates the three overarching theoretical dimensions of early algebraic thinking, presents a sampling of past empirical findings, and points to some of the more recent work in the field, including the contributions to this Special Issue. The paper concludes by highlighting the new directions of this domain of research and offering suggestions for further research.