Algebraic Tasks Research Articles

Analyses of Pre-Service Teachers’ Errors in Solving Algebraic Tasks in Ghana.

This study analyzed prospective basic education teachers’ errors in solving algebraic tasks using Newman’s Error Analysis (NEA). The study employed an interpretivist philosophical paradigm with a qualitative case study design. Data were collected from 250 prospective mathematics teachers randomly selected from four colleges of education in the eastern region of Ghana using super item tests and unstructured interview protocols. Thematic analysis was employed to analyze the qualitative data. The study findings revealed that transformation and encoding errors among prospective mathematics teachers are the most prevalent errors. Other error types encountered included comprehension errors and process skills errors. In order to reduce aspiring mathematics teachers’ errors, the study emphasizes the necessity of focused interventions and professional development programs to address transformation and comprehension errors in algebraic concepts.

The professional practice of designing tasks: how do pre-service early childhood teachers promote mathematical processes in early algebra?

AbstractSpanish educational curriculum adopts a mathematical process-based approach, which encompasses problem solving, reasoning and proof, communication, connections and representation. A fundamental role in the integration of these processes in mathematics teaching is played by teachers’ professional practice of designing tasks. According to this, our aim is to analyze the ways in which pre-service early childhood teachers understand the mathematical processes in the professional practice of designing early algebra tasks and to identify how they intend to promote these processes through the tasks. Content analysis techniques were used to examine the designed tasks. To illustrate the data analysis and results, six tasks are presented. As a result, pre-service early childhood teachers associate problem solving with challenges and questions. They understand problems as unfamiliar situations but ignore the relationships between students and tasks. Moreover, they do not encourage exploration of phases of problem solving and tend to use strategies more suitable for routine tasks. Communication is identified in all the tasks designed, encouraging interaction and discussion. However, only one task explicitly promotes mathematical language. For reasoning and proof, pre-service teachers begin to use questions to elicit explanations and justifications, but do not encourage verification strategies and various modes of reasoning. The process of connections is only present in one task, reflecting the fragmented nature of mathematics teaching. We conclude that the professional practice of designing mathematical tasks is a powerful in teacher education. However, training programs should place greater emphasis on the meaningful use of mathematical processes.

Algebraic Representations of a Splitting Task: Implications of Numerical Reasoning

ABSTRACT This qualitative research study uses middle-grades students’ numerical reasoning to model their symbolic representations of the relationship between two multiplicatively related unknowns on an algebra task. Students in sixth grade through ninth grade participated in clinical interviews that assessed their numerical reasoning using the Number Sequences framework, their interpretation of the equal sign, and their algebraic reasoning in the context of writing equations. Analysis used the mental structures that define their numerical reasoning to explain their algebraic capabilities, and changes in their reasoning about the equal sign in different contexts. Findings show that splitting, one mental structure related to the students’ numerical reasoning, is related to students’ symbolic representations of multiplicatively related unknowns. Furthermore, students who had not constructed a splitting operation tended to use numerical examples in place of reasoning about unknowns, and to reason operationally about the equal sign to compensate for this limitation on an algebra task.

Demetriou’s tests and levels of algebraic abilities and proportional reasoning in seventh, eighth, and ninth grades

Developing algebraic thinking is a key factor in learning mathematics. Despite its importance, many students still struggle with algebraic concepts. This research investigates students’ achievements in algebraic thinking using Demetriou’s test across 7<sup>th</sup> (approximately 12-13 years old), 8<sup>th</sup> (approximately 13-14 years old), and 9<sup>th</sup> (approximately 14-15 years old) grades. The study analyzes performance in different levels of algebraic tasks (i.e., [1] extrapolation of relationships, [2] coordinating simple structures, [3] operating with undefined symbolic structures, and [4] coordination with undefined structures), revealing a significant developmental leap in algebraic abilities during the 9<sup>th</sup> grade. While no statistically significant differences were found in the first level, 9<sup>th</sup> grade students demonstrated superior performance in levels 2, 3, and 4, suggesting cognitive readiness for abstract algebraic concepts around the age of 14. The research unveils a disjointed development in algebraic abilities, indicating a progression from basic arithmetic operations to proportional reasoning before the full integration of algebraic thinking. Notably, tasks involving variables in the third level pose persistent challenges for students. The findings contribute to understanding the optimal age for introducing algebraic concepts and underscore the importance of considering cognitive development in mathematics education. The study proposes implications for educators, such as emphasizing proportional reasoning in earlier grades and employing differentiated instruction based on individual students’ abilities.

Pengembangan e-modul fliphtml menggunakan media aplikasi cymath untuk meningkatkan kemampuan literasi matematika dan pengaruhnya terhadap antusiasme Mahasiwa dalam belajar materi aljabar

This research is a development research that develops modules that are used as e-modules using fliphtml and inside the module is inserted cymath which is an application that can help solve linear algebra tasks. It is hoped that from this research students can improve their mathematical literacy skills and can have an interest in learning linear algebra. Mixed method research research that combines qualitative and quantitative research types. The validity of the e-module 4.5 is in the very valid category, the Student Worksheet 4.7 is in the very valid category and the learning achievement test 4.6 is in the very valid category and for the learning outcomes test it is not only tested through a validator but also tested for validity and statistical reliability and shows data valid and reliable. Practical is said to be practical from activity observer data in the teacher's criteria obtained a score of 3.8 and meets good criteria. Effectively it is said that practically from student activity analysis is in good criteria. % of students who gave a positive response. There is an influence on the enthusiasm of students who are taught using e-modufliphtml using the cymath application media

Construct It! Building Painted Cubes Task: Serena’s Case

The Building Painted Cubes Task is a groupworthy algebraic task. Students build cubes using linking unit cubes, search for algebraic patterns, and report findings on posters. This task can create spaces for students to see themselves as doers of mathematics.

The hierarchical relations among mathematical competencies: From fundamental numeracy to complex mathematical skills.

Mathematical competencies can be conceptualized as layers of knowledge, with numeracy skills as the foundational core and more complex mathematical skills as the additional layers over the core. In this study, we tested an expanded hierarchical symbol integration (HSI) model by examining the hierarchical relations among mathematical skills. Undergraduate students (N = 236) completed order judgement, simple arithmetic, fraction arithmetic, algebra, and verbal working memory tasks. In a series of hierarchical multiple regressions, we found support for the hierarchical model: Additive skills (i.e., addition and subtraction) predicted unique variance in multiplicative skills (i.e., multiplication and division); multiplicative skills predicted unique variance in fraction arithmetic; and fraction skills predicted unique variance in algebra. These results support the framework of the HSI model in which mathematical competencies are related hierarchically, capturing the increasing complexity of symbolic mathematical skills. (PsycInfo Database Record (c) 2023 APA, all rights reserved).

Critical Thinking of Students with High and Low Mathematics Efficacy PISA Problem: A Case of Algebraic Task

This study aims to describe students' critical thinking on PISA problems on algebra content in terms of high and low mathematics efficacy. This descriptive qualitative research involves two junior high school students with high and low mathematics efficacy in one of the junior high schools in Jombang Regency. Data collection techniques were carried out by administering a mathematics efficacy questionnaire, Algebraic Content PISA Problems (ACPP), and interview guidelines. Data analysis was based on indicators of critical thinking ability (interpretation, analysis, evaluation, inference, explanation, and self-regulation). From the results of data analysis, critical thinking in the interpretation aspect shows that only students with high mathematics efficacy can write what is known and asked. Likewise, in the analysis aspect, it shows that only students with high mathematics efficacy can create a mathematical model of the given problem. In the evaluation aspect, students with high and low mathematical efficacy can write complete problem-solving. Likewise, in the inference aspect, students with high and low mathematics efficacy were able to draw conclusions logically. Likewise, in the explanation aspect, students with high and low mathematical efficacy can write down the final results and provide reasons for the conclusions drawn. Meanwhile, in the aspect of self-regulation, students with high and low mathematics efficacy have conducted a review. Therefore, understanding students' critical thinking is very important for students and teachers to improve students' ability to solve PISA problems.

Effects and challenges to implement differentiated mathematics teaching among fourth graders in Montenegro

In this quasi-experimental pretest-posttest study we examined the effects of differentiated instruction (DI) in within-class ability groupings of 246 Montenegrin fourth-graders and their ability to solve algebraic equations. We assessed 2 parallel student groups at equal achievement levels to compare DI, in which teaching and work modes were adapted to students’ grouping according to previous achievement and pretest scores, and traditional whole-class instruction. Pretest-posttest evaluations were administered to both groups, and observation indicators were evaluated to assess the level and type of student activities, engagement, and individualisation. Students in the homogeneous DI experimental groups with tailored instructions were significantly more successful at solving algebraic tasks than their peers in the traditional whole-class instruction control group. DI improved students’ results, but teachers required specific training and significantly more preparation time.

Investigating Pedagogical Content Knowledge and Decision-Making on Algebraic Tasks

Pedagogical Content Knowledge (PCK) tasks and decision-making are closely related to teaching and learning mathematics. This study aims to investigate teacher PCK in decision-making on different algebra tasks and to see activity in teacher-student interactions. This research is qualitative, with a research approach used as a case study. The participants in this study were only one teacher with different characteristics in mathematics tasks. Data collection consisted of primary research data from class observations, teacher interview transcripts, and researcher notes, while documents such as teacher notes were secondary data. Data analysis was carried out directly based on class observations and interviews. The findings in this study reveal teacher PCK through teacher decision-making regarding algebraic tasks in Swartz & Perkins (2016) three theories of thinking: generating ideas, clarifying ideas, and assessing the reasonableness of ideas.

Hierarchical off-diagonal low-rank approximation of Hessians in inverse problems, with application to ice sheet model initialization

Obtaining lightweight and accurate approximations of discretized objective functional Hessians in inverse problems governed by partial differential equations (PDEs) is essential to make both deterministic and Bayesian statistical large-scale inverse problems computationally tractable. The cubic computational complexity of dense linear algebraic tasks, such as Cholesky factorization, that provide a means to sample Gaussian distributions and determine solutions of Newton linear systems is a computational bottleneck at large-scale. These tasks can be reduced to log-linear complexity by utilizing hierarchical off-diagonal low-rank (HODLR) matrix approximations. In this work, we show that a class of Hessians that arise from inverse problems governed by PDEs are well approximated by the HODLR matrix format. In particular, we study inverse problems governed by PDEs that model the instantaneous viscous flow of ice sheets. In these problems, we seek a spatially distributed basal sliding parameter field such that the flow predicted by the ice sheet model is consistent with ice sheet surface velocity observations. We demonstrate the use of HODLR Hessian approximation to efficiently sample the Laplace approximation of the posterior distribution with covariance further approximated by HODLR matrix compression. Computational studies are performed which illustrate ice sheet problem regimes for which the Gauss–Newton data-misfit Hessian is more efficiently approximated by the HODLR matrix format than the low-rank (LR) format. We then demonstrate that HODLR approximations can be favorable, when compared to global LR approximations, for large-scale problems by studying the data-misfit Hessian associated with inverse problems governed by the first-order Stokes flow model on the Humboldt glacier and Greenland ice sheet.

Using the onto-semiotic approach to analyze novice algebra learners’ meaning-making processes with different representations

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.

What Can the Realization Tree Assessment Tool Reveal About Explorative Classroom Discussions?

One of the challenges of understanding the complexity of so-called reform mathematics instruction lies in the observational tools used to capture it. This article introduces a unique tool, drawing from commognitive theory, for describing classroom discussions. The Realization Tree Assessment tool provides an image of a classroom discussion, depicting the realizations of the mathematical object manifested during the discussion and the narratives that articulate the links between these realizations. We applied the tool to 34 classroom discussions about a growing-pattern algebraic task and, through cluster analysis, found three types of whole-class discussion. Associations with classroom-level variables (track, but not grade level or teacher seniority) were also found. Implications with respect to applications and usefulness of the tool are discussed.

Investigating two teachers’ development of combinatorial meaning for algebraic structure

This paper reports on the results of a four-day teaching experiment that supported two algebra teachers to develop a combinatorial meaning for algebraic structure. The purpose of the teaching episodes was to support the teachers (a) to establish a combinatorial understanding for algebraic structure (Tillema & Burch, 2022) by generalizing the cubic identity, a+b3=a3+3a2b+3ab2+b3, as a symbolization of quantitative and combinatorial relationships out of a contextualized problem (Tillema & Gatza, 2016) and (b) to develop a combinatorial meaning as a mobilization of their understanding through a series of algebraic tasks (cf. Thompson et al., 2014). The findings from this study contribute to research literature on teachers’ mathematical meanings within secondary algebra by investigating how teachers’ combinatorial meanings developed and how differences in their combinatorial meanings impacted their algebraic reasoning. The findings demonstrate a combinatorial pathway for supporting the development of expanding and factoring as reversible polynomial operations (cf. Sangwin & Jones, 2017).

The sequential power flow in complex plane for solving the VSC-MTDC hybrid AC/DC transmission grids

This paper presents new methodologies and modeling aimed at studying interwoven AC and DC subsystems of a power system in the complex plane. It is well known that the development of any former power flow application in real domain addressed to embedding FACTS devices is preceded by an arduous algebra task. To overcome this difficulty, a novel power flow solution method is proposed in this work. Thanks to the Wirtinger calculus, a Newton–Raphson method based on Taylor series expansions of nonlinear functions of complex variables and their complex conjugates is developed. Thus, the voltage-source-converter multi-terminal direct current is formulated in complex plane without any loss of accurateness and is termed as VSC-MTDC. Its performance is assessed through IEEE-test systems interconnected across a DC network prone to several scenarios, e.g., topology, loading magnitude and interchanging of active power.

Exploring high school teacher’s design of rich algebra tasks

The phenomenon in schools today is that teachers rarely change their tasks. However, the teacher's activities to change their tasks, build students' arguments, support solutions, and maintain arguments without long debates are important and exciting things to learn. This study explores the ideas of high school teachers about task design and practice in teaching mathematics. The authors surveyed twelve high school teachers who teach mathematics in East Java Province. First, the authors conducted preliminary observations to observe the design of mathematics teacher tasks for six months in eight schools. Second, the authors state that teachers are engaging and consistent in designing rich algebra tasks. Third, the authors examine the teacher's ideas through direct observation and unstructured interviews. The results show how teachers' ideas about task design enhance students' creative thinking by reforming tasks from textbooks into rich mathematics tasks. The design of the task carried out by the teacher is to create to stimulate creative thinking. The teachers also use their knowledge and understanding of the material and curriculum to modify mathematics tasks in students' mathematics books. The task given by the teacher is to improve students' reasoning, not just memorize formulas or properties.

Students’ problem-solving ability in solving algebra tasks using the context of Palembang

This descriptive research aimed to know students' problem-solving ability in arithmetic operations on algebra forms through an Indonesian realistic mathematics education, namely Pendidikan Matematika Realistik Indonesia (PMRI), approach in secondary school number 17 Palembang. The learning process, material, and assessment used were principles and characteristics of PMRI. The data collection technique was done by two students' activities and the written test to measure students' problem-solving abilities. The written test, which referred to the indicators of problem-solving ability, was given after the learning process. This study's findings indicate that Palembang's context helps students comprehend algebraic arithmetic operations. The principles and characteristics of PMRI play an essential role in enhancing students' problem-solving skills. To conclude, students develop and solve problems by modeling based on their mathematical ideas. In addition, students must be able to develop problem-solving strategies in which they employ a variety of procedures.

A Ruppert’s framework: How do prospective teachers develop analogical reasoning in solving algebraic problems?

It is widely agreed that knowing how prospective teachers develop analogical reasoning in solving problems is important. Some problem solving is domain specific and require particular ways of analogical reasoning skill, therefore this study aims to reveal the development of analogical reasoning and strategies used by prospective teachers. The research design use a qualitative method. As many as 69 mathematics prospective teachers were involved voluntarily to complete algebraic tasks and 12 of them were interviewed to investigate their analogical reasoning and solution strategies. The data analysis used the Ruppert’s framework consisting of four components: structuring, mapping, applying, and verifying. It was found that the first three components were fully performed by the prospective teachers. However, the verifying stage was applied by prospective teachers in different ways. The dominant strategy used was a combined strategy of multiplication and addition. Their strategies varied according to the subjects’ general ability. The more strategies employed in solving problems, the better their analogical reasoning becomes. This implies that instructional designs that will be developed by prospective teachers may vary. Therefore, during their candidature, they should be provided by many solving strategies in problem solving to develop students’ analogical reasoning.

Taking an Asset-Based Approach in the Use of a Culturally Located Task to Construct Functional Reasoning

Background: Algebra has traditionally been seen as a site of inequity and a gatekeeper for marginalized learners within both national and international studies. However, considerable research has shown that taking an asset-based approach in mathematics teaching improves learning for marginalized students, including Māori and Pāsifika learners. Purpose: The purpose of this research was to explore how a culturally located task set with a familiar Pāsifika context and within an algebraic frame was mathematized to support rich understandings. The focus was placed on how Pāsifika and Māori students drew on multimodal forms of communication (gesture, drawings, language, and symbols) to collectively make sense of a culturally located growing pattern task designed to develop functional reasoning. Research Design: The 12 student participants in the study were aged between 11 and 12 years and were of Māori and Pāsifika nations ethnicity, as were their teacher and the two researchers. The lesson reported on in this study was one of eight lessons and was representative of all lessons as the teacher and students drew on a variety of multimodal means of communication to construct functional reasoning. The design drew on both qualitative case study and design research. Data collection tools included interviews, video-recorded classroom observations, field notes, and photographs of work samples. The research design and analysis were informed through use of the Ula model as appropriate to all participants’ Pāsifika ethnicity. Similarly, storying was used, given that it is a traditional form of sharing used by indigenous peoples. Findings: Clearly evident in the results was how use of a culturally located task within a known context and the teacher’s asset-based approach supported students to engage in a challenging algebraic task. Multimodal forms of representation as reasoning and communication tools supported them to access more sophisticated forms of algebraic understandings as they began to generalize recursive, covarying, and correspondence relationships in increasingly sophisticated ways. Use of body language and gesturing was central to their communication, as was use of their first language and natural language. Conclusions/Recommendations: To change the gatekeeping role of algebra, teachers need to take a strength-based perspective and draw on what students bring to school as valued knowledge. For all students to access powerful ways of reasoning algebraically, the key role of many multimodal forms of communication needs to be recognized and affirmed.

A taxonomy of high school students’ levels of understanding in solving algebraic problems

Abstract 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.