Learning Of Algebra Research Articles

Znak jednakosti - problem učenja rane algebre i kako ga rešiti

The equals sign is one of the most important concepts and symbols in mathematics, the understanding of which is crucial for learning and understanding all mathematical content, especially algebra. The paper will draw attention to the problems related to the formation and understanding of this concept, and present a methodological approach to learning based on the context modeling of real-life situations (modeling length, balance, etc.), with the aim of overcoming this misconception. In the empirical section of the paper, we examined the effects of the methodological framework of learning algebra on the understanding of the equals sign through an experiment with parallel groups, and on a sample of the fourth-grade students of primary school (N = 257). The obtained results show that the methodological approach based on the context modeling of real-life situations improves student understanding of the equals sign, as a symbol of mathematical equivalence and their ability to solve problems containing this sign.

Digital Games Designed By Prospective Teachers in The Scratch Program Through The Eyes of Mathematics Teachers

The Scratch programming tool is an effective tool for the logical problem-solving skills of middle school students with its ease, richness of their visual characters and all their objects, as well as processing skills. In particular, the fact that research has shown that teaching with Scratch has a positive effect on students' mathematical achievements in algebraic expressions indicates that a different teaching method can be used in the field of algebra learning. Therefore, it is believed that the use of mathematics teachers in the lesson by learning this program before service will contribute to the students ' learning and love of mathematics. The aim of the research is to examine the opinions of mathematics teachers about the games that prospective teachers design in the Scratch program. The participants of this study, in which the case study was conducted, are six mathematics teachers. The research process was carried out in three stages. In the first stage, 13 games were designed by prospective teachers about achievements in algebraic expressions. In the second phase, six mathematics teachers were asked to score these games according to the revised rubric of Li et al (2013). In the third stage, teachers' opinions on the reasons for the scores they gave to the games were taken. Kappa coefficients were calculated for the scoring compliance among teachers. Qualitative data analysis techniques were used in the analysis of teacher opinions. In this study, it was revealed that most teachers found the games interesting, cle, and instructive/reinforcing. Some teachers found the games original, while others stated that they were classic games that they often encountered. In addition, some teachers emphasized that they can use these games in their lessons, while others stated that they cannot use them in their lessons for reasons such as missing instructions, code errors, and lack of program.

Development of Multiple Representation Translating Measurement Tool and Examination of 9th Grade Students’ Multiple Representations Translate Skills in Algebra

Abstract: In this study, it was aimed to examine 9th grade students' multiple representation translate skills in algebra learning area and the relationship between these representations translate skills. The study employs survey design. Selected through purposive sampling method, 637 ninth grade students in a socio-economically medium level province in Turkey participated to the study. Within the scope of the study, Multiple Representation Translating Measurement Tool (MRTMT) including four factors (verbal- graph- algebraic- table) was developed and data were collected through this instrument. The validity and reliability of the scale was tested with Rasch analysis and confirmatory factor analysis (CFA). Cronbach Alpha value was calculated as .88. The findings show that students’ skills of translating between representations level is low. In addition, students were most successful in expressing the situation with other representations when the table representation was given, but had difficulty in translating it to other representations when given an algebraic representation. Furthermore, it is found out that there is a significant relationship between students’ translating skills to verbal- graph- algebraic- table representations. Accordingly, it can be suggested that each representation should be used in an interrelated and holistic way in algebra teaching.

Slow down to speed up: Longer pause time before solving problems relates to higher strategy efficiency

We examined the influences of pre-solving pause time, algebraic knowledge, mathematics self-efficacy, and mathematics anxiety on middle-schoolers' strategy efficiency in an algebra learning game. We measured strategy efficiency using (a) the number of steps taken to complete a problem, (b) the proportion of problems completed on the initial attempt, and (c) the number of resets prior to completing the problems. Using the log data from the game, we found that longer pre-solving pause time was associated with more efficient strategies, as indicated by fewer solution steps, higher initial completion rate, and fewer resets. Higher algebraic knowledge was associated with higher initial completion rate and fewer resets. Mathematics self-efficacy and mathematics anxiety was not associated with any measures of strategy efficiency. The results suggest that pause time may be an indicator of student thinking before problem-solving, and provide insights into using data from online learning platforms to examine students' problem-solving processes.

The Use of Activity, Classroom Discussion, and Exercise (ACE) Teaching Cycle for Improving Students’ Engagement in Learning Elementary Linear Algebra

The main focus of the study was to identify the the effects of the ACE teaching cycle approach on students’ engagement in learning of Elementary Linear Algebra. The Elementary Linear Algebra course was delivered at the Mathematics Education Study Program of State University of Padang, Indonesia. An action research methodology was adopted for the study and the constructivist theory was used as a theoretical framework. This study used mixed methods to collect, describe, and interpret the data. The data were collected through focus group discussion, classroom observations, and questionnaire. The participants in this study consisted of 37 students enrolled in the Elementary Linear Algebra course during the January-June 2011 semester. The ACE teaching cycle approach that was implemented in this study consisted of three steps. The first step in this cycle was the concept maps activity. This activity encouraged students to prepare for the topics that would be discussed in the class. The next step was classroom discussion which provided a social context in which the students could work together to solve mathematics problems. Finally, exercises were assigned as homework. In this part, students practised solving mathematics problems and wrote reflective journal. Students’ involvement in the ACE teaching cycle approach in the Elementary Linear Algebra course improved their engagement in mathematics learning in the three domains of cognitive engagement, affective engagement and behavioural engagement.

Generalization of quadratic figural patterns: Shifts in student noticing

Quadratics, as the simplest non-linear function, are a foundational context for drawing together key algebraic concepts. Yet research has documented student difficulties in moving beyond rote procedures. This fine-grained study with Year 10 (15 or 16-year-old) students sought insights into how visualization and pair-work interactions might be harnessed for highlighting conceptual connections among quadratic concepts. Theorizations of symbolic generalization and student noticing were drawn on to investigate two pairs working on 19 quadratic generalization tasks. Different visual structures were found to elicit both productive and unproductive actions. Productive actions included building on each other’s visualizations, explaining disagreements verbally and with gestures, and verifying each other’s generalizations. Some unproductive actions were found that shifted focus away from symbolic generalization. An emergent framework to inform participative approaches to learning algebra is shared, along with suggested implications for including figural patterns when teaching quadratics and suggestions for further research.

Content and language integrated learning (CLIL) and the linguistic difficulties of Saudi students learning algebra

Abstract This study explores the type of linguistic difficulties that Saudi students encounter in Introductory Algebra courses at college level and how an integrated pedagogical approach, such as Content and Language Integrated Learning (CLIL), might impact students’ learning of both algebra and the language needed to learn it. A mixed-method design was implemented for 12 weeks to collect and analyze data in three phases. The study sample comprised of a study group of 28 students from an Introductory Algebra class. Findings identified and classified the types of difficulties that the students encountered and suggested a CLIL instructional model to overcome or minimize them.

An empirically based practical learning progression for generalisation, an essential element of algebraic reasoning

Generalisation is a key feature of learning algebra, requiring all four proficiency strands of the Australian Curriculum: Mathematics (AC:M): Understanding, Fluency, Problem Solving and Reasoning. From a review of the literature, we propose a learning progression for algebraic generalisation consisting of five levels. Our learning progression is then elaborated and validated by reference to a large range of assessment tasks acquired from a previous project Reframing Mathematical Futures II (RMFII). In the RMFII project, Rasch modelling of the responses of over 5000 high school students (Years 7–10) to algebra tasks led to the development of a Learning Progression for Algebraic Reasoning (LPAR). Our learning progression in generalisation is more specific than the LPAR, more coherent regarding algebraic generalisation, and enabling teachers to locate students’ performances within the progression and to target their teaching. In addition, a selection of appropriate teaching resources and marking rubrics used in the RMFII project is provided for each level of the learning progression.

Mathematical modelling routes supported by technology in the learning of linear algebra: a study with Costa Rican undergraduate students

Mathematical modelling routes supported by technology in the learning of linear algebra: a study with Costa Rican undergraduate students

Application of Topological Data Analysis to Multi-Resolution Matching of Aerosol Optical Depth Maps

Topological data analysis (TDA) combines concepts from algebraic topology, machine learning, statistics, and data science which allow us to study data in terms of their latent shape properties. Despite the use of TDA in a broad range of applications, from neuroscience to power systems to finance, the utility of TDA in Earth science applications is yet untapped. The current study aims to offer a new approach for analyzing multi-resolution Earth science datasets using the concept of data shape and associated intrinsic topological data characteristics. In particular, we develop a new topological approach to quantitatively compare two maps of geophysical variables at different spatial resolutions. We illustrate the proposed methodology by applying TDA to aerosol optical depth (AOD) datasets from the Goddard Earth Observing System, Version 5 (GEOS-5) model over the Middle East. Our results show that, contrary to the existing approaches, TDA allows for systematic and reliable comparison of spatial patterns from different observational and model datasets without regridding the datasets into common grids.

The effect of worked examples on student learning and error anticipation in algebra

The effect of worked examples on student learning and error anticipation in algebra

Students' Activeness in Mathematics Learning via Zoom on Algebra Learning Material

This research aimed to reveal the students’ activeness in mathematics learning on algebra using Zoom. This research is included in the qualitative research category with a descriptive approach. The collected data were analyzed using descriptive methods by showing the total score, percentage, or average score of each respondent's answer. The results showed that students were very active and motivated in achieving learning goals during learning via Zoom. This research implies that virtual learning is feasible to be encouraged.

Using the VA Framework to Teach Algebra to Middle School Students With High-Incidence Disabilities

Algebra is considered by many to be a gateway to higher-level mathematics and eventual economic success yet students with and without disabilities often struggle to develop algebra skills. This study builds on the limited understanding of how virtual manipulatives support students with disabilities in the area of algebra by investigating their use within the virtual-abstract (VA) framework. Using a multiple probe across behaviors, replicated across participant design, researchers found a functional relation between the VA framework and student algebraic learning. Mathematical behaviors based on grade-level curriculum included: one-step equations with positive and negative numbers, two-step equations with positive numbers, and two-step equations with positive and negative numbers. All three seventh-grade students with high-incidence disabilities improved their performance on each of the three algebra behaviors during intervention, and all participants maintained their accuracy after intervention, as compared to baseline to maintenance. Detailed results and their implications for practice are discussed further.

Stages of Ability to Recognize Number Concepts Through Modified Apron Media for Children with Special Needs

The concept of equality in mathematics is fundamental to be learned by elementary school level students. Knowledge of equations in mathematics is an essential prerequisite for learning algebra at a higher level. Especially in the ability to recognize the concept of numbers in children with special needs which can be developed using arithmetic apron media. This study uses a literature study approach. The subject is an extraordinary elementary school student. The data obtained by researchers came from various literatures that have been reviewed and analyzed by researchers. The results showed that students could reach the stage of introducing the concept of numbers

Effectiveness of the Metacognitive – Based Algebra Learning Model

Learning approaches are among the critical factors that determine learning effectiveness. A typical learning approach that promotes student awareness will result in successful learning. This article examines the effectiveness of a metacognitive-based algebra learning from three aspects, namely: 1) learning outcomes; 2) student responses; and 3) student activities in learning algebra. The results show that all three aspects meet the effectiveness criteria of metacognitive approach in learning algebra.

The Effectiveness of the Pipek Model (Concept Map-Based Interactive Learning) in Learning Modern Algebra

The major objective of this study was to investigate the effectiveness of the PIPEK model in learning modern algebra. This study was to identify the effects of the use of PIPEK model on students’ engagement and students’ learning in modern algebra course at Mathematics Department in Semester 2, 2020. During this study, the students completed achievement test and questionnaire. The study revealed that, in terms of surface strategy, students tended to memorize formulas and the method for solving problems. Moreover, students still relied heavily on the lecturer’s instructions; Particularly, related to the aspects of the attitude, student participant generally found modern algebra as a boring and difficult course. Futhermore, students felt stressed and anxious; The findings related to the behavioral aspect indicated that although student’s attention were high, they were not diligent in accomplishing independent work task. The finding also demonstrated that, the level of mastery achievement of student learning outcomes has not achieved classical completeness. However, the PIPEK model has given students the opportunity to engage more in student center learning process.

Undergraduate Students’ Learning of Linear Algebra Through Mathematical Modelling Routes

Mathematical modelling has acquired relevance at all educational levels in the last decades since integrating this activity in instruction provides significant contexts for improving students’ learning, including in linear algebra courses that have a notable presence in many undergraduate courses from different fields, including engineering and sciences. This paper reports a study aiming to characterise the distinct modelling routes performed by Costa Rican undergraduate students when solving a mathematical modelling task involving the concept of system of linear equations (SLE). In analysing those modelling routes, it was possible to identify their learning of linear algebra concepts and their modelling competencies as well as the associated difficulties that students faced. Data collection included participant observation, with audio recording of the students’ discussions, their written work on the task, and digital files of their work with technology. The results show that non-linear routes are associated with a greater mobilisation of students’ knowledge on SLE concepts and with their development of modelling competencies. The results also highlight the need to improve the students’ competency of validating results, an important step that they did not take, and suggest the need to make technology relevant to the students’ work on modelling tasks.

Students’ self-regulation learning ability in learning algebraic forms in wetland context with the help of interactive multimedia

One branch of mathematics that is very useful in solving problems is algebra. In fact, junior high school students still face difficulties in learning algebra. This study aimed to develop interactive multimedia on algebraic-form material with a wetland context, to describe student self-regulation learning (SRL) abilities in learning algebraic forms using interactive multimedia with wetland contexts, and to analyse the relationship between SRL ability with student learning outcomes after learning on algebraic form material in wetland context with interactive multimedia assistance. This research was a development research using the ADDIE method. It was tested in small groups of seventh grade students of SMP Negeri 9 Banjarmasin. The data collection techniques used tests and questionnaires. The data analysis technique used descriptive statistics and simple linear regression. The results showed that interactive multimedia on wetland context-algebraic form material was valid and could be used in learning algebraic forms in seventh grade. The trial results showed that the average student learning outcomes were in the sufficient category and as many as 80.0 percent of students had reached the KKM determined by the school. Student’s SRL ability was in the medium category, and there is no influence from SRL on student learning outcomes. The further research is needed in large groups to analyse how the influence of students’ SRL skills on learning outcomes in interactive multimedia assisted learning.

Automated Teaching System “Sets” (Research for Organizing the 1st Part of the Project)

The issues of building an automated learning system “Sets” which will allow students to master one of the important topics of the discipline “Discrete Mathematics” and to develop logical and mathematical thinking in this direction are studied. The corresponding topic of the 1st part of the project includes materials related to the concept of a set, operations on sets, algebra of sets, proofs of statements for sets, and the derivation of formulas for the number of set elements. The system is based on a construction of the statements proof editor for a set and of the formulas derivation editor for the number of set elements, both editors are to be used for teaching. The first of these allows students to split the original statement into a number of simpler statements, taken together equivalent to the original statement, to choose a method of proving each simple statement and to conduct their step-by-step proof. The second editor allows (using the inclusion-exclusion principle and the formula of the number of complement elements) to derive a step-by-step formula for the number of set elements through the specified numbers of elements for sets from which the resulting set is constructed. An important part of the system is to monitor the correctness of all actions of students, and on this basis the entire learning system is developed. The logical supervision over the correctness of the selected action in the first editor is performed by a Boolean function created by the system and corresponding to this action and by checking it for identical truth. In the second editor, invariants such as characteristic strings of the set and of its number of elements are used for verification. The rest of the system is related to learning of set algebra and to preparation to editors usage. The main focus here is on the learning strategy in which testing the understanding of the learned material is rather rigorous and eliminating the random choice of answers. The division of the material into sections with verification of the success of teaching not only by tests, but also by exercises and tasks, allows students to master the complex logical and mathematical techniques of proving statements for sets and derivation of formulas for the number of set elements.

Comparing and Discussing Multiple Strategies: An Approach to Improving Algebra Instruction

Productive learning of algebra is supported when students reflect on multiple strategies, compare them and discuss the rationale behind and relative merits of particular strategies. Comparison and Discussion of Multiple Strategies (CDMS) is an instructional approach designed to support these processes in math classrooms. In the current study, 16 Algebra I teachers received professional development and supplemental materials to support CDMS when teaching a unit on linear equation solving and 475 of their students completed assessments of their linear equation solving knowledge before and after the unit. Thirteen Algebra I teachers and their 359 students were the business-as-usual control group. CDMS increased how often teachers engaged their students in comparison of multiple strategies, sustained small group work, and sustained mathematical discussions. Students in CDMS classrooms also had higher knowledge of linear equations on the posttest, particularly procedural flexibility, even after controlling for pretest knowledge and school demographic differences. Thus, encouraging teachers to regularly compare and discuss multiple strategies increases students’ algebra learning. Findings highlight the need to expand theories of algebra learning to include attention to procedural flexibility, illustrate an instructional theory and method to promote broader learning about algebra, and provide evidence for effective instructional practices.