Search

While the coronavirus disease 2019 (COVID-19) is still considered as a pandemic in recent human history, evidence from World Health Organization (2021) so far has recorded a total of 116,521,281 confirmed cases of COVID-19 with 2,589,548 as a total of deaths from over 215 countries or territories worldwide. Recognizing that COVID-19 is not only pandemic since March 11, 2020, but spreading worldwide at unprecedented rate, number of sectors including schools and universities as a measure to minimize person-to-person transmission closed their services. Such an uncertain closure warranted restructuring of services provided by schools and universities. The challenges therefore have necessitated the current research to investigate and alleviate challenges brought about by the COVID-19. In essence, the present research's aim was to report on early mathematics learners (foundation phase) numerical errors, which is as a consequence of poor learners' comprehension and teachers' instructions. Based on the aim, the study was positioned within a cognitive theory in order to examine processing of numerical competence among early mathematics learners. A case study via 80 grade 3 learners with ages 8 and 9 was sampled. A textual analysis was used in unpacking and de-contextualizing processing of numerical competence by early mathematics learners. The evidence revealed learners' mathematical mistakes were caused from limited reading skills and ill-presented problems via teachers. Due to the need to teach children at home (home school) due to the COVID-19, it is hoped that the findings thus assist audience, including non-academic and parents, who grapple with poor instructions coupled with poor learners' comprehension.

Dynamic visual tools such as MATLAB have inbuilt features which are believed to be able to empower students to learn through the visualisation of three-dimensional objects. While student learning through MATLAB has been investigated regarding students in urban settings, only a handful of studies have investigated how MATLAB can assist students in rural settings. Spatial visualisation (SV) as a measure or reflection of one's cognitive reasoning is affected by family social economic status (SES). For instance, it is argued that SES in combination with other components, do enhance cognitive development in different ways. What is meant is that components such as but not exclusively, economic and occupational components of SES may vary and hence provide opportunities for generating better understanding of education (cognition). In this study, we randomly selected 100 second-year rural-based pre-service teachers in a vector calculus class at University of Zululand (UNIZULU). Students need SV skills to learn vector calculus and the Purdue spatial-visualization test/rotations (PSVT/R) is well established for measuring individuals' spatial reasoning. In this study, spatial reasoning skills were assessed through a vector calculus pre-test and through a post-test using the Purdue spatial-visualization test/rotations (PSVT/R). The experimental group of students learned the vector calculus topics supported by activities and investigations using MATLAB. Duval's Theory of Register of Semiotic Representation (TRSR) was employed to comprehend the impact of MATLAB on rural-based pre-service teachers' spatial-visualisation skills. From using an independent sample t-test, our findings indicated that, for participants in this study, using MATLAB had positive impact on the rural-based pre-service teachers' SV skills.

This study explores the relationships between students' perceptions of their teachers' and parents' attitudes toward them as learners of mathematics and mathematics and their attitudes related to mathematics learning through path analysis. Data was collected from 1,960 7th grade students enrolled in 19 public elementary schools in a city in Turkey. Results revealed that, through students' perceptions of their mothers, fathers, and teachers, their confidence in learning mathematics, belief about the usefulness of mathematics, and belief about the importance of mathematics were associated primarily with their mother's, father's, and teacher's positive or negative attitudes regarding the students as mathematical learners and their teacher's positive or negative views of teaching.

In the study reported on here, we investigated the effects of the use of dynamic geometry software (DGS) (i.e., GeoGebra) on learners' learning and a teacher's beliefs. The learners and teacher involved in the study were from a high-poverty, rural high school in South Africa. We compared grade 11 learners (N = 56) who used GeoGebra in the context of learning circle geometry with learners who experienced geometry through traditional lecture-based instruction. Participating learners were from classes in a public school located in the rural Umkhanyakude district of KwaZulu-Natal, Republic of South Africa. Results showed that learners using GeoGebra were more successful at solving problems and justifying their statements, while the other learners provided a limited justification for their answers. In a parallel and complementary investigation the teacher's attitudes toward using GeoGebra as an instructional tool were considered qualitatively. Results showed that even in high-poverty, rural settings where the availability of technological resources are limited, the use of GeoGebra affected learners' learning and had positive effects on the teacher's beliefs regarding teaching and learning.

Previous theoretical research has revealed conceptual similarities among a number of mathematical learning theories and theories regarding language acquisition. This intersection of ideas led to a novel framework defining four stages of mathematical learning: Receiving, Replicating, Negotiating Meaning, and Producing. Through qualitative research methods and transcripts of student communication and work, this study empirically investigates this theoretical construct. The findings herein demonstrate that this construct is helpful in characterizing where students are in the process of learning mathematics and how to help them attain the next level in the stages of learning.

This study applies the Mathematics Acquisition Framework (MAF) (Bossé, Ringler, Bayaga, Fountain, & Young, 2018) to investigate fifth- through seventh-grade students' comprehension of fractions and decimals and examines how students build understanding of mathematical principles and concepts regarding fractions and decimals. Based on case studies and discourse analysis, the results revealed students in different stages of the MAF with some students employing informal/social language in their mathematical communication and others using formal/academic language. Additionally, mathematically erroneous student communication was more positively interpreted through the lens of the MAF, which interpreted student errors, misunderstandings, and miscommunications as natural components to learning.

This study begins by connecting semantic elaboration with conceptual understanding and syntactic elaboration with procedural understanding in the context of fractions. Through case studies and discourse analysis, the work and communication of students in fourth through sixth grade is analyzed to determine the extent of their semantic and syntactic elaboration regarding fractions and fraction operations. Findings are that, while some students emphasized one form of elaboration over the other, some students demonstrated use of both forms of elaboration. Indeed, it is wondered if semantic and syntactic elaboration should be seen as more complementary than adversarial.

Students enrolled in post-secondary developmental mathematics courses hold mathematical misunderstandings that begin before their college careers. Specifically, these students struggle with conceptual understanding of fractions, ratios, and proportions. The Common Core State Standards in Mathematics outline standards for K-12 mathematics education. Standards for fractions, ratios, and proportions are found in grades 3-7. Through a research instrument, collegiate developmental mathematics students were investigated regarding their understanding of these standards. It was determined that participants showed evidence of limited knowledge of the examined areas involving fractions, ratios, and proportions. Because it seems that this lack of understanding is rooted early in their educational careers and is continued throughout students' academic careers, it is hoped that this work can be a starting point for future work in this field that may apply to investigating ways to bridge these gaps in mathematical understanding.

Most any students can explain the meaning of "a[superscript b]", for "a" ? R and for "b" ? Z. And some students may be able to explain the meaning of "(a + bi)[superscript c]," for "a, b" ? R and for "c" ? Z. However, few may be able to explain "a[superscript di]" or "a[superscript (c+di)]" let alone "(a + bi)[superscript (c+di)]" for "a, b, c, d" ? R. Through both mathematical explanation and reader investigations via a dynamic graphing applet, we will investigate the meaning of "(a + bi)[superscript (c+di)]". Additionally, we provide some course appropriate investigations which can be used by instructors in various grades 11-16 courses.

This study investigates college students' understanding of arithmetic fraction operations as defined by the Common Core State Standards in grades three through six Mathematics. This study is meant to extend upon an extensive body of research regarding elementary and middle grades students' understanding of fraction operations and see if recognized erroneous thinking among younger students persist among college students. Among these misconceptions include: fraction equivalence, common denominators, the algorithm and the concept of division, whole number bias, and incorrectly applying fraction operations. The results of this study suggest that college students have the same misunderstandings and misconceptions as elementary students in regards to fraction operations.

This paper examines university faculty perceptions on whether preservice teachers are sufficiently prepared to use information and communication technology (ICT) for teaching in early grades and barriers faced in incorporating these technological interventions in classroom settings. This study employed a qualitative research approach and semi-structured questionnaires to investigate faculty perceptions from a teacher education university and considered their responses through the lens of growth and fixed mindsets. The findings revealed that, despite preservice teachers being exposed to instructional technology as part of their academic preparation for early grades teaching, faculty perceived that challenges exist that inhibit preservice teachers' readiness to integrate technologies in their teaching of mathematics. In particular, many significant disagreements existed among the teacher education faculty as to how prepared they perceived their preservice teachers to be in the use of technology in their future classrooms. Findings from this study led to additional observations regarding university faculty opinions regarding their own use of ICT and its implementation in the department and programme.

In this paper we introduce the reader to a foundational topic of differential geometry: the curvature of a curve. To make this topic engaging to a wide audience of readers, we develop this intuitive introduction employing only basic geometry without calculus and derivatives. It is hoped that this introduction will encourage many more to both consider this mathematical notion and to develop enthusiasm for mathematical studies.

This study applies a framework for mathematical cognition and learning (Bossé, et al., 2018a; Bossé et al., 2019; Bossé et al., 2018b) to examples of students' comprehension of geometric concepts related to angles. This framework utilizes key concepts from second language acquisition (SLA) to elucidate processes involved in mathematical learning. The present study analyzes concrete examples of student-teacher and student-student discourse to illustrate how students at different stages of development build their understanding of mathematical principles and concepts regarding angles. The findings indicate that students at different framework levels learn, process, and communicate about angles in observably different ways that share similarities with mathematical learning and SLA.

Often, straightforward notions from one mathematical domain, when altered even slightly, can become rich and rewarding investigations involving numerous additional domains -- particularly when the investigation includes rigorous proof. This study begins with a familiar high school geometry problem (namely finding the circumcentre of a triangle), extends it to the study of equidistance and analyses the problem using geometry in concert with a number of other mathematical areas to construct a new region called the Circumregion. Altogether, the beautiful and intricate interconnectedness of mathematics is both employed and demonstrated.

In the context of an analytical geometry, this study considers the mathematical understanding and activity of seven students analyzed simultaneously through two knowledge frameworks: (1) the Van Hiele levels (Van Hiele, 1986, 1999) and register and domain knowledge (Hibert, 1988); and (2) three action frameworks: the SOLO taxonomy (Biggs, 1999; Biggs & Collis, 1982); syntactic and semantic elaborations (Kaput, 1987a, 1987b, 1989); and isomorphic, transcendent, and mixed connections (Adu-Gyamfi, Bossé, & Lynch-Davis, 2019). Along with producing a fuller analysis of student work and communication, the study found that for only the students with the lowest and highest scores regarding either their understanding or actions on the analytic geometry task might there be a predictive association between knowledge and action levels. For other students, a predictive association could not be determined. This may mean that the level of understanding a student possesses regarding a particular mathematical concept may not parallel the level of actions they use when working with an associated task.

Following a line of inquiry regarding the exact number of real roots of a real polynomial, this investigation considers: Descartes' Rule of Signs, the Budan-Fourier Theorem, and versions of Sturm's Method in contrast with the approximate root count gleaned from graphing utilities. Online applets are provided to allow the reader to freely experiment with different polynomial examples. Additionally, activities at the end of each section facilitate further investigations and deeper understanding of the topics.

While one branch of literature is replete with investigations of problem solving and another branch frequently investigates student use of dynamic mathematics environments (DMEs), most of the studies in both of these fields consider whether or not students can solve problems. Far fewer number of studies consider the cognitive processes associated during either problem-solving experiences or DME use and only a handful of studies consider cognitive processes associated with problem solving when working in a DME. This paper reports a novel approach to investigating, defining, and categorizing the cognitive processes used by students in mathematical problem-solving while working in a DME with examples found in student work. Using this approach, problem-solving is found to be nonlinear, iterative, and idiosyncratic. Insights gained by this analysis have both theoretical and practical applications in mathematics education.

The mathematical domain of fraction division continues to be an area of great difficulty for many teachers and preservice teachers (Lo and Luo, Journal for Mathematics Teacher Education 15:481-500, 2012; Newton, American Educational Research Journal 45:1080-1110, 2008; Rizvi and Lawson, International Education Journal 8(2):377-392, 2007; Young and Zientek, Investigations in Mathematics Learning 4(1):1-23, 2011). In this study, preservice teachers were given a fraction division task that sought to investigate their own personal approaches to solving a fraction division problem. Their results were contrasted with their ability to interpret student work on the same task. The purpose was to uncover information and gain greater insight into conceptualizations of fraction division they used and links or connections they made among verbal, diagrammatic, and algebraic representations as they solved the task themselves and then analyzed sample student solutions to the same task. Our findings have implications for future research and instruction on fraction division.

Previous research has investigated the representational translation practices of high school students, high school teachers, and college preservice teachers in various mathematical contexts including linear functions. Findings from qualitative research has frequently led to new notions about participant work and understanding. Many quantitative research has investigated the degree to which some in these populations correctly perform these translations. However, it seems that only infrequently have empirical research investigated findings from qualitative studies and vice versa, and findings regarding one population are rarely compared with findings of another population. This study (a) empirically explores the frequency of success of preservice teachers (N = 80) regarding representational translations in the context of linear functions, (b) quantifies results from previous qualitative, literature-based research regarding high school students and teachers, and (c) quantitatively compares the results. This study demonstrates that some mathematical representational translations are more difficult than others.

This study investigates representational code-switching (RCS) by considering three high school students' communications in the process of comparing and contrasting pairs of representations (e.g., equation and graph) in the context of rational functions. Supporting this study is research in the realms of students interacting with mathematical representations, practicing language learning, and linguistic code-switching (LCS). Attention is given to assessing student mathematical understanding based on frequency and type of RCS performed. This investigation concludes that RCS demonstrates either weaker or stronger mathematical performance depending on whether the RCS evidences semantic elaboration, isomorphic and transcendent connections, and domain register knowledge.

Rounding is a necessary step in many mathematical processes. We are taught early in our education about significant figures and how to properly round a number. So when we are given a data set and asked to find a regression line, we are inclined to offer the line with rounded coefficients to reflect our model. However, the effects are not as insignificant as they might seem at first. In this paper, we investigate some consequences of rounding the coefficients in a least squares linear regression with respect to the calculated value of R[superscript 2], and consider ways to minimize the amount of error that can arise.

Student problem solving in the context of a dynamic mathematics environment (DME) has previously been investigated primarily through the lens of whether or not the student could complete a problem-solving task. Herein, we investigate what trajectories students employ in the realms of mathematics, technology, and problem solving as they attempt to complete tasks and which of these trajectories are more helpful than others. Notably, it was determined that these trajectories are idiosyncratic, nonlinear, and iterative and that, while some trajectories help problem solving, others harm the problem solving process. Among other findings, it was determined that student access to technology may not assist their mathematical problem solving and may at times hinder it even further.

Research in second language acquisition and the learning of mathematics has matured through long, albeit disconnected, histories. This study examines how theories and models of second language acquisition can be applied to the learning of mathematics and, through this, develops a novel framework defining stages in the learning of mathematics. This framework considers dimensions of language (social and academic), cognitive level (undemanding and demanding), locus of activity (student and teacher), and primary mode of communication (listening, reading, speaking, and writing) and leads to the mathematical learning stages: receiving mathematics, reading and replicating mathematics, negotiating meaning; communicating mathematics; and producing mathematics. This framework has numerous implications for the learning and assessment of mathematics.

When establishing connections among representations of associated mathematical concepts, students encounter different difficulties and successes along the way. The purpose of this study was to uncover information about and gain greater insight into how student processes connections. Pre-calculus students were observed and interviewed while performing a task that required connections among graphical and algebraic representations of a polynomial relation. Their reasoning and processes on the task were examined. This revealed more detailed information about the nature of their connections and misconceptions. The study reveals different types of student connections among graphical and algebraic representations.

Humans possess a remarkable ability to recognise both simple patterns such as shapes and handwriting and very complex patterns such as faces and landscapes. To investigate one small aspect of human pattern recognition, in this study participants position lines of "best fit" to two-dimensional scatter plots of data. The study investigates the variation in participants' fits and whether there is some consistent metric being used in fitting the lines. For example, is there a natural tendency toward fitting lines similarly to one of the standard regression lines: vertical, horizontal, or orthogonal. This study also investigates the effect of outliers on the line a participant fits to a scatter plot with a strong linear trend and provides guidance for future inquiries. Finally, this investigation considers some practical aspects for classroom teachers.

Assessment and instruction are interwoven in mathematically rich formative assessment tasks, so employing these tasks in the classrooms is an exciting and time-efficient opportunity. To provide a window into how these tasks work in the classroom, this article analyzes summaries of student work on such a task and considers several students' solution strategies to exhibit the usefulness of these tasks in assessment, learning, and teaching in the classroom. This article also provides some guidance on implementing these tasks in the classroom.

This study investigates the nature of student errors in the context of problem solving and Dynamic Math Environments. This led to the development of the Problem Solving Action Identification Framework; this framework captures and defines all activities and errors associated with problem solving in a dynamic math environment. Found are three different types of errors: domain (mathematics, technology, and problem solving), process (interpretation, activity, and evaluation), and interaction (syntactic and semantic). Implications from this study address problem solving in general.

College algebra (a university freshman level algebra course) fulfills the quantitative literacy requirement of many college's general education programs and is a terminal course for most who take it. An online problem-based learning environment provides a unique means of engaging students in quantitative discussions and research. This article reports on a US-based study that examined a traditional, lecture-based college algebra section versus an online, quantitative literacy focused section; the latter included weekly news discussions as well as problem-based learning projects requiring data analysis. Students in each section responded to survey questions at the beginning and end of the semester. Qualitative analysis of pre- and post-survey responses revealed differences in students' mathematical disposition, attitude, and outlook on what utility mathematics has to offer, with the online section having the favorable outcome of each. Albeit modest, the results suggest that project-based learning in an online environment is a promising strategy for fostering the affective component of quantitative literacy in college algebra. More research is needed to capture the mechanisms through which such growth occurs.

Teachers and students commonly use various concrete representations during mathematical instruction. These representations can be utilized to help students understand mathematical concepts and processes, increase flexibility of thinking, facilitate problem solving, and reduce anxiety while doing mathematics. Unfortunately, the manner in which some instructionally employ representations potentially leaves students confounded and mathematical ideas unlearned. From the perspective of representational determinism, this paper explores the appropriate uses and misuses of concrete representations in respect to integer operations.

Dynamic mathematical environments allow users to reify mathematical concepts through multiple representations, transform mathematical relations and organically explore mathematical properties, investigate integrated mathematics, and develop conceptual understanding. Herein, we integrate Boolean algebra, the functionalities of a dynamic mathematical environment, and different mathematical topics to demonstrate investigations associated with dynamic Boolean mathematics.

While it is well recognized that students are prone to difficulties when performing linguistic-to-algebra translations, the nature of students' difficulties remain an issue of contention. Moreover, the literature indicates that these difficulties are not easily remediated by domain-specific instruction. Some have opined that this is the case because few frameworks exist which are sufficiently robust and revelatory to provide the insights needed to address these difficulties. This study uses a model situated in both a cognitive-oriented frame and the translation process itself to analyze student activity in the linguistic-to-algebra translation process, defines error types made, and recognizes the frequencies of such.

For high school and college instructors and students, this paper connects number systems, field axioms, and polynomials. It also considers other properties such as cardinality, density, subset, and superset relationships. Additional aspects of this paper include gains and losses through sequences of number systems. The paper ends with a great number of activities for classroom use. [ABSTRACT FROM AUTHOR]

Understanding how students translate between mathematical representations is of both practical and theoretical importance. This study examined students' processes in their generation of symbolic and graphic representations of given polynomial functions. The purpose was to investigate how students perform these translations. The result of the study suggests that students of different ability levels process translations differently and that students' apparent difficulties with translations may be directly connected with their processes and obstacles encountered during translations. The study's novel framework led to identifying aspects of student behavior that were not previously recognized in the literature. In addition to these findings, this study developed an evolution in theoretical frameworks through which to investigate and explain translation actions between representations. The findings of the study have implications for addressing student difficulties in translations among representations.

This study examined student actions, interpretations, and language in respect to questions raised regarding tabular, graphical, and algebraic representations in the context of functions. The purpose was to investigate students' interpretations and specific ways of working within table, graph, and the algebraic on notions fundamental to a conceptualization of linear functions. Through a case study method which investigated individual representations and student articulations within them, the study revealed that students can make a transition from a given representation of linear function to another and yet demonstrate limited understanding of linear functions.