Algebra Students Research Articles

An Evaluation of College Students’ Perceptions of Statisticians

The widely used Draw a Scientist Test was revised to focus on statistics and 110 Elementary Statistics students were asked to draw a statistician. In addition, to better understand students’ drawings and have some relative comparison, 173 College Algebra students were asked to draw a mathematician. A detailed analysis of students’ images and students’ demographic information was conducted using descriptive statistics, categorical data analysis, logistic regressions, and hierarchical cluster analysis. Results showed that students tend to perceive statisticians and mathematicians as primarily White and male. However, female students were more likely than male students to draw a female statistician and mathematician. Two themed clusters emerged from the hierarchical cluster analysis for both the math and statistics students. We discuss the implications of the results for teaching and future research.

Coordinating invisible and visible sameness within equivalence transformations of numerical equalities by 10- to 12-year-olds in their movement from computational to structural approaches.

“They are the same” is a phrase that teachers often hear from their students in arithmetic and algebra. But what do students mean when they say this? The present paper researches the notion of sameness within algebraic thinking in the context of generating equivalent numerical equalities. A group of Grade 6 Mexican students (10- to 12-year-olds) was presented with tasks that required transforming the given numerical equalities in such a way as to show their truth-value. The students initially indicated this by calculating and demonstrating that the total was the same on both sides. When asked not to calculate, their approaches evolved into more structural transformations involving decomposition so as to arrive at an equality with the same expression on each side. Students used the language of sameness—both visible and invisible—to describe the truth-value of their transformed expressions and equalities. The visible sameness referred to the resulting identical form of each side of the equality and the invisible sameness to the top–down equivalences that they had generated by their decomposing transformations—both types of sameness being characterized by their hidden numerical values. These findings suggest implications for transitioning to the algebraic domain of equations with their similarly hidden values.

Symbolizing lines and planes as linear combinations in a dynamic geometry environment

In this paper, we focus on students’ symbolizing activity and mathematization relating to linear combination and span in the context of a task sequence designed with digital tools. Considering tools and functions of a digital environment (specifically of GeoGebra) with design heuristics of Realistic Mathematics Education, we introduce four tasks (in R2 and R3) and results of pilot studies. Data sources come from six individual task-based interviews with two linear algebra students. We analyzed these with a lens of mathematization, with a particular focus on tool use. We found students’ use of sliders supported their reasoning about individual linear combinations; the trace function and slider animation supported their intuition for all possible linear combinations of vectors. After shifting from two to three dimensions, students symbolized all possible linear combinations with a parametric vector equation and reasoned about how this could represent particular points on the corresponding lines or planes.

Effect of Polya Problem Solving Teaching Method on the Achievement of Students in Algebra

The study investigated the effect of Polya problem solving teaching method on the achievement of students in algebra. The study employed quasi-treatment design. Three (3) research questions were asked and three (3) null hypotheses were formulated for the study. The population of the study was made up of 3600 SSII students in all the thirty one (31) secondary schools in Nsukka education zone of Enugu State. The sample for the study comprised of 300 SSII students drawn from four (4) co-educational schools out of 31 co-educational schools in Nsukka Education Zone of Enugu State drawn through simple random technique. In each of the 4 co-educational schools drawn 8 intact classes of SSII were randomly drawn. 4 intact classes out of the 8 intact classes were assigned to treatment group, while the 4 intact classes were assigned to control group. The treatment group consisted of 160 students who were taught algebra using Polya problem solving teaching method and the control group comprising 140 students who were taught algebra using the conventional method. Data were collected using algebra achievement test (AAT) for both pretest and post-test. Data were analyzed quantitatively using mean and standard deviation, while null hypotheses were tested at 0.05 level of significance using ANOVA. It was revealed from the study that polya problem solving teaching method increased the achievement of students in algebra and is superior to the conventional method of teaching algebra. Also, the study revealed that there was no significance difference in mean achievement scores of males an females in algebra. Based on the findings of the study, the researcher recommended among other things that polya problem solving teaching method should be used in teaching algebra and that mathematics teacher should be trained through workshops, in-service training, seminars etc on the uses of Polya problem solving teaching method approach in teaching algebra in all senior secondary schools for higher academic excellence mathematics generally and algebra in particular. In conclusion, students taught algebra with polya problem solving teaching method had higher mean achievement score than those taught with conventional teaching method. It was therefore concluded that Polya problem solving teaching method enhances students’ achievement in algebra. Keywords: Polya, problem solving, teaching method DOI: 10.7176/RHSS/12-6-04 Publication date: March 31 st 2022

Unboxing mathematics: creating a culture of modeling as critic.

From the socio-critical perspective of mathematical modeling, reflexive discussions about the nature, criteria, and consequences of mathematical models are not a natural consequence of modeling in school. This report is part of a larger study focused on stimulating reflexive discussions in practice employing constructivist grounded theory as a research method. Twenty-seven college algebra students engaged in a 3-week modeling project at a community college in the USA. Audio-recorded group discussions and written reflections were collected to determine how reflexive discussions were taking place. Analysis of students’ actions and reflexive discussions during the modeling project produced four concepts: voicing mathematics, personalizing mathematics, challenging mathematics, and negotiating mathematics. These concepts are integrated into an overall process for stimulating reflexive discussions and are conceptualized as unboxing mathematics. The overarching concept of unboxing mathematics represents one interpretation of how reflexive discussions may be constituted during modeling activities and identifies classroom mathematical practices specific to the socio-critical modeling context of this study.

Tense of Psychological Distancing Self Talk and Test Performance Among GT High School Students

Test taking is ubiquitous in academic life. Often, a student’s desire to perform well in these evaluative situations leads them to experience test-anxiety. However, test-anxiety has been repeatedly correlated with reduced test performance. Research suggests that reappraisal interventions promote the reduction of test anxiety which may result in improved test performance. But can linguistic tense mediate the effect of these interventions? If so, are these interventions helpful for GT students for whom test anxiety is a significant concern? This intervention study tested whether reading a message containing a type of reappraisal technique, objective psychological distancing, in a certain linguistic tense could reduce test anxiety and improve test performance for high school GT algebra students. Two hours before taking an exam, students read one of three messages: a first-person distancing, a third-person distancing, or a control message. There existed a slight improvement in student test performance from the control condition to the distancing conditions, yet this trend was not statistically significant.

Interactionist perspective on negotiation processes of students’ different understandings during small group work on linear algebra

Student group work represents a central learning setting within mathematics programs at the university level. In this study, a theoretical perspective on collaboration is adopted in which the differences between students’ interpretations of a mathematical concept are seen as an opportunity for individual restructuring processes. This so-called interactionist perspective is applied to student group work on linear algebra. The concepts of linear algebra at the university level are characterized by a versatility of different modes of expression and interpretation. For students of linear algebra, the flexible transitions between the different interpretations of linear algebra concepts usually pose a challenge. This study focuses on how students negotiate their different interpretations during group work on linear algebra and how transitions between interpretations might be stimulated or hindered. Video recordings of eight student groups working on a task that required flexible transition between interpretations of homomorphisms were sampled. The recordings were analyzed from an interactionist perspective, focusing on interaction situations in which the participating students expressed and negotiated different interpretations of homomorphisms. The analyses of students’ interactions highlight a phenomenon whereby differences in students’ interpretations remain implicit in group discussions, which constitutes an obstacle to the negotiation process.

An analytical comparison of students’ reasoning in the context of Inquiry-Oriented Instruction: The case of span and linear independence

In this report we analyze differences in reasoning about span and linear independence by comparing written work of 126 linear algebra students whose instructors received support to implement a particular inquiry-oriented (IO) instructional approach compared to 129 students whose instructors did not receive that support. Our analysis of students’ responses to open-ended questions indicated that IO students’ concept images of span and linear independence were more aligned with the formal concept definition than the concept images of Non-IO students. Additionally, IO students exhibited more coordinated conceptual understandings and used deductive reasoning at higher rates than Non-IO students. We provide illustrative examples of systematic differences in how students from the two groups reasoned about span and linear independence.

Identifying Student Responsibilities While Watching Mathematics Instructional Videos

This study investigated the ways in which college algebra students watch mathematics instructional videos about completing the square with the goal of identifying student responsibilities within a particular video and across different videos. Guided by the theory of didactic situations that have defined implicit teacher and student responsibilities within the context of the face-to-face mathematics classroom, participants watched three different videos about completing the square and answered interview questions. Using categories previously identified by the didactic contract for the face-to-face classroom, this study expanded the types of student responsibilities identified specifically for video watching and found that participants, regardless of overall prior knowledge but who had prior knowledge of completing the square, held a responsibility to use the specific set of steps they were taught by their teacher to solve problems. Findings may be useful to both mathematics teachers and video creators.

Reinventing Geometric Linear Transformations in a Dynamic Geometry Environment: Multimodal Analysis of Student Reasoning

This paper reports the multimodal resources that attached to students’ reasoning in the reinvention of specific geometric linear transformations (like reflections according to the axes, projections onto axes and composition of reflections) in a dynamic geometry environment. Following the design heuristics of Realistic Mathematics Education, we design a task (in ℝ2) referring to specific tools and functions of GeoGebra. Task-based interviews were conducted with a pair of linear algebra students, by way of a computer and a teacher. Data was collected using a video camera observing the students’ working environment, screen recorder software, student production and field notes. The data was analysed according to the multimodal paradigm focusing on all semiotic resources, such as gestures and artefact use, in addition to written signs. According to the findings, the artefact use, verbal and written mathematical expressions all interlaced with the emergence of gestures. The students mostly gestured when they faced a new reflection situation and when describing associated geometric actions. Finally, a shared environment with action, production and communication conveyed student reasoning and they managed to reinvent a number of geometric linear transformations.

Constructing and Adapting Technology-Infused Lessons

Design principles are used to construct and refine a technology-infused lesson for beginning algebra students learning about systems of linear inequalities.

Students’ structure sense ability in solving quadratic equation problems

Structure sense is the ability of students to see any relation between symbols, to manipulate, and to use algebraic structures. This is considered an important ability for students as algebra is the core topic within high school mathematics. An important skills need to be acquired by students in algebra is solving equations, such as solving quadratic equations. To address this, this research aims to understand students’ structure sense ability by investigating strategies used in solving quadratic equations of the form of a square of difference of two variables. To reach these aims, a qualitative descriptive method was conducted in the form of four quadratic equation tasks done by twelve Senior High School Students from one of senior high schools in Bandung. In this study, each student was required to work on each task using two different strategies if possible. The result from the students’ answers informed that the majority used structure sense strategy. However, there were mistakes conducted by fifty percent and more students completing both the procedural and structure sense strategy. In conclusion, we found out that students’ structure sense ability is good despite students experienced some difficulties, whether they used the procedural strategy or involved their structure sense.

Criação e usos do aplicativo LineAlg como objeto de aprendizagem na Educação Básica

RESUMO: O letramento matemático é indispensável aos processos educativos que têm por finalidade a formação de pessoas capazes de exercer plenamente sua cidadania. A escola contemporânea também não pode prescindir de tecnologias como a dos computadores e dos dispositivos eletrônicos móveis. Este trabalho articulou ambas as necessidades com o objetivo de desenvolver um aplicativo para auxiliar processos de ensino-aprendizagem de matrizes, determinantes e sistemas de equações lineares no Ensino Médio. Para tanto, houve a realização de pesquisas bibliográficas sobre o ensino desses tópicos da álgebra linear e, consequentemente, o desenvolvimento de uma aplicação móvel utilizando a linguagem Python e a biblioteca Kivy. Foram criadas funções que permitem calcular e verificar o passo a passo dos cálculos, bem como tomar lições por meio de um chatbot, ainda uma aplicação para o balanceamento de equações químicas. Além desses usos, sugere-se uma atividade com o aplicativo, a fim de evidenciar seu potencial de incrementar aulas de matemática e estudos individuais dos alunos em tópicos de álgebra, inclusive de forma remota.
 PALAVRAS-CHAVE: Ensino de Matemática, Tecnologias digitais, Álgebra linear.

Abstract algebra students’ evoked concept images for functions and homomorphisms

Homomorphism is a critical variety of function in undergraduate Abstract Algebra (AA) courses and function is one of the unifying concepts across many mathematical subject areas. However, despite homomorphism’s important place in the curriculum and its existence as a particular type of function, little is known of a student’s concept image of functions at advanced levels and the role this concept image may play in a students’ homomorphism-related activities. In this paper, we share cases that explore students’ concept image and treatment of functions at the undergraduate AA level. In particular, we focus on coherence of prior function understanding and functions in AA (homomorphisms), and how this coherence may account for student activity in tasks related to homomorphism. Our results reflect that even at the AA level, students may have limited concept images of functions and their understanding of function (and coherence with homomorphism) can serve as a support or obstacle to task performance in AA. We suggest that both instructors and researchers explicitly attend to the role of function and function understanding in student activity at advanced levels.

On Linear Functions and Their Graphs: Refining the Cartesian Connection

This study focuses on connections between linear functions and their graphs that were made by tertiary remedial algebra students. In particular, we describe students’ work on a Task designed to examine the connection between points on a graph and the equation of a line. The data consist of 63 responses to a written questionnaire and individual interviews with three participants. The results indicate that visual approaches impede students’ solutions and point to incomplete connections between algebraic and graphical representation. While algebraic approaches point to various connections used in approaching the Task, students’ ability to work with algebraic representation did not necessarily result in capitalizing on these connections. Furthermore, interpretations of the graph based on visual inspection appeared most useful when used in support of the algebraic approach.

Students’ Achievement Levels, Gender, and Learning Styles on Abstract Algebra: A Profile of Evidence Developing Ability

Mathematics education students’ ability on developing evidence needs to be reviewed by lecturers. 50% of the subjects in the mathematics education departement require students' accuracy in analyzing mathematical statements. This is increasingly important because the Indonesian National Qualifications Framework (KKNI) for the Bachelor level requires graduates to become technicians / analysts. The diversity of levels of students’ ability, gender, and learning styles are assumsed to be the discrepancy abilities to develop evidence. This study aims to describe the evidence developing ability based on the levels of achievement, gender, and student learning styles. The subjects of this study were abstact algebra students in the mathematics education department. Type of this research was descriptive qualitative with data collection techniques using test and non-test techniques. Students were given five questions about abstact algebra that demanded the evidence developing ability. The achivement levels and gender were obtained from students’ academic achievement data. The learning styles were attained from questionnaires. The results of the study is higher the students’ academic achievement, better the evidence developing ability on Abstract Algebra. The gender does not affect the evidence developing ability on Abstact Algebra. Students with visual learning styles have the ability to develop evidence better than students with other learning styles.

Analyzing Educational Objectives that Include Critical Thinking: Dot Product Problems in Vector Algebra

The primary and secondary school educational system should be stable and any upgrading reforms should be made gradually and consistently. This is especially important in mathematics education, since the element of logical reasoning while learning is more prominent there. Inconsistencies in reforms generate deficiencies in the higher levels of young students’ reasoning skills and this situation continues on the university stage of education. We will report our findings about the reasoning of first-year university students on elements of geometry and associated algebra. We conducted an experiment where students’ understanding of the definition of dot product of two vectors, cosine function and linear (in)dependence of vectors is evaluated, and address their mathematical activity to provide insight into the key elements of the problem they are solving. We use Bloom’s and SOLO taxonomy as a tool for the assessment of our findings. We obtained the data from written exams given to vector algebra students and also from individual interviews.

Interpretations of Slope Through Written and Verbal Interactions Between a Student and Her Tutors in Algebra 1

There is an ongoing need to support students’ learning of linear functions, and the study of slope makes up a foundational component of this learning. We applied techniques from systemic functional linguistics to document the meanings that were established through spoken interaction between a student and her tutors during discussions of slope. We found that, while fraction notation gave the student and tutors a common reference point to discuss slope, it also masked important differences in how the student interpreted slope compared to her tutors. The findings of this analysis imply the need not only to attend to how students quantify slope, but also whether students recognize slope as an attribute of a line.

Mistakes on display: Incorrect examples refine equation solving and algebraic feature knowledge

SummaryAlthough findings from cognitive science have suggested learning benefits of confronting errors (Metcalfe, 2017), they are not often capitalized on in many mathematics classrooms (Tulis, 2013). The current study assessed the effects of examples focused on either common mathematical misconceptions and errors or correct concepts and procedures on algebraic feature knowledge and solving quadratic equations. Middle school algebra students (N = 206) were randomly assigned to four conditions. Two errorful conditions either displayed errors and asked students to explain or displayed correct solutions and primed students to reflect on potential errors by problem type. A correct example condition and problem‐solving control group were also included. Studying and explaining common errors displayed in incorrect examples improved equation‐solving ability. An aptitude‐by‐treatment interaction revealed that learners with limited understandings of algebraic features demonstrated greater benefits. Theoretical implications about using examples to promote learning from errors are considered in addition to suggestions for educational practice.

Learning styles for algebra concepts: The case of engineering students at the Universidad de Santander (Valledupar, Colombia)

The objective of this research was to evaluate the learning styles for algebra concepts of the students of the industrial engineering program of the University of Santander, Valledupar, Cesar, Colombia. A descriptive study was developed, consisting of 40 algebra students and 6 mathematics teachers, the questionnaire is self-administered elaborated with a dichotomous scale for the student and for interpretation of the means, a scale was designed based on a scale between zero and five that reveals the score obtained in said test, the styles studied were active, reflective, theoretical and pragmatic. Concluding that the predominant style is the pragmatic and theoretical, which allowed to select and apply strategies aimed at generating interactivity between students and teachers around the knowledge of algebra.