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2. A Qualitative Comparison of Mathematics Teachers' and Students' Experiences in a Virtual Reality Algebra Application
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Julianna Washington, Taylor Darwin, Theodora Beauchamp, and Candace Walkington
- Abstract
Prisms VR, a secondary math learning application, allows for users to see, manipulate, and engage with mathematical concepts in an embodied way in Virtual Reality (VR) environment. We examine cases in which mathematics teachers and middle school students worked through Prisms and reflected upon their experiences. Findings indicate that VR environments can help teachers find alternative ways of presenting algebraic concepts in a hands-on, interactive, and embodied manner; and these findings were confirmed based on the student experience. VR allows teachers to make connections through scaffolding and personalized experiences, while also aiding students in the exploration and meaning-making. [This paper was published in: "Proceedings of the 18th International Conference of the Learning Sciences - ICLS 2024," edited by R. Lindgren et al., International Society of the Learning Sciences, 2024, pp. 1610-13.]
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- 2024
3. Creativity in High School through Drawing with Polynomials
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Rudy Baez, Henry Sanchez, and Duli Pllana
- Abstract
High school math projects center on solving real-world examples in a wide variety of situations that require a creative skill set. Creativity in high school mathematics takes place in a subtle form. In contrast, creativity presents itself conspicuously in art. Therefore, this paper will explore the utilization of polynomial equations in drawing various figures as part of algebra class projects. Additionally, these projects incorporate integrated digital tools in mathematics. The paper will analyze three algebraic projects: the first project involves drawing a bird, the second project focuses on drawing a butterfly, and the third project entails drawing a bat using polynomial equations with the technological tool Desmos. All three projects are products of student work that encompass figures requiring up to forty polynomial equations. The students' impressive work demonstrates the power of mathematics in pushing the boundaries of other subjects. As digital technology becomes increasingly integral, mathematical tools are positioned as universal tools for the future and are poised to rival artists in producing visual art. This raises the intriguing question: can a mathematician create better art with mathematical equations through digital technology than an artist?
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- 2024
4. Effect of Gamification on Gamers: Evaluating Interventions for Students Who Game the System
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Kirk P. Vanacore, Ashish Gurung, Adam C. Sales, and Neil T. Heffernan
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Gaming the system is a persistent problem in Computer-Based Learning Platforms. While substantial progress has been made in identifying and understanding such behaviors, effective interventions remain scarce. This study uses a method of causal moderation known as Fully Latent Principal Stratification to explore the impact of two types of interventions--gamification and manipulation of assistance access on the learning outcomes of students who tend to game the system. The results indicate that gamification does not consistently mitigate these negative behaviors. One gamified condition had a consistently positive effect on learning regardless of students' propensity to game the system, whereas the other had a negative effect on gamers. However, delaying access to hints and feedback may have a positive effect on the learning outcomes of those gaming the system. This paper also illustrates the potential for integrating detection and causal methodologies within educational data mining to evaluate effective responses to detected behaviors.
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- 2024
5. A Quasi-Experimental Study on the Effectiveness of a Supplemental Digital Mathematics Tool in Grade 9 Mathematics
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Montorya Lavonne Boswell
- Abstract
The problem addressed in this study was that teachers underestimate the benefits of technology usage in grade nine Algebra mathematics classrooms. The purpose of this quantitative quasi-experimental study was to examine how the use of a supplemental digital mathematics tool affected achievement scores among grade nine Algebra mathematics students over time. The quasi-experimental pretest/posttest model was used to investigate the connection between the independent variable-technology integration of a supplemental digital mathematics tool-and the dependent variable-student mathematics achievement. The participants in the study were from one of the high schools within the intended school district in central North Carolina. A convenience and purposive sampling were used to select the 54 participants. The use of IXL as the intervention was used instead of direct instruction to teach and monitor student progress towards understanding of Common Core standards. Prior to the six-week intervention, students were administered a 20-question paper version pretest to form a baseline score and a 20-question paper version posttest and the conclusion of the intervention. A paired samples t-test was conducted to determine what, if any, statistically significant differences existed in mathematics achievement scores in Common Core grade nine students using a supplemental digital mathematics tool over time. There was not a statistically significant difference between mean scores (p > 0.05) of the LEAP 2025 pretest and posttest therefore the null hypothesis was accepted. It is suggested that future research focus on two items: the relationship between technology and academic achievement in mathematics and to extend this study to explore the relationship between supplemental digital mathematics tools and students' mathematics acquisition in elementary and middle grades as this study was limited to ninth grade students. [The dissertation citations contained here are published with the permission of ProQuest LLC. Further reproduction is prohibited without permission. Copies of dissertations may be obtained by Telephone (800) 1-800-521-0600. Web page: http://www.proquest.com/en-US/products/dissertations/individuals.shtml.]
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- 2024
6. Trends, Insights, and Developments in Research on the Teaching and Learning of Algebra
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Amy B. Ellis and Zekiye Özgür
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This paper addresses the recent body of research in algebra and algebraic thinking from 2018 to 2022. We reviewed 74 journal articles and identified four clusters of content areas: (a) literal symbols and symbolizing, (b) equivalence and the equal sign, (c) equations and systems, and (d) functions and graphing. We present the research on each of these content clusters, and we discuss insights on effective teaching practices and the social processes supporting algebraic reasoning. The research base shows that incorporating algebraic thinking into the elementary grades, emphasizing analytic and structural thinking processes, and emphasizing covariational reasoning supports students' meaningful learning of core algebraic ideas. We close with a discussion of the major theoretical contributions emerging from the past five years, offering suggestions for future research.
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- 2024
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7. Recent Developments in Using Digital Technology in Mathematics Education
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Johann Engelbrecht and Marcelo C. Borba
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In this paper we review selected significant developments in the use of digital technology in the teaching and learning of mathematics over the last five years. We focus on a number of important topics in this field, including the evolvement of STEAM and critical making as well as the process of redefining learning spaces in the transformation of the mathematics classroom. We also address the increasing use of computer algebra systems and dynamic geometry packages; and the issue of student collaboration online, especially using learning environments and social media. We briefly touch on artificial intelligence systems, including hyper-personalisation of learning, multimodality and videos. We include a brief discussion on the impact of COVID-19 on mathematics education, and lastly on the more theoretical perspective of the epistemology of digital technology and the construct of humans-with-media. We conclude the discussion with some possible concerns and mentioning some possible new topics for research in the field.
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- 2024
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8. Analyzing the Structure of the Non-Examples in the Instructional Example Space for Function in Abstract Algebra
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Rosaura Uscanga and John Paul Cook
- Abstract
The concept of function is critical in mathematics in general and abstract algebra in particular. We observe, however, that much of the research on functions in abstract algebra (1) reports widespread student difficulties, and (2) focuses on specific types of functions, including binary operation, homomorphism, and isomorphism. Direct, detailed examinations of the function concept itself--and such fundamental properties as well-definedness and everywhere-definedness--are scarce. To this end, in this paper we examine non-examples of function in abstract algebra by conducting a textbook analysis and semi-structured interviews with abstract algebra instructors. In doing so, we propose four key categories based upon the definitive function properties of well-definedness and everywhere-definedness. These categories identify specific characteristics of the kinds of non-examples of function that abstract algebra instruction should emphasize, enabling us to hypothesize how students might be able to develop a robust view of function and explain in greater detail the nature of the reported difficulties that students experience.
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- 2024
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9. Exploring the Challenges of Learning Quadratic Equations and Reflecting upon Curriculum Structure and Implementation
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Bronwyn Reid O'Connor and Stephen Norton
- Abstract
An important topic of study in secondary mathematics is non-linear functions, including quadratic equations. In this study, findings from 25 Year 11 students indicated that difficulties with critical prerequisite concepts such as algebraic conventions impeded students' success in understanding and working with quadratics. Analysis of student errors identified misconceptions associated with the null factor law, and the nature of quadratic equations. This paper proposes that these findings are a result of limited timeframes nominated for learning quadratic topics outlined in the enacted curriculum. The implication of this is that the enactment of the Australian Curriculum: Mathematics F-10 did not support the development of conceptual understanding or procedural fluency with key mathematical concepts for these students. Without purposeful attention to prerequisite knowledge, and suitable time allocated to develop understanding and fluency, students' proficiency with topics such as quadratics is negatively influenced. A mastery approach to the hierarchically organised curriculum is supported by findings of this study.
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- 2024
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10. Linear Algebraic Nodal Analysis: An Applied Project for a First Course in Linear Algebra
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Jeffrey A. Anderson and Bryan B. Nguyen
- Abstract
Many students who enroll in a first course in linear algebra major in STEM disciplines other than mathematics. Teachers who serve such students may find it difficult to provide authentic problems from these broader areas that ignite students' interest in linear algebra. In this paper, we highlight an interdisciplinary learning activity that engages students in using linear systems of equations to model the behavior of practical electric circuits. This exercise fits nicely into standard introductory linear algebra curricula and is designed to excite students majoring in engineering, physics, or applied mathematics. We also include references to a collection of open-access resources to support instructors who want to use this material in project-based, flipped-learning, inquiry-oriented, or independent-study environments.
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- 2024
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11. Connecting the Threads: The Role of Multiplicative Thinking in Algebraic, Geometrical, and Statistical Reasoning
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Lorraine Day, Dianne Siemon, Rosemary Callingham, and Rebecca Seah
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Making connections within and between different aspects of mathematics is recognised as fundamental to learning mathematics with understanding. However, exactly what these connections are and how they serve the goal of learning mathematics is rarely made explicit in curriculum documents with the result that mathematics tends to be presented as a set of discrete, disconnected topics. Interest in establishing a more coherent approach to the teaching and learning of school mathematics has led to a focus on big ideas. That is, networks of related concepts, skills and ways of thinking that facilitate learning mathematics with understanding. Research on learning progressions has helped identify what these big ideas are and how they serve to build connections within and between different aspects of mathematics. This paper draws on research that provides an evidenced-based learning progression for multiplicative reasoning to illustrate the connective role of multiplicative thinking in the development of algebraic, geometrical, and statistical reasoning.
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- 2024
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12. Generalization Strategies and Representations Used by Final-Year Elementary School Students
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Jason Ureña, Rafael Ramírez-Uclés, María C. Cañadas, and Marta Molina
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Recent research has highlighted the role of functional relationships in introducing elementary school students to algebraic thinking. This functional approach is here considered to study essential components of algebraic thinking such as generalization and its representation, as well as the strategies used by students and their connection with generalization. This paper jointly describes the strategies and representations of generalization used by a group of 33 sixth-year elementary school students, with no former algebraic training, in two generalization tasks involving a functional relationship. The strategies applied by the students differed depending on whether they were working on specific or general cases. To answer questions on near specific cases they resorted to counting or additive operational strategies. As higher values or indeterminate quantities were considered, the strategies diversified. The correspondence strategy was the most used and the common approach when students generalized. Students were able to generalize verbally as well as symbolically and varied their strategies flexibly when changing from specific to general cases, showing a clear preference for a functional approach in the latter.
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- 2024
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13. Shortening the Math Sequence through a Corequisite College Algebra Course with Embedded Personalized Learning
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Roneet Merkin
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This paper reports on a novel corequisite design and implementation for College Algebra at Florida International University. The corequisite course uses online, just-in-time, prerequisite assignments delivered on an open-educational platform. Students get help from near-peer learning assistants inside a math emporium environment. The course remains three credits after the redesign and now supports all students who would otherwise be taking Intermediate Algebra as a prerequisite course.
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- 2024
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14. Alkaline: A Simplified Post-Quantum Encryption Algorithm for Classroom Use
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Joshua Holden
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This paper describes Alkaline, a size-reduced version of Kyber, which has recently been announced as a prototype NIST standard for post-quantum public-key cryptography. While not as simple as RSA, I believe that Alkaline can be used in an undergraduate classroom to effectively teach the techniques and principles behind Kyber and post-quantum cryptography in general. Classroom experiences with individual concepts used in Alkaline support this belief. In addition to cryptography, linear algebra and abstract algebra classes would be good candidates for the use of Alkaline. A few exercises suitable for use in these classes are included.
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- 2024
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15. Backward Transfer, the Relationship between New Learning and Prior Ways of Reasoning, and Action versus Process Views of Linear Functions
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Charles Hohensee, Laura Willoughby, and Sara Gartland
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Backward transfer is defined as the influence that new learning has on individuals' prior ways of reasoning. In this article, we report on an exploratory study that examined the influences that quadratic functions instruction in real classrooms had on students' prior ways of reasoning about linear functions. Two algebra classes and their teachers at two comprehensive high schools served as the participants. Both schools drew from low-socioeconomic urban populations. The study involved paper-and-pencil assessments about linear functions that were administered before and after a four- to five-week instructional unit on quadratic functions. The teachers were instructed to teach the quadratic functions unit using their regular approach. Qualitative analysis revealed three kinds of backward transfer influences and each influence was related to a shift in how the students reasoned about functions in terms of an action or process view of functions. Additionally, features of the instruction in each class provided plausible explanations for the similarities and differences in backward transfer effects across the two classrooms. These results offer insights into backward transfer, the relationship between prior knowledge and new learning, aspects of reasoning about linear functions, and instructional approaches to teaching functions.
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- 2024
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16. Classroom observational data: a professional development tool for introductory college mathematics instruction.
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Johnson, Patrick B., Holtzman, Nathalia, and Fernandez, Eva
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Two groups of mathematics faculty, one from a four-year college and one from a two-year college, redesigned their respective introductory college mathematics courses following presentation of observational data regarding how faculty had been teaching the courses. This presentation emphasised how infrequently faculty teaching introductory college mathematics employed recommended pedagogical practices. This work was part of a multi-year federal grant project designed to increase the numbers of underrepresented students majoring in STEM disciplines. While the two teams developed very different redesigned course activities, in both instances the primary motivation for initiating the work was the information provided to faculty in a professional development workshop regarding how they had previously been observed teaching the mathematics content and how infrequently they utilised the pedagogical practices recommended by STEM education experts. The paper also highlights faculty resistance to curricular reform and enumerates some ways of addressing this resistance. The paper also discusses why faculty from two-year and four-year institutions resisted working together on course redesign and provides recommendations for future efforts addressing course redesign. [ABSTRACT FROM AUTHOR]
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- 2024
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