The 50 most-cited articles in The Mathematics Enthusiast, 2023

The aim of this research was to analyze the level of mathematical understanding based on the mathematical connections made by a group of Mexican High School students when solving mathematical tasks related to the concepts of equation and linear function. The study employed the thinking aloud method to collect data, whereby students verbalized their thought processes while solving three mathematical tasks, followed by answering some final questions. The collected data were analyzed using a simplified version of thematic analysis based on a preliminary framework adopted in this research. The findings revealed that students demonstrated different levels of mathematical understanding depending on the types of mathematical connections they made. Moreover, these outcomes facilitated the refinement of the preliminary framework, providing a more precise set of indicators for each tier of mathematical understanding. These indicators can be valuable resources for educators and researchers, offering guidance in developing instructional strategies aimed at promoting higher levels of…

Productive struggle is an enigmatic concept in mathematics education. At first glance, the idea may be oxymoronic since the word struggle is not often associated with being productive. The two words may even seem counter-intuitive. The purpose of this article is to analyze productive struggle in the literature in order to better understand some if its essential features. The research method in this study is theoretical and includes a survey of relevant literature in mathematics and modelling education. This study attempts to explain the relationship between productive struggle and learning mathematics. Productive struggle can lead to enhanced mathematics understanding and sensemaking. Its role in enabling deeper and more flexible learning is set out. Findings suggest that productive struggle is underpinned by key elements such as the necessity for contextual tasks that enable multiple solutions and entry points. The type of tasks that engender productive struggle are discussed and finally, mathematical modelling is presented as an exemplar for tasks that encompass productive…

When solving problems, mathematically gifted individuals tend to internalize intuitive ideas and approaches, and to shorten their reasoning. Consequently, for teachers it is difficult to observe gifted students’ mathematical reasoning in the context of problem solving. In this paper we investigate nine gifted Swedish 9th grade students’ mathematical reasoning during problem solving in small groups at vertical whiteboards. The data consists of 5 filmed group-activities, that were analysed according to a framework of collaborative problem-solving (Roschelle & Teasley, 1995). The analysis shows that every group solved proposed problems successfully within different socially negotiated Joint Problem Spaces (JPS) and, importantly, that students were able to verbalize and display their mathematical reasoning. Additionally, it is indicated that using vertical whiteboards facilitated considerably the exhibition of students’ mathematical reasoning.

We mark the 50th anniversary of mathematical modeling education by reviving the term the spirit of mathematical modeling (SoMM), which idealistically reflects core aspects of mathematical modeling. The basis of our analysis is the notion of bildung, which is an educational philosophy that strives for harmonizing heart, mind, social life and culture. We built SoMM on five descriptions of mathematical modeling: two research studies from the 1970s, two studies about the work of professional modelers, and one about an environmental school project. We captured SoMM as a collection of aspects at the micro, meso and macro level: at the micro level, we found aspects such as agency, anticipating, scrutiny and critique as part of SoMM; at the meso level, we found collaborating, consulting and navigating social norms; and at the macro level, interdisciplinarity, relevance and social justice. Through the lens of bildung-based educational philosophies, we see that instruction and assessment traditions have transposed mathematical modeling into ‘teachable’ practices that drift away from SoMM. We…

Hermite polynomials arise when dealing with functions of normally distributed variables, and are commonly thought of as the analog of the simple polynomials on functions of regular variables. Therefore the Hermite expansion should be an analog of the Taylor expansion. Indeed there is a strong connection between the two – the general coefficient in the Hermite expansion is the weighted integral of the nth derivative, as compared to the nth derivative evaluated at zero in the case of Taylor. This fact can be used to derive the Hermite expansion for the integral and the derivative of a function. Furthermore, it provides a method of providing simple proofs of many of the Hermite identities. This connection is used to derive the Hermite expansions of the normal probability distribution function, the normal cumulative distribution function and the indicator function. Finally, an algorithm to numerically perform a Hermite expansion is presented, which is efficient in the sense that is only requires a single call to a quadrature method.

While slope is a topic in the algebra curriculum, having a robust understanding of slope is needed for students to truly understand several single and multivariable calculus topics with any depth. We begin with a review of the topic of slope and present what is known from its existing corpus of literature. We then outline the tenets of APOS theory. Building from there, we suggest what a robust, flexible understanding of slope involves, as well as how slope is used, with the APOS-slope framework acting as a theoretical lens. This is followed by the cases of two hypothetical students built from amalgamations of research and experience to emphasize why moving easily between different ways of thinking about and the various uses of slope is vital to successfully transition into calculus. We offer suggestions as to how university instructors might consider slope understanding when teaching calculus, then conclude with suggestions for future research on slope.

Separation of variables is a common method for producing an analytical based solution to partial differential equations. Despite the wide application of this method, often the physical phenomena described by the differential equations are not adequately involved in the discourse over the appropriate methods to solve a given problem, particularly in mathematics curricula. However, as mathematics is the tool to better understanding of the physical world, the meaning of the differential equation, boundary conditions, and initial conditions cannot be detached from the methods used to solve the differential equations. Failure to recognize the physical conditions being studied can lead to solution methods that are invalid or unphysical. This paper demonstrates how awareness of the physical nature of the system being investigated and its relationship to the mathematics can guide the selection of the relevant solution methods. To illustrate the importance of the comprehension of the physical meaning behind the mathematical equations and representations and the need to avoid rote application…

Figured numbers are sequences of natural numbers that, when represented by dots, form geometric configurations. The immediate sequence of figured numbers presents in its configuration the equilateral triangles, being the basis for finding several others, as long as a relationship between them is discovered. This article shows the construction of polygonal, stellar and three-dimensional figured numbers that can be constructed using the GeoGebra software as a resource to visualize the properties that relate these numbers to the triangular numbers. It is a way of observing the relationship that the figures shown have to each other, how we can understand the structure of these sequences, when presented in the form of figures, and the possibility of building figures that represent some of the most known sequences within the number’s figures such as 2D figured numbers such as square, pentagonal, centered triangular and star numbers within 3D figured numbers. Thus, this work presents such construction through the properties of pyramidal numbers, cubic numbers, octahedral numbers and two…

This article gives an overview of questions on diagnostics and procedures of high mathematical talent. Various methods such as intelligence tests, school achievement tests and checklists are presented and discussed. The conclusions favor multidimensional and multi-step approaches with a focus on special mathematical tests. As an example the approach of identifying children with a high mathematical potential used in the PriMa project at the University of Hamburg should illustrate an implementation of the issues raised.

Despite plethora of research that attends to the convincing power of different types of proofs, research related to the convincing power of counterexamples is rather slim. In this paper we examine how prospective and practicing secondary school mathematics teachers respond to different types of counterexamples. The counterexamples were presented as products of students’ arguments, and the participants were asked to evaluate their correctness and comment on them. The counterexamples varied according to mathematical topic: algebra or geometry, and their explicitness. However, as we analyzed the data, we discovered that these distinctions were insufficient to explain why teachers accepted some counterexamples, but rejected others, with seemingly similar features. As we analyze the participants’ perceived transparency of different counterexamples, we employ various theoretical approaches that can advance our understanding of teachers’ conceptions of conviction with respect to counterexamples.

Current mathematics students are members of Generation Z, a generation proving to be quite different than previous ones. Generation Z has never known a time without Google, nor a time of safety. Generation Z has a declining tendency to even attend college. If they do attend college, their expectations need to be met to keep them engaged in mathematics. Professors will need to adjust pedagogy.

This article describes the development of a coding scheme for analyzing mathematical tasks in Primary Source Projects (PSPs), curriculum materials based on primary historical sources designed for teaching standard topics from today’s undergraduate mathematics curriculum. Our scheme attends to social-cultural aspects of mathematical learning while focusing on the student actions expected as they work tasks. We exemplify our scheme with tasks drawn from a diverse set of PSPs and report results from its application to these projects. We conclude with comments on how our work can assist instructors, curriculum developers, and researchers, including those who are interested in other types of curricular materials.

Graph literacy, the ability to interpret and create graphical representations, is an important skill for students to learn mathematics and to succeed in STEM coursework and careers. Additionally, with the rapid development of technological devices and media, students are encountering increasingly more situations in which graph literacy is needed to make sense of and respond to information. In this paper, I present a conceptual analysis of what I consider the three layers constituting a graphical representation: frames of reference, coordinate system, and graph. Relatedly, I synthesize relevant literature and propose an analytical framework that could be used to make sense of students’ representations of spatial phenomena. The analytical framework was designed to answer the following question: How do students make sense of a spatial situation, establish frames of reference, and create re-presentations of the spatial situation via coordinated measurements? After I elaborate on the analytical framework, I draw data from teaching experiments to illustrate how the framework could be used…

Students’ development of reasoning through algebraic symbols is a crucial component of most mathematics courses. This paper reports the journey of one second-semester calculus student, Cedric, as he attempted to reason about and create algebraic representations for arbitrary partial sums and infinite series through two exploratory teaching interviews. We report Cedric’s symbolizing activity in terms of Eckman's (2023) expression framework, focusing on Cedric’s development and attribution of meaning to a personal expression template for denoting partial sums and series. Specifically, we describe how Cedric leveraged his initial meanings for partial sums to create personal expressions to reason about infinite series. We then describe Cedric's construction of a general personal expression template for partial sums and series, including cognitive and physical modifications he made to his personal expression template as he symbolized various series. Our results show potentially profitable task sequences and lines of questioning that instructors might utilize to help students be more…

This research aimed to determine how preservice secondary mathematics teachers supported students learning with multiple representations in their lesson plans. Participants were nine preservice secondary mathematics teachers who prepared lesson plans at the end of a 14- week mathematics methods course at a public university in Türkiye. The primary data source was the participants’ lesson plans. Semi-structured interviews were conducted with each participant regarding the details of their lesson plans. The analytic framework for meaningfulness in representational fluency was used as a lens to analyze the data. Findings revealed that the preservice teachers enriched their lesson plans with contextual, physical, verbal, visual, and symbolic representations. However, their support for students to develop representational fluency in lesson plans was limited to transpositions and translations. The preservice teachers neglected to include planning steps that would prompt students to articulate an invariant mathematical feature across multiple representations by making connections.

Relational and instrumental understanding of the Chain Rule can help teachers provide students a deeper and more meaningful calculus experience.

Agenda and meeting minutes from the December 13th 2023 meeting of the University of Montana Staff Senate. Highlights include Secretary election and Presentation by VP for Information Technology and Chief Information Officer Zach Rossmiller.

Congratulations Grads! -- Senior Capstone Proposal Highlight -- GLI Students Making a Splash! -- We want to treat YOU this finals week! -- Deadline Reminder: Studying abroad next Fall (2024)? -- School of Music at the Wilma, Dec. 16 -- Stress Less is HERE! -- Giving Opportunities -- Course Offerings -- AI Innovation Symposium -- Good Luck on Your Finals, GLI Students!

Student newspaper of the University of Montana, Missoula.

GLI Students Making a Splash! -- We want to treat YOU this finals week! -- Deadline reminder: Studying abroad next Fall (2024)? -- School of Music Tonight! -- End of Year Celebration! -- Stress Less is HERE! -- Giving Opportunities: Warm Clothing Drive and Holiday Helpers Gift Drive -- Scholarship Opportunity -- Good luck on your finals, GLI students!

Agenda and meeting minutes from the May 7, 2014 meeting of the Associated Students of the University of Montana (ASUM). The following resolutions were discussed during the meeting: SB14-23/24: Resolution Permitting ASUM to Open an Account with the UM Foundation SB13-23/24: Resolution Amending Article IV Section 7 of the ASUM Bylaws

GLI Students Making a Splash! -- Scholarship Info Sesh -- Event Opportunity: A Conversation on Myth, Misperception, and Reality in the West -- Course Offerings: IRSH 249, IRSH 391, IRSH 381 -- Scholarship Opportunity -- Professional Development: Max S. Baucus Institute -- Global Leadership Initiative: Visit our Website and Follow us on Instagram

We are thankful for YOU! -- GLI in the News: UM News Service -- Scholarship Opps -- Course Offering: IDS 191X: Global Justice and Sustainable Development -- Professional Development: Max S. Baucus Institute -- Global Leadership Development: Visit our Website and Follow us on Instagram

Student newspaper of the University of Montana, Missoula.

Agenda and meeting minutes from the November 15, 2023 meeting of the Associated Students of the University of Montana (ASUM). The following resolutions were discussed during the meeting: SB12-23/24: Resolution Amending Article IV Section 7 of the ASUM Bylaws

This article delves into an intervention designed to enhance a precalculus student’s understanding in constructing linear equations and in approximating function values. Before the intervention, the student primarily relied on the algebraic formula y2-y1/x2-x1 to determine slope, knew slope geometrically as Δy/Δx solely for integer values, and struggled to construct a linear equation without an explicitly shown y-intercept. Through the intervention, the student comprehended slope m as a unit rate and expanded this understanding to Δy = 𝑚Δ𝑥 for integer Δx by iterating m, Δ𝑥 number of times. She also successfully used this newfound understanding to construct linear equations, including tangent line equations, and approximate function values. Nonetheless, she faced some challenges in comprehending Δy = 𝑚Δ𝑥 for non-integer Δ𝑥 and in conceiving point as a multiplicative object with its coordinates representing covarying quantities. Further exploration is needed to validate these findings and address potential difficulties that students might face in grasping slope in the context of…

Connecting algebraic and graphical representations encompasses a large portion of mathematical activity for students in grades 8-14. In Calculus, the ability to represent distances on graphs using algebraic expressions is foundational for a wide range of results. However, research has shown that students may struggle to make such connections. In this article, we seek to answer the following questions: (1) What are the underlying conceptions critical to expressing distances on graphs of functions algebraically? and (2) What types of tasks may support students in developing this skill? We first offer a conceptual analysis of the connections between algebraic expressions and distances in graphs of functions. Next, we describe a hypothetical learning trajectory of tasks and associated learning goals we developed to support students in expressing distances. We then report the outcome of implementing the tasks with groups of Calculus students. While the results include some success in supporting students to express distances, they also pointed to some persistent obstacles as well as…

GLI in the News! Bloomberg, KPAX -- Course Offering: JRNL 102Y: Calling Bullsh*t -- Course Offering: IDS 191X: Global Justice and Sustainable Development -- Municipal Opportunity: Citizen's Law Enforcement Academy -- Professional Development: Max S. Baucus Institute -- Food Drive: Can the Cats -- Global Leadership Initiative: Visit our Website and Follow us on Instagram

Student newspaper of the University of Montana, Missoula.

Agenda and Minutes from the November 8th, 2023 meeting of the University of Montana Staff Senate.

Agenda and meeting minutes from the May 7, 2014 meeting of the Associated Students of the University of Montana (ASUM). The following resolutions were discussed during the meeting: SB11-23/24: Resolution Commending KBGA

A dramatic escape from a rehabilitation hospital, a high-speed car chase resulting in a crash, and an admission of fleeing to avoid jail. Despite fitting the description of an action movie, State v. Strizich presented a conundrum for the Montana Supreme Court, one that created substantial disagreement among the justices on whether evidence of the defendant’s flight should be admitted at trial. One major disagreement was whether to apply the four inferences test used by the federal courts, whereby courts determine the probative value of flight evidence—and therefore, its admissibility at trial—by assessing: the degree of confidence with which four inferences can be drawn: (1) from the defendant’s behavior to flight; (2) from flight to consciousness of guilt; (3) from consciousness of guilt to consciousness of guilt concerning the crime charged; and (4) from consciousness of guilt concerning the crime charged to actual guilt of the crime charged. The inferences test was first introduced by the Fifth Circuit in United States v. Myers and has since been widely adopted throughout the…

Reminder: Sign up for advising -- GrizPitch -- Course Offering: PEAS -- Grad Fair '23 -- Food Drive: Can the Cats -- Global Leadership Initiative: Visit our Website and Follow us on Instagram

Student newspaper of the University of Montana, Missoula.

Agenda and meeting minutes from the November 1, 2023 meeting of the Associated Students of the University of Montana (ASUM).

Happy Halloween from the Franke GLI! -- Beyond the Classroom: Plan and Scholarship Deadlines! -- Nominate an instructor to teach GBLD 194! -- Stolen Waters Summit -- President's Lecture Series: "Ancestors of the Future: Indigenous Resistance and the Climate Change" -- Spring 2024 Course CSD 491: Cultural Competence in Special Education -- Spring 2024 Course CSD 445: Social Communication Skills, Behavior, and Neurodiversity

Student newspaper of the University of Montana, Missoula.

Agenda and meeting minutes from the October 25, 2023 meeting of the Associated Students of the University of Montana (ASUM). The following resolutions were discussed during the meeting: Resolution Generally Revising Article IV, Section 4 of the Bylaws

Asia, The World, And Montana's Economic Future -- Panel Discussion: Held v. Montana -- Study Climate Change in the Mekong Delta this coming spring/summer! -- Low Income Home Energy Assistance Programs -- Affordable Connectivity Program

Student newspaper of the University of Montana, Missoula.

Agenda and meeting minutes from the October 18, 2023 meeting of the Associated Students of the University of Montana (ASUM).

GLI Peeps in action! Leadership Retreat no. 2 -- Advising Sessions -- Beyond the Classroom Plan -- Nominate a Professor for a GLI First-Year Seminar! -- IDS Career-Development Seminar Series: Overcoming Informality: Innovative Social Entrepreneurship to Advance Migrant Workers' Rights and Services in India -- New Course Offered Spring 2024: CSD 396: Inclusive Campus-Based Practices for Neurodiverse Persons/Service Learning -- Summer Wish List

Student newspaper of the University of Montana, Missoula.

Agenda and meeting minutes from the October 11, 2023 meeting of the Associated Students of the University of Montana (ASUM)

Student newspaper of the University of Montana, Missoula.

Agenda and meeting minutes from the October 4, 2023 meeting of the Associated Students of the University of Montana (ASUM). The following resolutions were discussed during the meeting: SB08-23/24: The Associated Students of the University of Montana1 Resolution Revising Article VII, Section 1 of the Bylaws SB09-23/24: The Associated Students of the University of Montana1 Resolution Revising Article IV, Section 2, of the ASUM Bylaws

Homecoming 2023! -- Coffee Talk with Kari Dilloo of Microsoft -- ISEP Study Abroad Info Session -- ELCS Presents the Conservation & Sustainability Careers & Opportunities Fair and the Government & Non-Profit Careers & Opportunities Fair -- Celebration of the 70th Anniversary of the U.S.-Korea Alliance -- The President's Lecture Series Presents: "Braiding Sweetgrass: Indigenous Wisdom, Scientific Knowledge and the Teaching of Plants" with Dr. Robin Wall Kimmerer -- Career Conversations