Digital Commons Network^™

Connecting Representations And Ways Of Thinking About Slope From Algebra To Calculus, Deborah Moore-Russo, Courtney Nagle

The Mathematics Enthusiast

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 …

An Analytical Framework For Making Sense Of Students’ Graphical Representations With Attention To Frames Of Reference And Coordinate Systems, Hwa Young Lee

The Mathematics Enthusiast

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 …

Fostering Students’ Development Of Productive Representation Systems For Infinite Series, Derek Eckman, Kyeong Hah Roh

The Mathematics Enthusiast

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. …

Mathematical Understanding Based On The Mathematical Connections Made By Mexican High School Students Regarding Linear Equations And Functions, Javier Garcia-Garcia

The Mathematics Enthusiast

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 …

Preservice Secondary Mathematics Teachers’ Ways To Support Learning With Multiple Representations In Their Lesson Plans, Hilal Gulkilik

The Mathematics Enthusiast

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, …

Guest Editorial: Connecting Mathematical Representations From Algebra To Calculus, Kyunghee Moon

The Mathematics Enthusiast

No abstract provided.

Understanding Point And Slope In Linear Equations And Approximations: A Case Study, Kyunghee Moon

The Mathematics Enthusiast

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 Δ_y/Δ_x 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 …

Expressing Distance In Graphs Of Functions In The Cartesian Plane: Obstacles And Interventions, Erika David Parr, Samuel Lippe

The Mathematics Enthusiast

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 …

The Spirit Of Mathematical Modeling – A Philosophical Study On The Occasion Of 50 Years Of Mathematical Modeling Education, Peter Frejd, Pauline Vos

The Mathematics Enthusiast

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 …

Questions About The Identification Of Mathematically Gifted Students, Marianne Nolte

The Mathematics Enthusiast

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.

Seeing Pascal’S Mystic Hexagon From A Different Angle, Anderson Norton

The Mathematics Enthusiast

Hexagons inscribed within circles have the mystic property that their three pairs of opposite sides (when extended) intersect on the same line. This surprisingly simple result seems to support the Platonic notion that elegant mathematical relationships exist in some perfect realm awaiting human discovery. To the contrary, this article introduces a transformational proof that provides for intuitive understanding of the mystic property while supporting the idea that mathematical objects, and relationships between them, arise from coordinations of our own mental actions.

The Tradition Of Large Integers In Historical Arithmetical Textbooks, Franka Miriam Bruckler, Vladimir Stilinovic

The Mathematics Enthusiast

After the Hindu-Arabic decimal positional system was introduced in Europe, throughout many centuries textbooks on elementary arithmetic, intended for beginners, had a more or less fixed organization of content, usually starting with chapters on numeration. These chapters, as a rule, contained one or more examples of large integers the purpose of which was simply to be named (read out loud), sometimes also vice versa. This tradition apparently began with the two first texts that significantly contributed to the spread of the decimal system in Europe—the Latin translations of al- Khwarizmi’s treatise on decimal arithmetic, and Leonardo’s Liber Abaci, containing examples …

Productive Struggle In Mathematical Modelling, Piera Biccard

The Mathematics Enthusiast

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 …

An Examination Of The Factors And Characteristics That Contribute To The Success Of Putnam Fellows, Robert A. J. Stroud, Thomas C. Defranco

The Mathematics Enthusiast

The William Lowell Putnam Mathematical Competition is an intercollegiate mathematics competition for students in the United States and Canada and is regarded as the most prestigious and challenging mathematics competition in North America (Alexanderson, 2004; AMS, 2020; Grossman, 2002; Reznick, 1994; Schoenfeld, 1985). Students who earn the five highest scores on the examination are named Putnam Fellows. Since its inception in 1938, only 306 individuals have won the competition and a select few have won multiple times. Clearly, being named a Putnam Fellow is a remarkable achievement and therefore, understanding the factors and characteristics that contribute to their success is …

On The Even Distribution Of Odd Primes: An On-Ramp To Mathematical Research, James Quinlan, Michael Todd Edwards

The Mathematics Enthusiast

The authors consider a conjecture by Chebyshev in 1853 on the distribution of odd primes among those that are one more than a multiple of four and those three more than a multiple of four—and use technology to explore the cardinality of these subsets. Generalizations are presented for student exploration along with several sources for more in-depth research.

Going Off On A Tangent: An Inclusion-Exclusion Identity, Aloysius Bathi Kasturiarachi

The Mathematics Enthusiast

Undergraduate research in tertiary education offers mathematics students a pathway to engage in high-impact practices. Using a simple sum equals product identity from number theory as a motivator, we build a series of inclusion-exclusion identities for convex polygons using the symmetry inherent in the tangent function. The techniques used are simple and accessible, illuminating and generalizable, in a manner that rejects a singular line of inquiry in favor of a plurality of mathematical ideas.

Horocycles, Horoscopes, Horizons…: A Review Of Petra Mikulan & Nathalie Sinclair’S (2023) Time And Education, Bharath Sriraman

The Mathematics Enthusiast

No abstract provided.

Stepping Stone Problem On Graphs, Rory O'Dwyer

The Mathematics Enthusiast

This paper formalizes the stepping stone problem introduced Ladoucer and Rebenstock [G] to the setting of simple graphs. This paper considers the set of functions from the vertices of our graph to N, which assign a fixed number of 1’s to some vertices and assign higher numbers to other vertices by adding up their neighbors’ assignments. The stepping stone solution is defined as an element obtained from the argmax of the maxima of these functions, and the maxima as its growth. This work is organized into work on the bounded and unbounded degree graph cases. In the bounded case, sufficient …

A General Expression For Hermite Expansions With Applications, Tom P. Davis

The Mathematics Enthusiast

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 …

Will They Automatically Work Together? Cooperation Among Non-Fools In Hobbes´S Leviathan, Karim Pluma

The Mathematics Enthusiast

Thomas Hobbes´s State of War is commonly imagined as a harrowing condition where hostile interactions are the rule and non-hostile encounters are the rare exception. However, while it is generally true that Hobbes purposely outlined his famed condition of anarchy as a condition of perennial conflict, it is also equally true that cooperative behavior was not uncommon. In fact, by only taking into account cooperative behavior, as presented in Leviathan, the anarchic humans leave the absolute uncertainties of the State of War and create the Commonwealth for their safety and well-being. Over the past fifty years or so, several exits …

Displaying Gifted Students’ Mathematical Reasoning During Problem Solving: Challenges And Possibilities, Attila Szabo, Ann-Sophie Tillnert, John Mattsson

The Mathematics Enthusiast

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 …

Tagging Opportunities To Learn: A Coding Scheme For Student Tasks, Janet Heine Barnett, Cihan Can, Daniel E. Otero

The Mathematics Enthusiast

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 …

The Chain Rule Does Not Have To Be A Pain Rule, Elizabeth Beeman, Jennifer Runge, Lauren Holden, Devon Maxwell, Yi-Yin (Winnie) Ko

The Mathematics Enthusiast

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

An Analysis Of Graduate Teaching Assistants’ Noticing Skills During Calculus And Physics Tutoring Scenarios, Lindsay Borger, M. J. Camarato, Molly H. Fisher

The Mathematics Enthusiast

Professional noticing of mathematical thinking, as defined by Jacobs, Lamb, and Philipp (2010) can be broken down into three components: attending to relevant cues, interpreting the mathematical understanding, and deciding the next best instructional steps. Most research on this topic has been conducted with elementary children. However, there is a gap in the research on professional noticing at more advanced levels, particularly college students. The purpose of this study was to take the concept of professional noticing and apply it to mathematics education at the post-secondary level. Specifically, the question we sought to answer in this study was: To what …

On Convincing Power Of Counterexamples, Orly Buchbinder, Rina Zazkis

The Mathematics Enthusiast

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, …

A Role For Affect In The Future Of Mathematics Education (With Thoughts On Intelligence), Marcelo Almora Rios

The Mathematics Enthusiast

A Review of:

1. Alan Schoenfeld, Heather Fink, Alyssa Sayavedra, Anna Weltman, & Sandra Zuñiga-Ruiz’s (2023) Mathematics Teaching on Target, Routledge, 164pp. ISBN (HB): 978-1-0324-4167-2;

2. Alan Schoenfeld, Heather Fink, Sandra Zuñiga-Ruiz, Siqi Huang, Xinyu Wei, & Brantina Chirinda’s (2023) Helping Students Become Powerful Mathematics Thinkers, Routledge, 272pp. ISBN (HB): 978-1-0324-4168-9.

Investigating Problems Posed By Pre-Service Mathematics Teachers For The Four Operations In Fractions, Yasemin Kiymaz

The Mathematics Enthusiast

This study investigates 40 preservice middle school teachers’ problems posed for two given fractions. Pre-service teachers were asked to pose three problems (farea, length and set model) for each of the operations of addition, subtraction, multiplication and division with these fractions. 12 problems posed by each pre-service teacher were examined in terms of their suitability for the given operation, the targeted model and real life relevance. Errors in the posed problems were also examined. As a result of the analysis, it was observed that the model in which pre-service teachers were the most unsuccessful in posing problems was the set …

Today’S Mathematics Student: Take Two, Carmen M. Latterell

The Mathematics Enthusiast

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.

Using Gardner's Three-Squares Problem For A Group Project In A Mathematical Problem Solving Module, Jonathan Hoseana

The Mathematics Enthusiast

Consider a 1 x 3 grid whose top-left vertex is connected to the bottom-right vertex of each of the unit squares. What is the sum of the three acute angles formed by the connecting segments with the unit squares' bases? This is the so-called three-squares problem, often attributed to Gardner. In a recent academic year, the author used a video on this problem, produced by the YouTube channel Numberphile, for a group project in a first-semester undergraduate module: Mathematical Problem Solving. The project involved collaborative writing on the problem and individual completion of a peer-assessment form. We report the outcomes …

From Zero To Epsilon: My Transformed Real Analysis Course, David Calvis

The Mathematics Enthusiast

In response to student evaluations I revised my undergraduate course in real analysis to a slides-and-worksheets model. This is the story of that revision, including why and how it was done, together with the results.