The 50 most-cited articles in The Mathematics Enthusiast, 2019

Creativity is a psychological construct that has gained research popularity (Akgul & Kaveci, 2016), however it remains a challenging one to define. The variety of definitions promulgated to understand creativity hints at the complexity of the mental process. Furthermore, as a subset of creativity, domain-specific mathematical creativity has also garnered a variety of definitions. The transdisciplinary research on creativity (Sriraman & Haavold, 2017) is seminal in this world of fast-paced innovation, invention, solution, and synthesis. Considering every human being with at least average cognitive abilities possesses the ability to think creatively (Baran, 2011), developing students’ creative talents and abilities must be high on a list of educational priorities. Much of the literature surrounding mathematical creative thinking is centered on trying to quantify an individual’s creative thinking abilities. There have also been studies conducted that enabled researchers to describe various traits and demonstrate multiple levels of creativity. The basis of this work will be to…

The present work offers a characterization of the development of mathematics education, based on scientometric indicators. The study analyzes 5633 documents registered in the Scopus database which are all related to research in mathematics education. The main results show an exponential increase on the scientific productivity, a tendency to increase collaborative work over individual work, a greater impact per index of the cited references during the last two decades, a high level of international collaboration, a certain concentration of the publications on a reduced group of investigators in some international impact journals, and the existence of 17 invisible colleagues that represent scientific communities with common research interests.

In this paper we present and discuss a designing experience carried out by prospective support teacher. They have been introduced to digital storytelling methodology and they are required to design story-problems in mathematics, devoted to support students with disabilities. Story-problems have been analyzed with respect to the narrative dimension and mathematical correctness. The teachers, who worked in groups, have been able to design and implement striped problems, i.e. stories with a deep integration between the mathematical and narrative aspects. However, most of the groups showed difficulties in creating well-posed problems from a mathematical point of view.

In this paper, we present a qualitative study in which we analyse the videorecordings of four groups of students solving Fermi Problems. Previous studies show that Secondary School students solve this type of problems using complex problem solving processes and developing mathematical models. In order to analyse the students’ problem solving processes, so-called Modelling Activity Diagrams were used. The results of the present study demonstrate that solving Fermi problems is a complex matter, and that some of the theoretical tools used in the field of Mathematical Education fail to adequately reflect this level of complexity. In addition, Modelling Activity Diagrams are presented as a more detailed analysis tool to characterise student choices and actions, as well as to make the structure of the Fermi problem addressed more visible.

The purpose of the investigation was to identify pre-service elementary teachers’ (PST) misconceptions at the culmination of their methods semester, prior to entering student teaching. Participants, n=134 (116 female) were pre-service elementary teachers from two universities in the intermountain region. The Statistical Reasoning Assessment (SRA) developed by Garfield (2003) was used to investigate student misconceptions in statistics and probability. Of the eight misconceptions, the Representativeness misconception and the Outcome orientation misconception were the least common (12.3 and 28.2% respectively) and the Comparing groups of the same size, Equi-probability bias, and Correlation implies causation misconceptions were the most common (70.2, 64.3, and 50.0% respectively). The confidence interval for the results was within a window of .389 to .427. Implications from the study are several, including a stand-alone statistics and probability course would likely improve PST’s understanding of concepts in the domains, misconceptions should be used to promote true understanding, and…

In this article the use of animations is explored to facilitate students acquiring of problem-solving. The goal is to spur interest in educators for further research on the use of animated videos, which combine contextualised stories, animated worked examples and practice questions on an interactive platform to teach learners mathematics concepts and problem solving skills.

This study examined a typical sample of K-8 pre-service teachers (PTs) enrolled in typical teacher-preparation programs (TPPs) to provide insight about where on the traditionalreform continuum PTs’ beliefs are positioned. To ensure accuracy in results, a sequential explanatory design was utilized in the examination of the PTs’ normative beliefs assessed three times throughout their TPPs (using Likert items and open-ended questions), relationships between other PTs’ and in-service teachers’ (ITs’) beliefs, and alignment between normative beliefs (what PTs believed they should do) and discursive claims about their teaching (what PTs claimed to do). Results of this study – particularly related to beliefs that are not positioned as far along the trajectory toward reform – are intended to assist mathematics educators and PD developers in targeting future instruction to meet PTs and ITs where they are. Beliefs related to “The power of students’ ideas”; “Critical thinking, problem solving, and understanding, justifying, and communicating processes and their connections to resulting…

Guided by the question “Which aspects could affect mathematics teachers’ identity, especially in the context of a postgraduate course that includes teaching practices as mathematics teacher educators?” we analyzed the transition between actual and designated mathematics teachers’ identity in a postgraduate training course. In particular, teaching practices during the course were oriented by emerging recommendations for mathematics teacher training. We concluded that the presence of an explicit intention to develop a specific practice plays a key role in the transition between actual and designated identity. In addition, the possession of methodological tools as a means for implementing renewed practices gave support and helped practitioners to plan classes promoting mathematical activity. In the process experienced by the practitioners, mentor teacher educators were clearly significant narrators because they reinforced or hinder the desire to reach the designated identity, through their coherence with practitioner’s goals or in contradiction with them respectively.

Developing students’ number sense is a critical area of research in mathematics education because of the role number sense plays in early mathematics learning. In particular, cognitive psychology research has pinpointed verbal counting as a number sense construct that is critical in later mathematics achievement. This study explored variations in 7- and 8-year-old students’ number sense outcomes as they engaged in a counting-focused instructional treatment for differing durations. Sixty students in three elementary classrooms in the United States participated in the counting-focused instructional treatment. A generalized estimating equations (GEE) analysis showed an associated average increase in test scores for students participating in 9 weeks of the counting-focused instructional treatment as compared to students participating in 3 weeks of the instructional treatment. This study provides preliminary results about an instructional practice, based in numerical cognition theory, for elementary mathematics teachers to facilitate opportunities for students to develop their number…

Studies on high ability in mathematics rarely take a teacher’s perspective. The purpose of our study is to add such a perspective, which we will do by using positioning theory to analyze elementary teachers’ discussions on mathematical tasks, aiming to challenge all pupils, including the highly able. The study is conducted in the context of a two-year long teacher professional development program on high ability. Teachers expressed both their teaching and the mathematical tasks as helpful in orchestrating teaching suitable for highly able pupils. They highlight the opportunities given by some tasks as well as the importance of guiding highly able pupils to go further in such tasks. However, they expressed their own limited mathematical knowledge and time needed for pupils with learning difficulties as obstacles to orchestrate teaching for the highly able. The results show that it is important to, in close cooperation with teachers, further explore how to orchestrate teaching that challenges highly able pupils.

Digital objects are composed of bitstreams, sequences of 1’s and 0’s, which require specific software (and in some cases hardware) to make the content understandable to human users. Digital objects, like word processing documents, digital images, websites, e-mails, datasets and so much more, are fragile, easy to modify, and susceptible to bit rot (loss or reordering parts of the bitstream) and obsolescence. Digital preservation is a combination of policies and workflows that dictate the active management of digital objects to ensure their continued authenticity and meaningful access over time. Obsolescence is one of the unending battles that digital preservationists fight. File formats are continually updated, hardware consistently replaced, and software abandoned. One common strategy to combat obsolescence is to migrate older digital objects into new formats. To keep these digital objects available for future users, long-term digital preservation is essential.

In this article I suggest some alternatives to the solution test for the understanding of graphs of inequalities in two variables. Based on the work done with two groups of preservice secondary mathematics teachers, and by using the idea of variation and the framework of action and process conceptions of inequalities, I explain how graphs of inequalities can be viewed as a collection of rays or curves. I also explain the potential benefits of these alternative explanations in the solving of optimization problems and in the instruction of functions as a medium of one variable and two variable functions.

The focus of this study is the mathematical reasoning of pre-service teachers. The author of this paper was the instructor of a class of pre-service teachers preparing to teach in lower secondary school. The instructor divided the class in small groups and gave each group some mathematical exercises to work on. The unit of analysis of this study is the video and audio-recorded dialogue of each group. The aim of this study is to characterize different aspects of pre-service teachers’ mathematical reasoning, as well as to indicate ways to develop their reasoning further. The basic idea of the study is that rote learning reasoning is imitative, and the opposite kind of reasoning is creative. However, the study indicates that the reasoning of the participating students sometimes could be neither imitative nor creative. Thus, some of the reasoning was in a grey zone somewhere between imitative and creative reasoning.

The Maths Arcade is a regular, optional drop-in session to play strategy games and puzzles, taking place at a range of U.K. universities. The main aims are to support students, create a maths-themed staff-student community and develop mathematical thinking through strategy game-play. At some universities, the Maths Arcade has additional aims, including motivating curricular work, linking to peer assisted learning and promoting cross-disciplinary interaction.The games used are strategy games with simple rules which have no explicit link to mathematics but appeal to logical thinking. The games and activities typically used at a Maths Arcade are described and an example of a deep investigation is given as a case study. An evaluation of the Maths Arcade at five U.K. universities was completed by distributing a questionnaire to attenders and non-attenders. Students who attend more sessions were more likely to report making friends at the Arcade, and a majority of students said that staff presence is helpful and they would like staff to attend more often, results which support the aim of…

Sorting is a process we encounter very often in everyday life. Additionally it is a fundamental operation in computer science. Having been one of the first intensely studied problems in computer science, many different sorting algorithms have been developed and analyzed. Although algorithms are often taught as part of the computer science curriculum in the context of a programming language, the study of algorithms and algorithmic thinking, including the design, construction and analysis of algorithms, has pedagogical value in mathematics education. This paper will provide an introduction to computational complexity and efficiency, without the use of a programming language. It will also describe how these concepts can be incorporated into the existing high school or undergraduate mathematics curriculum through a mathematical analysis of student-generated sorting algorithms.

Motorcycle tourism is a relatively understudied market in the U.S. but has the potential to assist rural communities in their efforts to develop a tourism economy. This report highlights the characteristics of motorcyclists and their travel needs as well as the potential for marketing to this niche of travelers. Highlights: • Motorcyclists tend to ride on scenic roads, eat and drink local cuisine, and stay away from the busy highways leading to the idea that ‘off the beaten’ path communities could benefit from motorcycle tourism. • Nonresident motorcyclists’ average age was 60, while resident motorcyclists’ average age was much younger at 48.7 years old. • All motorcyclists prefer scenic roads, lots of curves, safe parking places for their cycles at night, and a motel for their overnight stay. • Nonresident motorcyclists in Montana spend, on average, $110/day with the largest expenditure on restaurant and bar, followed by motel/hotel. • Alberta motorcyclists were the highest visitor to the state at 16% of all nonresident cyclists. • The most utilized entry points for motorcyclists…

In this paper the historical origins of fractional calculus are explored with respect to non-conformable derivatives. Implications are made for the teaching and learning of calculus and for mathematics education.

Until the seventeenth century, rational numbers were represented as fractions. It was starting from that century, thanks to Simon Stevin, that for all practical purposes the use of decimal notation became widespread. Although decimal numbers are widely used, their organic development is lacking, while a vast literature propagates misconceptions about them, among both students and teachers. In particular, the case of the period 9 is especially interesting. In this paper, changing the point of view, we propose substituting the usual definitions of the periodic decimal representation of a rational number - the one obtained through long division introduced in the secondary school and the one obtained through the series concept for undergraduate level - with one which, instead of infinite progressions, uses a known property of fractions deducible from Euler's Totient Theorem. The long division becomes a convenient algorithm for obtaining the decimal expansion of a rational number. The new definition overcomes the difficulties noted in the literature. An elementary proof of Euler's…

In 2015, a group of adolescents between the ages of eight and nineteen filed a lawsuit against the federal government for infringing upon their civil rights to a healthy, habitable future living environment. Those Plaintiffs in Juliana v. United States alleged that the industrial-scale burning of fossil fuels was causing catastrophic and destabilizing impacts to the global climate, threatening the survival and welfare of present and future generations. Seeking to reduce the United States’ contributions to atmospheric carbon dioxide, Plaintiffs demanded injunctive and declaratory relief to halt the federal government’s policies of promoting and subsidizing fossil fuels, due to the limited timeframe for addressing the impacts of climate change. While extensive motion practice has impeded a hearing on merits of Plaintiffs’ claims, Juliana v. United States addressed threshold questions concerning constitutional and procedural claims.

Research has shown that right triangle trigonometry poses a significant learning challenge for high school geometry students. One potential source of this challenge is the tendency for students to experience right triangle trig as an exclusively action-oriented set of strategies (i.e. SOHCAHTOA). This article describes a teaching experiment where students built understanding of right triangle trig using a unique set of manipulatives and direct measurement. Results of the experiment show that the lessons provided opportunities for students to develop their understanding beyond action-oriented strategies.

Teacher knowledge is a critical factor that influences pedagogical decisions. If we want teachers to make appropriate choices in the classroom we must know and understand the types of knowledge used during this decision-making process. To this end, we sought to understand how, and the extent to which, two 5th grade teachers drew upon and integrated their knowledge of mathematics, learners, and pedagogy while teaching. Stimulated recall interviews were analyzed to uncover the types of knowledge and interactions that occurred. Both teachers primarily used their knowledge of learners and pedagogy, with the knowledge of mathematics playing a supportive role. In addition, the teachers integrated their knowledge in one of two ways: a) one knowledge type was used to justify or explain a statement about a second knowledge type and b) a discussion of one knowledge type lead to an implication or reflection about a second knowledge type. These interactions allowed the teachers to use and build their connected knowledge. Understanding how teachers integrate and use their knowledge has…

In calculus-based statistics courses, the expected value of a random variable (EVORV) is discussed in relation to underlying mathematical notions. This study examines students’ understanding of the mathematical notions of EVORV in connection with its semiotic and lexical representations. It also assesses students’ computational competency revolving around EVORV. We collected qualitative data via surveys and interviews from eight students enrolled in a calculus-based university statistics course. The results suggest that while the students in general had the computational accuracy to correctly calculate EVORV, they struggled to understand the notion, and in particular to make sense of the term “random” in “random variable” and the symbol E(𝑋) in the mathematical context. The study provides a basis for understanding potential challenges to students’ learning of EVORV and other related statistics topics and how such challenges may emerge from the semiotic and lexical ambiguities inherent in terms and symbols used in statistics.

Educational institutions face increasing pressures to respond to the changing environment, including the rapid advances in technology. All curricula are subject to scrutiny, but arguably the sciences and mathematics curricula are under a special lens, given the impact of changing technology and the increasing importance accorded to the knowledge economy. This paper explores the continuing dialogue regarding the teaching and learning of mathematics including the transition from secondary to post-secondary and the emergent trends in the field of mathematics education. The recent evolution of mathematics teaching is described and the implications for the future of mathematics teaching are highlighted, including the need for an examination of the learning experiences afforded secondary and post-secondary mathematics students, as well as the need for dialogue between the secondary and post-secondary levels of mathematics education with the purpose of evidence-based development for mathematics teaching.

Few women and minorities work in STEM areas, making urgent appropriate role models, cultural barrier education, and growth mindset development. This study focused on a diverse group of 24 fifth grade girls at an urban school learning about successful mathematical careers of contemporary women of different races. Students viewed illustrated slide shows of the women’s lives and accomplishments, discussing their jobs and barriers they overcame. Students gained spatial skills constructing dioramas of the women’s lives and careers. Thematic analysis of student responses to questionnaires and sentences from reflection essays revealed aspects of a mathematician’s life and work that allowed students to feel inspiration or connection: concern for others and equity; professional trailblazer; persistence; mathematics passion; social life; and outside interests.

On the first page of his well-known “How to solve it” Pólya (1945) writes about problem-solving: "One of the most important tasks of the teacher is to help his students. This task is not quite easy; it demands time, practice, devotion, and sound principles. The student should acquire as much experience of independent work as possible. But if he is left alone with his problem without any help, he may make no progress at all. If the teacher helps too much, nothing is left to the student. The teacher should help, but not too much and not too little, so that the student shall have a reasonable share of the work." (p. 1) Striking in this quote is his balanced view on explication and exploration in the course of problem solving. The classic problem of designing a 3x3 magic square may be used as a didactic model to illustrate what Pólya exactly means with “not too much and not too little”. This model can also show how the teacher, without extensive explication, is able to put even young children on the right track of problem solving.

Employment rates have increased since the recession, but these gains haven’t reached all communities across the country, nor has everyone benefited from these gains. While overall employment rates have improved, employment for people with disabilities in rural areas has decreased. This fact sheet uses the recently released (December 2018) American Community Survey 5-year estimates to explore changes in employment rates for people with and without disabilities across the nine divisions of the United States, and for metropolitan, micropolitan, and non-core counties. This marks the first time longitudinal trends in ACS data can be explored across the rural-urban continuum using comparable definitions of disability.

The purpose of this study is to examine the experience of social isolation and loneliness among older adults living in Missoula, Montana, and the possibility that affection and gratitude may help ameliorate these negative states. The research questions that were created for this study are: How do older adults perceive social isolation and loneliness? What experiences do elderly people associate with affection and gratitude? Do elderly individuals who are socially isolated see themselves as such? What experiences of affection and/or gratitude do elderly individuals typically remember receiving? Which persons do elderly individuals associate most strongly with feelings of affection and/or gratitude? How do elderly individuals typically handle situations where they feel socially isolated and/or lonely? To answer these questions, the researchers created interview questions that answered these queries. Then the researchers created a flyer that was distributed to several older adults through Missoula Aging Services. Thus far, over 10 older adults have been interviewed for the study. The…

The use of journals in supporting student learning in elementary number theory is explored. Implications are made for the use of reflective writing for the teaching and learning of proofs and for undergraduate mathematics education.

In this paper, we discuss a novel approach for collaborative retrospective analysis. One researcher was directly involved in a classroom teaching experiment, adopting an emergent perspective as an interpreter-witness of classroom interactions during a four-week algebra instructional unit with sixth-grade students. The other researcher experienced and analyzed the data in reverse chronological order. We describe how this re-emergent perspective revealed aspects of students’ early algebraic reasoning.

Young adult books like The Art of Starving by Sam J. Miller carry a healing power when readers experience their pause and ponder moments. Because they impose reflection time, Pause and Ponder Moments reinforce reading as a deliberate and patient process. Imposing time to pause and think through points and concepts enables readers to truly grapple with content in meaningful ways. Pause and Ponder Moments also inspire attributional retraining and option awareness, an alternative to simply accepting the status quo. These moments carry promise for enriching how we read the world, how we respond to others, and how we live our lives.

Entomophagy is the human use of insects as food. This practice is exhibited around the world by a variety of cultures with densities highest in the continents of Africa, Australia, and South America. However, entomophagy is not commonly practiced in the United States and as a result many Americans lack the knowledge necessary for understanding, appreciating, and using this practice. As the value of finite natural resources and impacts of food production are recognized by Americans, entomophagy has become a more socially-viable alternative food source because its environmental benefits allow consumers to justify their interest and adoption of the practice. As acceptance of entomophagy improve, so do the opportunities and means to practice entomophagy in the US, but they are still limited. Despite improved access to insect - based products, development of consumer markets for insects in the US has been slow. One reason for this is a lack of usable and accessible information about the benefits and means of adopting entomophagy. Our aim is to improve access to information about…

Surveyors asked questions related to their travel and recreation plans for the upcoming Memorial Day Weekend and 4th of July holiday. 3,542 Montana residents (18 and older) representing every county and weighted to the population of the state were intercepted at gas stations from April 1 to May 15, 2019.

Juvenile recruitment is a key parameter in understanding ungulate population dynamics. Traditional methods in population composition surveys, such as estimating young: adult-female ratio’s, can be precluded by cost, safety, and feasibility. The use of remote cameras provides a potentially cutting-edge tool to apply to wildlife population estimation techniques. While the prevalence of remote cameras in ungulate studies has increased, few studies have used cameras to estimate vital rates, such as recruitment or survival. Here, we tested the potential of remote cameras to estimate calf: cow ratios and calf survival of elk (Cervus elaphus) using the Royle-Nichols (2003) occupancy model. Using the Royle-Nichols (2003) model, data collected from cameras on unmarked individuals can estimate detection probability and abundance. We compared camera-based estimates of calf: cow ratio to traditional ground-based estimates obtained from group classification surveys. We test this approach in a partially migratory elk population at the Ya Ha Tinda (YHT) Ranch, Alberta, Canada. We deployed cameras…

This fact sheet explores insurance rate changes associated with the Affordable Care Act (ACA) and Medicaid expansion on insurance coverage for rural and urban people with and without disabilities. Following the implementation of the ACA, rural people with disabilities had the greatest gains in insurance coverage. Rural people with disabilities in Medicaid expansion states saw a 10.7% increase in insurance coverage, compared to a 5.3% increase for those in non-expansion states. The loss of the ACA and Medicaid expansion could hurt rural people with disabilities more than their urban counterparts, or those without disabilities.

A Visitor Study by the National Park Service, Yellowstone National Park, and facilitated by Otak, RRC Associates, and the University of Montana Institute for Tourism and Recreation Research.

Annual Yellowstone National Park (Yellowstone) visitation has increased close to 45% since the early 2000s with much of the increase in the past five years. Yellowstone visitation rose to historic levels of use at 4.2 million visits in 2016. Based on historical visitation trends, growing international visitation, the retirement of Baby Boomers, and high levels of visitation from Gen X, demand for visitation is expected to continue to increase. In addition, a 2016 traffic mobility study (Otak) found the level of service on most Yellowstone roads rated a D during peak times, indicating serious platooning and delays causing safety issues for visitors. Using data garnered from the transportation mobility study, park management identified multiple areas of concern as “focal congested corridors.” These congested corridors are largely stemming from either dense vehicular traffic or wildlife jams. With this knowledge, Yellowstone managers identified a desire and need to evaluate experiences of different visitor segments at a subset of attractions within the park’s focal congested corridors.…

Meeting minutes from the December 19, 2019 meeting of the Executive Committee of the Clark Fork & Kootenai River Basins Council.

Roberta Evans describes her educational and professional background leading up to her obtaining her Ph.D. in Educational Leadership from the University of Nevada. Evans describes the job search process, which she endeavored with her husband at the time, Jerry Evans, and how they navigated the job market together, ultimately ending up at the University of Montana. She discusses her professional rise from Assistant Professor to her culmination as Dean of the Phyllis J. Washington College of Education and Leadership. Evans provides numerous examples of her struggles to find a balance between her work and family life, struggles which she encountered as the first woman hired into the Department of Educational Leadership at Montana, and also how she overcame these challenges. Evans then goes on to describe her own leadership style—agile leadership—and shares her advice to the next generation of women mentors in academia.

MISSOULA -For 16 years historian K. Ross Toole taught a course titled Montana and the West, a class that was frequently referred to as "one of the most popular and influential courses at the University of Montana."Between 1965 and 1981, hundreds of students took the course each semester, and nearly 1,700 students enrolled in Montana and the West in the final year Toole taught.That year Toole's lectures were recorded by MQTV, a nonprofit public interest media group, to be televised around the state.These recordings now are available online from UM's Mansfield Library at https://scholarworks.umt.

Student newspaper of the University of Montana, Missoula.

Student newspaper of the University of Montana, Missoula.

Student newspaper of the University of Montana, Missoula.

Student newspaper of the University of Montana, Missoula.

Student newspaper of the University of Montana, Missoula.

Student newspaper of the University of Montana, Missoula.

Student newspaper of the University of Montana, Missoula.

Student newspaper of the University of Montana, Missoula.

More importantly, organizers and participants hope that the workshop will create a lasting and cohesive network of practitioners who are dedicated to evaluating and addressing the risks and impacts of mining in salmonid-bearing watersheds."Many people, from community members to government officials, are asking what they need to know about mining impacts to rivers," Sergeant said."If we can arm people with the knowledge present in the room last week, that will be a big win for the special rivers in our region and the people that depend on them staying healthy."For more information about the workshop, please contact Chris Sergeant by email at