The 50 most-cited articles in The Mathematics Enthusiast, 2016

Although the field lacks a theoretically grounded, well-defined, and shared conception of mathematical knowledge required for teaching, there appears to be broad agreement that a specialized body of knowledge is vital to improvement. Further, such a construct serves as the foundation for different kinds of studies with different agendas. This article reviews what is known and needs to be known to advance research on mathematical knowledge for teaching. It argues for three priorities: (i) finding common ground for engaging in complementary studies that together advance the field; (ii) innovating and reflecting on method; and (iii) addressing the relationship of such knowledge to mathematical fluency in teaching and to issues of equity and diversity in teaching. It concludes by situating the articles in this special issue within this emerging picture.

The central theme in this article is that certain problem-solving frameworks (e.g., Polya, 1957; Carlson & Bloom, 2005) can be viewed within Wallas’ four stages of mathematical creativity. The author attempts to justify the previous claim by breaking down each of Wallas’ four components (preparation, incubation, illumination, verification) using both mathematical creativity and problem-solving/proving literature. Since creativity seems to be important in mathematics at the undergraduate level (Schumacher & Siegel, 2015), the author then outlines three observations about the lack of fostering mathematical creativity in the classroom. Finally, conclusions and future research are discussed, with emphasis on using technological advances such as Livescribe™ pens and neuroscience equipment to further reveal the mathematical creative process.

In the context of the increased mathematical demands of the Common Core State Standards and data showing that many elementary school teachers lack strong mathematical knowledge for teaching, there is an urgent need to grow teachers’ MKT. With this goal in mind, it is crucial to have research and assessment tools that are able to measure and track aspects of teachers’ MKT at scale. Building on the concept of “mathematical tasks of teaching” (Ball et al., 2008), we report on a new framework that unpacks the mathematical work of teaching that could serve as a scaffold for item writers who are developing assessments of MKT. We argue that this framework supports a focus on the mathematical work of teaching that moves beyond common content knowledge but without moving into a space of pedagogical choice. We also illustrate how the framework was constructed to highlight connections within and across the mathematical content of elementary school. The mathematical work of teaching framework has implications for assessment development at scale, and could be useful as an organizing tool in…

Finland has scored well in international assessments (e.g. PISA, TIMSS), and the pressure to attain excellent scores has activated a drive toward even more effective mathematics teacher education. This article presents the results of a qualitative assessment of the mathematics teacher education provided by the University of Eastern Finland. In this study, the views held by practicing teachers (N=101) and teacher educators (N=19) are compared so that the outstanding development needs of mathematics teacher education in terms of their contents can be revealed. The data was gathered via an electronic survey and was mainly analyzed using data-driven methods. In addition, framework provided by Mathematical Knowledge for Teaching (MKT) was used to categorize the respondents’ views regarding the contents of mathematics teacher education and to develop general guidelines for the reform of mathematics teacher education. The results indicate that mathematics teacher education should include pure mathematical content (Common Content Knowledge, CCK) and mathematical content that will have been…

This paper provides a historical account of the development of algorithmic complexity in a form that is suitable to instructors of mathematics at the high school or undergraduate level. The study of algorithmic complexity, despite being deeply rooted in mathematics, is usually restricted to the computer science curriculum. By providing a historical account of algorithmic complexity through a mathematical lens, this paper aims to equip mathematics educators with the necessary background and framework for incorporating the analysis of algorithmic complexity into mathematics courses as early on as algebra or pre-calculus.

Assessments of mathematical knowledge for teaching (MKT), which are often designed to measure specialized types of mathematical knowledge, typically include a representation of teaching practice in the assessment task. This analysis makes use of an existing, validated set of 10 assessment tasks to both describe and explore the function of the teaching contexts represented. We found that teaching context serves a variety of functions, some more critical than others. These context features play an important role in both the design of assessments of MKT and the types of mathematical knowledge assessed.

We discuss the adaptation and piloting of the previously developed U.S.-specific measures of mathematical knowledge for teaching to the Malawi context. The purpose is to produce measures that can be used to evaluate changes in mathematical knowledge for teaching gained through primary teacher education, thus informing teacher educators on the most effective evidence-based practices. By interviewing 14 teachers, we first examine whether the 16 recurrent mathematical tasks of teaching tasks identified in the U.S. are applicable to the Malawi context. This is followed by the discussion of the adaptability of the U.S. developed number concept and operations LMT measures. Next, we report on the item psychometric properties estimated from a pilot study in which 351 preservice primary school teachers participated at the end of their coursework. Our findings suggest that all the 16 tasks of teaching mathematics are applicable to the Malawi context, albeit to varying degrees, and should be complemented by additional tasks suggested by the Malawi teachers. For the LMT measures, we found that…

The student population in the U.S. and worldwide is becoming increasingly diverse, creating a need to support all learners, especially linguistically and culturally diverse subpopulations such as English language learners (ELLs). From a social equity standpoint, the need to support these learners is critical especially in mathematics classrooms. In the U.S, the demand for mathematics teachers who are adequately prepared to teach ELLs has in fact risen. Yet, little is known about what knowledge base is essential to teach mathematics to ELLs. Driven by the need to explore this knowledge base, in this paper I explore what is involved in reasoning about teaching mathematics to ELLs. To this end, a set of instructional scenarios illustrating the work of teaching mathematics to ELLs was utilized within an assessment environment. Interviews with 10 mathematics teachers reasoning about the scenarios showed that they drew on the information provided about ELLs’ proficiency levels while reasoning through the scenarios. Also, teachers’ reasoning seems to be qualified by the extent to which…

One area of study that has been gathering enthusiastic attention and interest is mathematical knowledge for teaching (MKT). How to research MKT, however, is still unsettled despite the plethora of unexamined areas of practice. As one of ways to unearth and measure MKT, this study uses interview prompts designed to providing written feedback, as a target area of practice. This study specifies in what ways the interview prompts are used in order to provide a comprehensive method to researching MKT. From interviews across professional communities with different kinds of mathematical expertise, the author develops a conceptual model based on the tasks of teaching, elements of pedagogical context, and domains of MKT. This model provides the fluent character of proficient MKT decision making in teaching practice and explains key features of the design of prompts for investigating and measuring MKT. From the analysis, two claims emerged: bidirectional approaches to investigating MKT and continuous and spontaneous aspects of MKT.

On August 19, 2016, Montana’s Department of Fish, Wildlife and Parks (FWP) instituted a temporary emergency closure on a large extent of the Yellowstone River and its tributaries between the northern Yellowstone National Park boundary near Gardiner, MT and Laurel, MT, about 183 miles downriver. The closure applied to all water based recreation uses on the affected rivers and streams. FWP’s actions followed the identification of the presence of an invasive parasite known to cause proliferative kidney disease in mountain whitefish. FWP’s decision to institute the temporary closure was based not only on the observed presence of the parasite, but also a number of confounding conditions that may threaten longer term impacts if not effectively addressed. The temporary closure is likely to significantly impact both the recreational fishing and the guided river excursion industries, as August and September are both busy seasons. This white paper provides a summary of the contribution of water based activities (fishing, rafting, floating, canoeing, and kayaking) to the Montana economy in…

This document is intended as a “How To” guide for managers and stakeholders wishing to implement the Individual, Clumps, and Openings (ICO) method for silvicultural prescriptions and/or monitoring. This guide is organized into stand-alone chapters. Managers should read and use chapters as they find useful to their own needs.

This paper reports the exploration of an aspect of knowledge needed for equitable mathematics teaching. Pedagogical Content Knowledge for Teaching Mathematics to English Language Learners (PCK-MELL) was proposed as a theoretical knowledge construct, a subdomain of MKT, and the construct was investigated through a process of survey instrument development and administration. The survey contained items intended to measure teachers’ knowledge of the obstacles encountered by ELLs in math classes, of the resources that ELLs draw upon, and of instructional strategies for teaching ELLs. Analysis of middle school mathematics teachers’ responses (N = 42) offered insights into how to improve the reliability and measurement validity of this sort of instrument, as well as directions for further theory development.

What does it mean to align an assessment to the domain of interest? In this paper, we analyze teachers’ performance on the Learning Mathematics for Teaching assessment of Proportional Reasoning. Using a mixture Rasch model, we analyze their performance on the entire assessment, then on two different subsets of items from the original assessment. We consider the affordances of different conceptualizations of the domain and consider the implications of the domain definition on the claims we can make about teacher performance. We use a single assessment to illustrate the differences in results that can arise based on the ways in which the domain of interest is conceptualized. Suggestions for test development are provided.

Earlier studies on improving classroom practice in mathematics have suggested a closer attention to nurturing an aesthetic appreciation for mathematics in students’ learning experiences. Recent evidence nonetheless reveals little indication of its presence. This article offers a potential model of the case for aesthetics in school mathematics. Central to this model is the harmonious hierarchy of necessity, existence, and uniqueness without any of which the case for aesthetics in student learning might be suboptimal, if not untenable. This article offers an example of the proposed model using a possible lesson designed to engage students aesthetically in the learning of mathematics. Pedagogical implications are discussed to reflect and revisit an interpretation of learning mathematics through problem solving.

The law governing the quantification and use of tribal water rights is complex and inconsistent, creating major challenges for tribes working to gain control over and make use of their water resources. There are even greater challenges a tribe must overcome if it wishes to safeguard non- consumptive water uses not generally protected under Western water law regimes. Non-consumptive water uses include any use that does not require removing water from the natural water body. Such uses include protecting in-stream water flows for fisheries, riparian habitat, traditional plants, ceremonial uses, or recreation. There are legal tools available to tribes, however, which can accomplish non-consumptive tribal water resource management goals. This article will describe and explore the pros and cons of several to those options.\nThis article will proceed by first describing the federal law that governs special tribal territory and resources, or Indian federally reserved rights. The article will then present the state of federal law governing tribal non-consumptive water use as it interacts…

This study examines the connection between relationship uncertainty, relationship satisfaction, and Facebook monitoring activity. We hypothesized that (1) people who monitor a romantic partner’s Facebook activity will also report feeling less satisfied in their current relationship, (2) increased monitoring of Facebook will correlate with greater uncertainty in romantic relationships, and that (3) greater uncertainty in a relationship will adversely affect relationship quality. Using an online survey taken by a volunteer sample of 77 participants across the United States, we found support for these hypotheses.

Questions and concerns have been raised about the crowding issues in Yellowstone and Glacier National Parks, as well as the perception of crowding on Montana’s rivers, trails, and other public resources. Crowding, or the perception of crowding, is both an individual evaluation and a cultural conditioning not easily understood from the psychological perspective. This white paper provides a summary of crowding research to help clarify perceptions of, reactions to, and solutions for crowding in natural areas.

This essay analyzes the Ninth Circuit Court of Appeal's rejection of the oral tradition evidence proffered by Native American claimants in Bonnichsen v. United States, 367 F.3d 864 (9th Cir. 2004), and the inconsistent treatment of oral-tradition evidence by courts in the United States. It discusses the harm to Native American claimants when courts summarily reject oral-tradition evidence, prohibiting a major source of knowledge from being meaningfully heard. Finally, this essay suggests four factors for courts to consider, on a case-by-case basis in the future, to evaluate the probative value of oral-tradition evidence, and it urges courts to employ a fair and balanced approach to evaluating oral tradition evidence in legal disputes involving Native American claimants.

Stress is one of the most common psychological conditions college students suffer. Stress is associated with anxiety, depression, high blood pressure, heart disease, disordered eating, irritability, decreased academic performance, and decreased life satisfaction. Intervention is needed. Essential oils are non-invasive and simple to use and were chosen for this study based on their potential ability to reduce stress. The aim of this study was to investigate the effectiveness of aromatherapy as a stress reduction tool for college students. This study took place in Aber Hall and involved residents of floors 6 and 7, as assigned by the residence life director. All residents residing on these floors were informed of the study. 10 subjects completed the study. All of the subjects were freshman females. Both groups were administered a pre-survey during the first week of the semester. The outcome measures were rated on a 1-10 scale, with 1 being the lowest and 10 being the highest level of stress, anxiety, energy, sleep quality, and muscle tension, aches, or stiffness. The treatment group…

In 2007 the governor’s Montana Tourism Advisory Council adopted the Montana Tourism and Recreation Charter. Its guiding principle of geotourism pledges to sustain and enhance the character of the places that the people of Montana call home. How can the state’s efforts to keep this pledge be assessed when “character of place” has never been defined in detail? To keep this pledge, and chart the future of geotourism in the state, it was important to clarify what is meant by character of place and identify the full range of present and potential problems that threaten the integrity of a character. Only by assessing character of place can we understand whether and how geotourism sustains the local environment and its aesthetic appeal, enhances the local community and its culture, and enriches the local heritage and well-being of its people.

Online career development refers to using online resources such as social media and job boards for finding employment and advancing careers. Social networking sites like Facebook, LinkedIn and Twitter allow users to connect with potential employers and advertise their strengths to a diverse audience. Online job boards connect employers and job seekers based on specific interests, skills and locations. Unfortunately, the same online tools that can help people find employment and advance their careers can also limit opportunities if the job seeker’s online presence is unprofessional or absent (Brown & Vaughn, 2011).

The Badger Two Medicine Area in the Lewis and Clark National Forest has faced conflict over management since the 1980s due to leasing of what is considered sacred land. Recently those leases were cancelled. However questions about how to manage the land still remain. This paper explores examples of comanagement between the federal government and Native American tribes in an effort to understand what options and obstacles the Blackfeet tribe will face in future management of the Badger Two Medicine Area. I examined the National Bison Range and Badlands National Park efforts at comanagement in depth and additional current comanagement situations with other federal agencies. Background information is provided on both of these topics. This policy piece found that comanagement suffered at both the National Bison Range and Badlands National Park due to poor communication, political and personal issues within agencies, and issues beyond agency control, such as funding. In situations where comanagement has been successful, strong interpersonal relationships and effective communication have…

During the funded period I continued what had been a preliminary investigation into a generalization of the central theory of extremal graph theory. With the help of funding from the UGP grant I was able to cover the travel expenses for several very productive visits.

The decimal expansions of the numbers 1/n (such as 1/3 = .03333..., 1/7 = 0.142857...) are most often viewed as tools for approximating quantities to a desired degree of accuracy. The aim of this exposition is to show how these modest expressions in fact have much more to offer, particularly in the case when the expansions are infinitely long. First we discuss how simply asking about the period (that is, the length of the repeating sequence of digits) of the decimal expansion of 1/n naturally leads to more sophisticated ideas from elementary number theory, as well as to unsolved mathematical problems. Then we describe a surprising theorem of K. Girstmair showing that the digits of the decimal expansion of 1/p, for certain primes p, secretly contain deep facts that have long delighted algebraic number theorists.

This article revisits an integral of radical trigonometric functions. It presents several methods of integration where the integrand takes the form 1+/- sin x or 1+/- cos x. The integral has applications in Calculus where it appears as the length of cardioid represented in polar coordinates.

behind the clear text and the intent of the actual lawmakers of Montana: the People of Montana.Part II of this article outlines the history of the 1972 Constitution through the documents generated and the surrounding events that shaped those documents and publications.Part III examines modern constitutional interpretation and, more particularly, focuses on the interpretive theories in state constitutional law.Part IV examines Montana's constitutional interpretive framework and offers commentary on the process used by the Montana Supreme Court.Part V outlines a proposed series of factors for conducting originalist jurisprudence and applies those factors to the documents available in Montana.Finally, Part VI concludes with an argument that the Montana Supreme Court should apply and use both the suggested framework and the wealth of ratification documents available when examining ambiguous language in the Montana Constitution.

A collection of 2015 full year and quarterly estimates of nonresident visitation, expenditures by nonresident travelers, and economic impact estimates, including expenditures by purpose of trip, airport of arrival, and international visitors.

INTRODUCTION " [O]nce in your life you need a doctor, a lawyer, a policeman and a preacher but every day, three times a day, you need a farmer."Brenda Schoepp 1During the 2013 Montana Legislative Session, Senator Eric Moore introduced Senate Bill 300, a constitutional referendum to create a right to farm and ranch. 2 The bill attempted to replicate a successful constitutional amendment in North Dakota in 2012, 3 but it was never heard at Moore's * The author is a third-year student at the Alexander Blewett III School of Law at the University of Montana.Prior to attending law school, the author served as communications director and lobbyist for the Montana Stockgrowers Association.She was present for many of the discussions about a possible constitutional amendment for a right to farm and ranch in Montana during the 2013 Legislative Session.

Given that depression is the “leading cause of disability worldwide,” and that less than 50% of people suffering from depression receive treatment, this study aims to provide support for a globally accessible depression treatment (WHO, 2012). The study conducted implemented an internet-based treatment for depression in which users were provided an opportunity to watch slam poetry videos related to mental health issues and write free responses regarding the content of the videos and their subjective experience of depression. Numerous studies provide support for the effectiveness of expressive writing, online mental health interventions, and slam poetry in particular for reducing symptoms of depression. Data collection occurred in two forms. Survey data about participants’ moods was collected before and after watching the slam poetry videos and again after providing the written response. Additionally, the researchers performed text analysis using Linguistic Inquiry and Word Count (LIWC) on the written responses to track use of negative and positive emotion words, among others, which…

The purpose of this study was to explore how visitors support Yellowstone National Park, differences in support among groups, and the relationships between support and various behavioral and psychological concepts. An on-site and mail-back survey was conducted. Results indicate that highly salient memories lead to place attachment and are significantly related to increasing park support. High park supporters tended to be older, more experienced with the park, have geotouristic tendencies, and are generally more involved in recreational activities. Practical implications from this research show a need to engage visitors and provide experiences that are likely to lead to park support.

Through the treaties we reserved that which is most important to us as a people: [t]he right to harvest salmon in our traditional fishing areas.But today the salmon is disappearing because the federal government is failing to protect salmon habitat.Without the salmon there is no treaty right.We kept our word when we ceded all of western Washington to the United States, and we expect the United States to keep its word."-Billy Frank, Jr. 1"The Lummi are salmon people; salmon is culture, and culture is salmon."As the effects of climate change manifest themselves in the Pacific Northwest, salmon, and salmon-dependent northwest tribes, will face the greatest hardships.3 For the tribes in and around the Salish Sea, salmon are inextricably linked to identity.Salmon are not simply a way of life-they are life.4 As United States District Judge George H.

This report focuses on ten Montana communities that have been designated resort areas by the Montana Department of Commerce and voters in those communities have approved a local Resort Tax. The Resort Tax, according to state statute, cannot exceed three percent, and can be collected on luxury items and on goods and services sold at establishments that cater to travelers, such as hotels and restaurants. The intent of the Resort Tax is to provide revenue to aid the communities in maintenance or improvements to infrastructure as well as having the potential to fund other community improvements or programs. Resort tax revenues can offset the burden of a small community hosting a large number of tourists, whether it be throughout the year, in a community such as Big Sky, or seasonally, such as in Craig.

In this article, the author examines the gendered experiences of twelve women who play Dungeons and Dragons, a popular roleplaying tabletop game. The author utilizes Peter Berger and Thomas Luckmann’s concept of the excursion to analyze how participants use roleplay in Dungeons and Dragons to achieve empowerment. Many of the women interviewed described being subjected to gendered behavioral expectations both in their everyday life and during gameplay. With an empowering gaming group, however, roleplay allowed the participants to challenge, and subvert, those gendered expectations. The author concludes that roleplay provides an empowering excursion from everyday life that allows women participants to deal with frustration about gendered treatment and challenge institutional gender roles.

Anglers in Montana are shifting towards a catch and release ethic. This shift is causing increased hook scarring in fish populations. Despite these increasing trends few studies have quantified the rate of hook scarring anglers observe, and their attitudes about hook scarring. We developed and conducted an angler survey on the West Fork of the Bitterroot, a section of river with over 30% scarring rates in Westslope Cutthroat in a 2014 Montana Fish, Wildlife and Parks (MFWP) electrofishing survey. We surveyed floating anglers as they pulled out at the end of the fishing day to get complete catch data, satisfaction information, observed hook scarring rates, attitudes about hook scarring, and equipment used. We surveyed 47 anglers of which 94% were fly-fishing and 72% were using barbless hooks. Anglers observed lower hook scarring rates than the previous MFWP electrofishing survey. On average anglers reported hook scarring rates to be very acceptable. There was no correlation between the rate of hook scarring anglers observed and satisfaction with their catch. As densities of anglers…

Meeting minutes from the December 8, 2016 meeting of the University of Montana faculty.

Student newspaper of the University of Montana, Missoula.

Quantifying the effects of spring freeze-thaw transitions and snowpack dynamics on surface albedo change using satellite optical and microwave remote sensing" (2016).

Schools also won awards at the conclusion of the conference.The 17 participating schools were divided into small and large delegations based on number of students attending the conference and judged on the overall preparedness, participation and excellence of their delegates.The three school award categories include Outstanding School for the top 10 percent, Distinguished School for the top 20 percent and Honorable School for the top 30 percent.The following small delegation schools won

Meeting minutes from the November 29, 2016 meeting of the University of Montana faculty.

Student newspaper of the University of Montana, Missoula.

Meeting minutes from the November 10, 2016 meeting of the University of Montana faculty.

Student newspaper of the University of Montana, Missoula.

In the Blackfeet Tribal Court of Appeals, the Court considered whether appellant, ARMC, was subject to the jurisdiction in the Blackfeet Tribal Court. The lower court erred in finding that it had jurisdiction in this matter under the arm of the tribe analysis it applied. The “arm of the tribe” analysis would have been more appropriate if ARMC was a section 16 entity. Rather, ARMC is a Section 17 Charter, which demands an analysis of whether the Charter was properly granted and ratified. ARMC does enjoy a congressional grant of sovereign immunity under 25 U.S.C. 477, but ARMC’s immunity defense was waived by the inclusion of an arbitration clause in the Participation Agreement. The Court ordered Blackfeet Housing and ARMC to undergo mediation, pursuant to their Participation Agreement.

MISSOULA -Nine fellows from Laos, as part of SportsUnited, learned about assistive technology during their visit to the University of Montana earlier this month.MonTECH, housed in UM's Rural Institute for Inclusive Communities, hosted the visit in partnership with the Maureen and Mike Mansfield

SB29-16/17: Resolution Amending Section 9.5 of ASUM Fiscal Policy. This resolution passed with unanimous consent at the October 19, 2016 meeting of the Associated Students of the University of Montana (ASUM).

SB28-16/17: Resolution Amending Section 9.6 of ASUM Fiscal Policy. This resolution passed 20Y-1N-0A at the October 19, 2016 meeting of the Associated Students of the University of Montana (ASUM).

Student newspaper of the University of Montana, Missoula.

Lucy E. Bailey, Oklahoma State University, pursues multiple theoretical frameworks for analyzing her personal collection of family letters.

is an Attorney practicing in Kalispell, Montana.She would like to thank Peter Schlemmer and her former Renewable Energy professor Sam Paneralla for their

Meeting minutes from the October 13, 2016 meeting of the University of Montana faculty.