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Teschner V. Commissioner, 38 T.C. ... No. 101 (1962), Harry A. Haines

Montana Law Review

Teschner v. Commissioner

Co-Creation Of Classroom Language: Supporting Learners To Develop Expressions Of Uncertainty And Statistical Concepts, Jioo Nimkar, Nayanatara Ruth Gautham

The Mathematics Enthusiast

This qualitative study explored how Grade 4 students and the researcher co-created language to express statistical concepts in a multilingual Indian classroom during a week-long instructional sequence. The analysis draws on a situated socio-cultural perspective on language in mathematics. Students’ language progressed from simple claims to more refined and sophisticated expressions, enabling them to discuss central tendencies and distributions while acknowledging uncertainty. Co-created phrases such as āsapās reshā (the nearby line) not only supported communication but also deepened reasoning. The article also discusses what role scaffolding played in refining these informal resources.

Evolution Of Pre-Service Teacher Beliefs Along The Traditional-Reform Continuum: An Examination Of Normative Beliefs And Discursive Claims A Decade Later, Mary Elizabeth Riley Lloyd, Adam Jordan

The Mathematics Enthusiast

This study replicated an earlier study that explored the beliefs of a sample of K-8 pre-service teachers (PSTs). Results of this study position PSTs’ beliefs along a traditional-reform continuum. The positioning from the current study is compared to that of the former study, to determine the level of progress made toward the National Council of Teachers of Mathematics’ commitment to high-quality instruction. To position beliefs, PSTs completed a Likert survey, including open-ended questions, three times throughout their teacher preparation program (TPP): before taking any of their mathematics education courses, after taking both required mathematics education courses, and after student teaching. …

Multimodal Interpretation Of Notation, Dave Hewitt, Jenni Ingram

The Mathematics Enthusiast

Challenges related to the teaching and learning of formal notation in school mathematics are widely documented, and specifically in relation to the underlying mathematical structures that the notation is intended to convey. In this article, we draw on embodied cognition to examine the interactions among three students working with the software Grid Algebra. Embodied cognition emphasises the role of gesture and movement in learning and understanding mathematics. Grid Algebra uses movement to direct students’ attention to mathematical operations on numbers and numerical expressions within the grid and the structure of these operations, while the software takes care of the …

Linguistic And Conceptual Transfer: Crosslinguistic Translanguaging In The English-French Bilingual Mathematics Classroom In The Uk, Jo Skelton, Jane Spiro

The Mathematics Enthusiast

This paper presents the findings of a study, conducted as part of a PhD thesis, exploring ways in which translanguaging can be used to support the development of conceptual understanding of division. The study focused on 18 8-9-year-old pupils learning mathematics through the instructional languages of English and French in an ‘immersion-style’ bilingual education context in England. The data was collected from four consecutive mathematics lessons focusing on division, the first two taught through English by a monolingual English teacher and the second two taught in French by a bilingual French teacher. Both teachers were interviewed directly following their second …

Pre-Service Teachers’ Noticing Of The Function Of Mathematical Naming In The Teaching Of Fractions, Juan Manuel González-Forte, Núria Planas

The Mathematics Enthusiast

In this paper, we present the rationale and results of a study conducted with 79 pre-service primary-school teachers on their sensitivities to notice the pedagogic function of naming practices in the language of teaching fractions. Our results are a contribution to noticing research on mathematics teacher education and language. Although participants were not familiar with discussions of the language of teaching mathematics in general and the language of teaching fractions in particular, they engaged in an intervention with a professional task that provided opportunities to notice potential relationships between mathematical naming and the conceptual understanding of fractions. The written answers …

From Words To Worlds: Concluding Reflections And Emerging Directions, Jenni Ingram, Núria Planas, David Pimm

The Mathematics Enthusiast

In concluding the second series of eight articles in the double special issue on ‘Engaging with communication and language in relation to mathematics and its education’, we take the opportunity to reflect across the articles to prompt us all to consider potential and emerging directions for future research. This final article is overall a commentary inspired by the communication-centred and language-centred contributions to mathematics education across the entire collection of articles, that is, all those comments that situate mathematics and its education as language and communication praxis. We explore this relationship between mathematics and its education with communication and language …

An Analysis Of The Pedagogical Price Of Mathematical Tricks: The Case Of Factoring, John Beam, Jason Belnap, Eric Kuennen, Amy Parrott, Jennifer E. Szydlik

The Mathematics Enthusiast

Students are exposed to a variety of mathematical tricks that ostensibly make it easier for them to solve problems. But tricks also undermine their understanding that mathematics is supposed to make sense; they rob students of opportunities to explore and struggle with ideas; and they obscure mathematical structure.

Affordances For Graphical Shape Thinking Through Vivid Informal Classroom Talk, Elizabeth Kimber, Cathy Smith

The Mathematics Enthusiast

The shape of a graph arising from the covariation of variables is a crucial idea in reasoning about functions and graphs. Students’ perspectives on graph shape may foreground visual features and consider the shape to be a fixed property of the function (static graphical shape thinking). In contrast, students can perceive the graph as a trace of covarying quantities, through which its shape arises (emergent graphical shape thinking). It is of interest to understand how these perspectives are supported and promoted by teachers in classroom discourse. We present findings from a study of classroom discourse in which we used systemic …

From Uncertainty To Dialogue: Visualising Collaborative Noticing And Mathematical Intuitions In Students' Narratives Within Probability Learning, Daniela Diaz-Rojas

The Mathematics Enthusiast

This study explores the critical role of language and communication in fostering collaborative noticing practices that display mathematical intuitions within probability learning. Through an analysis of student interactions, I show how the joint articulation of ideas and the exploration of shared doubt enable intuition, often perceived as an object that is held and a solitary experience, to become a socially shared and co-constructed process. The observed collaborative dialogue concerns shared content and form, constructing a joint narrative architecture that enables common understanding and allows the group to notice, examine, and reshape their intuitions. Collaborative noticing when working with probability problems …

Teachers’ Knowledge And Use Of Mathematics Register For Teaching And Communication, Pauline Tiong, Sheena Tan

The Mathematics Enthusiast

Research has progressively considered a resource-oriented rather than a deficit-based approach towards understanding the role of language in helping students learn mathematical concepts. Notably, there has been a call for greater attention to the notion of mathematics teaching talk, which focuses on teachers’ use of language as a resource to teach mathematics (and the language of mathematics). It is argued that mathematics teaching talk plays an important role in enabling teachers to communicate mathematical ideas or meanings of mathematical concepts and in turn, preparing students to participate better in mathematical discourse practices. Specifically, this paper seeks to exemplify what teachers …

Affectivity As A Form Of Communication During Interactions Between Parent, Child, And Touchcounts, Qiang Lin

The Mathematics Enthusiast

This qualitative study examines communication in family mathematics settings, focusing on how embodied communication is constituted during parent–child interaction with a multitouch technology application, TouchCounts. Moving beyond information–transmission perspectives, the study adopts an embodied and relational approach centered on affectivity—conceptualized as the circulation of attunement, resonance, and intensity of movement and feeling across human and non-human components of the parent–child–TouchCounts assemblage. Drawing on close qualitative analysis of two selected video-recorded excerpts, the study traces how mathematical events unfold moment by moment through bodily action and material engagement, and how affective flow shapes how participants respond to one …

Book Review: Irrationality, Transcendence And The Circle-Squaring Problem: An Annotated Translation Of J. H. Lambert’S Vorläufige Kenntnisse And Mémoire (2nd Ed.), Bharath Sriraman

The Mathematics Enthusiast

This review examines Irrationality, Transcendence and the Circle-Squaring Problem (2nd ed., 2024) by Eduardo Dorrego López and Elías Fuentes Guillén, offering the first complete annotated English translations of Johann Heinrich Lambert’s two foundational texts on the irrationality of π: the semi-popular Vorläufige Kenntnisse (1766/1770) and the rigorous Mémoire (1761/1768). The review situates these translations within Lambert’s remarkable and largely underappreciated life—a self-taught polymath who rose from a tailor’s workshop in Mulhouse to the Berlin Academy of Sciences—and gives careful attention to the proof of irrationality of π via continued fractions, placing it in dialogue with the Niven–Hermite integral approach, the …

Book Review: Mathematical Geography In The Eighteenth Century: Euler, Lagrange And Lambert, Bharath Sriraman

The Mathematics Enthusiast

This review examines Mathematical Geography in the Eighteenth Century: Euler, Lagrange and Lambert, edited by Renzo Caddeo and Athanase Papadopoulos (Springer, 2022), a volume bringing together ten historical and mathematical essays alongside complete English translations of six foundational memoirs on cartography by Euler (four), Lagrange (one), and Lambert (one). The review assesses the volume’s mathematical depth across its core topics: Euler’s system of differential equations establishing the non-existence of perfect maps, the derivation of the Mercator and conformal projections via partial differential equations, Lagrange’s introduction of the Cauchy–Riemann equations and holomorphic functions into cartographic theory, and Lambert’s construction of …

Bee Positive Missoula, Sophia Mahoney, Mckenna Hill, Olivia Cornwell, Lucas Hardoerfer, Brandon Quamme

Undergraduate Theses, Professional Papers, and Capstone Artifacts

Bees are an essential species that support the growth of plant communities and enhance human agricultural success. Urbanization is associated with increased impermeable surfaces, large-scale land-use change, and changes in plant and animal community assemblages. Urban expansion exacerbates habitat fragmentation and loss, which has a particularly negative impact on wild or native specialist bee pollinators and the plant species they pollinate. Global and local research indicate the widespread nature of these negative trends; however, there are abundant opportunities to mitigate the impacts on bee pollinators by restoring wild bee habitat in urban areas. Habitat includes nesting and floral resources, and …

Preview; Johnson V. Spire Home Inspection: A Debate On Judicial Resources Through Manufactured Finality, Madison Blank-Plautz

Montana Law Review Online

No abstract provided.

Montana Kaimin, September 10, 2026, Students Of The University Of Montana, Missoula

Montana Kaimin, 1898-present

Student newspaper of the University of Montana, Missoula.

Montana Kaimin, September 3, 2026, Students Of The University Of Montana, Missoula

Montana Kaimin, 1898-present

Student newspaper of the University of Montana, Missoula.

Stat 216.51: Introduction To Statistics, Blake P. Miller

University of Montana Course Syllabi, 2026-2030

No abstract provided.

Revisiting The Secretary Problem, Steven J. Brams, D. Marc Kilgour

The Mathematics Enthusiast

The venerable Secretary Problem asks how a decision maker (DM) should select one of n candidates, who come up randomly, when the only information available is each candidate’s strict rank within the set of previous applicants. Moreover, DM may select only the current candidate. Our illustrative case has n = 9 candidates, for which the Standard Method is to reject outright the first 3 candidates and then choose the first of the 4th through 8th candidates who is better than all of the first three, or the 9th candidate if none of them is. We compare the Standard Method with …

The Jefferson-D’Hondt Method And Majorization, Stanley Chang

The Mathematics Enthusiast

This article investigates the interplay between apportionment functions and order-theoretic structures on population and allocation vectors. We define a class of functions, called apportionment maps, that respect population monotonicity, house monotonicity, and size monotonicity. A central focus is placed on the preservation or violation of majorization under these functions, with particular attention to the Jefferson method and related divisor schemes. By applying tools from convex analysis and the theory of Schur-convex functions, we characterize the extent to which fairness criteria are preserved under apportionment.

Ranked Choice Voting And Condorcet Failure In The Alaska 2022 Special Election: How Might Other Voting Systems Compare?, Jeanne Clelland

The Mathematics Enthusiast

The August 2022 special election for the U.S. House of Representatives in Alaska featured three official candidates and was conducted by the single-winner ranked choice voting system known as “Instant Runoff Voting.” The results of this election displayed a well-known but relatively rare phenomenon known as “Condorcet failure:” Nick Begich was eliminated in the first round despite being more broadly acceptable to the electorate than either of the other two candidates. More specifically, Begich was the Condorcet winner of this election: Based on the Cast Vote Record, he would have defeated each of the other two candidates in head-to-head contests, …

The Topology Of Simple Games, Leah Valentiner, Ismar Volić

The Mathematics Enthusiast

We initiate the study of simple games from the point of view of combinatorial topology. The starting premise is that the losing coalitions of a simple game can be identified with a simplicial complex. Various topological constructions and results from the theory of simplicial complexes then carry over to the setting of simple games. Examples are cone, join, and the Alexander dual which have interpretations as familiar game-theoretic notions and objects such as the arithmetic of games and dual games. We also provide some new topological results about simple games, most notably in applications of homology of simplicial complexes to …

Redistributing Votes When Candidates Drop Out Of An Election, Stanley R. Huddy, Michael A. Jones, Jennifer M. Wilson

The Mathematics Enthusiast

The U.S. Presidential Primary is a series of statewide primaries and caucuses used by the Democratic and Republican parties to winnow a pool of candidates to arrive at their nominees. As unsuccessful candidates leave the race, there is a question as to how voters who planned to vote for the candidate reassign their votes in the upcoming primaries/caucuses. When a profile of voters’ preferences is independent of elimination under plurality, then voters whose first-place candidate has dropped out of the race have their votes distributed to the remaining candidates in the same proportion of the remaining candidates’ first-place votes. For …

Instructive Misadventures When Calculating Election-Related Probabilities, David Mccune

The Mathematics Enthusiast

This article describes an interesting error I committed in the course of my research in voting theory. While embarrassing, the error has led to productive discussions about mathematical failure with my students in my probability and statistics course. I describe my mistake and discuss how the incorporation of such mistakes into the classroom can enhance student trust and engagement.

States Of Disarray: Cleaning Data For Gerrymandering Analysis, Ananya Agarwal, Fnu Alusi, Arbie Hsu, Arif Syraj, Ellen Veomett

The Mathematics Enthusiast

The mathematics of redistricting is an area of study that has exploded in recent years. In particular, many different research groups and expert witnesses in court cases have used outlier analysis to argue that a proposed map is a gerrymander. This outlier analysis relies on having an ensemble of potential redistricting maps against which the proposed map is compared. Arguably the most widelyaccepted method of creating such an ensemble is to use a Markov Chain Monte Carlo (MCMC) process. This process requires that various pieces of data be gathered, cleaned, and coalesced into a single file that can be used …

The Irrationality Of The California Gubernatorial Recall System, Wesley H. Hollday, Eric Pacuit

The Mathematics Enthusiast

We analyze California’s gubernatorial recall system using basic ideas from the mathematical theory of social choice. We expose some pathologies of the system, use computer simulations to explore how often these pathologies might occur, investigate the effects of strategic voting on the analysis, and suggest changes to the system that cure the pathologies.

Stat 216.50: Introduction To Statistics, Scott Davis

University of Montana Course Syllabi, 2026-2030

No abstract provided.

Drone Remote Sensing For Forestry: An Economic Assessment, Micah G. Scudder

University Grant Program Reports

This project addressed a critical knowledge gap in modern forest management: the lack of a standardized, rigorous economic framework to evaluate the true financial viability of Unmanned Aerial Systems (UAS/drones). Supported by this seed grant, an economic evaluation protocol rooted in engineering economics was developed and translated into a functional financial template tool. When external structural funding changes at a partner flight agency restricted upfront fixed-wing hardware access, the research design was pivoted from a single-site case study at Lubrecht Experimental Forest to a multi-project empirical validation. The framework was applied across nine distinct drone remote sensing projects managed by …

2026 Annual Report – Celebrating 40 Years Of Research Excellence, Melissa Weddell

Institute for Tourism and Recreation Research Publications

This report marks the 40th anniversary of the Institute for Tourism and Recreation Research (ITRR) at the University of Montana, founded in 1987 as the research arm of the state's tourism and recreation industries and funded through a small share of the Lodging Facility Use Tax.

It covers the 2025 travel year, when 13.2 million nonresident visitors spent $5.6 billion in Montana, supporting 52,610 jobs and $360 million in state and local tax revenue. Featured studies cover the state's museums and paleontological sites, its $1.27 billion outfitting and guiding industry, and tourism's growing role in Eastern Montana's rural economies. Additional …