The 50 most-cited articles in The Mathematics Enthusiast, 2018

Mathematics has a history of being a two-dimensional inscribing practice. We describe the potential evolution in doing, thinking, and learning mathematics with the emergence of a technological innovation that enables real-time 3D virtual and material interactions. Using the 3D drawing pen as a simple and recently available technology, we highlight how it helps re-think long-standing assumptions and dichotomies in mathematics education including, for example, the material distinction between diagram and manipulative, the semiotic distinction between icon and index, and the developmental progression of action-icon-symbol. We then speculate on the future possibilities of the shift in technological infrastructure that 3D pens and similar technology may give rise to.

Julie Kessel has 10 years of experience working with children as a preschool teacher for both the Head Start and Bright Horizon organizations.Additionally, Julie has served children and families as a Parent's as Teachers instructor for three years through the Head Start home visiting program.

In this article, we explore affordances of adopting the framework of praxeology by Chevallard in the analysis of mathematics classroom communication in relation to the communication in a textbook. While adopting praxeology, we carried out detailed analysis of communication in both classroom data and textbooks. The construed praxeologies describe the organisation of knowledge expressed for the same type of task in both classroom and textbook. The praxeologies were compared, with specific attention to the teacher’s practice. This analysis illuminates how teachers’ practices, realised in classroom communication, may be compared to other texts describing the same topics, with a focus on procedures, explanations, theoretical aspects, et cetera. Hence, praxeology as a framework enabled an analytical structuring of classroom and textbook communication, and consequently a systematic comparison. In other studies about the use of mathematics textbooks the teaching frequently is categorised as regulated by the textbook, and in this article, we problematize this. The teaching practice was, in…

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In the early 1960s, Ursula Le Guin wrote ‘The Masters’, a short novel that offers a sharp contrast to the ‘maths for all’ discourse of contemporary mathematics education reforms. Le Guin writes of a world – Edun – where ‘mathematical prohibition’ is law. Mathematical reason is banned for all people by the Priests of Edun, and failure to obey is punishable by death. Despite the threat of this totalitarian anti-math regime, some citizens create a collective heterotopia in which they practice mathematics in secret. Le Guin’s story is an opportunity to conduct a thought experiment: ‘what if maths became forbidden?’ This ‘what if’ experiment (Haraway 2016) allows us to consider how statements such as ‘maths for all’ or ‘no to maths’ are grounded in rationalisations that construe mathematical subjectivity as a determined actor for citizen agency in contemporary societies. The paper suggests that we need to move beyond a ‘maths for all’ or ‘no to maths’ dichotomy by interrogating how they both operate as ‘states of exception’ around politics of fear producing in/exclusions.

The notion of function is considered one of the most difficult parts of the common lower secondary curriculum. In this paper we discuss the potential role played by linear functions, invariably used as first examples of this new notion. As empirical basis we use a praxeological analysis of the function chapter in four Indonesian lower secondary textbooks. The main point of our analysis is that the class of functions of type 𝑓(𝑥)=𝑎𝑥 (where a is some given number) does not appear explicitly at the level of theory, neither in the sense of being given a name, or in the sense that properties of the class is studied. We discuss the implications of this for students’ learning of the more general (theoretical) notion of function.

I propose a new construct, teacher noticing of students’ prior knowledge, and provide examples of a professional development intervention with the goal of supporting teacher learning. The examples demonstrate that combining discussions of animations of classroom instruction, video clubs, and lesson study can help teachers to attend to students’ prior knowledge and anticipate actions that use that knowledge for promoting mathematical understanding. I discuss challenges and new questions for mathematics education researchers brought about by the new construct. I suggest that the construct is valuable for developing teachers’ professional knowledge.

This paper features an anecdotic narration in which a fictive narrator, addressing the author of this paper, explains the functional role of mathematics education in the maintenance of the bureaucratic society. Drawing from critical philosophy and critical mathematics education, it is argued that the objective of mathematics education as realised in schools is to teach students to understand, accept, follow or at least ignore pre-defined rules as they are required for the bureaucratic administration of modern society. The theory of the narrator is underpinned with explanations that students offered in an interview study conducted by the author. Rather than supporting his alternative theory on a strict methodical basis, the narrator illustrates its explanatory strength in its confrontation with the experiences of students. The ‘provocation’ does not only lie with the narrator’s theory about the nature of mathematics education but with the fact that reality, as it is presented by the students, fits in all too well with the narrator’s explanations.

This article undertakes an ideological critique of mathematics education from a capitalist perspective. By replacing ‘society’ with ‘education’ and ‘the figure of the Jew’ with ‘mathematics’ in quotations from philosopher Slavoj Žižek, we characterize mathematics as the symptom of educational ideology. From such substitutions, we get statements like: Education does not exist and Mathematics is its symptom. In order to explore the kernel of truth in these statements, we introduce two concepts: identity-quilted-speech, to specify the so-called certainty of twentieth century mathematics (M20), and qualified-labor-power, to characterize the commodity that results from school production. Through the development of these concepts, we show how M20 actualizes Kant’s radical ethics. We indicate the need to consider the mathematics classroom from the sociological perspective of jouissance. We present three instances of the inescapable production of meaning imposed on us by what we call the juggernaut of capitalist society. This inescapable production leaves us no apparent alternative but to…

In this paper we consider motion of an object in a plane to provide a mechanical interpretation of Euler's formula. We believe that this approach contributes to a deeper and more intuitive understanding of Euler's formula and can especially be useful for students learning it and encountering complex numbers for the first time. Euler's formula can be introduced in many ways and using various approaches, but not all of these approaches help develop an intuition for where it comes from and why it works. We believe that the emphasis on physical interpretations and connections to motion can contribute to a more natural introduction and easier understanding of this fascinating formula. In this paper, we describe and consider motion in a plane, using it to give a detailed explanation of Euler's formula. Also, our paper points to the need for an integrated approach to teaching mathematics and physics. In our opinion the interpretation of mathematical results based on the physical phenomena and processes has an important methodological and motivational role in the process of learning. We…

Since its independence, a country desiring to overcome its poverty and misery aimed at producing the competent citizens capable of pursuing glory and boosting the economy. This country wanted to escape from ignorance, wanted to be wiser, wanted to be wealthier. Dr. Chile was looking for the key to unlock the path to success and growth. Experiments were conducted to fine-tune the features of the MINDNIAC: the creature created for achieving economic productivity and prosperity. The idea of more mathematics for greater scientific skills opened the path for believing that advanced mathematics would potentially create the perfect human for society. Dr. Chile was ready to advertise this product as a program of perfecting humans. The MINDNIAC rose as the redeemer of its nation.

This article sheds light on a number of discursive conditions relating to being researchers in mathematics education and with an interest in diversity. The data derived from a self-reflective trialogue (dialogue of three people) between the three authors, three researchers. Two of Foucault’s governing technologies were adopted: technologies of power and technologies of the self. By exploring regularities between these in our trialogue we construed formations of governing technologies in relation to subjectification and subjectivation. We uncovered five formations: “Tensions between mathematics education (ME) researchers from different traditions through processes of normalization and othering”, “Limiting space between ME researchers within the sociopolitical through dismissal of knowledge”, “The socio-political tradition of a need for theory connects theory and ME researcher's’ self-cultivation”, “The researchers’ processes of self-cultivation connect theory and compassionate research practices”. and “Research on policy statements as resistance towards technologies of domination in…

The contribution introduces the concept of epistemological violence from critical psychology into the discourse of mathematics education research. The concept is specific to violence produced through ‘knowledge’. It addresses the negative impact of research on the Other – the group being studied as distinct from the ones studying. It holds the possibility to link research ethics and the idea of scientific correctness to each other, by focussing on the relationship between theoretical propositions about the Other and practices of data interpretation in empirical research products. An exemplary study in the Journal of Mathematics Teacher Education illuminates how easily epistemological violence is (re-)produced in the dissemination of research results when it goes unreflected. Finally, the scope and limits of this concept in mathematics education research are discussed.

The mathematical continuum has a number of formulations and technical definitions. Two of these reference the geometric line and the real number system. This conceptual coupling of line and number has been an enduring source for mathematical invention and paradox. The continuum captures the monstrous desire of mathematics, a desire to re-assemble the point with the line, the discrete with the continuous, the finite with the infinite. This paper explores how the continuum is a source of fundamental ambiguity fueling our desires and fears about mathematics.

After a short survey of the genesis of the ideas behind “functional reasoning” in mathematics education at the beginning of the 20th century, the development and implementation of these ideas in both parts of Germany and in Hungary in the second half of the 20th century are discussed and compared. The ideas of Felix Klein had a strong influence in both countries. However, from the 1950’s to the 1970’s (the period of “New Math”) an attempt was made to teach functions and mappings in a more abstract sense, involving both elementary (real-valued) functions and geometric transformations. In the 1970’s and 1980’s there were lots of discussions about the notion and notation of functions. Finally, dynamic aspects of functions – one of the main ideas of the reforms at the beginning of the 20th century – came into focus again in the 1980’s and remained essential to this day.

I am keen to offer as little explanation as possible for the above work. I can say that it is intended to be a meditation on mathematics inside and outside school and a critique of a) mathematics, b) mathematics education, c) mathematics educational research, and, d) critiques of mathematics education and mathematics educational research. The work also critique’s itself and then critiques that critique (maybe). It is deliberately contradictory.

Now, what do you make of that? We want to discuss mathematics education, and research in mathematics education (we don’t really need to specify, hence the parenthesis). And we want to address teaching and learning (mathematics, but we skipped the parenthesis this time around because we used the word just before) in terms of not-being. You will notice, however, the provocative presence of “liberated”, a strong verb which alludes to the idea of freedom while acknowledging certain constraints: captivity, dependence, and even liability. Its selection was a bold choice. But having made such a suggestive statement (without actually saying things upfront), we hastily print a colon to bring in a potential alternative – more positive, one should hope. This alternative mentions to be about going “towards” something. If we care enough to read the next parenthesis (yep, another set!), we realize that this something does not exist yet, since it lies in the future. This something we finally name with the title’s last breath: “doing|mathematics”. For those who haven’t read our recent articles…

The stereotypical mathematical identity story of the ‘mad mathematician’ (consumed by mathematics and socially inept) is widely reported as problematic to understanding what it means to be mathematical, and to engaging students in mathematics. This problem is explored in the improvisational parody ‘Math Therapy’, which occurred during my filming of ‘Performatics’(https://vimeo.com/147449932); an Arts Based Research (ABR), using filmed-drama. ‘Performed’ and informed by the lived experiences of three post graduate students studying courses with mathematical content, their mathematical experiences are juxtaposed with those of a socially imagined ‘mad mathematician’ in a fictional, generic therapy context entitled ‘Math Therapy’. The ABR facilitated a performance of embodied ways of imagining and performing mathematical identity; subverting, making ridiculous, and disordering the stereotypical ‘mad mathematician’ identity, so as to critically question and challenge its authority. Through the performance, the mathematical identities available to us are also revealed as a parody; between…

In this experimental paper, we document an experience of artistic research, initiated by searching an answer for the question "Is it possible to learn mathematics through dance?". We reflect on three different pathways towards experiencing mathematics in dance: theory, methodology and artistic experience. Following these pathways requires us to challenge established assumptions about mathematics and research and finally allows for exploring the dancing body itself as the potential space of mathematical knowledge. While we remain far from our initial aim of being able to make concrete suggestions about how to integrate dance as an integral part of school mathematics, this article reveals the emancipatory potential of dancing mathematics, facilitating positive experiences with mathematics that hitherto appeared impossible to us.

This article analyses the philosophical underpinnings of mathematical modelling with the focus on the modelling cycles in mainstream mathematics education. Due to its background of epistemological representation, the main focus is set on Kantian philosophy. I argue that representational thinking leads to a problematic view on mathematics and its relation to reality. Finally, I outline an alternative approach to mathematical modelling according to Deleuze’s differential ontology.

There were two major goals of this mentored grant. One was to examine how technology with shared display and multi-user features influences pre-service teachers’ participation pattern and social dynamics in collaborative learning activities. The second goal was to help the Principal Investigator receive mentoring on applying for research grants from external funding agencies. Both goals were successfully achieved at the completion of this grant.

In MEIC v. U.S. Office of Surface Mining, the cost of coal mining’s climate impacts and the agency’s NEPA review obligations are at issue. The United States District Court for the District of Montana found that the Office of Surface Mining and Enforcement failed to adequately consider the need for an EIS and to take a hard look at the indirect, cumulative, and foreseeable impacts of a proposed coal mine expansion in central Montana. In its NEPA analysis, the court concluded that if the benefits of a carbon-intensive project are quantified, then the costs to the climate should be also, using the Social Cost of Carbon Protocol tool. The court held that the Enforcement Office was arbitrary and capricious in its EA of the proposed coal mine expansion and in its consideration of an EIS.

Hydroelectric power is an efficient and clean source of power. In an era when air emissions dominate public concern about the environmental effects of the energy sector, it is a paradox that among the most highly regulated energy projects are hydroelectric dams, which do not combust fuel. This is partly due to a failure of successive statutory enactments,which have transformed hydroelectric licensing from a regulatory “one-stop shop” with a single regulator, to a process chained to a bewilderingnumber of often conflicting regulatory agencies, often riven with delay. Hydroelectric licensing has also failed because its capacious standard of review encourages special-interest groups with even a marginal interest to seek rents from regulated users. Even with the introduction of dispute resolution, and the possibility of obtaining exemptions, the barriers to hydroelectric permitting and relicensing are excessive. Fortunately, a number of reforms would ease the regulatory burden on hydroelectric power expansion, while still considering and addressing environmental concerns.

We construct nine lines in the plane over the integers having the property that for any three general points in the plane over any field, at last one of the constructed lines avoids the three points. We then show that nine lines is the fewest number possible having this property.

Avoiding complex geometric and analytic constructions, this paper considers techniques for visualizing the location of complex roots of quadratic, cubic, and quartic real polynomial functions. This provides teachers and students of mathematics with a better understanding of the nature of these functions and their respective real and complex roots.

Our contribution to this special issue is not intended to offer a theoretical argument in conversation with the papers which form The Disorder on Mathematics Education (Straehler-Pohl, Bohlmann & Pais 2017). It is informed by much of the thinking contained therein and driven by a similar concern with the institutionalisation (and therefore the inevitable co-option and colonisation) of the socio-political dimensions of academic research in mathematics education. However, it is intended to sit alongside as a disorderly and comparatively uninvited guest at the conversation. Rather than advocating a specific set of approaches to the teaching and learning of mathematics for social justice, we are striving after a disorderliness of format to allow the advancement of a (somewhat utopian) imagination and hope, to unsettle ourselves and others and to offer the occasional, penumbrian glimpse of 'the speculative could' (Straehler-Pohl, Pais & Bohlmann 2017, p. 3). We present research fragments collected as part of an activist project without introduction or comment; however, alongside these,…

In this essay I offer some reflections on the field of mathematics education, and particularly the sociopolitical analysis of mathematics education that has emerged in contemporary scholarship. Here I attempt to do two things. First of all I respond to a recent book on “disorder” in mathematics education, identifying some themes and problematics that I find intriguing and generative from my perspective outside the field. Here I reflect on the way that mathematics is positioned in educational discourse generally as a proxy for human capital and general intelligence. Next I relate stories from my life and practice as a primary school teacher in which mathematics, as I understood it, bumped productively against problems in everyday life. Finally, I conclude with a reflection on the productive tension between naïve place-based mathematical understandings and abstract context-bridging mathematical knowledge forms.

Wildland firefighting represents a high risk occupation, and exposes those who engage in it to a significant amount of danger and uncertainty. Decision making and cognitive function take on added importance as firefighters attempt to manage these risks. Canadian researcher Dr. Michael Williams-Bell from Durham College joins the podcast, along with Drs. Chuck Dumke and Charlie Palmer from the University of Montana, to discuss recent research they have completed in cognitive function, and what these findings mean for wildland firefighters.

The downloadable file includes the slides about the Collaborative Language Planning Project that were presented at the Stabilizing Indigenous Language Symposium in June 2018.

Meeting minutes from the December 6, 2018 meeting of the University of Montana faculty.

Agenda and meeting minutes from the December 5, 2018 meeting of the Associated Students of the University of Montana (ASUM). The following resolutions were discussed: SB17-18/19: Resolution Prompting Family Rooms SB18-18/19: Resolution Requesting that Business Services Bill Graduate Students Enrolled in Two Programs Once for the ASUM Fee SB19-18/19: Resolution Regarding Primary Elections

The University of Montana will hold its Spring New Student Orientation Tuesday through Wednesday, Jan. 8-9, for new undergraduates starting in the spring

Agenda and meeting minutes from the November 28, 2018 meeting of the Associated Students of the University of Montana (ASUM). The following resolutions were discussed: SB14-18/19: Resolution Regarding Summer Appointments to Senate SB15-18/19: Resolution Regarding Student Group Resource Center

Student newspaper of the University of Montana, Missoula.

Student newspaper of the University of Montana, Missoula.

Agenda and meeting minutes from the November 14, 2018 meeting of the Associated Students of the University of Montana (ASUM). The following resolutions were discussed: SB12-18/19: Resolution Regarding Seats on the Elections Committee SB13-18/19: Resolution Amending the Student at Large Status

Meeting minutes from the November 8, 2018 meeting of the University of Montana faculty.

Agenda and meeting minutes from the November 7, 2018 meeting of the Associated Students of the University of Montana (ASUM). The following resolution was discussed: SB11-18/19: Resolution Regarding Campaign Reimbursements

Student newspaper of the University of Montana, Missoula.

Campus Map projects and activities, including the internships, we've been able to ensure students get the knowledge, readings and theories they need."Students and employers alike agree that the course has provided positive, real-world experience."The UMEM student interns were absolutely amazing, and I could not have pulled off the Monopoly Ball fundraiser without them," said Kris Holmes, community development

Student newspaper of the University of Montana, Missoula.

Agenda and meeting minutes from the October 31, 2018 meeting of the Associated Students of the University of Montana (ASUM).

Agenda and meeting minutes from the October 24, 2018 meeting of the Associated Students of the University of Montana (ASUM). The following resolution was discussed: SB10-18/19: Resolution Regarding the Open Space Bond and City Stewardship Levy

Student newspaper of the University of Montana, Missoula.

Agenda and meeting minutes from the October 17, 2018 meeting of the Associated Students of the University of Montana (ASUM). The following resolution was discussed: SB09-18/19: Resolution Regarding Administrative Transparency

Student newspaper of the University of Montana, Missoula.

Meeting minutes from the October 11, 2018 meeting of the University of Montana faculty.

Agenda and meeting minutes from the October 10, 2018 meeting of the Associated Students of the University of Montana (ASUM).