Mathematical Inquiry Research Articles

Education

Long ago, I forgave Arthur Conan Doyle for making the greatest literary detective of all time a chemist and his arch villain a mathematician, but there was a logic behind it. After all, Sherlock Holmes was a man of action and intellect. He collected data before he interpreted it. His nemesis, Moriarty, is characterized as an organizer, acting by proxy and letting others be his eyes and ears. Doyle's fiction was a reflection of his time, but times have changed. There was a time when gathering data often required special skills, equipment, and a great deal of time. Now, inexpensive cameras and sensors are ubiquitous, as is the software to extract and manipulate data, and our two Education section contributions this issue exemplify this component of modern mathematical inquiry. The first is a piece about Bayesian inversion, a category of problems where a well-established model is available, but some internal parameters are unknown. The mathematical challenge is to map a noisy data set to the probability distributions of the parameters so that they best explain the data. The second contribution surveys models for describing angles of repose and free surfaces of granular materials. In both cases, the authors demonstrate the value of their ideas by comparing their mathematical results with simple, inexpensive, self-contained experiments. In “Estimating Parameters in Physical Models through Bayesian Inversion: A Complete Example” authors Moritz Allmaras, Wolfgang Bangerth, Jean Marie Linhart, Javier Polanco, Fang Wang, Kainan Wang, Jennifer Webster, and Sarah Zedler put forth a true team effort and walk the reader through an example of Bayesian inversion from start to finish. The probabilistic model is relatively simple, but as the title states, we face an inverse problem in trying to determine the parameters. The goal is to determine a probability density of the parameters given a set of measurements with known or assumed probability densities. In this case, the authors walk us through an example using data extracted from video footage of a falling object to determine the gravitational acceleration and the object's viscous drag coefficient. The inversion for a small number of parameters can be tackled directly with numerical integration, but as the authors note, larger numbers of parameters curse us with a greater number of dimensions, a topic beyond the scope of this introductory communication. Like any good detective story, things are not as they seem. The authors find that either their assumptions are a bit off or gravity is weaker in College Station than in other towns in Texas (or anywhere else in the world). To resolve the discrepancy, the authors propose an additional measurement error that produces a more believable set of parameter distributions, highlighting one of the main points of the article: No amount of data will help unless we carefully choose where and how we take measurements. In the second Education section offering, authors Rachael Gordon-Wright and Pierre A. Gremaud explore free surfaces of granular materials. Unlike motions of simple gases and liquids which in most circumstances can be explained adequately by the Navier--Stokes equations, continuum models of granular materials pose a greater challenge. As the authors suggest, a mountain of potatoes may have a different shape than a pile of sugar grains. In their article “Granular Free Surfaces,” they limit the discussion to steady free surfaces, a rich topic because unlike a fluid, granular materials can sustain a shear stress while at rest and so there are many possible free surfaces. For these problems, the relevant mathematical structure is not a differential equation but rather a differential inclusion. For linear slopes, a direct calculation will reveal the angle of repose if one knows a few internal parameters. Next, the authors put a “spin” on the problem by exploring free surfaces in rotating cylinders. Finally, they compare their mathematical results with experimental observation and in this case find qualitative agreement. While we feature two very different contributions in this issue, they share some important features beyond the use of home-brew data collection. Both articles would fit well with an upper division undergraduate or introductory graduate modeling course. Both hint at the challenges posed to mathematicians when theoretical predictions provide qualitative but not quantitative agreement. Finally, like all great detective stories, both articles describe a great problem, round up the usual theoretical suspects, and bring us to a satisfying resolution.

Artifacts as sources for problem-posing activities

The problem-posing process represents one of the forms of authentic mathematical inquiry which, if suitably implemented in classroom activities, could move well beyond the limitations of word problems, at least as they are typically utilized. The two exploratory studies presented sought to investigate the impact of problem-posing activities when they are implemented in meaningful situations involving the use of cultural artifacts, with its associated mathematics, and particular teaching methods. These situations fall under those defined by Stoyanova and Ellerton (1996) as semi-structured situations. Furthermore, these studies investigated the potential that problem-posing activities have for identifying and stimulating critical and creative thinking in mathematics.

Thinking outside the Cube

Mathematical Lens uses photographs as a springboard for mathematical inquiry and appears in every issue of Mathematics Teacher. all submissions should be sent to the department editors. For more background information on Mathematical Lens and guidelines for submitting a photograph and questions, please visit http://www.nctm.org/publications/content.aspx?id=10440#lens.

The Pragmatic Maxim and the Idea of Necessity in Mathematical Inquiry

Abstract In this paper we examine the pragmatic meaning of mathematical necessity. While Peirce writes extensively on the pragmatic meaning of a "would be", he does not provide a developed account of the pragmatic meaning of a mathematical "must be". After examining several possibilities such as information relative accounts, conceivability, the ideal world, and possible worlds accounts, we consider mathematical necessity in terms of his overall pragmatism whereby metaphysical notions are understood in terms of their pragmatic effects, and, in particular, their role in inquiry. We argue that mathematical necessity should be considered a regulative hypothesis. Peirce often addresses questions of modality, such as whether to posit real possibility or physical necessity, by appealing to what is good for inquiry. Just as the hypothesis of the real world constrains and directs inquiry in science, positing the reality of mathematical possibility (or what Peirce sometimes calls " substantive possibility") can also be seen to constrain and direct mathematical inquiry as well as natural science. The necessity of mathematics may, in the end, be stipulated in the form of a hypothesis, just as the reality of the natural world functions as a hypothesis in Peirce's system. The effect of positing this ideal world of substantive possibility lies both in its effects within mathematical inquiry, and ultimately within the natural sciences as mathematical conclusions constrain the hypotheses we see fit to entertain.

Book Review: Adding depth to portraits of mathematical inquiry Karin Brodie (2010) Teaching mathematical reasoning in secondary school classrooms

Book Review: Adding depth to portraits of mathematical inquiry Karin Brodie (2010) Teaching mathematical reasoning in secondary school classrooms

Preservice Teachers’ Changing Conceptions About Teaching Mathematics in Urban Elementary Classrooms

In this article, the authors report changes in mathematical disposition, participation, and competencies within a group of Pasifika students as a teacher established the discourse of mathematical inquiry and argumentation. Within a classroom-based design approach, the teacher used a communication and participation framework tool to support students to engage in a range of collective mathematical practices. Drawing on analyses of student interviews conducted over one school year, the authors provide a narrative that illustrates how changes in agency and accountability accompanied shifts in the mathematical inquiry discourse. The results show positive learning outcomes for Pasifika students when the general and mathematical obligations attend to the cultural, social, and mathematical well being of all students in mathematics classrooms.

Forging Mathematical Relationships in Inquiry-Based Classrooms With Pasifika Students

In this article, the authors report changes in mathematical disposition, participation, and competencies within a group of Pasifika students as a teacher established the discourse of mathematical inquiry and argumentation. Within a classroom-based design approach, the teacher used a communication and participation framework tool to support students to engage in a range of collective mathematical practices. Drawing on analyses of student interviews conducted over one school year, the authors provide a narrative that illustrates how changes in agency and accountability accompanied shifts in the mathematical inquiry discourse. The results show positive learning outcomes for Pasifika students when the general and mathematical obligations attend to the cultural, social, and mathematical well being of all students in mathematics classrooms.

Mathematical Lens: Pumped Up about Mathematics

“Mathematical Lens” uses photographs as a springboard for mathematical inquiry and appears in every issue of the Mathematics Teacher. All submissions should be sent to the department editors. For more background information on “Mathematical Lens” and guidelines for submitting a photograph and questions, please visit http://www.nctm.org/publications/content.aspx?id=10440#lens.

Mathematical Lens: The Spider–Trickster Does Mathematics

“Mathematical Lens” uses photographs as a springboard for mathematical inquiry and appears in every issue of the Mathematics Teacher. All submissions should be sent to the department editors. For more background information on “Mathematical Lens” and guidelines for submitting a photograph and questions, please visit http://www.nctm.org/publications/content.aspx?id=10440#lens.

The emergence of mathematical structures

We present epistemological ruptures that have occurred in mathematical history and in the transformation of using technology in mathematics education in the twenty-first century. We describe how such changes establish a new form of digital semiotics that challenges learning paradigms and mathematical inquiry for learners today. We focus on drawing analogies between the emergence of non-Euclidean geometry with recent advances in technological environments that are dynamic and interactive both visually and haptically. This analysis yields a new digital semiotic theory.

Mathematical Inquiry and Its Educational Values from a Lakatosian Fallibilist Perspective : Focusing on Proof and Proving

Mathematical Inquiry and Its Educational Values from a Lakatosian Fallibilist Perspective : Focusing on Proof and Proving

ECONOMIC APPLICATIONS OF FUZZY SUBSET THEORY AND FUZZY LOGIC: A BRIEF SURVEY

Traditional mathematics is the language of precision. Statements are either true or false. But, in reality there are few things that are truly simply true or false. Life is full of shades of grey. Capturing these shades of grey has been problematic using traditional mathematics. Mathematics can model the uncertainty itself but rarely incorporates it into the model. However, an emerging area of mathematical inquiry known as the mathematics of uncertainty seeks to overcome some of these problems by integrating the uncertainly, the shades of grey, directly into the model. Fuzzy theory is one of those emerging ideas. This paper is simply an overview of some of the basic mechanics of fuzzy theory and recent economic applications in both micro and macro-economics.

Understanding Peirce’s Mathematical Inquiry as a Practice: Some Ontological Implication

Abstract One of Peirce's most important and influential contributions to pragmatism is his reconceptualization of traditional metaphysical ideas like reality in terms of human inquiry. Whether we consider his definition of reality as the external permanence which gives science its selfcorrective, or his definition of reality as the object of the final opinion in the long run, Peirce clearly intends to connect his idea of reality to the practice of inquiry. In this paper, we will consider some of the difficulties of applying the same pragmatic argument when it comes to the reality of mathematics. First, there is the difficulty of discerning what Peirce's position is regarding the reality of mathematics, since there are several disparities in his writings. Second, there is the difficulty of which view regarding the reality of mathematics serves inquiry best. In other words, must a mathematician suppose he studies real entities in order to practice mathematics? In considering this problem, we will examine the Quine-Putnam Indispensability thesis, as well as Angus Kerr-Lawson's bicategorial view. We will conclude that Peirce's philosophy of mathematics only apparently commits him to a bicategorial ontological dualism, and that is more monistic view-which is more agreeable to the Indispensability Thesis-is ultimately preferable.

Cas-enabled technologies as ‘agents provocateurs’ in teaching and learning mathematical modelling in secondary school classrooms

This paper draws on a one year study of three secondary school classrooms to examine the nature of student-student-technology interaction when working in partnership with computer algebra systems (CAS) on mathematical modelling tasks and the classroom affordances and constraints that influence such interaction. The analysis of these data indicates that CAS enabled technologies have a role to play as provocateurs of productive student-student-teacher interaction in both small group and whole class settings. Our research indicates that technologies that incorporate CAS capabilities have the potential to mediate collaborative approaches to mathematical enquiry within life-related mathematical tasks.

Changing roles and identities in the construction of a community of mathematical inquiry

In the current shifts in mathematics classrooms, teachers are challenged to use effective pedagogy to develop inquiry communities in which all participants are offered opportunities to engage in the reasoning discourse of proficient mathematical practices. The challenge for teachers is to know how to implement these pedagogical changes. This article outlines how a teacher and researcher worked together in a collaborative partnership using a purposely designed communication and participation framework as a tool to scaffold development of collective reasoning in the inquiry community. The findings illustrate the interconnections between the teacher’s beliefs, past experiences, and current, and future expectations for her diverse students. Explanations are provided of how different, often conflicting, voices emerged including one that drew on the teacher’s cultural knowledge. This provided many learning opportunities for the researcher as the teacher developed her students’ voices in culturally appropriate ways.

Peirce’s Philosophy of Mathematical Education: Fostering Reasoning Abilities for Mathematical Inquiry

I articulate Charles S. Peirce’s philosophy of mathematical education as related to his conception of mathematics, the nature of its method of inquiry, and especially, the reasoning abilities required for mathematical inquiry. The main thesis is that Peirce’s philosophy of mathematical education primarily aims at fostering the development of the students’ semeiotic abilities of imagination, concentration, and generalization required for conducting mathematical inquiry by way of experimentation upon diagrams. This involves an emphasis on the relation between theory and practice and between mathematics and other fields including the arts and sciences. For achieving its goals, the article is divided in three sections. First, I expound Peirce’s philosophical account of mathematical reasoning. Second, I illustrate this account by way of a geometrical example, placing special emphasis on its relation to mathematical education. Finally, I put forth some Peircean philosophical principles for mathematical education.

Zen and the art of neriage: Facilitating consensus building in mathematics inquiry lessons through lesson study

One danger of integrating inquiry-based problem-solving activities into mathematics lessons is that different strategies could be accepted without in-depth discussions on the cogency and efficiency of the strategies. To overcome this issue, Japanese teachers typically go through a series of lesson-study-based teacher learning sessions and learn how to help students build consensus on the best mathematical strategy and think deeply about problem solving (neriage in Japanese). Assuming that this can also be an effective model in other cultural contexts, a video-based lesson study was conducted for a group of US teachers to effectively incorporate consensus building discussions in their mathematical inquiry lessons. Through the lesson study, the teachers learned to release control of class discussions to their students and help them discuss and examine different strategies. This article concludes with various aspects that the teachers learned for effectively implementing neriage in their lessons.

Assessing a Mathematical Inquiry Course: Do Students Gain an Appreciation for Mathematics?

This study evaluates changes in students' performance and appreciation for mathematics as the result of taking a required general education course designed for calculus-ready students. The purpose of the study is to determine whether the goals of this rigorous general education course are being met. Changes in students' performance are measured by comparing grades on a pre-course and a post-course test. Changes in students' appreciation of mathematics are measured by scores on a pre-course and post-course survey, including open-ended questions concerning beliefs about mathematics. The study reveals a significant increase in grades on the post-course test, indicating gains in students' performance. It also reveals a significant increase in scores concerning creativeness, indicating a greater appreciation for the creativity involved in doing mathematics. The study reveals a significant decrease in scores on questions concerning attitudes, indicating a decrease in students' feelings about mathematics as it relates to them personally. The open-ended responses show an increase in students' appreciation for proofs and for the role of mathematics in the world. This report concludes with a discussion of some of the perceived benefits and challenges of including a rigorous, non-calculus mathematics course in the general education requirements for calculus-ready students.

Teachers’ perceptions regarding the importance of mathematical inquiry activities for infants and toddlers and the efficacy of current practices

Teachers’ perceptions regarding the importance of mathematical inquiry activities for infants and toddlers and the efficacy of current practices

Boundaries and roles: Positioning and social location in the Virtual Math Teams (VMT) online community

As research in Computer-Supported Collaborative Learning (CSCL) expands its understanding of joint knowledge building, new perspectives on how social reality is constructed become necessary. Our research concentrates on the longitudinal or diachronic trajectories of Virtual Math Teams (VMT) at The Math Forum, an online community supporting mathematical inquiry. We investigate how these virtual teams constitute themselves while engaged in building collaborative knowledge. We describe as well, through the lens of positioning theory, the interactional activities that allow participants to situate themselves, others, and their collective knowledge resources over time. Our analysis suggests that positioning work is central to constructing a sustained problem space and illustrates how the configurations of positions and resources that co-participants put forward through interaction might change across a team’s trajectory. These changes constitute and are sensitive to the participants’ evolving reasoning routines and other forms of joint participation uniquely related knowledge building. In addition, we show that the VMT activity system affords participants a level of disciplinary engagement which is partly illustrated by active engagement in positioning work. Finally, we suggest that an interactional approach to roles and participation provides a fruitful framework for researchers, designers, and practitioners interested in understanding and creating engaging CSCL interactions.