Eliminating counterexamples: A Grade 8 student’s learning trajectory for contrapositive proving

This study uses a teaching experiment and retrospective analysis to develop a learning trajectory for improving a Grade 8 student’s ability to construct, critique, and validate contrapositive arguments. The study is predicated on the hypothesis that adolescents perform poorly on contrapositive reasoning tasks because they lack sufficient ways of justifying contrapositive argumentation as a viable mode of argumentation. By studying a student’s actions and comments as she develops, critiques, and validates not-the-conclusion-implies-the-conditions-are-impossible arguments for conditional claims, a promising learning trajectory for contrapositive argumentation is developed. The student’s learning trajectory demonstrates how a conception of contrapositive proving as eliminating counterexamples can be useful in developing, critiquing, and validating contrapositive arguments.     

Validating a concept inventory for measuring students’ probabilistic reasoning: The case of reasoning within the context of a raffle

Assessing students’ conceptions related to independence of events and determining probabilities from a sample space has been the focus of research in probability education for over 40 years. While we know a lot from past studies about predictable ways students may reason with well-known tasks, developing a diagnostic assessment that can be used by teachers to inform instruction demands the use of familiar and unfamiliar contexts. This paper presents the current work of a research team whose aim is to create a formative concept inventory with strong evidence of validity that uses a psychometric model to confidently predict whether a student exhibits one or more misconception across many items. We illustrate this process in this paper using a particular item with a context of a raffle aimed to measure whether a student reasons with misconceptions related to independence or equiprobability. The results of two aspects of the validity process: cognitive interviews to assess response processes on individual items, and a large-scale administration to examine internal structure of the concept inventory revealed difficulties in assessing students’ reasoning about these key probability concepts and trends in the prevalence of misconceptions across grades. Results can provide guidance for others aiming to develop assessments in mathematics education and also support further possibilities for research into understanding students’ reasoning about independence and sample space.     

An exponential lower bound for Cunningham’s rule

In this paper we give an exponential lower bound for Cunningham’s least recently considered (round-robin) rule as applied to parity games, Markov decision processes and linear programs. This improves a recent subexponential bound of Friedmann for this rule on these problems. The round-robin rule fixes a cyclical order of the variables and chooses the next pivot variable starting from the previously chosen variable and proceeding in the given circular order. It is perhaps the simplest example from the class of history-based pivot rules. Our results are based on a new lower bound construction for parity games. Due to the nature of the construction we are also able to obtain an exponential lower bound for the round-robin rule applied to acyclic unique sink orientations of hypercubes (AUSOs). Furthermore these AUSOs are realizable as polytopes. We believe these are the first such results for history based rules for AUSOs, realizable or not. The paper is self-contained and requires no previous knowledge of parity games.     

An axiom system for Lingenberg’s metric-Euclidean planes

We show that Lingenbergs metric-Euclidean planes are the rectangular planes of Karzel and Stanik which satisfy the axiom If two of the perpendicular bisectors of a triangle exist, then so does the third.This paper was written while the author was at the Institute of Mathematics of University of Biaystok with a Fulbright grant. I thank the Polish-U.S. Fulbright Commission for the grant, Professor Krzysztof Pramowski for the hospitality, and Ewa Walecka for drawing the figures.     

An extension of Babbage’s criterion for primality

Let n > 1 and k > 1 be positive integers. We show that if $$\left( {\begin{array}{*{20}c} {n + m} \\ n \\ \end{array} } \right) \equiv 1 (\bmod k)$$ for each integer m with 0 ≤ m ≤ n ? 1, then k is a prime and n is a power of this prime. In particular, this assertion under the hypothesis that n = k implies that n is a prime. This was proved by Babbage, and thus our result may be considered as a generalization of this criterion for primality.     

An analytical comparison of students’ reasoning in the context of Inquiry-Oriented Instruction: The case of span and linear independence

In this report we analyze differences in reasoning about span and linear independence by comparing written work of 126 linear algebra students whose instructors received support to implement a particular inquiry-oriented (IO) instructional approach compared to 129 students whose instructors did not receive that support. Our analysis of students’ responses to open-ended questions indicated that IO students’ concept images of span and linear independence were more aligned with the formal concept definition than the concept images of Non-IO students. Additionally, IO students exhibited more coordinated conceptual understandings and used deductive reasoning at higher rates than Non-IO students. We provide illustrative examples of systematic differences in how students from the two groups reasoned about span and linear independence.     

An analogue of Edmonds’ Branching Theorem for infinite digraphs

We extend Edmonds’ Branching Theorem to locally finite infinite digraphs. As examples of Oxley or Aharoni and Thomassen show, this cannot be done using ordinary arborescences, whose underlying graphs are trees. Instead we introduce the notion of pseudo-arborescences and prove a corresponding packing result. Finally, we verify some tree-like properties for these objects, but give also an example that their underlying graphs do in general not correspond to topological trees in the Freudenthal compactification of the underlying multigraph of the digraph.     

An Analogue of Bochner’s Theorem for Damek-Ricci Spaces

We characterize the image of radial positive measures θ’s on a harmonic NA group S which satisfies ∫ _S ? ₀(x)?dθ(x)<∞ under the spherical transform, where ? ₀ is the elementary spherical function.     

An extension of Fujita’s non-extendability theorem for Grassmannians

In this paper we study smooth complex projective varieties X containing a Grassmannian of lines ${{\mathbb G}(1, r)}$ which appears as the zero locus of a section of a rank two nef vector bundle E. Among other things we prove that the bundle E cannot be ample.     

An Analogue of Beurling’s Theorem for NA Groups

In this paper, we prove Beurling’s theorem for NA groups, from which we derive some other versions of uncertainty principles.     

An Analogue of Nakayama’s Conjecture for Johnson Schemes

Nakayamas Conjecture is one of the most famous theorems for representation theory of symmetric groups. Two general irreducible characters of a symmetric group belong to the same p-block if and only if the p-cores of the young diagrams corresponding to them are the same. The conjecture was first proven in 1947 by Brauer and Robinson. We consider an analogue of Nakayamas Conjecture for Johnson schemes.Received January 28, 2004     

An exponential lower bound for Zadeh’s pivot rule

Mathematical Programming - The question whether the Simplex Algorithm admits an efficient pivot rule remains one of the most important open questions in discrete optimization. While many natural,...     

Every rose has its thorn: secondary teachers’ reasoning about statistical models

Statistical modeling is a core component of statistical thinking and has been identified by several countries as a curricular goal for secondary education. However, many secondary teachers have minimal preparation for teaching this topic. The goal of this research study is to learn about teachers’ perceptions of the role statistical models play in statistical inference and how these perceived purposes affect their reasoning about statistical models and inference. Problem-solving interviews were conducted with four in-service teachers who had recently taught a modeling and simulation-based introductory statistics course. Teachers’ responses suggest they may not see modeling variation as the primary purpose of statistical modeling and instead substitute two other purposes: making a decision and replicating the data collection process. Suggestions for how to build on teachers’ transitional conceptions and refocus attention on modeling variation are discussed.     

Integer comparisons across the grades: Students’ justifications and ways of reasoning

This study is an investigation of students’ reasoning about integer comparisons—a topic that is often counterintuitive for students because negative numbers of smaller absolute value are considered greater (e.g., −5 > − 6). We posed integer-comparison tasks to 40 students each in Grades 2, 4, and 7, as well as to 11th graders on a successful mathematics track. We coded for correctness and for students’ justifications, which we categorized in terms of 3 ways of reasoning: magnitude-based, order-based, and developmental/other. The 7th graders used order-based reasoning more often than did the younger students, and it more often led to correct answers; however, the college-track 11th graders, who responded correctly to almost every problem, used a more balanced distribution of order- and magnitude-based reasoning. We present a framework for students’ ways of reasoning about integer comparisons, report performance trends, rank integer-comparison tasks by relative difficulty, and discuss implications for integer instruction.     

An examination of pre-service mathematics teachers’ ethical reasoning in big data with considerations of access to data

Implementations of Big Data analysis are reshaping society. The novel ways mathematics operate in society warrants new efforts for mathematics education, both in teaching the new technology and in providing an ethical and critical awareness of its implications. This interview study investigates pre-service teachers' ethical reasoning in data science contexts, focusing on aspects of access to the data that underpin the technology. Findings show that pre-service teachers offer a wide array of ethical arguments related to access to data, that informs their effort to think critically on oppressive situations. However, there is also an indication that their reasoning can be limited by lacking understanding of the related data science methodology, implying that mathematics teacher education should encompass more of this.     

An Inverse Problem for Maxwell’s Equations in Anisotropic Media

The authors consider Maxwell's equations for an isomagnctic anisotropic and inhomogeneous medium in two dimensions, and discuss an inverse problem of determining the permittivity tensor (ε1ε2ε2ε3) and the permeabilityμin the constitutive relations from a finite number of lateral boundary measurements. Applying a Carlcman estimate, the authors prove an estimate of the Lipschitz type for stability, provided thatε1,ε2,ε3,μsatisfy some a priori conditions.     

An analogue of Beurling’s theorem for the Jacobi transform

In this paper, we prove Beurling＇s theorem for the Jacobi transform, from which we derive some other versions of uncertainty principles.