Projective background of the infinitesimal rigidity of frameworks

We present proofs of two classical theorems. The first one, due to Darboux and Sauer, states that infinitesimal rigidity is a projective invariant; the second one establishes relations (infinitesimal Pogorelov maps) between the infinitesimal motions of a Euclidean framework and of its hyperbolic and spherical images. The arguments use the static formulation of infinitesimal rigidity. The duality between statics and kinematics is established through the principles of virtual work. A geometric approach to statics, due essentially to Grassmann, makes both theorems straightforward. Besides, it provides a simple derivation of the formulas both for the Darboux-Sauer correspondence and for the infinitesimal Pogorelov maps. The research for this article was supported by the DFG Research Unit “Polyhedral Surfaces”.     

App-based scaffolds for writing two-column proofs

This paper presents apps designed to assist students in understanding and developing proofs in geometric theorems. These technologies focus on triangle congruence, triangle similarity and properties of parallelograms. Focus group discussions and initial testing of the apps revealed that the apps offered a more engaging medium for learning proving and were capable of facilitating proof-writing skills in geometry.     

Are self-constructed and student-generated arguments acceptable proofs? Pre-service secondary mathematics teachers’ evaluations

This study examined 14 pre-service secondary mathematics teachers’ productions and their evaluations of self-constructed and student-generated arguments in the domains of algebra, geometry, and number theory. Pre-service secondary mathematics teachers’ (PSMTs) evaluations of their own arguments indicate if they considered self-productions as proofs from a learner perspective. Similarly, PSMTs’ evaluations of student-generated arguments indicate if they decided given students’ productions could be counted as proofs from a teacher perspective. Our results show that the majority of PSMTs suspected that their invalid productions did not qualify as proofs. Furthermore, the PSMTs who were confident with their work and claimed that they had constructed a proof were more likely to make a correct judgment on four of the six student-generated arguments. We discuss implications of these findings for supporting PSMTs’ learning of proof and future research on the construction-evaluation activity.     

Proof validation and modification in secondary school geometry

Proof validation is important in school mathematics because it can provide a basis upon which to critique mathematical arguments. While there has been some previous research on proof validation, the need for studies with school students is pressing. For this paper, we focus on proof validation and modification during secondary school geometry. For that purpose, we employ Lakatos’ notion of local counterexample that rejects a specific step in a proof. By using Toulmin’s framework to analyze data from a task-based questionnaire completed by 32 ninth-grade students in a class in Japan, we identify what attempts the students made in producing local counterexamples to their proofs and modifying their proofs to deal with local counterexamples. We found that student difficulties related to producing diagrams that satisfied the condition of the set proof problem and to generating acceptable warrants for claims. The classroom use of tasks that entail student discovery of local counterexamples may help to improve students’ learning of proof and proving.     

Using dynamic geometry to explore non-traditional theorems

The purpose of this article is to provide examples of ‘non-traditional’ theorems that can be explored in a dynamic geometry environment by university and high school students. These theorems were encountered in the dynamic geometry environment. The author believes that teachers can ask their students to construct proofs for these theorems. The Appendix contains proof outlines for each theorem.     

Triangles II: Complex triangle coordinates

Summary This paper is the second in a series of three examining Euclidean triangle geometry via complex cross ratios. In the first paper of the series, we examined triangle shapes. In this paper, we coordinatize the Euclidean plane using cross ratios, and use these triangle coordinates to prove theorems about triangles. We develop a complex version of Ceva's theorem, and apply it to proofs of several new theorems. The remaining paper of this series will deal with complex triangle functions.     

Macros and modules in geometry

The paper highlights the importance of “macros” or modules for teaching and learning Geometry using Dynamical Geometry Software (DGS). The role of modules is analyzed in terms of “writing” and “reading” Geometry. At first, modules are taken as tools for geometrical construction tasks and as tools to describe and analyze these constructions. For proofs, decomposing a given geometrical statement may be supported by using prototypical pictures representing theorems of geometry (“modules”). Reading theorems into geometry and constructing proofs is still a major achievement of the student—which may be reached by using macros and modules as a major heuristic strategy     

Geometrical Solution to the Fermat Problem with Arbitrary Weights

The prime motivation for the present study is a famous problem, allegedly first formulated in 1643 by Fermat, and the so-called Complementary Problem (CP), proposed but incorrectly solved in 1941 by Courant and Robbins. For a given triangle, Fermat asks for a fourth point such that the sum of its Euclidean distances, each weighted by +1, to the three given points is minimized. CP differs from Fermat in that the weight associated with one of these points is –1 instead of +1. The geometrical approach suggested in 1998 by Krarup for solving CP is here extended to cover any combination of positive and negative weights associated with the vertices of a given triangle. Among the by-products are surprisingly simple correctness proofs of the geometrical constructions of Torricelli (around 1645), Cavalieri (1647), Viviani (1659), Simpson (1750), and Martelli (1998). Furthermore, alternative proofs of Ptolemy's theorem (around A.D. 150) and an observation by Heinen (1834) are provided.     

Relations among five radii of circles in a triangle,its sides and other segments

Students use GeoGebra to explore the mathematical relations among different radii of circles in a triangle (circumcircle, incircle, excircles) and the sides and other segments in the triangle. The more formal mathematical development of the relations that follows the explorations is based on known geometrical properties, different formulas relating the radii to the sides and the inequalities between the different averages. The activities described were conducted with pre-service teachers of mathematics, with empirical investigation of the relations using dynamical geometry software, and formal presentation of proofs.     

Are You Convinced? Middle‐Grade Students' Evaluations of Mathematical Arguments

Students learn norms of proving by observing teachers generating proofs, engaging in proving, and generalizing features of proofs deemed convincing by an authority, such as a textbook. Students at all grade levels have difficulties generating valid proof; however, little research exists on students' understandings about what makes a mathematical argument convincing prior to more formal instruction in methods of proof. This study investigated middle‐school students' (ages 12–14) evaluations of arguments for a statement in number theory. Students evaluated both an empirical and a general argument in an interview setting. The results show that students tend to prefer empirical arguments because examples enhance an argument's power to show that the statement is true. However, interview responses also reveal that a significant number of students find arguments to be most convincing when examples are supported with an explanation that “tells why” the statement is true. The analysis also examined the alignment of students' reasons for choosing arguments as more convincing along with the strategies they employ to make arguments more convincing. Overall, the findings show middle‐school students' conceptions about what makes arguments convincing are more sophisticated than their performance in generating arguments suggests.     

Dynamic investigation of triangles inscribed in a circle,which tend to an equilateral triangle

We present a geometrical investigation of the process of creating an infinite sequence of triangles inscribed in a circle, whose areas, perimeters and lengths of radii of the inscribed circles tend to a limit in a monotonous manner.

First, using geometrical software, we investigate four theorems that represent interesting geometrical properties, after which we present formal proofs that rest on a combination between different fields of mathematics: trigonometry, algebra and geometry, and the use of the concept of standard deviation that is taken from statistics.     

Soap films and GeoGebra in the study of Fermat and Steiner points

We discuss how mathematics and secondary mathematics education majors developed an understanding of Fermat points for the triangle as well as Steiner points for the square and regular pentagon, and also of soap film configurations between parallel plates where forces are in equilibrium. The activities included the use of soap films and the interactive geometry program GeoGebra. Students worked in small groups using these tools to investigate the properties of Fermat and Steiner points and then justified the results of their investigations using geometrical arguments. These activities are specific approaches of how to encourage prospective teachers to use physical experiments to support students’ development of mathematical curiosity and mathematical justifications.     

The Isis problem as an experimental probe and teaching resource

The Isis problem, which has a link with the Isis cult of ancient Egypt, asks: “Find which rectangles with sides of integral length (in some unit) have area and perimeter (numerically) equal, and prove the result.” Since the solution requires minimal technical mathematics, the problem is accessible to a wide range of students. Further, it is notable for the variety of proofs (empirically grounded, algebraic, geometrical) using different forms of argument, and their associated representations, and it provides an instrument for probing students’ ideas about proof, and the interplay between routine and adaptive expertise. A group of 39 Flemish pre-service mathematics teachers was confronted with the Isis problem. More specifically, we first asked them to solve the problem and to look for more than one solution. Second, we invited them to study five given contrasting proofs and to rank these proofs from best to worst. The results highlight a preference of many students for algebraic proofs as well as their rejection of experimentation. The potential of the problem as a teaching tool is outlined.     

Classroom sociomathematical norms for proof presentation in undergraduate in abstract algebra

This paper is a case study of the teaching of an undergraduate abstract algebra course with a particular focus on the manner in which the students presented proofs and the class engaged in a subsequent discussion of those proofs that included validating the work. This study describes norms for classroom work that include a set of norms that the presenter of a proof was responsible for enacting, including only using previously agreed upon results, as well as a separate set that the audience was to enact related to developing their understanding of the presented proof and validating the work. The study suggests that the students developed a sense of communal and individual responsibility for contributing to growing the body of mathematical knowledge known by the class, with an implied responsibility for knowing the already developed mathematics. Moreover, the behaviors that norms prompted the students to engage were those that literature suggests leads to increased comprehension of proofs.     

Harris-type results on geometric and subgeometric convergence to equilibrium for stochastic semigroups

We provide simple and constructive proofs of Harris-type theorems on the existence and uniqueness of an equilibrium and the speed of equilibration of discrete-time and continuous-time stochastic semigroups. Our results apply both to cases where the relaxation speed is exponential (also called geometric) and to those with no spectral gap, with non-exponential speeds (also called subgeometric). We give constructive estimates in the subgeometric case and discrete-time statements which seem both to be new. The method of proof also differs from previous works, based on semigroup and interpolation arguments, valid for both geometric and subgeometric cases with essentially the same ideas. In particular, we present very simple new proofs of the geometric case.     

Estimation of the offspring mean in a supercritical branching process with non-stationary immigration

In this paper, we consider the conditional least squares estimator (CLSE) of the offspring mean of a branching process with non-stationary immigration based on the observation of population sizes. In the supercritical case, assuming that the immigration variables follow known distributions, conditions guaranteeing the strong consistency of the proposed estimator will be derived. The asymptotic normality of the estimator will also be proved. The proofs are based on direct probabilistic arguments, unlike the previous papers, where functional limit theorems for the process were used.     

A Finite Non-Unit Free Triangle Order

The notion of a free triangle representation of a partially ordered set was first introduced by Josh Laison [4] as a generalization of the ideas of interval and trapezoid representations. A free triangle representation assigns a triangle to each element of a partially ordered set, with all triangles having one vertex on each of two parallel baselines and a third ‘free’ vertex between the two baselines. In a previous paper [1] we presented an example of an infinite non-unit free triangle order. In this paper we use some of the same ideas to construct an example of a finite, albeit more complicated, non-unit free triangle order.The majority of the content in this paper (theorems, proofs, etc.) was prepared before Ken's untimely death in March of 2005.     

On the Davenport–Heilbronn theorems and second order terms

We give simple proofs of the Davenport–Heilbronn theorems, which provide the main terms in the asymptotics for the number of cubic fields having bounded discriminant and for the number of 3-torsion elements in the class groups of quadratic fields having bounded discriminant. We also establish second main terms for these theorems, thus proving a conjecture of Roberts. Our arguments provide natural interpretations for the various constants appearing in these theorems in terms of local masses of cubic rings.     

Teaching-Learning of Evaluative Criteria for Mathematical Arguments Through Classroom Discourse: A Cross-National Study

Videotaped lessons of 5th graders on equivalent fractions from 7 American and 6 Japanese classrooms were analyzed in terms of a recurrent pattern in public discourse among a teacher and students. This pattern—called inquiry, response, feedback—occurs when a teacher initiates discourse (mostly with an inquiry), a student or students respond (often with an answer to the teacher inquiry), and the teacher provides feedback to the student's response. We found2 approaches to the teaching-learning of the criteria for evaluating mathematical arguments. In the Japanese classroom, students were encouraged to offer their own argument to the whole class and evaluate arguments proposed by other students. They seldom were given direct evaluation by their teacher. In contrast, American teachers often gave individual elaboration as well as direct evaluation to the student's responses, and some of the teachers offered their own opinions about mathematics, about valid ways of argumentation, or about both. The Japanese approach would help students acquire evaluative criteria indirectly through participating in mathematical discourse, whereas the American approach would help students learn modes of arguments through direct instruction.