Preservice teachers learning to teach proof through classroom implementation: Successes and challenges

Proof and reasoning are central to learning mathematics with understanding. Yet proof is seen as challenging to teach and to learn. In a capstone course for preservice teachers, we developed instructional modules that guided prospective secondary mathematics teachers (PSTs) through a cycle of learning about the logical aspects of proof, then planning and implementing lessons in secondary classrooms that integrate these aspects with traditional mathematics curriculum in the United States. In this paper we highlight our framework on mathematical knowledge for teaching proof and focus on some of the logical aspects of proof that are seen as particularly challenging (four proof themes). We analyze 60 lesson plans, video recordings of a subset of 13 enacted lessons, and the PSTs’ self- reported data to shed light on how the PSTs planned and enacted lessons that integrate these proof themes. The results provide insights into successes and challenges the PSTs encountered in this process and illustrate potential pathways for preparing PSTs to enact reasoning and proof in secondary classrooms. We also highlight the design principles for supporting the development of PSTs’ mathematical knowledge for teaching proof.     

A proof of the arithmetic-geometric mean inequality using non-Euclidean geometry

This note presents a proof of the arithmetic-geometric mean inequality that uses basic facts about the upper half-plane model of hyperbolic plane geometry. This material could find use as enrichment material in any model-oriented course on the classical geometries.     

A quadratic growth learning trajectory

This paper introduces a quadratic growth learning trajectory, a series of transitions in students’ ways of thinking (WoT) and ways of understanding (WoU) quadratic growth in response to instructional supports emphasizing change in linked quantities. We studied middle grade (ages 12–13) students’ conceptions during a small-scale teaching experiment aimed at fostering an understanding of quadratic growth as phenomenon of constantly-changing rate of change. We elaborate the duality, necessity, repeated reasoning framework, and methods of creating learning trajectories. We report five WoT: Variation, Early Coordinated Change, Explicitly Quantified Coordinated Change, Dependency Relations of Change, and Correspondence. We also articulate instructional supports that engendered transitions across these WoT: teacher moves, norms, and task design features. Our integration of instructional supports and transitions in students’ WoT extend current research on quadratic function. A visual metaphor is leveraged to discuss the role of learning trajectories research in unifying research on teaching and learning.     

A story-based dynamic geometry approach to improve attitudes toward geometry and geometric proof

This work is a part of a larger study, which presents geometry through a daily life story using dynamic geometry software. It aimed in particular to enable students to feel the importance of geometry in daily life, to share in the process of formulating geometric statements and conjectures, to experience the geometric proof more than validating the correctness of geometric statements and to start with a real-life situation and go through seven steps to geometric proof. The content of the suggested approach was organized so that every activity was a prerequisite for entering the next one, either in the structure of geometric concepts or in the geometric story context. Some indications will be presented according to three Likert-type questionnaires, which were prepared by the researcher with the purpose of assessing students?? attitudes toward geometry and geometric proof, using computers in mathematics learning and the suggested approach. The analysis of single responses to questionnaire items showed significant changes in students?? beliefs about geometry, importance and functions of geometric proof and toward using the suggested approach.     

Reasoning and proof in geometry: effects of a learning environment based on heuristic worked-out examples

We analyze heuristic worked-out examples as a tool for learning argumentation and proof. Their use in the mathematics classroom was motivated by findings on traditional worked-out examples, which turned out to be efficient for learning algorithmic problem solving. The basic idea of heuristic worked-out examples is that they encourage explorative processes and thus reflect explicitly different phases while performing a proof. We tested the hypotheses that teaching with heuristic examples is more effective than usual classroom instruction in an experimental classroom study with 243 grade 8 students. The results suggest that heuristic worked-out examples were more effective than the usual mathematics instruction. In particular, students with an insufficient understanding of proof were able to benefit from this learning environment.     

A short proof of Hua's fundamental theorem of the geometry of hermitian matrices

We present a new and simple proof of Hua's fundamental theorem of the geometry of hermitian matrices which characterizes bijective maps preserving adjacency in both directions on the real vector space of all n × n hermitian matrices.     

The standard proof,the elegant proof,and the proof without words of tasks in geometry,and their dynamic investigation

Solution of problems in mathematics, and in particular in the field of Euclidean geometry is in many senses a form of artisanship that can be developed so that in certain cases brief and unexpected solutions may be obtained, which would bring out aesthetically pleasing mathematical traits. We present four geometric tasks for which different proofs are given under the headings: standard proof, elegant proof, and the proof without words. The solutions were obtained through a combination of mathematical tools and by dynamic investigation of the geometrical properties.     

Cultivating inquiry about space in a middle school mathematics classroom

During 46 lessons in Euclidean geometry, sixth-grade students (ages 11, 12) were initiated in the mathematical practice of inquiry. Teachers supported inquiry by soliciting student questions and orienting students to related mathematical habits-of-mind such as generalizing, developing relations, and seeking invariants in light of change, to sustain investigations of their questions. When earlier and later phases of instruction were compared, student questions reflected an increasing disposition to seek generalization and to explore mathematical relations, forms of thinking valued by the discipline. Less prevalent were questions directed toward search for invariants in light of change. But when they were posed, questions about change tended to be oriented toward generalizing and establishing relations among mathematical objects and properties. As instruction proceeded, students developed an aesthetic that emphasized the value of questions oriented toward the collective pursuit of knowledge. Post-instructional interviews revealed that students experienced the forms of inquiry and investigation cultivated in the classroom as self-expressive.     

Cognitive trajectory of proof by contradiction for transition-to-proof students

History and research on proof by contradiction suggests proof by contradiction is difficult for students in a number of ways. Students’ comprehension of already-written proofs by contradiction is one such aspect that has received relatively little attention. Applying the cognitive lens of Action-Process-Object-Schema (APOS) Theory to proof by contradiction, we constructed and tested a cognitive model that describes how a student might construct the concept ‘proof by contradiction’ in an introduction to proof course. Data for this study was collected from students in a series of five teaching interventions focused on proof by contradiction. This paper will report on two participants as case studies to illustrate that our cognitive trajectory for proof by contradiction is a useful model for describing how students may come to understand the proof method.     

A learning trajectory for university students regarding the concept of vector

This paper presents and evaluates a hypothetical learning trajectory by which students bridge the transition from elementary to university-level instruction regarding the concept of vector. The trajectory consists of an instructional sequence of five tasks and begins with a problem in context. Each task is carried out with the support of a Virtual Interactive Didactic Scenario, accompanied by exploration and guided learning sheets, in which the problem is introduced through the simulation of the movement of a robotic arm. This proposal was implemented at the beginning of the SARS-CoV-2 pandemic using various digital media. Two teaching experiments were carried out with engineering students at a Mexican public university. We present the hypothetical learning trajectory that should be followed toward solving the task, and contrast it in each case with the students’ actual learning trajectory. The results show that more than 70 % of the students successfully transitioned from the geometrical vector representation of elementary physics to the algebraic one.     

Independent learning and operational research in the classroom

While Operational Research and Education are common keywords in OR journals, most work focuses on course content. Relatively little has been written about processes of education and learning in OR. This paper adds to this important area by considering the use of more Independent Learning (IL) in Operational Research education. Included is an explanation of IL and justification for using it. Activities which encourage IL are described and supported by three case studies which give examples of how an academic might introduce IL into a range of OR education settings. Other topics include IL and assessment, arguments against the use of IL, and critical success factors.     

Mathematical videos,social semiotics and the changing classroom

With the advancement of digital technology, the roles of teachers and students are slowly changing. The classroom is on the verge of becoming a new, more o     

Impacts of inquiry pedagogy on undergraduate students conceptions of the function of proof

Mathematicians and mathematics educators agree that proof is an important tool in mathematics, yet too often undergraduate students see proof as a superficial part of the discipline. While proof is often used by mathematicians to justify that a theorem is true, many times proof is used for another purpose entirely such as to explain why a particular statement is true or to show mathematics students a particular proof technique. This paper reports on a study that used a form of inquiry-based learning (IBL) in an introduction to proof course and measured the beliefs of students in this course about the different functions of proof in mathematics as compared to students in a non-IBL course. It was found that undergraduate students in an introduction to proof course had a more robust understanding of the functions of proof than previous studies would suggest. Additionally, students in the course taught using inquiry pedagogy were more likely to appreciate the communication, intellectual challenge, and providing autonomy functions of proof. It is hypothesized that these results are a response to the pedagogy of the course and the types of student activity that were emphasized.     

Group inquiry to aid organisational learning in enterprises

This paper describes a method for surfacing and exploring ‘situated knowledge’ in medium-sized organisations, with employee groups utilising a ‘low impact’ form of group support system (GSS), based on wireless handsets. Some results of piloting this method are summarised and one intervention is presented in detail. The method encouraged organisational members to give voice to the emotions and politics of leadership and learning in organisations, and helped to articulate how situated knowledge was ignored, as well as utilised. The method is practical, and may be used by organisations for themselves to aid the development of group as well as individual reflection, to stimulate the consideration of change.     

A fixed point formula of Lefschetz type in Arakelov geometry I: statement and proof

We consider arithmetic varieties endowed with an action of the group scheme of n-th roots of unity and we define equivariant arithmetic K ₀-theory for these varieties. We use the equivariant analytic torsion to define direct image maps in this context and we prove a Riemann-Roch theorem for the natural transformation of equivariant arithmetic K ₀-theory induced by the restriction to the fixed point scheme; this theorem can be viewed as an analog, in the context of Arakelov geometry, of the regular case of the theorem proved by P. Baum, W. Fulton and G. Quart in [BaFQ]. We show that it implies an equivariant refinement of the arithmetic Riemann-Roch theorem, in a form conjectured by J.-M. Bismut (cf. [B2, Par. (l), p. 353] and also Ch. Soulé’s question in [SABK, 1.5, p. 162]). Oblatum 22-I-1999 & 20-II-2001?Published online: 4 May 2001     

In search of more triangle centres. A source of classroom projects in Euclidean geometry

A point E inside a triangle ABC can be coordinatized by the areas of the triangles EBC, ECA, and EAB. These are called the barycentric coordinates of E. It can also be coordinatized using the six segments into which the cevians through E divide the sides of ABC, or the six angles into which the cevians through E divide the angles of ABC, or the six triangles into which the cevians through E divide ABC, etc. This article introduces several coordinate systems of these types, and investigates those centres of ABC whose coordinates, relative to a given coordinate system, are linear (or quasi-linear) with respect to appropriate elements of ABC, such as its side-lengths, its angles, etc. This results in grouping known centres into new families, and in discovering new centres. It also leads to unifying several results that are scattered in the literature, and creates several open questions that may be suitable for classroom discussions and team projects in which algebra and geometry packages are expected to be useful. These questions may also be used for Mathematical Olympiad training and may serve as supplementary material for students taking a course in Euclidean geometry.     

A hypothetical learning trajectory for conceptualizing matrices as linear transformations

In this paper, we present a hypothetical learning trajectory (HLT) aimed at supporting students in developing flexible ways of reasoning about matrices as linear transformations in the context of introductory linear algebra. In our HLT, we highlight the integral role of the instructor in this development. Our HLT is based on the ‘Italicizing N’ task sequence, in which students work to generate, compose, and invert matrices that correspond to geometric transformations specified within the problem context. In particular, we describe the ways in which the students develop local transformation views of matrix multiplication (focused on individual mappings of input vectors to output vectors) and extend these local views to more global views in which matrices are conceptualized in terms of how they transform a space in a coordinated way.