Exploring Project-Based Teaching for Engaging Students' Mathematical Learning

The declining interest of learners in mathematics in the learning process has resulted in poor achievement (Yeh et al., 2019). To get rid of these poor achievements, we explored project-based teaching in four topical areas (e.g., mathematical concepts of coordinate geometry, trigonometry, sequence, and series) in the school mathematics. This paper results from observing changes in engagement of learners in learning mathematics by motivating them through the project-based learning (PBL) guided by two theories – knowledge constitutive interests (Habermas, 1972), and collaborative and cooperative learning under the paradigms of interpretivism and criticalism. In this ethos, PBL is an “engaging and learner-directed approach that provides equal opportunities for students to explore their knowledge and understanding” (Thomas, 2000, p. 12). More specifically, we adopted the 'action research' method with the secondary level students (Grade IX) of one of the institutional schools in their classrooms. The information was collected by observing and recording the changes seen in consecutive seventeen days. The research landed that project-based learning is an appropriate pedagogy for engaged learning. The study revealed that the students were motivated while they got opportunities to interact in the projects. Moreover, the findings show that PBL is helpful to engage the learners through questioning, pair/group discussion, discovery learning, and concept mapping.

Visual-spatial Skills and Mathematics Content Conceptualisation for Pre-service Teachers

Empirical evidence in literature identified significant association between spatial ability and educational performance particularly in science, technology, engineering and mathematics (STEM). The purpose of this study was to explore pre-service teachers’ spatial skills in solving mathematics problems, in the context of coordinate geometry. It is envisaged that spatial skills allow for the perception of visual information and, therefore, spatial cognition has been considered as a key skill in teaching mathematics. However, literature asserts that teachers are ill prepared to teach mathematics, hence there is limited use and misuse of spatial skills in teaching the subject. This study, therefore, examines the spatial orientation of pre-service teachers in teaching coordinate geometry. This is a mixed methods study in which pre-service teachers answered a coordinate geometry test to explore their content knowledge and their ability to interpret, analyse and apply visual spatial models to solve mathematical problems in coordinate geometry. The study established that the spatial orientation skills of pre-service teachers determine their performance in mathematics, especially coordinate geometry.

On the Exponential Diophantine Equation (132m) + (6r + 1)n = z2

Nowadays, mathematicians are very interested in discovering new and advanced methods for determining the solution of Diophantine equations. Diophantine equations are those equations that have more unknowns than equations. Diophantine equations appear in astronomy, cryptography, abstract algebra, coordinate geometry and trigonometry. Congruence theory plays an important role in finding the solution of some special type Diophantine equations. The absence of any generalized method, which can handle each Diophantine equation, is challenging for researchers. In the present paper, the authors have discussed the existence of the solution of exponential Diophantine equation (132m) + (6r + 1)n = Z2, where m, n, r, z are whole numbers. Results of the present paper show that the exponential Diophantine equation (132m) + (6r + 1)n = Z2, where m, n, r, z are whole numbers, has no solution in the whole number.

Low-Valent Tungsten Redox Catalysis Enables Controlled Isomerization and Carbonylative Functionalization of Alkenes

Tungsten catalysis has played an instrumental role in the history of organometallic chemistry, with electrophilic, fully oxidized W(VI) catalysts featuring prominently in olefin polymerization and metathesis reactions. Here, we report that the simple W(0) precatalyst, W(CO)₆, catalyzes the isomerization and hydrocarbonylation of alkenes via a W(0)/W(II) redox couple. The 6- to 7-coordinate geometry changes associated with this redox process are key in allowing isomerization to take place over multiple positions and stop at a defined unactivated internal site that is primed for <i>in situ</i> functionalization. DFT studies and crystallographic characterization of multiple directing-group-bound W(II) model complexes illuminate potential intermediates of this redox cycle and showcase the capabilities of the 7-coordinate W(II) geometry to facilitate challenging alkene functionalizations.

Low-Valent Tungsten Redox Catalysis Enables Controlled Isomerization and Carbonylative Functionalization of Alkenes

Tungsten catalysis has played an instrumental role in the history of organometallic chemistry, with electrophilic, fully oxidized W(VI) catalysts featuring prominently in olefin polymerization and metathesis reactions. Here, we report that the simple W(0) precatalyst, W(CO)₆, catalyzes the isomerization and hydrocarbonylation of alkenes via a W(0)/W(II) redox couple. The 6- to 7-coordinate geometry changes associated with this redox process are key in allowing isomerization to take place over multiple positions and stop at a defined unactivated internal site that is primed for <i>in situ</i> functionalization. DFT studies and crystallographic characterization of multiple directing-group-bound W(II) model complexes illuminate potential intermediates of this redox cycle and showcase the capabilities of the 7-coordinate W(II) geometry to facilitate challenging alkene functionalizations.

Vector Arithmetic in the Triangular Grid

Vector arithmetic is a base of (coordinate) geometry, physics and various other disciplines. The usual method is based on Cartesian coordinate-system which fits both to continuous plane/space and digital rectangular-grids. The triangular grid is also regular, but it is not a point lattice: it is not closed under vector-addition, which gives a challenge. The points of the triangular grid are represented by zero-sum and one-sum coordinate-triplets keeping the symmetry of the grid and reflecting the orientations of the triangles. This system is expanded to the plane using restrictions like, at least one of the coordinates is an integer and the sum of the three coordinates is in the interval [−1,1]. However, the vector arithmetic is still not straightforward; by purely adding two such vectors the result may not fulfill the above conditions. On the other hand, for various applications of digital grids, e.g., in image processing, cartography and physical simulations, one needs to do vector arithmetic. In this paper, we provide formulae that give the sum, difference and scalar product of vectors of the continuous coordinate system. Our work is essential for applications, e.g., to compute discrete rotations or interpolations of images on the triangular grid.

Regulating catalytic activity of multi-Ru-bridged polyoxometalate based on active sites differential environment with six-coordinate geometry and five-coordinate geometry transition

Five-coordinate geometry around ruthenium with highly exposed active sites have attracted intensive scientific interests due to their superior properties and extensive applications. Herein, we report a series of structurally controllable...

Conformational polymorphism in a cobalt(II) dithiocarbamate complex

Two conformational polymorphs of (N,N-dibutyldithiocarbamato-κ2 S,S′)[tris(3,5-diphenylpyrazol-1-yl-κN 2)hydroborato]cobalt(II), [Co(C45H34BN6)(C9H18NS2)] or [TpPh2Co(S2CNBu2)], 1, are accessible by recrystallization from dichloromethane–methanol to give orthorhombic polymorph 1a, while slow evaporation from acetonitrile produces triclinic polymorph 1b. The two polymorphs have been characterized by IR spectroscopy and single-crystal X-ray crystallography at 150 K. Polymorphs 1a and 1b crystallize in the orthorhombic space group Pbca and the triclinic space group P-1, respectively. The polymorphs have a trans (1a) and cis (1b) orientation of the butyl groups with respect to the S2CN plane of the dithiocarbamate ligand, which results in an intermediate five-coordinate geometry for 1a and a square-pyramidal geometry for 1b. Hirshfeld surface analysis reveals minor differences between the two polymorphs, with 1a exhibiting stronger C—H...S interactions and 1b favouring C—H...π interactions.