Students discussing mathematics in small-group interactions: opportunities for discursive negotiation processes focused on contentious mathematical issues

Small-group discussions involving students and their teacher that focus on meanings constructed during the mathematics lessons or solutions to problems produced in these lessons offer great potential for debate and argument. An analysis of the epistemological nature of knowledge can give deeper insight, to gain a better understanding of the emerging discontinuities in argumentations, negotiations, and clarifications about contentious meaning differences that arise. In most cases mathematical interactions between students and a teacher about contentions are very fragile and seem to be handled more or less directly—by side-stepping to another topic or by resolving via the teacher’s authority, for example. Therefore, the maintenance of such negotiation processes in mathematics teaching is a specific challenge for students and the teacher. The type of closure of these processes seems to be related to the emerging maintenance processes. In this paper, small-group discussions are interpretatively analyzed in the three steps “Initiation—Maintenance—Closing” with the focus on fundamental (dialogical) learning.     

Visual representations as objects of analysis: the number line as an example

Our paper examines the representational nature of number lines as they are used in instructional tasks. The examination is informed by a so-called mathedidactical analysis of the number line as a tool used in teaching students mathematics. This analysis led to the identification of a family of number line models, based on visual aspects of number lines each reflecting different forms and functions. In the article, number line tasks are unpacked to illustrate the visual representational components of particular number line models. We illuminate how these components of the models provide tools to locate whole numbers and integers, operate with them, and facilitate reasoning and understanding of underlying mathematical concepts.     

On the number of representations in the ternary Goldbach problem with one prime number in a given residue class

For

     

An upper bound for the chromatic number of line graphs

It was conjectured by Reed [B. Reed, ω,α, and χ, Journal of Graph Theory 27 (1998) 177–212] that for any graph G, the graph’s chromatic number χ(G) is bounded above by , where Δ(G) and ω(G) are the maximum degree and clique number of G, respectively. In this paper we prove that this bound holds if G is the line graph of a multigraph. The proof yields a polynomial time algorithm that takes a line graph G and produces a colouring that achieves our bound.     

On the number of representations of integers by some quadratic forms in ten variables

A method of finding the so-called Liouville's type formulas for the number of representations of integers by $$a_1 (x_1^2 + x_2^2 ) + a_2 (x_3^2 + x_4^2 ) + a_3 (x_5^2 + x_6^2 ) + a_4 (x_7^2 + x_8^2 ) + a_5 (x_9^2 + x_{10}^2 )$$ quadratic forms is developed.     

On a number of poisson matrices in Bang-Bang representations for 3 × 3 embeddable matrices

We give a counterexample to the Strong Bang-Bang Conjecture according to which any 3 × 3 embeddable matrix can be expressed as a product of six Poisson matrices. We exhibit a 3 × 3 embeddable matrix which can be expressed as a product of seven but not six Poisson matrices. We show that an embeddable 3 × 3 matrix P with det $\text{P ≥}\text{1}\text{8}$ can be expressed as a product of at most six Poisson matrices and give necessary and sufficient conditions for a 3 × 3 stochastic matrix P with det $\text{P ≥}\text{1}\text{8}$ to be embeddable. For an embeddable 3 × 3 matrix P with det $\text{P <}\text{1}\text{8}$ we give a new bound for the number of Poisson matrices in its Bang-Bang representation.     

A simple formula for the number of spanning trees of line graphs

Suppose is a loopless graph and is the graph obtained from G by subdividing each of its edges k () times. Let be the set of all spanning trees of G, be the line graph of the graph and be the number of spanning trees of . By using techniques from electrical networks, we first obtain the following simple formula: Then we find it is in fact equivalent to a complicated formula obtained recently using combinatorial techniques in [F. M. Dong and W. G. Yan, Expression for the number of spanning trees of line graphs of arbitrary connected graphs, J. Graph Theory. 85 (2017) 74–93].     

Local search algorithms for finding the Hamiltonian completion number of line graphs

Given a graph G=(V,E), the Hamiltonian completion number of G, HCN(G), is the minimum number of edges to be added to G to make it Hamiltonian. This problem is known to be -hard even when G is a line graph. In this paper, local search algorithms for finding HCN(G) when G is a line graph are proposed. The adopted approach is mainly based on finding a set of edge-disjoint trails that dominates all the edges of the root graph of G. Extensive computational experiments conducted on a wide set of instances allow to point out the behavior of the proposed algorithms with respect to both the quality of the solutions and the computation time.     

Minimizing the number of workers in a paced mixed-model assembly line

We study a problem of minimizing the maximum number of identical workers over all cycles of a paced assembly line comprised of m stations and executing n parts of k types. There are lower and upper bounds on the workforce requirements and the cycle time constraints. We show that this problem is equivalent to the same problem without the cycle time constraints and with fixed workforce requirements. We prove that the problem is NP-hard in the strong sense if $m=4$ and the workforce requirements are station independent, and present an Integer Linear Programming model, an enumeration algorithm and a dynamic programming algorithm. Polynomial in k and polynomial in n algorithms for special cases with two part types or two stations are also given. Relations to the Bottleneck Traveling Salesman Problem and its generalizations are discussed.     

On the number of line separations of a finite set in the plane

Let S denote a set of n points in the Euclidean plane. A subset S′ of S is termed a k-set of S if it contains k points and there exists a straight line which has no point of S on it and separates S′ from S?S′. We let f_k(n) denote the maximum number of k-sets which can be realized by a set of n points. This paper studies the asymptotic behaviour of f_k(n) as this function has applications to a number of problems in computational geometry. A lower and an upper bound on f_k(n) is established. Both are nontrivial and improve bounds known before. In particular, $\text{f}{}_{\mathrm{k}}\text{(n) = f}{}_{\mathrm{n?k}}\text{(n) = Ω(n}\text{log}\text{k)}$ is shown by exhibiting special point-sets which realize that many k-sets. In addition, $\text{f}{}_{\mathrm{k}}\text{(n) = f}{}_{\mathrm{n?k}}\text{(n) = O(nk}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{)}$ is proved by the study of a combinatorial problem which is of interest in its own right.     

Distribution of the minimum number of points in a scanning interval on the line

Let N points be distributed at random on [0,1), and let y(t) be the number of points in [t,t+p), when p? (0,1). For certain step functions, g(t), for all t in [0,1 ? p) is found. Choosing g(t)≡m results in a multiple comparison test, which may be used to test for differences between any m independent normally distributed means at a given experiment-wide level of significance.     

Locating negative decimals on the number line: Insights into the thinking of pre-service primary teachers

This paper explores misconceptions of the number line which are revealed when pre-service primary teachers locate negative decimals on a number line. Written test responses from 94 pre-service primary teachers provide an initial data source which is supplemented by group responses to worksheets completed during a lesson and individual interviews. Two main misconceptions leading to incorrect placement of negative decimals on a number line are identified. One relates to having separate number ‘rays’ for positive and negative numbers, which are aligned according to context. The other (with several variations) results from creating the negative part of the number line by amalgamating translated positive intervals. These misconceptions explain a large percentage of wrong answers. The most important implication for education at school, as well as in teacher education, is that the teaching of negative numbers and of the number line must not be confined to integers, as is frequently the case, but must also include negative fractions and decimals.     

Explicit realization of irreducible representations of classical compact lie groups in the spaces of sections of line bundles

We use the Borel-Weil scheme for the construction of irreducible representations of compact Lie groups in the spaces of holomorphic sections of line bundles over homogeneous manifolds. We find the explicit form of the space of sections and construct an invariant scalar product. We show that the space of holomorphic sections locally satisfies the Zhelobenko indicator system. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 10, pp. 1316–1323, October, 1998.