MATHEMATICS STUDENT TEACHERS’ RESPONSES TO INFLUENCES AND BELIEFS

This article forms part of an ongoing study of student teachers of secondary mathematics. The aspect reported on in this article is an analysis of the effects of the influences brought to bear upon four individual student teachers of secondary mathematics as they progress through a one-year postgraduate course of teacher training (PGCE) based at a British University. The students have differing initial beliefs about teaching, learning and mathematics. As anticipated in the literature, the student's initial beliefs survive virtually intact throughout the year. However, the study suggests that the link between initial beliefs and teaching approach is not deterministic. The study suggests ways of encouraging student teachers to employ a range of pupil activities in their teaching.     

OVERCONVERGENCE AND ZEROS OF SEQUENCES OF RATIONAL FUNCTIONS

In the present paper we consider sequences of rational functions with a bounded number or free poles con-verging uniformly with a maximum geometric rate to a holomorphic function on a regular set,and we examinethe limiting distribution of the Zeros and its relations with the phenomenon of overconvergence.Our resultsfurther extend the well known classical theory of overconvergence and the zeros of sections of Taylor series.     

ENTIRE CHROMATIC NUMBER AND △-MATCHING OF OUTERPLANE GRAPHS

Let G be an outerplane graph with maximum degree A and the entire chromatic number Xvef（G）. This paper proves that if △ ≥6, then △＋ 1≤Xvef（G）≤△＋ 2, and Xvef （G） = △＋ 1 if and only if G has a matching M consisting of some inner edges which covers all its vertices of maximum degree.     

RATIONAL FUNCTIONS OF BEST UNIFORM APPROXIMATION AND HOLOMORPHIC CONTINUATION OF FUNCTIONS

Let f be a function, continuous and real valued on the segment △,△ (-∞,∞) and {Rn} be the sequence of the rational functions of best uniform approximation to fon △ of order (n,n). In the present work, the convergence of {Rn} in the complex plane is considered for the special caseswhen the poles (or the zeros, respectively) of {Rn} accumulate in the terms of weak convergence of measures to acompact set of zera capacity.As a consequence, sufficient conditions for the holomorphic and the meromorphic continuability of fare given.     

THE BINDING NUMBER OF A GRAPH AND ITS TRIANGLE

In this paper we prove the following conjecture of Woodall:if bind(G)≥3/2,then Gcontains a triangle.Moreover,we also prove that if bind(G)≥3/2,then each vertex is contained ina 4-cycle,each edge is contained in a 5-cycle when V(G)≥11,and there exists a 6-cycle in G.     

ENTIRE CHROMATIC NUMBER AND Δ-MATCHING OF OUTERPLANE GRAPHS

1IntroductionWe only consider simple graphs in this paper unless otherwise stated.For a plane graphG,we denote its vertex set,edge set,face set,minimum degree,and maximum degree by V(G),E(G),F(G),δ(G),and?(G),respectively.For v∈V(G),let dG(v)denote the …     

THE INFLUENCE OF TEACHERS’ PRESENTATIONS ON PUPILS’ MENTAL REPRESENTATIONS

Teachers use a variety of external representations to communicate mathematical ideas to their pupils. This paper reports a preliminary study of the internal mental representations that 6- and 7- year-old pupils form as a result of their interactions with the teacher's verbal, written, pictorial and concrete material representations, involving two-digit numbers and operations on them. The results presented here concern the picture-like mental representations that pupils use in performing two-digit calculations mentally. The evidence suggests that pupils seldom spontaneously visualise teachers’ representations or attempt mental manipulation of visual images to help with calculation. Pupils can, however, have mental representations which reproduce some aspects of the teachers’ representations.     

APPROXIMATE COMMON DIVISORS OF POLYNOMIALS AND DEGREE REDUCTION FOR RATIONAL CURVES

This paper deals with how to perturb a given set of polynomials so as to include a common linear factor. An algorithm is derived for determining such a set of perturbation polyrlomials which are subject to certain constrains at the endpoints of a prescribed parametric interval and minimized in a certain sense. This result can be combined with subdivision technique to obtain a continuous piecewise approximation to a rational curve.     

METHODS IN GHANAIAN PRIMARY MATHEMATICS TEXTBOOKS AND TEACHERS’ CLASSROOM PRACTICE

The official school mathematics curriculum has a powerful influence on classroom practice in a developing country like Ghana, where many teachers with low teaching qualifications hardly ever have access to other sources of information and activity for their teaching. The study reported in this paper investigated the congruence between the teaching methods presented in the official curriculum materials and teachers’ classroom practice. The study involved a range of methods: classroom observation and content and discourse analyses of lessons in mathematics. The study found that, though there was rhetoric in the introduction of the curriculum materials on the use of discovery teaching methods, few learning/teaching activities that would encourage the use of such discovery methods were included in the materials. It was observed that both the official curriculum and the teachers who implement it emphasise expository teaching methods.     

THE ROTATION NUMBER AND DIRECTION OF A CONTINUOUS FLOW ON THE TORUS

In this paper the rotation direction r_f and rotation number p_f for any continuous flowon the torus are defined by applying Weil's theorem proved by Markley.The followingresults are obtained:(i)r_f=0 iff all orbits of f are proper and each limit set of f is homotopic to zeo onT~2;(ii)if r_f≠0,then p_f is irrational iff f has at least one non-trivial P stable orbit,p_f isrational iff f has at least one non-zero-homotopic closed or singular closed orbit.Then a method of computing the rotation number of certain flows is given.     

EXISTENCE AND THE NUMBER OF PERIODIC SOLUTIONS ABOUT ONE CLASS DIFFERENTIAL EQUATION

ＥＸＩＳＴＥＮＣＥＡＮＤＴＨＥＮＵＭＢＥＲＯＦＰＥＲＩＯＤＩＣＳＯＬＵＴＩＯＮＳＡＢＯＵＴＯＮＥＣＬＡＳＳＤＩＦＦＥＲＥＮＴＩＡＬＥＱＵＡＴＩＯＮＦｅｎｇＺｈａｏｓｈｅｎｇ（冯兆生）（ＮｏｒｔｈｅｒｎＪｉａｏｔｏｎｇＵｎｉｖｅｒｓｉｔｙ，北方交通大学...     

THE STRONG LAW OF LARGE NUMBER AND PARAMETER ESTIMATION OF ONE CLASS OF NON-NEGATIVE INTEGER-VALUED TIME SERIES

1.IntroductionLet{xt}beatimeseriesdefinedonprobabilityspace{fi,F,P),satisfyingwhere(1)p(t,s,j)isarandomfieldwithValueIor0,t,8EI={0,11,f2,'.},jEN={1,2,'.},andsatisfies(i)p{p(t,s,j)=0}=1whens>t;(1.2)(if)ac(tl,sl,jl)andp(tz,sa,j~)aremutuallyindependentwhenif/iZor81/s2;(ill)FOrfords,j,p(t,s,i)fort2sisanonhomogeneousMarkovChainwithtransitionprobabilityandinitialdistributiont=0T=0(2){Mt}isani.i.d.non-negativeinteger-valuedtimeseriesandindependentofn(t,sli).Inthefollowing,weassumeEMtz相似文献   

HAMILTONIAN STRUCTURE AND INFINITE NUMBER OF CONSERVATION LAWS FOR THE COUPLED DISCRETE KdV EQUATIONS

A new discrete isospectral problem is introduced, from which the coupled discrete KdV hierarchy is deduced and is written in its Hamiltonian form by means of the trace identity.It is shown that each equation in the resulting hierarchy is Liouville integrable. Furthermore,an infinite number of conservation laws are shown explicitly by direct computation.     

QUALITATIVE THEORY OF THE KUKLES SYSTEMS:(Ⅰ) NUMBER OF CRITICAL POINTS

The special cubic system (a natural and simPlest generaIization of thequadratic system of C1ass (l) to cubic system)f.'x = yl y = --x 6y a,x' a2xy a,y' a#x' a,x'y a.xy' a,y' (1)was first studied by the Russian Mathematician I.S.KukIes [ll on the Ilecessaryand sufficient conditions fot O(0,0) to be a ccnter of (1). After [ll, the sameproblem and also the order of O as a fine fOcus were also studied iIl [2--l0] and[13-19], so (l) is now called as a "Kukles system". In l988-…     

MODELLING AND NUMERICAL SOLUTIONS OF A GAUGE PERIODIC TIME DEPENDENT GINZBURG-LANDAU MODEL FOR TYPE-Ⅱ SUPERCONDUCTORS

1.IntroductionCentraltothetheoryoftype-IIsuperconductorsisAbrikosov'schaxacterizationofthemixedstateasalattice-likearrangementofquantizedfluxlines,oryorticesofsuperconductingelectronpairs.TheAbrikosov'svortexlattice,whichhasalsobeenobservedinexperiments,isthesolutionsoftheGinzburgLandau(GL)equationswithatypeofspatialperiodicity.Recentlytherehavebeenseveralauthor8studiedthegaugeperiodicsolutionsoftheGLsuperconductivitymodelfromdifferentpointof.iews[1'1o)11'17].Roughlyspeaxing,gaugeperiodics…     

L~p CONTINUITY OF HRMANDER SYMBOL OPERATORS OpS_（0,0） ~m AND NUMERICAL ALGORITHM

If we use Littlewood-Paley decomposition, there is no pseudo-orthogonality for Ho¨rmander symbol operators OpS m 0 , 0 , which is different to the case S m ρ,δ (0 ≤δ < ρ≤ 1). In this paper, we use a special numerical algorithm based on wavelets to study the L p continuity of non infinite smooth operators OpS m 0 , 0 ; in fact, we apply first special wavelets to symbol to get special basic operators, then we regroup all the special basic operators at given scale and prove that such scale operator’s continuity decreases very fast, we sum such scale operators and a symbol operator can be approached by very good compact operators. By correlation of basic operators, we get very exact pseudo-orthogonality and also L 2 → L 2 continuity for scale operators. By considering the influence region of scale operator, we get H 1 (= F 0 , 2 1 ) → L 1 continuity and L ∞→ BMO continuity. By interpolation theorem, we get also L p (= F 0 , 2 p ) → L p continuity for 1 < p < ∞ . Our results are sharp for F 0 , 2 p → L p continuity when 1 ≤ p ≤ 2, that is to say, we find out the exact order of derivations for which the symbols can ensure the resulting operators to be bounded on these spaces.     

ON WEAK MINIMUM ABERRATION AND NUMBER OF CLEAR TWO-FACTOR INTERACTION COMPONENTS IN s_Ⅳ～(m-p) DESIGNS

This article obtains some theoretical results on the number of clear two-factor interaction components and weak minimum aberration in an s_Ⅳ～(m-p) design,by considering the number of not clear two-factor interaction components of the design.     

VISUALISATION AND THE INFLUENCE OF TECHNOLOGY IN ‘A’ LEVEL MATHEMATICS: A CLASSROOM INVESTIGATION

The study reported here is part of a wider study, which aims to investigate the potential of the graphical calculator for mediating the development of students’ abilities to visualise the graphs of functions at GCE Advanced level. This paper focuses on how the graphical calculator influenced six particular students’ work with functions. Initial results have illuminated ways in which the technology can have a positive impact on students’ visualisation capabilities. It is proposed that visual thinking forms a significant part of many students’ mathematical reasoning, enabling students to derive richer meaning from given problems. It is suggested further that use of the technology mediates the development of students’ visual capacities, by helping to highlight the links between complementary modes of representation.