Series and Sequences

This paper presents a computational arrangement of the classical summation formula n(n + 1)/2 using a purely geometric argument based on discrete area. We prove its algebraic equivalence to Gauss's formula through a direct, two-step operational rule: multiply the number by its half, then add its half. This formulation, Ghaly's Method, is designed for pedagogical simplicity and provides a faster intuitive understanding for mental arithmetic. The proof is constructed from first principles and begins with Gauss's formula to demonstrate equivalence.

∀∃vs∃∀ A central lesson of modern set theory is that size is not absolute: whether a set is countable, and what the cardinality of the continuum is, depends on the surrounding universe in which these claims are interpreted. The Löwenheim-Skolem phenomenon, the independence of the Continuum Hypothesis (CH), and multiverse perspectives make this model-dependence explicit. Motivated by this relativity of cardinality, and by Gödel's general idea of building richer domains from simpler ones via structured, definable expansions, we study two complementary constructions that formalize the separation between external size and internal completeness.

We present a novel derivation of the classical Leibniz series for π using complex integration techniques. Starting from the well-known Gregory-Leibniz series π/4 = Σ((-1)^n)/(2n+1), we develop an alternative derivation pathway through complex integration from 0 to i in the complex plane. Our approach systematically transforms the infinite series into a complex integral, ultimately expressing the result in terms of complex logarithms: π/4 = (1/2i)ln((1+i)/(1-i)) = (1/2i)ln(e^(iπ/2)). This derivation provides a direct connection between the Leibniz series and complex logarithmic functions through contour integration, offering new geometric and analytical insights into this classical result.

This paper provides an in-depth insight into basic operations with inequalities that are carefully crafted for the illustration of monotonicity and boundedness characterization of the general recursive radical sequence. Essentially such properties are necessary and sufficient in the proof of their convergence. Furthermore, other hybrid forms of this sequence type are extensively explored taking into consideration the leading term to appropriately obtain well-defined intervals, where convergence is permitted, using simple but highly effective and concise short proves without any recourse to the application of other mathematical techniques and particularly in general the use of prove by mathematical inductions which are best tailored for recursive sequences involving the square root. In furtherance to this approach, the paper gave an illustrative prove by showing an extension of the method to cubic radicals which are also recursively defined as a sequence of real numbers in terms of its leading term.

In this paper we study some basic properties of rough $I$-convergent double sequences in the line of D$\ddot{u}$ndar [8]. We also study the set of all rough $I$-limits of a double sequence and relation between boundedness and rough $I$-convergence of a double sequence.

The cardinality of sets is a model-dependent concept; it is not an absolute measure. In one model, it may not be the same as in another, even if both models are 'complete' within their respective universes. This relativity is central to modern set theory and explains why cardinality and CH are tied to the framework of the model. Thus the cardinality of real numbers is contingent upon the chosen axiomatic system and the specific models of set theory. According to the diagonal argument within the Zermelo-Fraenkel (ZF) framework, real numbers are non-denumerable. However, their precise cardinality remains a topic of debate, varying with different extensions of the ZF axioms. For example, under the inner-model axiom 'V-ultimate L', the Continuum Hypothesis (CH) holds true. In contrast, it is negated if Martin's Axiom is adopted. Additionally, there is an orthogonal view that assumes there are myriad mathematical universes, some in which the continuum hypothesis is true and others in which it is false, but all equally worth of exploring. This model-dependent viewpoint is underlined by the Löwenheim-Skolem theorem, which asserts that no theory can have only non-denumerable models. Accepting any form of axiomatization implies that terms like 'denumerable' and 'non-denumerable' are relative. This necessitates the acceptance of the relativization of cardinal numbers, as initially suggested by Skolem and Carnap. To investigate this uncertainty in cardinality, we analyzed two analogous sets, one within the frameworks of the ZFC (ZF and Axiom of Choice (AC)) model, and another within Wang's Σ model, which is different from ZFC. Both sets are formed following Gödel idea of building larger sets starting from smaller ones. The first set is created from the set S p , which comprises all closed intervals with rational endpoints. Then in every interval marked is one irrational number, creating set S m that contains all rational endpoints from set S p , and irrational numbers marked in each interval. A key attribute of set S m in the ZF framework is that it is everywhere dense, and as it will be shown it is complete. Set S p encompasses all possible intervals with rational endpoints. This characteristic enables to identify or construct any Cauchy sequence representing any number, by identifying all

In this paper we study some basic properties of rough $I$-convergent double sequences in the line of D$\ddot{u}$ndar [8]. We also study the set of all rough $I$-limits of a double sequence and relation between boundedness and rough $I$-convergence of a double sequence.

A fundamental problem in mathematics education is “What can we do for students to enhance their mathematical ability and achievement and how can we do it?” This paper searches for a promising solution to this problem. To that end, the author focuses on the lesson study approach, specifically adopting the same approach adopted by the mathematician George Polya, who reflected on his experience and described his methods of mathematical problem solving. As a mathematics educator, the author shares his experience of two types of lesson study of primary school mathematics, and proposes a framework of a dynamic cycle in the lesson study of school mathematics for promoting the professional development of mathematics teachers and educators. This professional development may improve the mathematical ability and achievement of students at schools.

In this paper, we present the concept of statistically (λ-statistically) convergent sequences for rough variables. Furthermore, the relation between convergence statistically in trust and converges λ −statistically in trust is given. Also, some properties of statistically (λ-statistically) convergent sequences are discussed. In addition, we introduce statistically Cauchy sequence in rough spaces.

Two concepts—one of statistical convergence and the other of de la Vallée-Poussin mean—play an important role in recent research on summability theory. In this work we define a new type of summability methods and statistical completeness involving the ideas of de la Vallée-Poussin mean and statistical convergence in the framework of probabilistic normed spaces.

The purpose of this article was to evaluate the impact of SMASSE Program on students’ attitude and academic performance in secondary schools in Kenya. The government of Kenya and Japan jointly initiated SMASSE INSET program since 2004 with the aim of changing teachers’ attitude and teaching approaches of science and mathematics which was expected to translate into changing students’ attitude hence improving their academic achievements. Teachers were expected to apply the new skill ‘ASEI-PDSI’ in the curriculum implementation at school level, while the QASO and head teachers were expected to supervise its implementation. The objective of the study was to evaluate the extent to which the training has changed the teachers’ teaching approaches and students’ attitude towards the teaching of biology in secondary schools. The study was conducted in Bomet district in Kenya. A sample of 360 from three students, 24 science HODs, 24 Biology SMASSE teachers and 24 public secondary schools were ...

The purpose of the study was an evaluation of the effectiveness of SMASSE program in performance of science and mathematics in primary schools in Kenya. SMASSE was introduced due to consistently poor performance in science and mathematics became a matter of serious concern (waititu & Orando 2009). Although dismal performance in these subjectshad already been accepted as the norm in some schools, the government of Kenya and otherstakeholders felt there had to be an intervention, hence the Strengthening of Mathematics andScience in primary schools (Waititu & Orando 2009). The INSET Curriculum was thus designed to improve andstrengthen the teacher’s competent in the teaching of Science and Mathematics subjects. The programactivities are centred on the ASEI (Activity, Student, Experiment, and Improvisation) & PDSI (Plan, Do, See and Improve) approach, which emphasize on learner–centered preparation and presentation of lessons. The study was an evaluation using documented data that shows...

In this work it is shown that semi-orthogonal functions are suitable tool to obtain new results about some classic topics of Orthogonal Polynomials on the unit circle. More precisely, it is raised the study of the orthogonality mesure through the zeros of semi-orthogonal functions. Besides, it can be chosen a suitable basis relates with semi-orthogonal functions, so that the representation of the multiplication operator which can be obtained is a five-diagonal matrix. 2000 Mathematics Subject Classification. 42C05.

In this paper we introduce the notion of rough statistical convergence in the fuzzy setting, which generalizes rough convergence of sequences of fuzzy numbers. We define the set of rough statistical limit points of a sequence of fuzzy numbers and prove some results associated with these notions. MSC 2010. 40A05, 03E72.

The purpose of the study was an evaluation of the effectiveness of SMASSE program in performance of science and mathematics in primary schools in Kenya. SMASSE was introduced due to consistently poor performance in science and mathematics became a matter of serious concern (waititu & Orando 2009). Although dismal performance in these subjectshad already been accepted as the norm in some schools, the government of Kenya and otherstakeholders felt there had to be an intervention, hence the Strengthening of Mathematics andScience in primary schools (Waititu & Orando 2009). The INSET Curriculum was thus designed to improve andstrengthen the teacher’s competent in the teaching of Science and Mathematics subjects. The programactivities are centred on the ASEI (Activity, Student, Experiment, and Improvisation) & PDSI (Plan, Do, See and Improve) approach, which emphasize on learner–centered preparation and presentation of lessons. The study was an evaluation using documented data that shows...

The notion of statistical convergence was defined by Fast (Colloq. Math. 2:241-244, 1951) and over the years was further studied by many authors in different setups. In this paper, we define and study statistical τ-convergence, statistically τ-Cauchy and S * (τ)-convergence through de la Vallée-Poussin mean in a locally solid Riesz space. MSC: 40A35; 40G15; 46A40

The importance of mathematics education can best be acknowledged through the multiple roles it plays both in the life and development of an individual and of the society, hence the need to make mathematics education more accommodating of student needs. In the attempt to improve performance in mathematics, it is no longer a matter of having trained teachers in a school but teachers whose classroom practices can enhance student achievement. Quality teaching and learning is reflected in high student achievement in an increasingly competitive society where good performance in mathematics is imperative as a prerequisite for any significant advancement. The Kenyan government has continued to intervene so as to enhance achievement in mathematics. One major intervention by the government is the Strengthening Mathematics and Science in Secondary Education (SMASSE) project. This joint initiative between the governments of Kenya and Japan was intended to provide quality In-Service Training (INSET) for teachers in order to enhance teaching and learning hence, achievement in mathematics and sciences. Teachers have had to keep abreast by attending the INSET. Since the INSET's launch, significant student achievement particularly in mathematics is yet to be realized in national examinations. The purpose of this study was to investigate the application of the principles of Activity-Student-Experiment-Improvisation/Plan-Do-See-Improve (ASEI/PDSI) by mathematics teachers in secondary schools of Nakuru District, with the aim of facilitating greater understanding about the use of principles for enhanced student achievement. Specific objectives of the study were: to assess school preparedness in terms of resources; to investigate how actual instructional sessions are conducted, and to determine prevailing views on ASEI/PDSI principles. Reviewed literature covered aspects of the objectives of this study. The descriptive survey design was used for this study whose target population was 4,040 subjects among them 45 principals, 45 mathematics Heads Of Department (HOD), 118 mathematics teachers and 3,826 form three students. Convenience, stratified and simple random sampling techniques were used to select a total sample of 540 respondents (10 secondary schools, 10 principals, 10 HODs, 40 teachers and 480 students. Questionnaires, interview schedules and class observation schedules were used to collect data. Piloting of instruments was done and the split-half method followed by the Spearman-Brown prophecy formula was used to determine the reliability of the instruments. Instrument validity was determined through appraisal by experts. Data obtained from the study was analyzed using Statistical Package for Social Sciences (SPSS) computer software, version 17.0 and presented using simple descriptive statistics such as tables, graphs, charts and figures. Data obtained through interviews was reviewed, transcribed, organized into coherent categories and coded for analysis using the SPSS computer software and then presented and discussed thematically. Findings from the study revealed that the schools had adequate, professional and SMASSE trained teachers, school facilities and teaching-learning resources were fairly adequate while mathematics instructional sessions were teacher-dominated with little or no active involvement of students. Prevalent during lessons was the use of text books and the chalkboard, lessons lacked extensive student activities. Application of ASEI/PDSI principles was invisible, teachers' and students' attitude towards mathematics and ASEI/PDSI principles was relatively positive albeit factors that hinder their application.

The main purpose of this work is to define rough statistical convergence in probabilistic normed spaces. We have proved some basic properties as well as some examples which shows this idea of convergence in probabilistic normed spaces is more generalized as compared to the rough statistical convergence in normed linear spaces. Further, we have shown the results on sets of statistical limit points and sets of cluster points of rough statistically convergent sequences in these spaces.

Mathematical and pedagogical awareness is crucial for a teacher to notice, articulate, interpret aspects of classroom practice, and make on the spot decisions. Awareness is related to teacher knowledge and is rooted in the context of the actual classroom practice. Mason (2002) talks about different levels of awareness both in mathematics and in mathematics teaching and relates them to the process of noticing that involves systematic reflection on acts or issues. A number of studies at both pre-service and in-service levels that have been published in this Journal of Mathematics Teacher Education [JMTE] focus on what the teachers ''notice'' in a classroom while teaching or reflecting on it and investigate ways that can support them in the process of developing this ability (e.g., Mellon 2011; Scherer and Steinbring 2007). Approaches that seem to be effective in teacher education and professional development for supporting the development of teacher awareness include teacher interaction and collaboration and the use of theoretical tools in analyzing and studying mathematics teaching. This theme of teacher awareness is reflected in the articles of this issue of JMTE. The three articles in the research category and one article in the Mathematics Teacher Education around the World category provide further insights into the nature and development of mathematics teachers' awareness or noticing during practice. The article by Jimmy Sherrer and Mary Kay Stein and the article by Shari Stockero and Laura Van Zoest focus on teachers' noticing of critical classroom interactions that involves students' contributions and teachers' reactions. Sherrer and Stein propose and evaluate an intervention where a coding scheme was given to the teachers to analyze teaching while Stockero and Van Zoest examine teachers' actions and decisions in the context of actual teaching when pivotal teaching moments emerge. The article by Jennifer Tobias focuses on the development of prospective teachers mathematical awareness related to the fraction concept in the context of a teaching experiment that allowed mathematical communication and argumentation. Finally, the article by Su Liang, Sarah Glaz, Thomas DeFranco, Charles Vinsonhaler, Robin Grenier, and Fabiana Gardetti reports on mathematics teacher education and professional development in China and indicates ways that mathematical and pedagogical awareness of

The purpose of the study was an evaluation of the effectiveness of SMASSE program in performance of science and mathematics in primary schools in Kenya. SMASSE was introduced due to consistently poor performance in science and mathematics became a matter of serious concern (waititu & Orando 2009). Although dismal performance in these subjectshad already been accepted as the norm in some schools, the government of Kenya and otherstakeholders felt there had to be an intervention, hence the Strengthening of Mathematics andScience in primary schools (Waititu & Orando 2009). The INSET Curriculum was thus designed to improve andstrengthen the teacher’s competent in the teaching of Science and Mathematics subjects. The programactivities are centred on the ASEI (Activity, Student, Experiment, and Improvisation) & PDSI (Plan, Do, See and Improve) approach, which emphasize on learner–centered preparation and presentation of lessons. The study was an evaluation using documented data that shows...

The purpose of this article was to evaluate the impact of SMASSE Program on students’ attitude and academic performance in secondary schools in Kenya. The government of Kenya and Japan jointly initiated SMASSE INSET program since 2004 with the aim of changing teachers’ attitude and teaching approaches of science and mathematics which was expected to translate into changing students’ attitude hence improving their academic achievements. Teachers were expected to apply the new skill ‘ASEI-PDSI’ in the curriculum implementation at school level, while the QASO and head teachers were expected to supervise its implementation. The objective of the study was to evaluate the extent to which the training has changed the teachers’ teaching approaches and students’ attitude towards the teaching of biology in secondary schools. The study was conducted in Bomet district in Kenya. A sample of 360 from three students, 24 science HODs, 24 Biology SMASSE teachers and 24 public secondary schools were ...

The purpose of the study was an evaluation of the effectiveness of SMASSE program in performance of science and mathematics in primary schools in Kenya. SMASSE was introduced due to consistently poor performance in science and mathematics became a matter of serious concern (waititu & Orando 2009). Although dismal performance in these subjectshad already been accepted as the norm in some schools, the government of Kenya and otherstakeholders felt there had to be an intervention, hence the Strengthening of Mathematics andScience in primary schools (Waititu & Orando 2009). The INSET Curriculum was thus designed to improve andstrengthen the teacher's competent in the teaching of Science and Mathematics subjects. The programactivities are centred on the ASEI (Activity, Student, Experiment, and Improvisation) & PDSI (Plan, Do, See and Improve) approach, which emphasize on learner-centered preparation and presentation of lessons. The study was an evaluation using documented data that shows the performance of Science and Mathematics. The study adopted the Kirkpatrickmodel of training evaluation. Although there was some significant improvement in performance of Science and Mathematics subjects, a lot need to be done to improve the attitude of teachers. It clearly emerged that, majority of teachers are coerced to attend the INSET training. Few teachers are willing to beobserved by others during teaching and learning while the modern technology using ICT is rarely used.

In [1] we uniquely introduced ultra exponential functions (uxpa) and defined next step of the serial binary operations: addition, multiplication and power. Also, we exhibited the topic of limit summability of real functions in [2,3]. In this paper, we study limit summability of the ultra exponential functions and prove some of their properties. Finally, we pose an unsolved problem about them.

The topic of gamma type functions and related functional equation f(x+1) = g(x)f(x) has been seriously studied from the first half of the twentieth century till now. Regarding unique solutions of the equation the asymptotic condition lim x→∞ g(x+ w) g(x) = 1, for each w > 0, is considered in many serial papers including R. Webster’s article (1997). Motivated by the topic of limit summability of real functions (introduced by the author in 2001), we first show that the asymptotic condition in the Webster’s paper can be reduced to the convergence of the sequence g(n+1) g(n) to 1, and then by removing the initial condition f(1) = 1 we generalize it. On the other hand, Rassias and Trif proved that if g satisfies another appropriate asymptotic condition, then the equation admits at most one solution f , which is eventually log-convex of the second order. We also sow that without changing the asymptotic condition for this case, a uniqueness theorem similar to the Webster’s result is als...

Limit summability of real functions was introduced by M.H. Hooshmand in 2001. In order to study derivation of the limit summand function, he has introduced a functional sequence corresponding to a given function f with D f ⊇ N * that is related to the Euler-type constants. In the way, we prove two main criteria for its convergence together with an extensive inequality between the limit summand function and the generalized Euler-type constants. The main inequality is also extended whenever f is a convex or concave function. Among other things, we obtain some inequalities for many special functions such as the gamma, digamma and zeta functions.

The word tetration (was coined by Reuben Louis Goodstein) is the next hyper operator after exponentiation, and is defined as iterated exponentiation. The first uniqueness theorem for approximation of tetration was proved by the author in 2006. As a consequence of the researches the ultra power (a x), ultra exponential (uxp a (x)) and infra logarithm (Iog a (x)) functions have been uniquely introduced and studied by many recent papers. They are unique solutions of the functional equations f (x) = a f (x−1) and the Ablel functional equation f (a x) = f (x) + 1, under some conditions. In this paper we want to uniquely introduce ultra exponential functional sequence, give all the next hyper operators and obtain the serial binary operations: multiplication, power, ultra power, ultra power of order 2, and so on. Finally, we study many important mathematical analysis properties of ultra powers and ultra exponential functional sequences.

Let f be a real (or complex) function with domain D f containing the positive integers. We introduce the functional sequence {fσ n (x)} as follows:fσ n (x) = xf (n) + P n k=1 (f (k) − f (x + k)) and say that the function f limit summable at the point x0 if the sequence {fσ n (x0)} is convergent, (fσ n (x0) → fσ(x0)) as n → ∞, and we call the function fσ(x) as the limit summand function (of f). In this article, we first give a necessary condition for the limit summability of functions and present some elementary properties. Then we prove some tests about limit summability of functions and consider the relation between f (x) and fσ(x). One of the main theorems in this paper gives a uniqueness conditions for a function to be a limit summand function. Finally, as a consequence of this theorem we deduce a generalization of a result due to Bohr-Mollerup [1]. 1 Preliminaries Definitions and theorems in this article are for complex functions, except when the real case is explicitly mentioned. In general we assume f : D f → C, where D f ⊆ C. In the real case we take the function f : D f → R, where D f ⊆ R. For a function with domain D f ,we put Σ f = {x|x + N * ⊆ D f };

tions on processes and objects as different sides of the same coin', Educational Studies in Mathematics 22, 1-36 Tall, D. (1999) 'Reflections on APOS theory in elementary and advanced mathematical thinking', in Zaslavsky, 0. (ed), Proceedings of the 23rd Conference of the

In this paper, we propose extensions for the classical Kummer test, which is a very far-reaching criterion that provides sufficient and necessary conditions for convergence and divergence of series of positive terms. Furthermore, we present and discuss some interesting consequences and examples such as extensions of the Olivier’s theorem and Raabe, Bertrand and Gauss’s test.

The space of linearly recursive sequences of complex numbers admits two distinguished topologies. Namely, the adic topology induced by the ideal of those sequences whose first term is 0 and the topology induced from the Krull topology on the space of complex power series via a suitable embedding. We show that these topologies are not equivalent.

The paper describes the results of a research study aimed at preparation of preservice primary school teachers for inquiry based m athematics education (IBME). The teaching experiment involved pre-service teache rs’ work using a variety of techniques (with strictly formulated questions and open problems) in various environments suitable for this approach. We concent rated on pre-service teacher content knowledge and ability to apply the gained k nowledge in problem solving.

In Japan, assessment in classroom teaching cannot be considered apart from classroom lessons. This paper describes typical mathematics classroom situations and some of the developments in classroom assessment underway in Japan. The first section of the paper discusses classroom teaching, including whole-class instruction, discussion, objectives, and the importance of lesson plans and records. Next, assessment practices during and after lessons are discussed. Two long-term research projects to assess higher-order thinking are then considered. The Open-End Approach to Teaching Mathematics Project included three types of open-ended mathematics problems: how to find rules or relations, how to classify, and how to measure. Students' achievement was evaluated in terms of fluency, flexibility, and originality. The Developmental Treatment of Mathematics Problems Project had features similar to the Open-End project, except that students formulated or posed problems of their own. Student ...

Improving the quality of mathematics and science (M&S) education by improving the quality of teaching of teachers is the main approach of the technical cooperation of JICA in the basic education sector1 up until now (JICA IFIC 2007b). Characteristics of major approaches in M&S education cooperation have been analyzed to draw some lessons learnt for project formulation and implementation (JICA 2004, JICA IFIC 2007b). Analyses based on the perspectives of capacity development (CD) of projects for M&S education and in-service teacher training (INSET) are also reported (Mabuchi and Yokozeki 2004、JICA IFIC 2006). The Strengthening of Mathematics and Science in Secondary Education (SMASSE) Project implemented in Kenya is regarded as one of the most successful projects in the M&S education cooperation in JICA. Some of the success factors of SMASSE are already reported (JICA Social Development Cooperation Department 2002, JICA IFIC 2007a). Taking SMASSE as a case study, this paper analyzes ...

The strengthening mathematics and science education in-service education and training (SMASE INSET) in primary school has been initiated to improve pedagogical methods used by teachers teaching mathematics and science in public schools. It also addresses issues of attitude towards both teachers and pupils both teachers and pupils, how the content can be delivered within speculated time as well as use of resources mobilization and management. The proposed study was to establish the perceptions of teachers in the implementation of SMASE INSET in public primary schools; because good performance in our primary schools depends on teachers using guiding principles of SMASE INSET. Their perception has shed light so as to improve the implementation of SMASE. The Activity Student Experiment Improvisation (ASEI-PDSI) paradigm of the trends in the teaching and learning science and mathematics and according to the finding of several, need surveys. The study employed survey research design and u...

A non-terminating sequence like 31, 331, 3331, 33331, … starts with first seven terms as prime numbers, while the 8th term, which is 333333331, can be expressed as 17 x 19607843. Using Fermat’s Little Theorem, it can be easily proved that there are many more terms in this sequence that are not prime numbers. This paper puts forward a solution to find factors of composite numbers in all such sequences without using Fermat’s Little theorem or divisibility tests. The solution uses a prime number only once to scan all the terms of the unending sequence together, to check if any term is divisible by that prime number instead of checking every term separately, hence reduces the computational complexity. The solution finds the smallest number of the sequence which is divisible by a particular prime number and also proves that it cannot be assumed that all the terms of such sequences will be prime numbers.

The following text includes the preface and a short preview of all chapters of the book “On Research in Mathematics Teacher Education. From a Study of Teaching Practices to Issues in Teacher Education”. This book is a product of Thematic Group 3 “From a Study of Teaching Practices to Issues in Teacher Education” of the First Conference of the European Society for Research in Mathematics Education (CERME 1), held in Osnabrück, Germany, from 27 th till 31 st August, 1998. The editors gratefully thank Elmar Cohors-Fresenborg and Inge Schwank from the University of Osnabrück for their collaboration.

Over the years performance of mathematics at secondary level has dropped significantly causing a major concern to society. It is against this background that this study sought to investigate the effect of Strengthening Mathematics and Sciences in Secondary Education, In-Service Education Training project on performance of mathematics at Kenya Certificate of Secondary Examinations, with regard to the subject content, pedagogy/methodologies, attitude and teaching-learning material. The study area was in Kisii Central District in Kisii County. The sample size comprised of 20 secondary schools, 41 mathematics teachers and 380 students. Both probability and nonprobability Sampling techniques were adopted to determine sample size. Qualitative data were collected through interview and document analysis. Questionnaires were used to collect quantitative data from students, teachers of mathematics and Mathematics Head of Departments. Quantitative data were analyzed by both descriptive and inferential statistics (chi-square). Qualitative analysis involved categorization and development of themes. The research found out that Strengthening Mathematics and Sciences Secondary Education, In-Service Education Training project had significantly improved performance of mathematics. The study established that Strengthening Mathematics and Sciences Secondary Education, In-Service Education Training has changed attitude of teachers and students, improved Pedagogical Content Knowledge and Teaching/Learning Materials, and thus some improvement in performance of mathematics. The study will be useful to the mathematics teachers in evaluating their work against the SMASSE INSET methodologies in relation to their performance. The study will also be useful to DQASO's for their supervision processes from one school to another. The findings should also

In May 1999, the International Conference on Mathematics Teacher Education (ICMTE) was held in Taipei, Taiwan. The conference program consisted of plenary talks, research reports, panel discussions, and a workshop on multimedia interactive learning environments in teacher education. This report highlights issues raised by presenters during the conference. PERSPECTIVES ON TEACHER EDUCATION Cooney opened the conference with a plenary address in which he considered various contradictions within teacher education. He noted that prospective teachers often enter teacher education programs in search of teaching methods that help ensure certainty and predictability in the classroom. These expectations are in contrast to reform-oriented teacher education programs that encourage reflection, highlight attention to context, and characterize mathematics and its teaching as problematic. Cooney emphasized that mathematical activities that integrate content and pedagogy in a reflective way can serve as powerful entry points for preservice teachers' reconceptualization of the enterprise of teaching mathematics. Lerman provided a review of different research perspectives on mathematics teacher education, particularly as they related to teachers' beliefs and the contexts in which teachers teach. He claimed that the lack of an adequate or appropriate theory impedes progress in mathematics teacher education. Lerman suggested that theories that address the complexity of motives, goals, knowledge/power, and the forming of identity in social practices, need to be included in the study of teacher education.

This paper discusses the conceptions, practices and reflections about practices of a mathematics teacher, Maria, with respect to classroom communication and their change during the activity of a collaborative project involving a researcher and two other mathematics teachers. The case study of this teacher, who teaches at grades 5-6, draws on interviews and participant observation of the collaborative project meetings. The results show the relevance of the project to develop the teacher's understanding of communication issues in her classroom, putting her practices under scrutiny, and developing richer communication processes between her and her students.

Perennial poor performance in Mathematics at National examinations has elicited concern by the Government of Kenya. This dismal performance was attributed to broad curriculum, lack of facilities, inadequate staffing, poor pedagogical approaches and lack of hands on activities. Mathematics, which is crucial in Kenya's realization of vision 2030, hampers its achievement. Studies have shown that interventions so far adopted had yielded negligible impact. The acronym ASEI stands for Activity, Students, Experiment, Improvisation whereas PDSI stands for Plan, Do, See and Improve, hence the ASEI-PDSI movement. Therefore, there was need to establish the extent of practice and implementation of ASEI-PDSI pedagogical approach in Vihiga County. This work established the preparedness of Stakeholders in their roles to implement the ASEI-PDSI movement; and the extent of practice of this approach. The data was collected using questionnaires and interview schedules; with principals, Heads of Mathematics department, teachers and Education officers as the respondents. The data was analysed statistically. The study ascertained the practice of PDSI approach, rated at 4.332 on a 5-point Likert scale, impacting positively on performance and attitude. There were improved teaching methodologies, improvisation of resources, and use of lesson rationale in Mathematics. It was noted that 95.89 % of the teachers employed the ASEI approach. Also, 67.12 % and 10.96 % of the teachers improvised teaching and learning aids in the absence of the conventional ones mostly and always respectively. Teachers engaged students in varied activities in class and at home. However, 27.40 % rarely improvised since 35.71% of the teachers were new in the profession or experienced burn out syndrome due to overstaying in a station or missed critical SMASSE (strengthening of mathematics and science in secondary education) cycles. The study showed that 97.26 % of the teachers had schemes of work, affirming the importance of the tool for planning in any teaching and learning setup. The schemes of work were always used by 45.83 % and mostly used by 40.28 % of the sample respondents. However, 12.50 % used them rarely. These results need to inform stakeholders on effectiveness of the SMASSE in-service education training and make a follow up with a view of strengthening the Programme.

The Strengthening of Mathematics and Science in Secondary Education (SMASSE) initiative in Kenya was in response to the continuous poor performance in Mathematics and Science despite the effort of employing qualified teachers, increasing salary, providing equipments and constructing laboratories and Mathematics classrooms. SMASSE-INSET program was made available and compulsory to all serving Mathematics and Science teachers in the country in 2004 at the district level. The Activity, Student, Experiment and Improvisation (ASEI) through Planning, Doing, while Seeing then Improving (PDSI) approach introduced through the SMASSE project is an attempt to improve performance. The purpose of this study was to evaluate the attitudes of teachers toward SMASSE-INSET and establish the relationship between attitudes and implementation of SMASSE-INSET resolutions. Evaluation research design was adopted and the target population included all mathematics teachers of Nandi Central District secondary schools. Sixty (60) mathematics teachers participated in the study. The data were collected using questionnaires and analyzed using descriptive and inferential statistics. The findings of the study show that Mathematics teachers tend to have a positive attitude toward the SMASSE-INSET project and despite the challenges they face, they work hard to implement ASEI/PDSI pedagogy, which is a studentcentered approach. Teachers with a more positive attitude toward SMASSE-INSET program tend to apply the ASEI/PDSI pedagogy.

The Strengthening of Mathematics and Science in Secondary Education (SMASSE) initiative in Kenya was in response to the continuous poor performance in Mathematics and Science despite the effort of employing qualified teachers, increasing salary, providing equipments and constructing laboratories and Mathematics classrooms. SMASSE-INSET program was made available and compulsory to all serving Mathematics and Science teachers in the country in 2004 at the district level. The Activity, Student, Experiment and Improvisation (ASEI) through Planning, Doing, while Seeing then Improving (PDSI) approach introduced through the SMASSE project is an attempt to improve performance. The purpose of this study was to evaluate the attitudes of teachers toward SMASSE-INSET and establish the relationship between attitudes and implementation of SMASSE-INSET resolutions. Evaluation research design was adopted and the target population included all mathematics teachers of Nandi Central District secondary...