The influence of religious beliefs to several leading mathematicians in early Soviet years, especially among members of the Moscow Mathematical Society, had drawn the attention of militant Soviet marxists, as well as Soviet authorities. The issue has also drawn significant attention from scholars in the post-Soviet period. According to the currently prevailing interpretation, reported purges against Moscow mathematicians due to their religious inclination are the focal point of the relevant history. However, I maintain that historical data arguably offer (...) reasons to cast reasonable doubts on this interpretation. In this paper, by reviewing the relevant literature, I raise some methodological and philosophical concerns, in an attempt to contribute to a better understanding of the issue. I maintain that an efficient line of reasoning is to discuss issues in the context of their making, taking into consideration the specific features of each era’s culture. Thus, by focusing on P.A. Nekrasov’s case, I attempt to point to an alternative interpretation, in which the different treatment of religious inclined mathematicians by Soviet authorities is explained in the context of the ideological confrontation between two contrasting worldviews, as part of the ongoing class war in the several phases of Soviet history. (shrink)

This paper challenges the first Gettier counterexample to the tripartite account of knowledge. Noting that 'the man who will get the job' is a description and invoking Donnellan's distinction between their 'referential' and 'attributive' uses, I argue that Smith does not actually believe that the man who will get the job has ten coins in his pocket. Smith's ignorance about who will get the job shows that the belief cannot be understood referentially, his ignorance of the coins in his pocket (...) shows that it cannot be understood attributively. An explanation for why Smith appeared to have justified true belief is given by distinguishing between 'belief' and 'belief in truth'. Smith believes the sentence 'the man who will get the job has ten coins in his pocket' to be true (he mistakenly believes that Jones will get the job, of whom he knows that he has ten coins in his pocket) (hence his 'belief'), the sentence is true (hence 'truth'), and he has sufficient reason to assent to it (hence his 'justification'). But he does not believe the proposition expressed. Hence he does not know it either. (shrink)

The period from 1900 to 1935 was particularly fruitful and important for the development of logic and logical metatheory. This survey is organized along eight "itineraries" concentrating on historically and conceptually linked strands in this development. Itinerary I deals with the evolution of conceptions of axiomatics. Itinerary II centers on the logical work of Bertrand Russell. Itinerary III presents the development of set theory from Zermelo onward. Itinerary IV discusses the contributions of the algebra of logic tradition, in particular, Löwenheim (...) and Skolem. Itinerary V surveys the work in logic connected to the Hilbert school, and itinerary V deals specifically with consistency proofs and metamathematics, including the incompleteness theorems. Itinerary VII traces the development of intuitionistic and many-valued logics. Itinerary VIII surveys the development of semantical notions from the early work on axiomatics up to Tarski's work on truth. (shrink)

This is not a mathematics book, but a book about mathematics, which addresses both student and teacher, with a goal as practical as possible, namely to initiate and smooth the way toward the student’s full understanding of the mathematics taught in school. The customary procedural-formal approach to teaching mathematics has resulted in students’ distorted vision of mathematics as a merely formal, instrumental, and computational discipline. Without the conceptual base of mathematics, students develop over (...) time a “mathematical anxiety” and abandon any effort to understand mathematics, which becomes their “traditional enemy” in school. This work materializes the results of the inter- and trans-disciplinary research aimed toward the understanding of mathematics, which concluded that the fields with the potential to contribute to mathematics education in this respect, by unifying the procedural and conceptual approaches, are epistemology and philosophy of mathematics and science, as well as fundamentals and history of mathematics. These results argue that students’ fear of mathematics can be annulled through a conceptual approach, and a student with a good conceptual understanding will be a better problem solver. The author has identified those zones and concepts from the above disciplines that can be adapted and processed for familiarizing the student with this type of knowledge, which should accompany the traditional content of school mathematics. The work was organized so as to create for the reader a unificatory image of the complex nature of mathematics, as well as a conceptual perspective ultimately necessary to the holistic understanding of school mathematics. The author talks about mathematics to convince readers that to understand mathematics means first to understand it as a whole, but also as part of a whole. The nature of mathematics, its primary concepts (like numbers and sets), its structures, language, methods, roles, and applicability, are all presented in their essential content, and the explanation of non-mathematical concepts is done in an accessible language and with many relevant examples. (shrink)

This study aims to find out how high the level and trends of student misconceptions experienced by high school students in Indonesia. The subject of research that is a class XI student of Natural Science (IPA) SMA Negeri 1 Anjatan with the subject matter limit function. Forms of research used in this study is a qualitative research, with a strategy that is descriptive qualitative research. The data analysis focused on the results of the students' answers on the test essay (...) subject matter limit function with the number of students by 16 (sixteen). Data collection techniques used are shaped test methods essay, interview method to students who have misconceptions, and methods of documentation of the test answers. Examination of the validity of the data using a triangulation technique that compares the data written test results with data from interviews. The results of this study can be described as follows; (1) The level of misconceptions experienced by students belonging to the category of low, amounting to 12.18%. However, students who do not understand the concept quite high at 40.38%, and the others are students who understand the concept that is equal to 47.44%. (2) The misconception most experienced students lie in subconcepts explain the meaning of limit function at one point through the calculation of values around that point, that is equal to 20.51%. The tendency misconceptions experienced by students is located on errors and operating concepts that misconceptions students that there should be no limit in the completion of the writing of the emblem and misconceptions about the limit students to conclude if the limit value of 0 is no limit to the value of the function. (shrink)

This study assessed the administration of punishment, students’ test anxiety, and performance in mathematics in secondary schools of Cross River State. Three null hypotheses were formulated following a correlational research design. Proportionate stratified random sampling technique was employed in selecting a sample of 2,554 respondents representing 5% of the population of 51,097 junior secondary school students distributed across 271 public secondary schools in Cross River State. Administration of Punishment Questionnaire (APQ) designed by the researchers, Test Anxiety Inventory (TAI) (...) developed by Dawood et al., (2016), and Mathematics Achievement Test (MAT) designed by Owan (2012), were used as instruments for data collection. The null hypotheses were tested using one-way MANOVA), Pearson Product Moment Correlation, and Multiple regression analyses with the aid of SPSS v23. Findings revealed amongst others that administration of punishment has a significant influence on students’ test anxiety and performance in mathematics F (6, 5098) = 24.149, p<.05; Wilk’s Λ = 0.945, partial η2 = .028 on a joint basis; administration of punishment has a significant influence respectively on students’ test anxiety F (3, 2550) = 22.697, p<.0005, partial η2 =.026 and performance in mathematics F (3, 2550) = 23.090, P<.0005, partial η2 =.026; there is an inverse significant relationship (r= -339, p<0.05) between students’ tests anxiety and performance in mathematics. It was recommended that the use of corporal punishment should be totally abolished on strict grounds that it hurts the psychological disposition of learners resulting in a dip in their performance consequently. (shrink)

Teaching in- effectiveness of most mathematics teachers has been attributed a negative attitude of the teachers and their inability to utilize modern ICT facilities in the teaching of the subject. The study was on the influence of teachers’ attitude towards ICT variables, the utilization of ICT facilities and the teaching effectiveness of mathematics teachers in public senior secondary schools in Cross River State. Two hypotheses were drawn from the stated research questions built around the dependent and independent variables. (...) Literature review was carried out according to the variables in the study. A survey design was adopted based on the descriptive nature of the study, while the random sampling technique was used. Two sets of structured questionnaires, titled; ‘Teacher Effectiveness Questionnaire’ (TEQ) and ‘ICT Evaluation Test Questionnaire’ (ICTETQ) were designed to obtain data from respondents. Data collected were analysed using the t-test statistic for the first hypothesis and Analysis of Variance (ANOVA) statistic for the last hypothesis. The hypotheses were tested at .05 levels of significance. The results of the findings were as follows: There is no significant influence of mathematics teachers’ attitude towards ICT facilities and their teaching effectiveness in public senior secondary schools in Cross River State; facilities utilization has no significant influence on mathematics teachers’ effectiveness in public senior secondary schools in Cross River State. Based on the findings, it was recommended among other things that basic training and other facilities like electricity be made available in schools to stimulate a positive attitude in Mathematics teachers and to able them (mathematics teachers), utilize the available ICT facilities and others facilities that should be provided in schools while at the same time teachers’ attitude should be channelled in the direction of imbibing and integrating ICTs into teaching so as to ensure an effective mathematics teaching in public senior secondary schools in Cross River State. (shrink)

This study assessed students’ perception of teachers’ effectiveness and learning outcomes in mathematics and economics in secondary schools of Cross River State, Nigeria. Two null hypotheses were formulated to direct the study. The factorial research design was adopted for the study. Cluster and purposive sampling techniques were however employed in selecting a sample of 1,800 students from the three education zones in Cross River State. “Students’ Perception of Teachers Effectiveness Questionnaire (SPTEQ)”, Mathematics Achievement Test (MAT), and Economics Achievement (...) Test (EAT) were used as instruments for data collection. Reliability estimates of .86, .91, and .81 were obtained for the three instruments using Cronbach alpha. Collected data were analysed using descriptive statistics while the research question was answered and the null hypothesis tested using Pearson Correlation matrix and multiple regression analyses (where applicable), with the aid of Minitab v18. Findings revealed amongst others that, there is a significant composite influence of students’ perception of teachers’ motivational skills, teaching methods, relationship with students, and communication level on learning outcomes in Mathematics (F = 142.46, p < .05) and Economics (F = 150.31, p < .05) respectively. It was recommended amongst others that secondary school teachers should adopt motivational techniques such as praises, rewards, and smiles, in strengthening the interest of learners during mathematics and Economics lessons. Keywords:. (shrink)

I claim that a relatively new position in philosophy of mathematics, pluralism, overlaps in striking ways with the much older Jain doctrine of anekantavada and the associated doctrines of nyayavada and syadvada. I first outline the pluralist position, following this with a sketch of the Jain doctrine of anekantavada. I then note the srrong points of overlaps and the morals of this comparison of pluralism and anekantavada.

The study determined the influence of innate mathematical characteristics on the number sense competencies of junior high school students in a Philippine public school. The descriptive-correlational research design was used to accomplish the study involving a nonrandom sample of sixty 7th-grade students attending synchronous math sessions. Data obtained from the math-specific Learning Style and Self-Efficacy questionnaires and the modified Number Sense Test (NST) were analyzed and interpreted using descriptive statistics, Pearson’s Chi-Square, and Simple Linear Regression analysis. The research (...) instruments and statistics were all validated and tested for reliability. The analysis revealed that the students are visual learners, had no or slight self-efficacy, and their number sense competency level is poor. They encountered difficulty in all the components and domains of the NST. Moreover, the students’ mathematical self-efficacy is significantly related and may influence their number sense competency level. Building upon the learners’ self-efficacy to further their understanding and skills in number sense is necessary. (shrink)

A new AI system, called AlphaGeometry, trained under synthetic data has demonstrated the ability to solve geometric problems at the International Olympiad level. This essay considers the fact that human abilities to learn and do math as well as many other tasks are increasingly augmented with AI. Clearly, smart technologies like AlphaGeometry are redefining a number of concepts and institutions such as learning, schools, education, teacher-student relationships, creativity, etc, which are so fundamental for what we’ve thought of as modern society, (...) economy, and culture. What are we preparing for in mathematics education in the face of the possibility of the continued emergence of disruptive technologies, signaling the beginning of the end, if a new reality of mathematics is not taken into account? (shrink)

The sudden shift in the education system during the pandemic and its subsequent evolution during the post-pandemic era have been pivotal in fostering significant educational development and growth. However, this paradigm shift has not been without challenges. This paper aims to investigate the challenges faced by 68 mathematics teachers in four public elementary schools in the Philippines. The respondents were purposively selected to answer the study’s instrument. Using a descriptive survey research methodology, this study explored the five domains in (...) teaching mathematics, namely planning lessons, executing lessons, assessment, mathematical knowledge, and using technology tools. This paper revealed that novice, experienced, and highly experienced teachers alike had moderate difficulty enriching mathematical knowledge and utilizing technology in teaching the subject matter. Moreover, teachers with more than five years’ experience had moderate difficulty in the five domains of teaching mathematics in the post-pandemic era. This finding emphasizes the widespread impact of the education shift for teachers of all levels of experience. The paper further emphasizes the importance of supporting experienced teachers during this transformative phase of formulating innovative learning activities and adapting to reimagined pedagogical concepts. Consequently, this study underscores the need for additional training interventions to enhance their mathematical knowledge and use of technology as an integral teaching tool. (shrink)

The study investigated the effects of computer-assisted instruction on mathematics achievement among secondary school students in Rivers State. Two research questions and two hypotheses guided the study. The design was quasi-experimental. The population of the study was 215 students in a senior secondary school Kpor in Gokana. The sample of the study was 35 students. The sample size was drawn using simple random sampling technique. The instrument used to collect data was multiple choice achievement test. The instrument (...) was validated by experts in computer science education and the researchers, while the reliability of the instrument was established at 0.81. Mean and standard deviation was used to analyze pre-test and post-test scores to answer research questions, while z-test was used to test the hypotheses at 0.05 significant level. The findings of the study showed that students taught with computer assisted instruction performed higher than those taught with traditional instructional methods. Findings also revealed that there is a significant difference between the pretest and post-test achievement scores of students taught Mathematics with computer assisted instructional method. The study recommended among others that government should integrate computer and educational software into the mainstream of mathematics curriculum from the junior secondary school level as this will help to prepare students for stellar performance in Mathematics at senior secondary level. (shrink)

The study investigated the effect of cognitive restructuring on junior secondary school mathematics test anxiety in Oshimili south L.G.A of Delta State. Two research questions and two hypotheses tested at 0.05 level of significance guided the study. Quasi-experimental research design was adopted for this study. The population for this study was a total of 1224 students. These comprised of all the JSS 2 students from Oshimili South Local Government Area of Delta State. Research sample consisted of 120 JSS (...) 2 students with high level of mathematics test anxiety selected through purposive sampling technique. The instrument adopted for this study was Maths anxiety rating scale-R. Data collected from the study were analyzed using mean and ANCOVA. Results obtained from the study indicated that Cognitive restructuring therapy (CRT) was effective on mathematics test anxiety of junior secondary school students. The results equally showed that Cognitive restructuring therapy was more effective on the female students’ mathematics anxiety than their male counterparts. Furthermore, the results indicated a significant difference in the post-test mathematics test anxiety mean scores of students treated with CRT and those in the control group. Also, there was significant difference in the post-test mathematics test anxiety mean scores of male and female students treated with CRT. Based on the findings of this study, some recommendations were noted. The researcher recommended, among others that Schools’ Guidance Counselors should be empowered to make good use of the cognitive restructuring technique, in counseling students who find it difficult to comprehend mathematics skills. (shrink)

The purpose of this study was to investigate the relationships between the students’ self-efficacy, test anxiety and performance on Mathematics of Saint Joseph College, Junior High School Department for SY 2019- 2020. The respondents of the study were randomly chosen junior high school students. Descriptive correlational method of research was used. The data gathered were statistically treated using the Excel and IBM SPSS programs, which summarize the percentage scores, weighted means, correlation coefficients, independent t-test values, and corresponding (...) p-values of relevant statistical tools. Findings indicated that, at average, the students were not confident on their mathematical self-efficacy, moderately anxious on their mathematical test anxiety and satisfactory on their mathematical performance. More findings showed that the following variables have significant relationships: mathematical anxiety and mathematical self–efficacy; mathematical self-efficacy and mathematical performance; and mathematical test anxiety and mathematical performance. The higher the self-efficacy level, the lower the degree of testanxiety; the higher the self-efficacy, the better the mathematics performance; and the lower the test-anxiety, the better the performance. The study recommends regular conduct of diagnostic tests on mathematical self-efficacy and test anxiety and design programs to address the enhancement of learners’ mathematical self-efficacy and reduce their mathematical test anxiety. The researcher recommends Mathematics Learning Development Program (MLDP) as the output of the study. (shrink)

The aim of this research was to x-ray some causes of poor performance of pupils in primary school mathematics. Specifically, the study examined the use of instructional materials and pupils’ academic performance in mathematics; parents’ socio-economic background and pupils’ academic performance in mathematics; compared the performance of private and public primary school pupils in mathematics; examined ways in which teachers contribute to pupils’ poor performance in mathematics. The study employed a correlational and quasi- (...) experimental research designs. A simple random sampling technique was used to select a sample of 270 pupils and 45 teachers drawn from nine primary schools in Akamkpa local government area of Cross River State. A questionnaire and a mathematics achievement test (MAT) were instruments used for data collection. The collected data was analysed using a simple percentage (%) and arithmetic mean (X). The findings of the study revealed that the use of instructional materials adequately led to pupils’ poor performance in mathematics; parents’ socio-economic status contributed to the pupils’ performance in mathematics; pupils in private primary school perform better than their colleagues in public schools in mathematics; and teachers contribute to the poor performance of pupils in mathematics. Based on this results, conclusions and recommendations were made. (shrink)

This study explores the needs of Senior High School (SHS) Mathematics Teachers in Cabanatuan City’s Schools Division Office, focusing on professional qualifications, familiarity with essential Mathematics knowledge, and employed pedagogical strategies. Using descriptive research approach, the study examines the correlation between respondent profiles, Mathematics knowledge, pedagogical approaches, and the adequacy of educational infrastructure. Results indicate that most respondents demonstrate excellent familiarity with essential Mathematics knowledge, predominantly employing assessment as a pedagogical strategy. No significant relationship exists (...) between respondent profiles and General Mathematics essential knowledge competence. Respondents highly rate the DepEd-issued Teacher’s Guide, and Mathematics materials’ availability and sufficiency in classrooms are satisfactory. The study recommends providing more usable Mathematics tools and materials, emphasizing targeted Mathematics-related seminars for SHS Mathematics teachers in public secondary schools within Cabanatuan City. (shrink)

Students do not necessarily use the definitions presented to them when determining examples or non-examples of given mathematical ideas. Instead, they utilize the concept image they carry with them as a result of experiences with such examples and nonexamples. Hence, teachers should try exploring students‟ images of various mathematical concepts in order to improve communication between students and teachers. This suggestion can be addressed through error analysis. This study therefore is a descriptive-qualitative type that looked into the errors committed by (...) senior high school students in dealing with mathematical functions, explored the underlying reasons for such errors, and provided recommendations on how students could learn to manage their errors and how teachers could realize how to manage students‟ errors. Initial findings showed that the students generally demonstrated mathematical, logical and strategic errors. Mathematical error was generally exhibited in their inability to use properties and operations properly. Logical error was exhibited in their false arguments, rearranging concepts, improper classifications, arguing cyclically, and using equivalent transforms. Strategic error was manifested in their not being able to distinguish patterns, lack of integral concept, and not being able to transform a word problem into symbols. Errors prevailed despite the students‟ positive view and confidence in mathematics not only because of poor image concepts but because of lack of exposure to certain important mathematical tasks. Further investigation on the other sources of these errors is therefore necessary. (shrink)

The study sought to identify and evaluate the classroom assessment thoughts, practices, and skills of secondary mathematics teachers in Bulacan. The study revealed that there are no significant relationships between teachers’ thoughts of classroom assessments and practices, and classroom assessment practices and skills. However, there is significant relationship between teachers’ thoughts of classroom assessments and skills. There are no significant differences between the teachers’ thoughts of classroom assessments and their age, educational attainment, teaching experience, number of years in teaching (...) mathematics and in-service training. There is a significant relationship between the teachers’ thoughts of classroom assessment and their classroom assessment skills. In the light of the findings, the researcher recommends that: Secondary mathematics teachers should adopt varied and innovative ways of assessing students’ learning in mathematics. School administrators should ensure that all teachers with appropriate certificates must take more training courses in assessment to improve their skills and use of desirable classroom assessment practices. Further research may focus on the relationship between mathematics teachers’ assessment skills and thoughts, practices and skills, and thoughts and practices. (shrink)

This study aimed to investigate the longitudinal trends in mathematical competencies of Grade 8 students in a public high school located in Zamboanga del Sur, Philippines. The study collected data over a period of six academic years, allowing for a comprehensive analysis of students' performance in 16 distinct mathematical competences of basic education curriculum. These topics include, but are not limited to, special products and factors, factoring, and basic concepts of probability. Using a quantitative research design, the study analyzed (...) both numerical and secondary data to identify patterns or trends in students' mathematical competencies. Trends were analyzed through regression analysis using slopes. The significance of these trends was then evaluated using t-tests for the standard error of slopes. The statistical analysis revealed significant upward trends in certain mathematical topics, indicating an improvement in the students' competencies over the six academic years. However, certain topics displayed downward or stable trends, indicating areas for improvement in the teaching methods used for these topics. The study highlights the importance of flexible and adaptable approaches to mathematics education. It emphasizes the need for constant evaluation and improvement of teaching methods to enhance student learning outcomes. The findings of this study provide valuable insights for educators and policymakers to guide evidence-based decision-making and improve the quality of mathematics education in the new normal setup. Implications and future research directions are incorporated in the study. (shrink)

This study developed an instrument that measures the attitude of Filipino high school students towards mathematics, with reliable predictors and factors. Using the responses of 300 high school students from Zamboanga Sibugay, the validity and reliability of the Mathematics Attitude Scale (MAS) was tested using Exploratory Factor Analysis (EFA) and reliability analyses. The EFA showed that four-factor structures of the instrument, regarding the mathematics attitude for high school students, explained 27.48% of the variance in (...) the pattern of relationships among the items. The Average Variance Extracted (AVE), Composite Reliability, and Cronbach’s Alpha coefficients were reported. They proved that the extracted constructs have obtained and satisfied convergent validity. Thirty-three items remained in the final questionnaire after deleting the twenty-seven items with factor loadings of less than 0.4 (Students’ Perceived Motivation and Support in Learning Mathematics: twelve items; Students’ Perceived Anxiety in Learning Mathematics: ten items; Students’ Perceived Self-Efficacy in Learning Mathematics: six items; and Teachers and Parents’ Influences to Students in Learning Mathematics: five items). This study has confirmed the four-factor structure of the MAS. Educators and researchers can use the MAS to better understand the attitudes of Filipino high school students towards mathematics. (shrink)

If there is an area of discourse in which disagreement is virtually absent, it is mathematics. After all, mathematicians justify their claims with deductive proofs: arguments that entail their conclusions. But is mathematics really exceptional in this respect? Looking at the history and practice of mathematics, we soon realize that it is not. First, deductive arguments must start somewhere. How should we choose the starting points (i.e., the axioms)? Second, mathematicians, like the rest of us, are fallible. (...) Their ability to recognize whether a putative proof is correct is not infallible. In most cases, disagreement over the correctness of a putative proof is, however, evanescent. Once an error is spotted and communicated, the disagreement disappears. But this is not always the case. Sometimes it is recalcitrant; that is, it persists over time. In order to zoom in on this type of disagreement and explain its very possibility, we focus on a single case study: a decades-long (1921-1949) controversy between Federigo Enriques and Francesco Severi, two prominent exponents of the Italian school of algebraic geometry. We suggest that the instability of the mathematical community to which they belonged can be explained by the gap between an abstract criterion of rigor and local criteria of acceptability. It is this instability that made the existence of recalcitrant disagreement over putative proofs possible. We do not condemn speculative mathematics but rather its pretense of being rigorous mathematics. In this respect, we show that the overly self-confident Severi and the more intuitive, visionary Enriques had a completely different attitude. (shrink)

This study aimed to construct structural equation model of students’ competence in mathematics through selected students profile variables. The structural model revealed interesting influence of the profile variables to the competency in mathematics. It can be conveyed that better mother’s work status, higher educational level expected to complete, more confident and did not repeat kinder, have better competency in mathematics. The four variables that directly influenced the competence variables were also influenced with other profile variables such as (...) family background. The family background and confidence level consistently had the highest total effect and indirect effect to the competence in mathematics. Hence, this model can serve as a guide in making programs in the classroom or curriculum in mathematics. (shrink)

The purpose of this study was to determine the form of teacher readiness and difficulty when implementing the mathematics curriculum in high school, measured from four teacher competency assessments. Schools in Indonesia are still 50% learning from home until 2022, this situation has an impact on the achievement of student learning outcomes. In 2020, the ministry conducted a survey of 4000 students and the results of the survey and out of 100% of the participants, 58% were of the (...) opinion that online learning was ineffective, 38% lacked attention from the teacher, 35% had poor communication, 62% had poor internet access, and 80% had difficulty learning mathematics online. The method in this research is descriptive qualitative. The technique of collecting data is by means of field observations, interviews and collecting documentary evidence. Data analysis techniques by presenting data, data reduction and intersecting data are drawn into conclusions. The number of people who were observed was 2 with a total of 58 and 13 people interviewed sources. Findings, difficulties arise from unprepared pedagogic competence and professional competence. Teachers have no self-constructed modules, difficulty writing symbols and mathematical proofs online. Students provide an assessment of teacher competence online, from 100% of respondents 80% disagree with pedagogic and 90% disagree with professional competence. This is due to the lack of readiness of the two competencies. In conclusion, online learning is expected by students and teachers to have their own modules. Students argue that the teacher's difficulties when writing symbols, providing examples of questions and proving mathematical formulas can be written in modules and modules can be given to students to study at home. (shrink)

The prediction potential of the model "emotional intelligence and self-reliance" to students' mathematical performance was investigated in this study. This research was conducted in the third and fourth quarters of the academic year 2021-2022. The quantitative research design, specifically comparative and regression analysis, was used in this study. The comparative design was utilized to assess the differences in emotional intelligence and self-reliance between male and female students, and the regression analysis was performed to see if the model "emotional intelligence and (...) self-reliance" can predict students' mathematical performance. In terms of emotional intelligence and self-reliance, the findings show no significant difference between male and female students. Furthermore, the student's emotional intelligence and self-reliance were strong predictors of mathematics performance. It implies that emotional intelligence and self-reliance are essential factors in better math learning. The study suggested that teachers may improve their students' emotional intelligence and self-reliance by integrating social and emotional learning programs into their classes. (shrink)

A basic thesis of Neokantian epistemology and philosophy of science contends that the knowing subject and the object to be known are only abstractions. What really exists, is the relation between both. For the elucidation of this “knowledge relation ("Erkenntnisrelation") the Neokantians of the Marburg school used a variety of mathematical metaphors. In this con-tribution I reconsider some of these metaphors proposed by Paul Natorp, who was one of the leading members of the Marburg school. It is shown (...) that Natorp's metaphors are not unrelated to those used in some currents of contemporary epistemology and philosophy of science. (shrink)

The previous Part I of the paper discusses the option of the Gödel incompleteness statement (1931: whether “Satz VI” or “Satz X”) to be an axiom due to the pair of the axiom of induction in arithmetic and the axiom of infinity in set theory after interpreting them as logical negations to each other. The present Part II considers the previous Gödel’s paper (1930) (and more precisely, the negation of “Satz VII”, or “the completeness theorem”) as a necessary condition for (...) granting the Gödel incompleteness statement to be a theorem just as the statement itself, to be an axiom. Then, the “completeness paper” can be interpreted as relevant to Hilbert mathematics, according to which mathematics and reality as well as arithmetic and set theory are rather entangled or complementary rather than mathematics to obey reality able only to create models of the latter. According to that, both papers (1930; 1931) can be seen as advocating Russell’s logicism or the intensional propositional logic versus both extensional arithmetic and set theory. Reconstructing history of philosophy, Aristotle’s logic and doctrine can be opposed to those of Plato or the pre-Socratic schools as establishing ontology or intensionality versus extensionality. Husserl’s phenomenology can be analogically realized including and particularly as philosophy of mathematics. One can identify propositional logic and set theory by virtue of Gödel’s completeness theorem (1930: “Satz VII”) and even both and arithmetic in the sense of the “compactness theorem” (1930: “Satz X”) therefore opposing the latter to the “incompleteness paper” (1931). An approach identifying homomorphically propositional logic and set theory as the same structure of Boolean algebra, and arithmetic as the “half” of it in a rigorous construction involving information and its unit of a bit. Propositional logic and set theory are correspondingly identified as the shared zero-order logic of the class of all first-order logics and the class at issue correspondingly. Then, quantum mechanics does not need any quantum logics, but only the relation of propositional logic, set theory, arithmetic, and information: rather a change of the attitude into more mathematical, philosophical, and speculative than physical, empirical and experimental. Hilbert’s epsilon calculus can be situated in the same framework of the relation of propositional logic and the class of all mathematical theories. The horizon of Part III investigating Hilbert mathematics (i.e. according to the Pythagorean viewpoint about the world as mathematical) versus Gödel mathematics (i.e. the usual understanding of mathematics as all mathematical models of the world external to it) is outlined. (shrink)

Mathematics has always been a core part of western education, from the medieval quadrivium to the large amount of arithmetic and algebra still compulsory in high schools. It is an essential part. Its commitment to exactitude and to rigid demonstration balances humanist subjects devoted to appreciation and rhetoric as well as giving the lie to postmodernist insinuations that all “truths” are subject to political negotiation. In recent decades, the character of mathematics has changed – or rather broadened: it (...) has become the enabling science behind the complexity of contemporary knowledge, from gene interpretation to bank risk. Mathematical understanding is all the more necessary for future jobs, as well as remaining, as ever, a prophylactic against the more corrosive philosophical views emanating from the humanities. (shrink)

J. N. (Jitendra Nath) Mohanty (1928–2023). -/- Professor J. N. Mohanty has characterized his life and philosophy as being both “inside” and “outside” East and West, i.e., inside and outside traditions of India and those of the West, living in both India and United States: geographically, culturally, and philosophically; while also traveling the world: Melbourne to Moscow. Most of his academic time was spent teaching at the University of Oklahoma, The New School Graduate Faculty, and finally Temple University. (...) Yet his preeminent work in Husserlian phenomenology developed alongside his eminent work in Indian philosophy: describing his interests as “a fusion of disparate horizons.” -/- J. N. Mohanty was born September 26, 1928, in Cuttack (Odisha, East India). After graduating from high school, he went on to study both Indian and Western philosophy in Calcutta, earning bachelor’s and master’s degrees. There he read Whitehead and Kant’s First Critique; although he wanted to include mathematics in his curriculum, he was led instead to include Indian philosophy and Sanskrit. On the shelves of his teacher, Ras Vihar Das, he came upon a copy of the English translation (Ideas I, 1931) of Edmund Husserl’s classic Ideen I (1913), which presented Husserl’s ground-breaking conception of phenomenology. In 1952–1954 he left India for the first time, reaching Göttingen to study mathematics and philosophy, which earned him a doctorate in mathematics and “philosophy of mathematical sciences” (in his own words). In Göttingen, Mohanty found the powerful mathematical world that Husserl himself had earlier interacted with, where several Husserl students had formed the Göttingen school of phenomenology. During these years Mohanty studied primarily mathematics, alongside Kant and also Vedic Sanskrit. From his friend Günter Patzig, interpreter of Aristotle and Frege, Mohanty was drawn to Frege in relation to mathematical logic. He attended lectures of Heidegger, intrigued by his ontological thinking. Yet, despite the Husserlian legacy, Mohanty was completely self-taught in his studies of Husserl (as he has reported). With a doctorate in mathematics, and ideas from Kant and Frege in his philosophical background, Mohanty set about crafting his own conception of philosophy grounded in phenomenology, drawing on Husserl’s extensive work, critically sifting through Husserl’s texts and their emerging concepts of intentionality, meaning, subject, intersubjectivity, and world. In between he wrote his first book-length study: on phenomenological insights in Nicolai Hartmann and A. N. Whitehead (1958). -/- Over many decades Mohanty formulated and argued, in analytical detail, for a conception of phenomenology and its place in philosophy, later presented in a clear and concise book titled Transcendental Phenomenology: An Analytic Account (1989). Over these very decades the same scholar explored classical and recent Indian philosophy, thinking through kindred ideas of consciousness, self, and knowledge drawn from the Indian philosophical contexts. While writing on Nyāya theory of truth, he also pondered whether the world and finite individual are illusory or real, and whether Marx, Arendt, Gandhi (whom he heard speak in Calcutta), or Vinoba Bhave (with whom he marched across India for the land-grant movement) could best navigate post-Independent India’s social and svarāj or self-rule reforms. The two Mohantys, thinking through a vision of self and world, turned out to be “non-different” or “non-dual” as they each practiced critical phenomenology from both inside and outside the respective philosophical and cultural traditions. Numerous students, fellows, and colleagues or collaborators have benefited immensely from this infusion and unified approach to diversity in philosophical thought. In the 2000s, moving into retirement, Mohanty wrote two long books devoted to his understanding of Husserl and phenomenology and the calling of philosophy itself. This two-volume study shows Mohanty himself thinking through Husserl, critically, in The Philosophy of Edmund Husserl: A Historical Introduction (2008), and Edmund Husserl’s Freiburg Years: 1916–1938. As Mohanty worked on Husserl, he carefully indicated where he agreed and where he rejected or changed ideas, all part of his practice in phenomenology of “description and interpretation.” It was the same pattern he used in addressing the thought of Husserl vis-à-vis Kant or Frege or even Quine. -/- As Mohanty developed his understanding of phenomenology over the years, he wrote books on theory of meaning and the concept of intentionality, developing a model of ideal meaning and its foundation in intentionality, drawing on Husserl’s results. He followed these with the book Husserl and Frege (1982), linking the thought of those foundational figures for the “continental” and “analytic” traditions, respectively, in twentieth-century Western philosophy. Over his long career Mohanty addressed both traditions in his clear and accessible writing style. While developing his views on phenomenology, Mohanty regularly looked to “Husserl and his others,” evaluating views in Heidegger, Sartre, Merleau-Ponty, Ricoeur, and then Derrida and others in the wake of Husserl. With his background in Kant, Mohanty also looked toward Heidegger and Hegel in relation to Husserl’s later work in the Crisis (1935–38). Similarly, he looked to contemporary analytic philosophers, adjudicating his own, oft-wise Husserlian views in relation to Frege, Nagel, and others. Amid his active scholarly career, Mohanty co-founded the journal Husserl Studies, and was editorial advisor to Philosophy & Phenomenological Research, Philosophy East & West, Journal of Indian Philosophy, Sophia, among others. -/- Yet all this while Mohanty was also thinking and writing about Indian philosophy and its relation to phenomenology. In his own retrospective, Indian philosophy is “the permanent background” of his Husserlian thinking, while Kant is the recurrent Western background of his Husserlian phenomenology. Mohanty’s form of “transcendental phenomenology” evolved, in his own perspective, against the background of his studies of Navya-Nyāya on logic and Vedānta on consciousness, in Indian philosophy, and against the background of Kant’s First Critique on the transcendental, in Western philosophy. Accordingly, Mohanty’s study of logical form and of the intentionality of consciousness seeks a fusion of East and West in the conception of transcendental phenomenology. (Cf. Mohanty’s apt response to critics in The Empirical and the Transcendental (2000).) Even in the context of North American Husserl scholarship, Mohanty has exercised an earnest fusion of East and West. For the so-called East Coast and West Coast interpretations of Husserl’s crucial notion of noema both find a sympathetic spirit in J. N. Mohanty’s careful and nuanced interpretation of Husserlian transcendental phenomenology. With Mohanty the two faces of the noema are the logical (Fregean) and the phenomenal (Kantian), and these views of intentional structure join in consciousness—in a way resonant with Indian thought. -/- - David Woodruff Smith (UC Irvine) and Purushottama Bilimoria (SFSU, San Francisco; University of Melbourne) -/- For photo of Prof Mohanty visit APA online memorial_minutes2023 Courtesy of APA Memorial Minutes (& Proceedings) 2023 . (shrink)

This study aimed to trace the 2015 to 2019 MAME graduates in the College of Teacher Education for Graduate Studies of the University of Northern Philippines in terms of their personal profile, their work-related profile before and after taking their master’s degree, reasons of taking up the program, competency level before and after taking the program, appraisal of the most useful courses offered in the program, evaluation on the contribution of the program to their personal and professional growth and assessment (...) on the features of the program and suggestions of how to improve the program. The researcher utilized mixed method of research. The subjects were the MAME graduates, and a questionnaire and open-ended questions were sent to them thru various online flat forms. Based on the findings of the study, the following conclusions are derived: pursuing graduate studies chooses no sex, civil status or age as long as there is a will to improve one's self; students have math- related educational background and finished the program within the prescribed number of years; having master’s degree could contribute greatly in landing a permanent teaching job in public schools; professional and personal growth and development is the strongest drive to pursue advance education; the program offers subjects that are useful which enhanced the teaching competencies of the graduates and consequently contribute to the holistic development of the graduates; and that there are areas needing improvement in the MAME program. (shrink)

This study investigated problems of school management and secondary school students' academic performance in Calabar Education Zone of Cross River State. Two null hypotheses were formulated to guide the study. The study adopted an ex-post facto research design. Proportionate stratified sampling technique was employed in selecting a sample of 3616 students out of the population of 18, 078 students. "Problems of School Management Questionnaire (PSMQ)," and Senior Secondary Mathematics Achievement Test (SSMAT) were the instruments designed by (...) the researcher, were used for data collection. The data collected were analysed using descriptive statistics; while the null hypotheses were tested at .05 level of significance using population t-test and multiple regression analyses where applicable. Microsoft Excel 2016 spreadsheet program was used in the analysis of data. Findings from the study revealed among others that; secondary school students' academic performance in Calabar Education zone is significantly high, disciplinary control, classroom management, and teachers' motivation significantly influence secondary school students' academic performance respectively. Based on these results, conclusions were drawn and recommendations were made. (shrink)

This study primarily assessed students' achievement in mathematics using a mobile educational application to help them learn and adapt to changes in education. The study involved selected Grade 9 students at a public high school in Nueva Vizcaya, Philippines. This study used a quasi-experimental method, particularly a post-test control group design. Descriptive statistics such as frequencies, percent, mean, and standard deviation were used to describe the achievement of the students in mathematics. A t-test for independent samples was (...) also computed to determine if there is a significant difference between the students’ achievement in mathematics after using the mobile educational application by the control and experimental groups. Results revealed that students' achievement in mathematics was very proficient after using the application. Moreover, the usage of educational mobile applications has a better impact on students' achievement in mathematics than the traditional approach to teaching. Teachers are encouraged to place priority on solving problems. Overall, the use of the mobile educational application can enhance learning in mathematics, thus the continued use of the said application is advised. (shrink)

Gaisi Takeuti (1926–2017) is one of the most distinguished logicians in proof theory after Hilbert and Gentzen. He extensively extended Hilbert's program in the sense that he formulated Gentzen's sequent calculus, conjectured that cut-elimination holds for it (Takeuti's conjecture), and obtained several stunning results in the 1950–60s towards the solution of his conjecture. Though he has been known chiefly as a great mathematician, he wrote many papers in English and Japanese where he expressed his philosophical thoughts. In particular, he used (...) several keywords such as "active intuition" and "self-reflection" from Nishida's philosophy. In this paper, we aim to describe a general outline of our project to investigate Takeuti's philosophy of mathematics. In particular, after reviewing Takeuti's proof-theoretic results briefly, we describe some key elements in Takeuti's texts. By explaining these texts, we point out the connection between Takeuti's proof theory and Nishida's philosophy and explain the future goals of our project. (shrink)

This paper deals with Natorp’s version of the Marburg mathematical philosophy of science characterized by the following three features: The core of Natorp’s mathematical philosophy of science is contained in his “knowledge equation” that may be considered as a mathematical model of the “transcendental method” conceived by Natorp as the essence of the Marburg Neo-Kantianism. For Natorp, the object of knowledge was an infinite task. This can be elucidated in two different ways: Carnap, in the Aufbau, contended that this endeavor (...) can be divided into two distinct parts, namely, a finite “constitution” of the object of knowledge and an infinite incompletable empirical description. In contrast, and more in the original spirit of Cohen and Natorp, the physicist and philosopher Margenau in The Nature of Physical Reality (Margenau. 1950) conceived the infinity of this “Aufgabe” as an infinite dialectical process, in which relative “data” and “conceptual constructs” determine each other. This dialectical process eliminates the dichotomy between Anschauung and Begriff that distinguished the Marburg Neo-Kantianism from Kantian orthodoxy, namely, the abandonment of the difference between intuition and concept. Finally, the paper deals with the non-Archimedean geometrical systems that played a central role in Natorp’s defence of Cohen’s “infinitesimal” metaphysics. (shrink)

intro to Part 1 - -/- Most people disliked mathematics when they were at school and they were absolutely correct to do so. This is because maths as we know it is severely incomplete. No matter how elaborated and complicated mathematical equations become, in today's world they're based on 1+1=2. This certainly conforms to the world our physical senses perceive and to the world scientific instruments detect. It has been of immeasurable value to all knowledge throughout history and (...) has elevated science to the lofty status it enjoys. Science is now striving towards Unification - where the subatomic realm, all matter, energy, forces, space and time will be seen as entangled parts of one universe. While 1+1=2 has been vital in getting humanity to this point, it's time to suppress our attachments to the past and realize that whereas 1+1 will always equal 2, it's also capable of equalling the 1 which represents unification. -/- intro to Part 2 -/- b) Division by zero is accepted, in Newtonian maths, to be impossible. But we can regard division by zero as division by nothing i.e. division that has no effect. In this case, 1 divided by 0 is 1. However, to a physicist there is no such thing as nothing (even empty space contains energy). What could the something called 0 actually be? It could be a binary digit. If we use the base of ten (for simplicity) and attach one and zero to it as exponents, we get 10^1 divided by 10^0 = 10^1. If we then cancel 10 from each factor in the expression, we get 1 divided by 0 = 1. At the start of the paragraph, this was referred to as division by nothing. Then 0 was called a binary digit and division by nothing became division by something. The 1 that the division equals is the unified field of space-time. Division by 0 is impossible in Newtonian maths because the result can be infinity. But the word “infinity” can, as the last section of this book shows, apply to the unified field of spacetime. So division by zero is not impossible because it results in the universe, which is obviously possible … a possibility that has always been, and always will be, realized. -/- intro to Part 3 -/- If quantum entanglement has existed in the entire universe forever, everything would be everywhere and everywhen. Space, time and 5th-dimensional hyperspace would not be restricted to certain parts of the Mobius Universe but would exist in every particle. Past, present and future would not exist as the distinct periods which everyday life assumes. All instants of all periods would exist eternally, permitting time travel to any point in the past and to any point in the future. Entanglement may be created by simply zipping along at close to the speed of light - “Quantum entanglement of moving bodies” by Robert M. Gingrich and Christoph Adami in Physical Review Letters 89, 270402 (issue of 30 December 2002) – which might be achieved, according to this book, by warping space so it’s either a fraction of the 90 degrees allowing instantaneous travel or almost at 270 degrees to space as we know it. (shrink)

The study evaluated primary texts' availability and utilisation status in core subjects (English Language, Mathematics, Social Studies and Basic Science) in primary schools’ libraries in Obubra Local Government Area of Cross River State. The researchers formulated six null hypotheses to guide the study. The study adopted the descriptive survey research design. The target population of this study comprised a total of 30,036 teachers and pupils, distributed across the 73 public primary schools. A proportionate stratified sampling technique was used to (...) select 30% and 5% of the available teachers and pupils in each school, resulting in a sample of 270 respondents. Two instruments were used for data collection: Availability and Pupils Utilisation of Primary Texts in Core Subjects Questionnaire (APUPTCSQ) and Teachers Utilization of Primary Texts in Core Subjects Questionnaire (TUPTCSQ). Collected data were analysed using descriptive statistics, while the null hypotheses were all tested at the .05 level of significance using population t-test and independent t-test statistical techniques. Findings revealed that textbooks in core subjects are available to a significant extent; the utilisation status of texts in core subjects is not significantly low; the available books in core subjects are not considerably utilised by the pupils and teachers respectively; factors affecting availability are not substantially different from those affecting utilisation of textbooks in core subjects in the area of study. Based on the study's findings, it was recommended, among others, that textbooks be procured and distributed evenly across all subjects and schools in the Local Government Area by the Government. Outdated books should also be eliminated from the school libraries frequently by the primary school librarians or other people assigned to do so. (shrink)

This study assessed the mode of test administration, birth variables, and students’ academic performance in Mathematics in Obubra Local Government Area of Cross River State, Nigeria. The study was guided by three null hypotheses following a quasi- experimental research design. Simple random and purposive sampling techniques were employed in selecting a sample of 250 respondents from a population of 5,792 students. Birth Order Checklist (BOC) and Mathematics Achievement Test (MAT) were used for data collection. The experimental group was (...) assigned to take a Computer-Based Test (CBT) while the control group took a Paper-Pencil Test (PPT) in mathematics. The null hypotheses were tested at .05 alpha level using two-way analysis of variance. Major findings revealed that, the interactive effects of gender and mode of test administration on the academic performance of students is non- significant [F(1,249)=0.028, p>.05, partial η2=.000]; the interactive effects of gender and birth order on the academic performance of students was not statistically significant [F(3,249)=0.782, p>.05, partial η2=.010]; the interactive effect of gender and birth order on the academic performance of students in mathematics is statistically significant [F(3,249)=2.854, p<.05, partial η2 =.034]. Based on these findings conclusions and recommendations were made for improved academic performance and school system effectiveness. (shrink)

It has become a fact that fluency and competency in utilizing the advancement of technology, specifically the computer and the internet is one way that could help in facilitating learning in mathematics. This study investigated the effects of Dynamic Geometry Software (DGS) and Computer Algebra Systems (CAS) in teaching Mathematics. This was conducted in Zamboanga del Sur National High School (ZSNHS) during the third grading period of the school year 2015-2016. The study compared the achievement and (...) attitude towards Mathematics between those students taught with DGS and CAS and those students who were taught without DGS and CAS. A quasi-experimental pretest-posttest control group design was used with two groups which was matched according to their mathematical level. An achievement test was developed and administered to assess the students’ achievement in Mathematics. Aiken’s attitude scale was also used to assess the students’ attitude towards Mathematics. The data were treated with One-Way Analysis of Covariance (ANCOVA) and t-test at 0.05 level of significance. The study revealed a significant difference between the mathematics achievement between students taught with DGS and CAS and those who were taught without it. The integration of DGS and CAS in Mathematics lessons leads to better academic achievement result compared with that of the conventional method of teaching. Moreover, there was no significant difference on the attitude level of the students between the two groups. (shrink)

This study used a path analytic approach to examine the composite interaction of supervisory and records management with secondary school system effectiveness in terms of students' academic performance, teachers' job effectiveness and principals' administrative effectiveness. Two research questions were answered while one null hypothesis was tested. The study adopted a factorial research design. Convenience sampling technique was adopted by the researchers in selecting a sample of 1,200 respondents which comprised 271 principals and 929 teachers from a population of 271 (...) principals, and 4,878 teachers. Principal Innovative Management Practices and Administrative Effectiveness Questionnaire (PIMPAEQ), Teachers work effectiveness questionnaire (TWEQ), and Students Mathematics Achievement Test (SMAT) were all used as instruments for data collection. Collected data were analyzed using descriptive statistics, the research questions and null hypothesis were answered and tested (at .05 level of significance) respectively using multiple regression and path analyses with the aid of SPSS v25 and Amos v22. Findings from the study established amongst others that; supervisory management and records management practices have a significant joint contribution to students’ academic performance by 66.4%, teachers’ job effectiveness by 71%, and principals’ administrative effectiveness by 86.8%. Supervisory management and records management practices had a significant influence on students’ academic performance (F= 1183.641, p<.05), teachers’ job effectiveness (F= 1465.615, p<.05), and principals’ administrative effectiveness (F=3924.763, p<.05). It was recommended amongst others that; secondary school principals should jointly use supervisory and records management practices as innovative techniques for improving students’ academic performance, teachers’ job effectiveness and their own administrative effectiveness. KEYWORDS:. (shrink)

This is a collection of new investigations and discoveries on the history of a great tradition, the Lvov-Warsaw School of logic , philosophy and mathematics, by the best specialists from all over the world. The papers range from historical considerations to new philosophical, logical and mathematical developments of this impressive School, including applications to Computer Science, Mathematics, Metalogic, Scientific and Analytic Philosophy, Theory of Models and Linguistics.

Much problem solving and learning research in math and science has focused on formal representations. Recently researchers have documented the use of unschooled strategies for solving daily problems -- informal strategies which can be as effective, and sometimes as sophisticated, as school-taught formalisms. Our research focuses on how formal and informal strategies interact in the process of doing and learning mathematics. We found that combining informal and formal strategies is more effective than single strategies. We provide a theoretical (...) account of this multiple strategy effect and have begun to formulate this theory in an ACT-R computer model. We show why students may reach common impasses in the use of written algebra, and how subsequent or concurrent use of informal strategies leads to better problem-solving performance. Formal strategies facilitate computation because of their abstract and syntactic nature; however, abstraction can lead to nonsensical interpretations and conceptual errors. Reapplying the formal strategy will not repair such errors; switching to an informal one may. We explain the multiple strategy effect as a complementary relationship between the computational efficiency of formal strategies and the sense-making function of informal strategies. (shrink)

The Introduction outlines, in a concise way, the history of the Lvov-Warsaw School – a most unique Polish school of worldwide renown, which pioneered trends combining philosophy, logic, mathematics and language. The author accepts that the beginnings of the School fall on the year 1895, when its founder Kazimierz Twardowski, a disciple of Franz Brentano, came to Lvov on his mission to organize a scientific circle. Soon, among the characteristic features of the School was its (...) serious approach towards philosophical studies and teaching of philosophy, dealing with philosophy and propagation of it as an intellectual and moral mission, passion for clarity and precision, as well as exchange of thoughts, and cooperation with representatives of other disciplines.The genesis is followed by a chronological presentation of the development of the School in the successive years. The author mentions all the key representatives of the School (among others, Ajdukiewicz, Lesniewski, Łukasiewicz,Tarski), accompanying the names with short descriptions of their achievements. The development of the School after Poland’s regaining independence in 1918 meant part of the members moving from Lvov to Warsaw, thus providing the other segment to the name – Warsaw School of Logic. The author dwells longer on the activity of the School during the Interwar period – the time of its greatest prosperity, which ended along with the outbreak of World War 2. Attempts made after the War to recreate the spirit of the School are also outlined and the names of followers are listed accordingly. The presentation ends with some concluding remarks on the contribution of the School to contemporary developments in the fields of philosophy, mathematical logic or computer science in Poland. (shrink)

The first learning game to be developed to help students to develop and hone skills in constructing proofs in both the propositional and first-order predicate calculi. It comprises an autotelic (self-motivating) learning approach to assist students in developing skills and strategies of proof in the propositional and predicate calculus. The text of VALIDITY consists of a general introduction that describes earlier studies made of autotelic learning games, paying particular attention to work done at the Law School of Yale University, (...) called the ALL Project (Accelerated Learning of Logic). Following the introduction, the game of VALIDITY is described, first with reference to the propositional calculus, and then in connection with the first-order predicate calculus with identity. Sections in the text are devoted to discussions of the various rules of derivation employed in both calculi. Three appendices follow the main text; these provide a catalogue of sequents and theorems that have been proved for the propositional calculus and for the predicate calculus, and include suggestions for the classroom use of VALIDITY in university-level courses in mathematical logic. (shrink)

The potential benefits of cognitive skills in enhancing mathematics ability have been claimed by numerous researchers. Since mathematics requires a complete understanding and grasp of abstract concepts, it is essential to explore how learning with metacognitive skills affects mathematics learning. Thus, the study investigates the students' metacognitive awareness and conceptual understanding of integers. A descriptive-correlational method approach was utilized, and it was carried out on 303 seventh-grade students. The data were obtained using a metacognitive awareness inventory and (...) achievement test on integers. It was further revealed that the students have average metacognitive awareness and performed well in the fundamental operation of integers. Follow-up qualitative analysis revealed that students who were high achievers had the best understanding, average achievers had corrected or incomplete understanding, and low achievers had a functional misconception of integers. Moreover, the student's metacognitive awareness was significantly related to their conceptual understanding of integers. This indicates that student's higher-order thinking skills, such as metacognition, are essential since they are associated with building conceptual skills. Thus, teachers should encourage students' metacognitive awareness to improve students' conceptual understanding of integers. The study provides relevant information for educational managers on the potential factors to be considered in improving mathematics education practices, particularly in promoting metacognition among high school students. (shrink)

As the international momentum of the school effectiveness movement continues, its exponents remain largely impervious to criticism. This paper argues that while they may not readily align themselves with the individualistic aspects of Conservative social philosophy, their methodology necessarily secretes an atomised social ontology. The charge of ideological commitment rests on the fact that the essentially positivist epistemology employed by school effectiveness researchers presupposes an ontology of closed systems and atomistic events. Thus any notion of the structuring of (...) life‐chances is automatically ruled out of court. The reciprocity of social ontology and methodology is explored in order to account for the myopia of the school effectiveness movement. (shrink)

The Introduction outlines, in a concise way, the history of the Lvov-Warsaw School—a most unique Polish school of worldwide renown, which pioneered trends combining philosophy, logic, mathematics and language. The author accepts that the beginnings of the School fall on the year 1895, when its founder Kazimierz Twardowski, a disciple of Franz Brentano, came to Lvov on his mission to organize a scientific circle. Soon, among the characteristic features of the School was its serious approach (...) towards philosophical studies and teaching of philosophy, dealing with philosophy and propagation of it as an intellectual and moral mission, passion for clarity and precision, as well as exchange of thoughts, and cooperation with representatives of other disciplines. The genesis is followed by a chronological presentation of the development of the School in the successive years. The author mentions all the key representatives of the School, accompanying the names with short descriptions of their achievements. The development of the School after Poland’s regaining independence in 1918 meant part of the members moving from Lvov to Warsaw, thus providing the other segment to the name—Warsaw School of Logic. The author dwells longer on the activity of the School during the Interwar period—the time of its greatest prosperity, which ended along with the outbreak of World War 2. Attempts made after the War to recreate the spirit of the School are also outlined and the names of continuators are listed accordingly. The presentation ends with some concluding remarks on the contribution of the School to contemporary developments in the fields of philosophy, mathematical logic or computer science in Poland. (shrink)

The paper explicates the stages of the author’s philosophical evolution in the light of Kopnin’s ideas and heritage. Starting from Kopnin’s understanding of dialectical materialism, the author has stated that category transformations of physics has opened from conceptualization of immutability to mutability and then to interaction, evolvement and emergence. He has connected the problem of physical cognition universals with an elaboration of the specific system of tools and methods of identifying, individuating and distinguishing objects from a scientific theory domain. The (...) role of vacuum conception and the idea of existence (actual and potential, observable and nonobservable, virtual and hidden) types were analyzed. In collaboration with S.Crymski heuristic and regulative functions of categories of substance, world as a whole as well as postulates of relativity and absoluteness, and anthropic and self-development principles were singled out. Elaborating Kopnin’s view of scientific theories as a practically effective and relatively true mapping of their domains, the author in collaboration with M. Burgin have originated the unified structure-nominative reconstruction (model) of scientific theory as a knowledge system. According to it, every scientific knowledge system includes hierarchically organized and complex subsystems that partially and separately have been studied by standard, structuralist, operationalist, problem-solving, axiological and other directions of the current philosophy of science. 1) The logico-linguistic subsystem represents and normalizes by means of different, including mathematical, languages and normalizes and logical calculi the knowledge available on objects under study. 2) The model-representing subsystem comprises peculiar to the knowledge system ways of their modeling and understanding. 3) The pragmatic-procedural subsystem contains general and unique to the knowledge system operations, methods, procedures, algorithms and programs. 4) From the viewpoint of the problem-heuristic subsystem, the knowledge system is a unique way of setting and resolving questions, problems, puzzles and tasks of cognition of objects into question. It also includes various heuristics and estimations (truth, consistency, beauty, efficacy, adequacy, heuristicity etc) of components and structures of the knowledge system. 5) The subsystem of links fixes interrelations between above-mentioned components, structures and subsystems of the knowledge system. The structure-nominative reconstruction has been used in the philosophical and comparative case-studies of mathematical, physical, economic, legal, political, pedagogical, social, and sociological theories. It has enlarged the collection of knowledge structures, connected, for instance, with a multitude of theoreticity levels and with an application of numerous mathematical languages. It has deepened the comprehension of relations between the main directions of current philosophy of science. They are interpreted as dealing mainly with isolated subsystems of scientific theory. This reconstruction has disclosed a variety of undetected knowledge structures, associated also, for instance, with principles of symmetry and supersymmetry and with laws of various levels and degrees. In cooperation with the physicist Olexander Gabovich the modified structure-nominative reconstruction is in the processes of development and justification. Ideas and concepts were also in the center of Kopnin’s cognitive activity. The author has suggested and elaborated the triplet model of concepts. According to it, any scientific concept is a dependent on cognitive situation, dynamical, multifunctional state of scientist’s thinking, and available knowledge system. A concept is modeled as being consisted from three interrelated structures. 1) The concept base characterizes objects falling under a concept as well as their properties and relations. In terms of volume and content the logical modeling reveals partially only the concept base. 2) The concept representing part includes structures and means (names, statements, abstract properties, quantitative values of object properties and relations, mathematical equations and their systems, theoretical models etc.) of object representation in the appropriate knowledge system. 3) The linkage unites a structures and procedures that connect components from the abovementioned structures. The partial cases of the triplet model are logical, information, two-tired, standard, exemplar, prototype, knowledge-dependent and other concept models. It has introduced the triplet classification that comprises several hundreds of concept types. Different kinds of fuzziness are distinguished. Even the most precise and exact concepts are fuzzy in some triplet aspect. The notions of relations between real scientific concepts are essentially extended. For example, the definition and strict analysis of such relations between concepts as formalization, quantification, mathematization, generalization, fuzzification, and various kinds of identity are proposed. The concepts «PLANET» and «ELEMENTARY PARTICLE» and some of their metamorphoses were analyzed in triplet terms. The Kopnin’s methodology and epistemology of cognition was being used for creating conception of the philosophy of law as elaborating of understanding, justification, estimating and criticizing legal system. The basic information on the major directions in current Western philosophy of law (legal realism, feminism, criticism, postmodernism, economical analysis of law etc.) is firstly introduced to the Ukrainian audience. The classification of more than fifty directions in modern legal philosophy is suggested. Some results of historical, linguistic, scientometric and philosophic-legal studies of the present state of Ukrainian academic science are given. (shrink)

Curriculum integration is frequently promoted as a means of enabling deeper and more authentic learning, with Mathematics often considered a suitable subject for doing so. This review investigates which elements contribute to the effectiveness of Mathematics integration in secondary education. Teacher factors include attitudes towards integrative practices and knowledge of both disciplinary content and curriculum integration theory. Pedagogy factors concern utilising activities that best synthesise concepts from multiple subjects to enhance learning, especially projects. Institutional factors relate to curriculum (...) and assessment requirements as well as school-level variables such as access to planning time and professional development. Limitations in the research literature include a narrow focus on connecting Mathematics with scientific disciplines, a reliance on qualitative methods and small sample sizes, and unknown generalisability to education systems other than those in which studies were conducted. Even so, attempting to navigate the factors involved in effective curriculum integration with Mathematics appears worthwhile. (shrink)

Aims: The purpose of this study is to identify the factors associated with the academic performance of Grade 10 students during their First Quarter of school as well as the significant correlation between the factors and student academic performance in Mathematics. -/- Study Design: Descriptive Correlation Design. -/- Place and Duration of Study: The study was conducted at Agusan National High School in Cagayan de Or City's East 1 District during the school year: 2022 – 2023. (...) -/- Methodology: The respondents were Two hundred thirty-one (231) students in Grade 10 at Agusan National High School in Cagayan de Oro City's East 1 District. This study used a researcher-made questionnaire that underwent validity and reliability testing and the academic performance of the students. -/- Results: The results showed that students agree with routines related to mastering Mathematics at a high level. Furthermore, there is no correlation between student study habits and Mathematics performance in terms of self-confidence, but there is a substantial positive association between the student's study habits and performance in terms of attitude. The study habits and learning techniques used by students were found to be important determinants of how well they performed in Mathematics. The researcher strongly suggested using the enhancement plan in teaching Mathematics to Junior High school students. -/- Conclusion: Students have a very positive attitude toward their study habits when learning Mathematics. Students felt that studying and learning Mathematics was essential. Students' success and academic growth depend heavily on their Mathematics performance. It is essential for pupils to master and comprehend its concepts. Students' study habits in terms of attitudes have an impact on how they learn Mathematics. (shrink)