In this multi-disciplinary investigation we show how an evidence-based perspective of quantification---in terms of algorithmic verifiability and algorithmic computability---admits evidence-based definitions of well-definedness and effective computability, which yield two unarguably constructive interpretations of the first-order Peano Arithmetic PA---over the structure N of the natural numbers---that are complementary, not contradictory. The first yields the weak, standard, interpretation of PA over N, which is well-defined with respect to assignments of algorithmically verifiable Tarskian truth values to the formulas of PA under the interpretation. (...) The second yields a strong, finitary, interpretation of PA over N, which is well-defined with respect to assignments of algorithmically computable Tarskian truth values to the formulas of PA under the interpretation. We situate our investigation within a broad analysis of quantification vis a vis: * Hilbert's epsilon-calculus * Goedel's omega-consistency * The Law of the Excluded Middle * Hilbert's omega-Rule * An Algorithmic omega-Rule * Gentzen's Rule of Infinite Induction * Rosser's Rule C * Markov's Principle * The Church-Turing Thesis * Aristotle's particularisation * Wittgenstein's perspective of constructive mathematics * An evidence-based perspective of quantification. By showing how these are formally inter-related, we highlight the fragility of both the persisting, theistic, classical/Platonic interpretation of quantification grounded in Hilbert's epsilon-calculus; and the persisting, atheistic, constructive/Intuitionistic interpretation of quantification rooted in Brouwer's belief that the Law of the Excluded Middle is non-finitary. We then consider some consequences for mathematics, mathematics education, philosophy, and the natural sciences, of an agnostic, evidence-based, finitary interpretation of quantification that challenges classical paradigms in all these disciplines. (shrink)

Du Châtelet holds that mathematical representations play an explanatory role in natural science. Moreover, she writes that things proceed in nature as they do in geometry. How should we square these assertions with Du Châtelet’s idealism about mathematical objects, on which they are ‘fictions’ dependent on acts of abstraction? The question is especially pressing because some of her important interlocutors (Wolff, Maupertuis, and Voltaire) denied that mathematics informs us about the properties of material things. After situating Du Châtelet (...) in this debate, this chapter argues, first, that her account of the origins of mathematical objects is less subjectivist than it might seem. Mathematical objects are non-arbitrary, public entities. While mathematical objects are partly mind-dependent, so are material things. Mathematical objects can approximate the material. Second, it is argued that this moderate metaphysical position underlies Du Châtelet’s persistent claims that mathematics is required for certain kinds of general knowledge, including in natural science. The chapter concludes with an illustrative example: an analysis of Du Châtelet’s argument that matter is continuous. A key premise in the argument is that mathematical representations and material nature correspond. (shrink)

For over thirty years I have argued that all branches of science and scholarship would have both their intellectual and humanitarian value enhanced if pursued in accordance with the edicts of wisdom-inquiry rather than knowledge-inquiry. I argue that this is true of mathematics. Viewed from the perspective of knowledge-inquiry, mathematics confronts us with two fundamental problems. (1) How can mathematics be held to be a branch of knowledge, in view of the difficulties that view engenders? What could mathematics be knowledge (...) about? (2) How do we distinguish significant from insignificant mathematics? This is a fundamental philosophical problem concerning the nature of mathematics. But it is also a practical problem concerning mathematics itself. In the absence of the solution to the problem, there is the danger that genuinely significant mathematics will be lost among the unchecked growth of a mass of insignificant mathematics. This second problem cannot, it would seem, be solved granted knowledge-inquiry. For, in order to solve the problem, mathematics needs to be related to values, but this is, it seems, prohibited by knowledge-inquiry because it could only lead to the subversion of mathematical rigour. Both problems are solved, however, when mathematics is viewed from the perspective of wisdom-inquiry. (1) Mathematics is not a branch of knowledge. It is a body of systematized, unified and inter-connected problem-solving methods, a body of problematic possibilities. (2) A piece of mathematics is significant if (a) it links up to the interconnected body of existing mathematics, ideally in such a way that some problems difficult to solve in other branches become much easier to solve when translated into the piece of mathematics in question; (b) it has fruitful applications for (other) worthwhile human endeavours. If ever the revolution from knowledge to wisdom occurs, I would hope wisdom mathematics would flourish, the nature of mathematics would become much more transparent, more pupils and students would come to appreciate the fascination of mathematics, and it would be easier to discern what is genuinely significant in mathematics (something that baffled even Einstein). As a result of clarifying what should count as significant, the pursuit of wisdom mathematics might even lead to the development of significant new mathematics. (shrink)

I argue that certain species of belief, such as mathematical, logical, and normative beliefs, are insulated from a form of Harman-style debunking argument whereas moral beliefs, the primary target of such arguments, are not. Harman-style arguments have been misunderstood as attempts to directly undermine our moral beliefs. They are rather best given as burden-shifting arguments, concluding that we need additional reasons to maintain our moral beliefs. If we understand them this way, then we can see why moral beliefs are (...) vulnerable to such arguments while mathematical, logical, and normative beliefs are not—the very construction of Harman-style skeptical arguments requires the truth of significant fragments of our mathematical, logical, and normative beliefs, but requires no such thing of our moral beliefs. Given this property, Harman-style skeptical arguments against logical, mathematical, and normative beliefs are self-effacing; doubting these beliefs on the basis of such arguments results in the loss of our reasons for doubt. But we can cleanly doubt the truth of morality. (shrink)

Mathematics is an important subject taught in primary and secondary schools that equips students with foundational knowledge and skills for organizing their lives. This study determined the Mathematics proficiency level among the Grade Three pupils in Cagayan de Oro City in School Year 2022-2023. Specifically, it sought to determine the respondents’ profile in terms of language used at home, study habits, parental involvement, and attitude towards Mathematics; find out the proficiency level in Mathematics; and determine the significant relationship between the (...) respondents’ proficiency level in Mathematics when grouped according to their profile. Data were gathered from one hundred fifty (150) respondents. The researcher-made questionnaire and Mathematics Test patterned from the Division of Cagayan de Oro Periodical Tests for the School Year 2022-2023 were utilized in the study. It used a descriptive research design to analyze the data using frequency, mean, standard deviation, correlation analysis, and P-value. Findings revealed a significant relationship between respondents’ proficiency level in Mathematics and their study habits, parental involvement, and attitude towards Mathematics. In contrast, no significant relationship exists between the language used at home. It is recommended that teachers incorporate literacy in teaching Mathematics, use game-based strategies, and review Mathematics lessons to increase the chances of obtaining high scores. (shrink)

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)

The city of Kruja dates back to its existence in the 5th and 6th centuries. In the inner city are preserved great historical, cultural, and architectural values that are inherited from generation to generation. In the city interact and coexist three different typologies of dwellings: historic buildings that belong to the XIII, XIV, XV, XIII, XIX centuries (built using the foundations of previous buildings); socialist buildings dating back to the Second World War until 1990; and modern buildings which were built (...) from 1990 onwards. According to the questionnaires, the creation of mathematical models applied to each category will result in contradictory attitudes but also fairy ones based on different percentages. There are underlined 5 quality of life indicators(questions) from a total of 30 questions of the questionnaire, which are involved in mathematical regression and are statistically significant with a significance level p<5% (with a reliability of 95%), which interact for all three categories of buildings. The quality of life indicators: dwelling area, heating mode during winter, level of dwelling improvement, time spent in the dwelling, and monthly electricity payments, are the main actors who will be compared in order to draw conclusions. According to the statistical calculations trend, it is noted that the socialist buildings category does not directly participate in the debate between historic and modern buildings by means of the quality of life indicators (questions). (shrink)

Mathematics is obviously important in the sciences. And so it is likely to be equally important in any effort that aims to understand God in a scientifically significant way or that aims to clarify the relations between science and theology. The degree to which God has any perfection is absolutely infinite. We use contemporary mathematics to precisely define that absolute infinity. For any perfection, we use transfinite recursion to define an endlessly ascending series of degrees of that perfection. That series (...) rises to an absolutely infinite degree of that perfection. God has that absolutely infinite degree. We focus on the perfections of knowledge, power, and benevolence. Our model of divine infinity thus builds a bridge between mathematics and theology. (shrink)

The published works of scientists often conceal the cognitive processes that led to their results. Scholars of mathematical practice must therefore seek out less obvious sources. This article analyzes a widely circulated mathematical joke, comprising a list of spurious proof types. An account is proposed in terms of argumentation schemes: stereotypical patterns of reasoning, which may be accompanied by critical questions itemizing possible lines of defeat. It is argued that humor is associated with risky forms of inference, which (...) are essential to creative mathematics. The components of the joke are explicated by argumentation schemes devised for application to topic-neutral reasoning. These in turn are classified under seven headings: retroduction, citation, intuition, meta-argument, closure, generalization, and definition. Finally, the wider significance of this account for the cognitive science of mathematics is discussed. (shrink)

A new trend in the philosophical literature on scientific explanation is that of starting from a case that has been somehow identified as an explanation and then proceed to bringing to light its characteristic features and to constructing an account for the type of explanation it exemplifies. A type of this approach to thinking about explanation – the piecemeal approach, as I will call it – is used, among others, by Lange (2013) and Pincock (2015) in the context of their (...) treatment of the problem of mathematical explanations of physical phenomena. This problem is of central importance in two different recent philosophical disputes: the dispute about the existence on non-causal scientific explanations and the dispute between realists and antirealists in the philosophy of mathematics. My aim in this paper is twofold. I will first argue that Lange (2013) and Pincock (2015) fail to make a significant contribution to these disputes. They fail to contribute to the dispute in the philosophy of mathematics because, in this context, their approach can be seen as question begging. They also fail to contribute to the dispute in the general philosophy of science because, as I will argue, there are important problems with the cases discussed by Lange and Pincock. I will then argue that the source of the problems with these two papers has to do with the fact that the piecemeal approach used to account for mathematical explanation is problematic. (shrink)

The study ascertained mathematical problem-solving skills and self-efficacy as correlates of Pre-service NCE Mathematics Teachers’ Performance in South-East, Nigeria. Seven research questions guided the study while seven hypotheses were tested at 0.05 level of significance. Correlation research design was used for the study. The population of the study was 197 pre-service NCE Mathematics teachers in South-East, Nigeria. All the population of 197 was studied as sample because, it is small and manageable. Mathematics Problem-Solving Skill Test (MPSST) and Pre-Service (...) Teachers’ Self-Efficacy Questionnaire (PSTSEQ) were used as instruments for data collection. Both instruments were validated by three experts in science Education while reliabilities of MPSST and PSTSEQ were established using Cronbach Alpha method and coefficient values of 0.82 and 0.80 respectively were obtained which were considered high enough to confirm the instruments as reliable. Data were collected and analyzed using Pearson product moment correlation coefficient. The findings revealed that a high positive and significant relationship existed between mathematical problem-solving skills scores and academic performance scores and between self-efficacy scores and academic performance scores of pre-service NCE mathematics teachers. It was recommended among others that Mathematics lecturers should make conscious effort towards improving the mathematical problem-solving skills and self-efficacy of the pre-service NCE mathematics teachers through modeling, conditioning, and behaviour modification for improved academic performance in Mathematics. (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 significance and use of absence of a thing is highlighted taking examples from mathematics, physics, semi-conductor electronics, computer science and cognitive science. The profundity of absence is discussed.

Mathematics is everywhere and yet its objects are nowhere. There may be five apples on the table but the number five itself is not to be found in, on, beside or anywhere near the apples. So if not in space and time, where are numbers and other mathematical objects such as perfect circles and functions? And how do we humans discover facts about them, be it Pythagoras’ Theorem or Fermat’s Last Theorem? The metaphysical question of what numbers are and (...) the epistemological question of how we know about them are central to the philosophy of mathematics. These and related philosophical questions are of particular interest because of mathematics’ unusual status. Mathematics is exceptional in that, on the one hand, it appears unhesitatingly true—no one doubts that 2 + 3 = 5—but on the other, as just noted, it is not about the physical world. This ambivalent status is what gives the philosophy of mathematics its special interest. The philosophy of mathematics is also one of the oldest academic fields, more or less coeval with philosophy itself. But contemporary philosophy of mathematics is rather different from its pre-twentieth-century antecedents, largely for three reasons. The first is that since the seventeenth century, mathematics has become integral to science. Science has over the past few centuries become increasingly mathematical, and indeed the fundamental science of nature, physics, is today recognised as a branch of applied mathematics. The second is that mathematics underwent a transformation in the course of nineteenth century: having started the century as a rather traditional-looking science of quantity it emerged a hundred years later a radically transformed abstract theory of structure. The final factor in the transformation of the philosophy of mathematics is the rise of modern logic. Developed by Frege, Cantor and others in the late nineteenth century, modern logic pervades contemporary mathematics, philosophy and computer science, and has had an immeasurable effect on the philosophy of mathematics. These volumes will collect the major works in this major field, with a focus on the last few decades. The anthology will include technical work, which interprets philosophically significant mathematical results or subfields of mathematics, as well as purely philosophical writing, aimed at those without advanced mathematics. The collection should be of interest to both philosophers and mathematicians, as well as to anyone who is susceptible to wondering what the main intellectual tool used in science, economics and finance, and indeed everyday life is ultimately about. (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)

Andr ́e Weil viewed mathematics as deeply intertwined with metaphysics. In his essay ”From Metaphysics to Mathematics,” he illustrates how mathematical ideas often arise from vague, metaphysical analogies and reflections that guide researchers toward new theories. For instance, Weil discusses how analogies between different areas, such as number theory and algebraic functions, have led to significant breakthroughs. These metaphysical underpinnings provide a fertile ground for mathematical creativity, eventually transforming into rigorous mathematical structures. -/- Alexander Grothendieck’s work, particularly (...) in ”R ́ecoltes et Semailles,” resonates with Weil’s ideas by emphasizing the organic and generative aspects of mathematical creation. Grothendieck sees mathematical work as akin to cultivating a garden, where ideas grow and develop in a nurturing environment. He describes the process of mathematical discovery as a deeply personal and creative journey, involving both solitary reflection and collaborative effort. -/- Grothendieck’s concept of creating mathematical ”houses” aligns with his broader philosophical view that mathematics provides structures within which new ideas can flourish. His work in category theory and algebraic geometry exemplifies this approach, where he developed vast, interconnected frameworks that have become foundational in modern mathematics. Grothendieck’s analogy of constructing houses for others to live in highlights the mathematician’s role in creating abstract frameworks that others can inhabit and explore. This metaphor emphasizes the constructive nature of mathematics, where researchers build theoretical structures that form the basis for further exploration and application by the mathematical community. (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)

Nietzsche has a surprisingly significant and strikingly positive assessment of mathematics. I discuss Nietzsche's theory of the origin of mathematical practice in the division of the continuum of force, his theory of numbers, his conception of the finite and the infinite, and the relations between Nietzschean mathematics and formalism and intuitionism. I talk about the relations between math, illusion, life, and the will to truth. I distinguish life and world affirming mathematical practice from its ascetic perversion. For Nietzsche, (...) math is an artistic and moral activity that has an essential role to play in the joyful wisdom. (shrink)

Mathematics is considered one of the most challenging courses for many elementary pre-service teachers (EPTs). In hopes for improvement, teacher educators have been incorporating a jigsaw strategy to equip future teachers with the necessary competencies. Hence, the study aimed to determine the effectiveness of the jigsaw strategy with EPTs’ level of mathematics achievement and interest in an online cooperative learning environment. The research utilized a pre-experimental design with 40 participants. The researchers used pretest and posttest questions covering fraction operations and (...) a mathematics interest questionnaire before and after the intervention. With the implementation of the jigsaw strategy, results showed that the EPTs’ mathematics achievement improved while their interest in mathematics improved to some extent. Likewise, a significant difference between the pre-and post-intervention concerning the EPTs’ mathematics achievement and interest in mathematics was found. Furthermore, correlation analysis revealed that study time is negatively related to mathematics achievement, as is the overall level of interest in mathematics. These findings imply that a jigsaw strategy is a useful tool in teaching mathematics online. However, several modifications may be considered to recognize that the EPTs’ interest in mathematics varies from one person to another. Teacher education institutions are encouraged to apply this strategy not only for EPTs to increase their mathematical know-hows but also to serve as an example for them to employ in their future teaching careers. (shrink)

Errett Bishop's work in constructive mathematics is overwhelmingly regarded as a turning point for mathematics based on intuitionistic logic. It brought new life to this form of mathematics and prompted the development of new areas of research that witness today's depth and breadth of constructive mathematics. Surprisingly, notwithstanding the extensive mathematical progress since the publication in 1967 of Errett Bishop's Foundations of Constructive Analysis, there has been no corresponding advances in the philosophy of constructive mathematics Bishop style. The aim (...) of this paper is to foster the philosophical debate about this form of mathematics. I begin by considering key elements of philosophical remarks by Bishop, especially focusing on Bishop's assessment of Brouwer. I then compare these remarks with ``traditional'' philosophical arguments for intuitionistic logic and argue that the latter are in tension with Bishop's views. ``Traditional'' arguments for intuitionistic logic turn out to be also in conflict with significant recent developments in constructive mathematics. This rises pressing questions for the philosopher of mathematics, especially with regard to the possibility of offering alternative philosophical arguments for constructive mathematics. I conclude with the suggestion to look anew at Bishop's own remarks for inspiration. (shrink)

Mathematical confidence and critical thinking are essential in preparing preservice teachers. Thus, this study explored the perceived confidence and critical thinking levels in mathematics of elementary and secondary preservice teachers. A descriptive-correlational-comparative research design was employed, with a sample of 107 randomly selected preservice teachers enrolled in the Bachelor in Elementary and Secondary Education programs of a state university in West Philippines. The study used arithmetic mean, standard deviation, Spearman’s rank-order correlation, and independent samples t-test to analyze and draw (...) conclusions from the data. The findings revealed that the preservice teachers have high confidence and critical thinking skills. Their program significantly correlates with their perceived critical thinking and confidence level. Besides, the preservice teachers’ confidence levels and perceived critical thinking skills significantly correlate. Further analysis found significant confidence and critical thinking differences favoring the secondary over the elementary preservice teachers. These findings provide insights that would benefit mathematics educators in providing priority programs to enhance the preparation of future math teachers. (shrink)

This study investigated the parental involvement in learning Mathematics of students in relation to attitude and academic performance. The respondents of this study were the three-hundred fifty-six grade 9 students at 9 secondary schools in the Division of Bago City. Results showed that most of the respondents were female, belonged to income range Php 12,082 and below, their parents attained high school level, and had other works aside from being mentioned in the option of occupation. Level of parental involvement when (...) taken as a whole was moderate; in terms of its areas, motivator and nature of Mathematics were interpreted as high and resource provider, monitor, Mathematics content advisor, and Mathematics learning counselor were all interpreted as moderate; level of attitude of students towards Mathematics was moderate; academic performance of students was very satisfactory; there existed significant difference on the parental involvement in learning Mathematics in terms of six areas. Results also revealed that there was a significant relationship between parental involvement in learning Mathematics and attitude of students; significant relationship also existed between parental involvement in learning Mathematics and academic performance; significant relationship was also noted between attitude of students and academic performance. Intervention plan/program was also provided based on the results of the study. (shrink)

In this paper, I study how mathematicians are presented in western popular culture. I identify five stereotypes that I test on the best-known modern movies and television shows containing a significant amount of mathematics or important mathematician characters: (1) Mathematics is highly valued as an intellectual pursuit. (2) Little attention is given to the mathematical content. (3) Mathematical practice is portrayed in an unrealistic way. (4) Mathematicians are asocial and unable to enjoy normal life. (5) Higher mathematics is (...) ... (shrink)

Mathematical modeling offers a promising approach to improving mathematics education. This study aims to determine if the concept of metacognitive awareness in the learning process is associated with mathematical modeling. This study also considers the interaction effect of sex and academic year level on both variables. Focusing the study on preservice elementary teachers might address potential issues and targeted intervention in their preparation program concerning their ability to teach and guide young learners in modeling activities. The research sample (...) includes 140 preservice elementary teachers at Central Luzon State University, Philippines. Data collection used an adapted metacognitive awareness inventory and a validated researcher-made mathematical modeling competency test aligned with the K-12 mathematics curriculum in the Philippines. Results revealed that the preservice elementary teachers had a high metacognitive awareness and mathematical modeling competency, ranging from 22 to 31 out of 36 points. Besides, Factorial ANOVA indicates that academic year level positively affects both variables regardless of sex, and stepwise regression analysis unveiled that information management, declarative knowledge, and planning significantly predict 41.4% of the mathematical modeling competency variance. This suggests that developing metacognitive awareness supports preservice elementary teachers in performing modeling tasks that improve their competency level in mathematics. (shrink)

While the consequences of mathematically-based software, algorithms and strategies have become ever wider and better appreciated, ethical reflection on mathematics has remained primitive. We review the somewhat disconnected suggestions of commentators in recent decades with a view to piecing together a coherent approach to ethics in mathematics. Calls for a Hippocratic Oath for mathematicians are examined and it is concluded that while lessons can be learned from the medical profession, the relation of mathematicians to those affected by their work is (...) significantly different. There is something to be learned also from the codes of conduct of cognate but professionalised quantitative disciplines such as engineering and accountancy, as well as from legal principles bearing on professional work. We conclude with recommendations that professional societies in mathematics should sponsor an (international) code of ethics, institutional mission statements for mathematicians and syllabuses of ethics courses for incorporation into mathematics degrees. (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)

This study delved into the perceptions of Mathematics’ student teachers regarding the implementation of gamification in secondary schools at Nasugbu, Batangas. This research investigates the rising global trend of implementing gamification in education, particularly in Mathematics teaching, to address contemporary learner needs by examining student teachers' use of gamified activities, their design factors, encountered challenges, and perceived benefits. Purposive sampling was utilized in a multiple-case study approach to select ten (10) secondary school Mathematics’ student teachers engaged in practice teaching in (...) Nasugbu, Batangas. Qualitative findings revealed common gamified activities employed by student teachers, emphasizing the importance of student-centered design, alignment with learning objectives, student engagement, assessment, and adaptation to the classroom context. Challenges faced by student teachers in integrating gamification included technical limitations, sustaining engagement, time management, and addressing diverse learning styles. Despite these challenges, integrating gamification showed benefits such as increased engagement, improved learning outcomes, reduced math anxiety, and enhanced attitudes towards Mathematics. In conclusion, gamified activities play a significant role in enhancing student engagement and learning outcomes in Mathematics classrooms. The findings suggest student teachers to explore and integrate gamified activities into Mathematics lessons to create engaging environments supporting student success. (shrink)

The main purpose of the study is to understand the mathematical problem-solving strategies among student teachers. This study used both quantitative and qualitative type of research. Aside from the semi-structured interviews, data were gathered through participant's actual mathematical problem-solving outputs and the videotaped interviews. Findings revealed that the problem-solving strategies among student teachers in the Problem-Solving subject are cognitive, metacognitive and other strategies. The cognitive strategies used by the student teachers are rehearsal, elaboration, and organization. The metacognitive strategies (...) are critical thinking and self-regulation. Other strategies are overlapping the cognitive and metacognitive strategies. These are prediction/orientation, planning, monitoring, and evaluating. The findings also suggest a significant influence of the strategies to the academic performance of the student teachers. (shrink)

The role of mathematics in the development of Gilles Deleuze's (1925-95) philosophy of difference as an alternative to the dialectical philosophy determined by the Hegelian dialectic logic is demonstrated in this paper by differentiating Deleuze's interpretation of the problem of the infinitesimal in Difference and Repetition from that which G. W. F Hegel (1770-1831) presents in the Science of Logic . Each deploys the operation of integration as conceived at different stages in the development of the infinitesimal calculus in his (...) treatment of the problem of the infinitesimal. Against the role that Hegel assigns to integration as the inverse transformation of differentiation in the development of his dialectical logic, Deleuze strategically redeploys Leibniz's account of integration as a method of summation in the form of a series in the development of his philosophy of difference. By demonstrating the relation between the differential point of view of the Leibnizian infinitesimal calculus and the differential calculus of contemporary mathematics, I argue that Deleuze effectively bypasses the methods of the differential calculus which Hegel uses to support the development of the dialectical logic, and by doing so, sets up the critical perspective from which to construct an alternative logic of relations characteristic of a philosophy of difference. The mode of operation of this logic is then demonstrated by drawing upon the mathematical philosophy of Albert Lautman (1908-44), which plays a significant role in Deleuze's project of constructing a philosophy of difference. Indeed, the logic of relations that Deleuze constructs is dialectical in the Lautmanian sense. (shrink)

Mathematics has a long track record of refining the concepts by which we make sense of the world. For example, mathematics allows one to speak about different senses of "sameness", depending on the larger context. Phenomenology is the name of a philosophical discipline that tries to systematically investigate the first-personal perspective on reality and how it is constituted. Together, mathematics and phenomenology seem to be a good fit to derive statements about our experience that are, at the same time, well-defined, (...) precise, and significant to our inner lives. However, there are difficulties stemming from the fact that phenomenology deals with inherently subjective things. Phenomenological investigations seem to lose their appeal when trying to approach them in the clear-cut way afforded by mathematics. How to overcome this obstacle? We argue that the Arts play a special role in mediating between the precise statements of mathematics and the sometimes fuzzy nature of our experience. Mathematics and art are complementary ways to come to a comprehensive understanding of reality. (shrink)

The study aimed to determine if differentiated instruction effectively addresses learning gaps in mathematics. In particular, it explored how it can improve the student’s learning gaps concerning mathematical performance and confidence. The study employed a quasi-experimental design with 30 purposively-selected Grade 10 participants divided into differentiated (n = 15) and control groups (n = 15), ensuring the utmost ethical measures. The mean and standard deviation were used to describe the participants’ performance and confidence. Independent samples t-tests were used to (...) determine the significant differences in the performance and confidence between the two groups. In contrast, dependent samples t-tests were used to determine the significant differences in each group’s pre and posttest performance and confidence. Findings bared that the differentiated instruction successfully addressed students’ performance in mathematics even in a short period. It also increased the participants’ confidence when answering fundamental problems. Continuing differentiated instruction activities are recommended since it benefits students who struggle in mathematics, particularly in answering fundamental operations. Differentiated teaching activities in mathematics can boost academic achievement and engagement and prepare students for future success while fostering a positive and inclusive classroom culture that values individual learning needs and preferences. (shrink)

The primary purpose of this study was to examine the intricate relationship between enrichment programs, pedagogical preparedness, and the mathematical proficiency of pre-service teachers. Employing a quantitative research design, the study utilized a descriptive-correlational method to establish the relationship between variables. A comprehensive sampling was used and adapted well-structured survey questionnaires were distributed to gather relevant data from the participants. Correspondingly, the data analysis was performed using statistical tools such as mean and standard deviation, Pearson’s correlation coefficient (r), and (...) regression analysis to distill complex data into meaningful insights and answer the research objectives. The findings revealed that the levels of both enrichment programs and pedagogical preparedness among pre-service teachers were very high, indicating that they are always manifested and consistently implemented. Additionally, the level of mathematical proficiency was found to be high, suggesting that the pre-service teachers frequently exhibited strong mathematical skills. There is a confirmation that the levels of enrichment programs and pedagogical preparedness have a significant relationship with mathematical proficiency. Therefore, this connects to the idea that enrichment programs and pedagogical preparedness influence the mathematical proficiency of pre-service teachers. These findings emphasize the importance of integrating comprehensive enrichment programs and adaptive pedagogical strategies into teacher training, enabling pre-service teachers to develop advanced mathematical understanding and effective teaching practices. Continuous professional development tailored for pre-service teachers, complemented by peer mentoring, seminars, and collaborative training within and beyond schools, can further refine teaching skills and enhance mathematical proficiency. By fostering this holistic approach, the study contributes to improving teaching effectiveness and student outcomes, promoting a culture of excellence in mathematics education. (shrink)

Cantor argued that absolute infinity is beyond mathematical comprehension. His arguments imply that the domain of mathematics cannot be grasped by mathematical means. We argue that this inability constitutes a foundational problem. For Cantor, however, the domain of mathematics does not belong to mathematics, but to theology. We thus discuss the theological significance of Cantor’s treatment of absolute infinity and show that it can be interpreted in terms of negative theology. Proceeding from this interpretation, we refer to (...) the recent debate on absolute generality and argue that the method of diagonalization constitutes a modern version of the vianegativa. On our reading, negative theology can evoke an attitude of humility with respect to the boundedness of the human condition. Along these lines, we think that the foundational problem of mathematics concerning its domain can be addressed through a methodological attitude of humility. (shrink)

Indigenous communities have a rich cultural heritage encompassing diverse ways of knowing, learning, and understanding the world around them. This mixed methods study utilized the explanatory sequential design to determine the level and relationship of the IP mathematics teachers' beliefs and teaching practices and gain a deeper insight into these beliefs and attitudes. There are 115 respondents for the quantitative phase, while 10 participants in the qualitative phase. Data were collected through survey and key informant interviews and were analyzed through (...) mean, standard deviation, Pearson product moment correlation, and thematic analysis. Results showed that the IP mathematics teachers possess high levels for both beliefs and teaching practices, and a significant moderate positive relationship exists between the two variables. Qualitative data analysis revealed that teachers' beliefs include the ideal qualities of an effective mathematics teacher, respect for students' culture and background, and a constructivist approach to teaching. Teaching practices include discussing student performance and intervention during meetings, implementing culturally relevant approaches, technology, varied activities, and peer mentoring and coaching. Further, it is concluded that teachers’ belief, which influences teaching practices, is shaped by different environmental factors. It is recommended that IP mathematics teachers continue to participate in personal and professional development activities and strengthen collaboration and reflection. (shrink)

The origins of testing scientific models with statistical techniques go back to 18th century mathematics. However, the modern theory of statistical testing was primarily developed through the work of Sir R.A. Fisher, Jerzy Neyman, and Egon Pearson in the inter-war period. Some of Fisher's papers on testing were published in economics journals (Fisher, 1923, 1935) and exerted a notable influence on the discipline. The development of econometrics and the rise of quantitative economic models in the mid-20th century made statistical (...) class='Hi'>significance testing a commonplace, albeit controversial tool within economics. -/- In the debate about significance testing, methodological controversies intertwine with epistemological issues and sociological developments. Our aim in this chapter is to expound these connections and to show how the use of, and the debate about, significance testing in economics differs from other social sciences, such as psychology. (shrink)

This article surveys recent literature by Parsons, McGee, Shapiro and others on the significance of categoricity arguments in the philosophy of mathematics. After discussing whether categoricity arguments are sufficient to secure reference to mathematical structures up to isomorphism, we assess what exactly is achieved by recent ‘internal’ renditions of the famous categoricity arguments for arithmetic and set theory.

It is a striking fact from reverse mathematics that almost all theorems of countable and countably representable mathematics are equivalent to just five subsystems of second order arithmetic. The standard view is that the significance of these equivalences lies in the set existence principles that are necessary and sufficient to prove those theorems. In this article I analyse the role of set existence principles in reverse mathematics, and argue that they are best understood as closure conditions on the powerset (...) of the natural numbers. (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)

Symmetry is not an inherent characteristic of mathematical proofs; instead, it is a property that arbitrarily manifests in different modes of presentation. This arbitrariness leads to the conclusion that symmetry cannot be part of the defining or essential properties that characterize proofs. Consequently, contrary to some authors’ claims, symmetry does not significantly contribute to the validity, accuracy, or soundness of mathematical proofs. What is more, it does not even play any critical role in heuristic aspects such as explanatory (...) power. The examples developed in this paper constitute compelling evidence supporting these claims. (shrink)

In this chapter we use methods of corpus linguistics to investigate the ways in which mathematicians describe their work as explanatory in their research papers. We analyse use of the words explain/explanation (and various related words and expressions) in a large corpus of texts containing research papers in mathematics and in physical sciences, comparing this with their use in corpora of general, day-to-day English. We find that although mathematicians do use this family of words, such use is considerably less prevalent (...) in mathematics papers than in physics papers or in general English. Furthermore, we find that the proportion with which mathematicians use expressions related to ‘explaining why’ and ‘explaining how’ is significantly different to the equivalent proportion in physics and in general English. We discuss possible accounts for these differences. (shrink)

Computation is central to contemporary mathematics. Many accept that we can acquire genuine mathematical knowledge of the Four Color Theorem from Appel and Haken's program insofar as it is simply a repetitive application of human forms of mathematical reasoning. Modern LLMs / DNNs are, by contrast, opaque to us in significant ways, and this creates obstacles in obtaining mathematical knowledge from them. We argue, however, that if a proof-checker automating human forms of proof-checking is attached to such (...) machines, then we can obtain apriori mathematical knowledge from them, even though the original machines are entirely opaque to us and the proofs they output are not human-surveyable. (shrink)

This chapter argues that the standard conception of Spinoza as a fellow-travelling mechanical philosopher and proto-scientific naturalist is misleading. It argues, first, that Spinoza’s account of the proper method for the study of nature presented in the Theological-Political Treatise (TTP) points away from the one commonly associated with the mechanical philosophy. Moreover, throughout his works Spinoza’s views on the very possibility of knowledge of nature are decidedly sceptical (as specified below). Third, in the seventeenth-century debates over proper methods in the (...) sciences, Spinoza sided with those that criticized the aspirations of those (the physico-mathematicians, Galileo, Huygens, Wallis, Wren, etc) who thought the application of mathematics to nature was the way to make progress. In particular, he offers grounds for doubting their confidence in the significance of measurement as well as their piece-meal methodology (see section 2). Along the way, this chapter offers a new interpretation of common notions in the context of treating Spinoza’s account of motion (see section 3). (shrink)

In an attempt to curtail examination malpractice, the Joint Admission and Matriculation Board (JAMB) has been generating different paper types with a different order of test items in the Unified Tertiary Matriculation Examination (UTME). However, the permutation of test items may compromise students’ performance unintentionally because constructive suggestions in theory and practice recommend that test items be sequenced in ascending order of difficulty. This study used data collected from a random sample of 1,226 SSIII students to ascertain whether the permutation (...) of test items has any effect on the performance of students in two different subjects (Use of English and Mathematics). The study adopted the Equivalent Groups Quasi-Experimental Research Design with three independent groups. Findings emerged, amongst others, that there is a significant difference in the performance scores of prospective university students' in use of English and Mathematics examintaions arranged in three different orders (ED, DE, R). There are no significant gender differences in the performance of students in Use of English and Mathematics based on test item permutation. However female students perform better than male students when test items are arranged in ascending order of difficulty while males perform better when test items are arranged in descending order of difficulty. It was concluded that the permutation of test items in UTME examination tends to affect the performance of students in Use of English and Mathematics. This finding has implications for the future conduct of UTME examinations and enrolment into higher education as the randomization of UTME test items changes the difficulty order of different paper types. It was recommended that other measures of curtailing examination malpractices that would not affect students’ academic performance should be adopted. (shrink)

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)

The objective of this case study concentrated on examining the learning gap, going through some components of the transformation process, and coming up with some ways for aiding students who were experiencing lost learning. A qualitative research design was utilized by the researchers to understand and solve the cases related to Mathematics learning difficulties. Creswell (2008) asserts that qualitative research can be used to discover and comprehend the significance that certain people or groups assign to social or human issues. (...) The researchers used purposive sampling to determine 15 student participants to understand the cases of learning loss and prescribe better interventions to the students. To solve the problems associated with loss of learning experienced by the learners after remote learning, a case study will be utilized as the strategy of inquiry since it fits the situation that is to focus the said key cases, discuss some transforming process, and develop interventions. After analyzing the interviews, researchers identified three common themes across the three cases which serve to clarify the learners' experiences after the post-remote learning: the learners' learning experiences during remote learning, factors that caused learning loss in Mathematics, and learners' coping strategies. In a deeper analysis, the impact of distance learning on students during the pandemic goes beyond just the difficulties in learning and understanding lessons which resulted in difficulty in learning math during in-person classes. The lack of in-person interaction with classmates and teachers also contributes to a sense of isolation and boredom, negatively affecting students' overall well-being and mental health. (shrink)

Computation and Metacognitive skills are essential sub-skills under the domain of Critical Thinking which is a 21 st Century Skill. Having acquired these skills can greatly help students to have a better performance in the Mathematics course. The purpose of this study was to determine whether computation and metacognitive skills are significant predictors of students' performance in Mathematics. Students from four sections of the course Mathematics in the Modern World which was offered during the first semester of the academic year (...) 2018-2019 were selected as the participants of this study which was conducted at Gov. Alfonso D. Tan College, Tangub City, Misamis Occidental, Philippines. The results of this study showed that students' scores in computation and metacognitive test of skills significantly correlate with the students' performance in Mathematics in terms of Grade Point Average (GPA) and a positive direct relationship has been established. Using the Regression Analysis, the result revealed that both computation and metacognitive skills of the students were significant predictors of their performance in Mathematics. These research outcomes affirm that if students acquire these skills, they tend to perform better in Mathematics. This suggests that teachers are encouraged to be more strategic in training the students to acquire and develop computation and metacognitive skills by administering various mathematical problems that encourage students to think critically. (shrink)

To make sense of what Gilles Deleuze understands by a mathematical concept requires unpacking what he considers to be the conceptualizable character of a mathematical theory. For Deleuze, the mathematical problems to which theories are solutions retain their relevance to the theories not only as the conditions that govern their development, but also insofar as they can contribute to determining the conceptualizable character of those theories. Deleuze presents two examples of mathematical problems that operate in this (...) way, which he considers to be characteristic of a more general theory of mathematical problems. By providing an account of the historical development of this more general theory, which he traces drawing upon the work of Weierstrass, Poincaré, Riemann, and Weyl, and of its significance to the work of Deleuze, an account of what a mathematical concept is for Deleuze will be developed. (shrink)

This study investigates the effect of mobile app on students’ Behavioral engagement, Confidence with technology, Mathematics confidence, Affective engagement, and Mathematics with technology. Grade 9 students were provided with mobile apps to support them in studying mathematics during distance learning. Post-test control group was utilized in the study to compare the mathematics and technology attitudes of the students in the control and treatment groups. This study used the Mathematics and Technology Attitudes Scale (MTAS) questionnaire adapted from Pierce et al. (2007). (...) Majority of students have a favorable attitude about the use of mobile educational applications, ranging from neutral to positive. Although there were no statistically significant differences in several of the subscales, the experimental group showed greater optimism than the control group. Teachers should continue incorporating mobile educational applications into the classroom to help and enhance the learning process given the good experiences reported by students. To further enhance students' attitudes toward the topic, it should be considered to address the negative attitude toward mathematical confidence. (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)

Learning mathematics through distance education can be challenging, with the “logical” and “rewarding” nature proving difficult to measure. This article aimed to articulate an argument explaining the “logical” and “rewarding” nature of online mathematics learning, elucidating their causal factors. Existing data from the literature that involving students at Visayas State University, Philippines, were utilized in this study. The study used statistical measures to capture descriptions from the data, and quantile regression analysis was employed to forecast the predictors of the logicality (...) and rewarding nature of learning mathematics at a distance. Results indicate that learning mathematics in the new normal is perceived as “logical” and moderately “rewarding”. The regression and correlation analyses revealed a significant positive association between the perceived “logical” nature and the rewarding experience of learning mathematics. The constructed statistical models depicted that the determinants of logicality in learning mathematics include family income, money spent on the internet, learning environment, household size, social status, and health. Moreover, causal factors such as family income, money spent on the internet, learning environment, leisure activities, social status, and health significantly determine the rewarding nature of learning mathematics online. Conclusively, institution must support college students with their online learning needs. Furthermore, mathematics instructors should create lively and exciting online discussions that boost their logical thinking. Providing problem-solving task that are intrinsically rewarding can contribute to a more fulfilling learning experience. (shrink)