This study examines the associations of constructivist beliefs and classroom climate on teachers' self-efficacy in instruction, classroom management, and student engagement among Australian secondary mathematics teachers. To do this, it uses the integrated model of teachers' self-efficacy with the concept of analysis of teaching tasks. The study uses structural equation modeling to analyze data from 495 mathematics teachers in the Teaching and Learning International Survey 2013. The results reveal the integrated model is a valid theoretical (...) framework to explain Australian secondary mathematics teachers' self-efficacy. Teachers' constructivist beliefs and classroom climate are positively and statistically significantly related to teachers' self-efficacy in instruction, classroom management and student engagement. In contrast, constructivist beliefs have no significant correlation with classroom climate. (shrink)

ABSTRACT: The Smith Report has generated central questions about the mathematics education of UK adolescents. This paper highlights the close match between the goals of school mathematics, adolescence and exploratory pedagogy. This is contrasted with the prescriptive nature of current regimes. In particular, without careful attention to pedagogy it is possible that the introduction of different pathways may lead to a failure to achieve the outcomes desired by employers and universities, and to inequity in provision for students.

Dialogic Teaching is effective in fostering student learning; yet, it is hard to implement. Little research focused on secondary teachers’ learning of DT and on the link between teachers’ understanding and practices, although these two are usually strongly intertwined. Using a wide range of evidence, this case study systematically investigated and compared two secondary teachers’ understanding and practice of DT during their participation in a continuing professional development programme. The CPDP appeared effective to some extent. The History (...) teacher’s understanding of DT, i.e. being a co-learner, appeared highly effective in implementing DT, whereas the Mathematics teacher’s understanding of DT, i.e. creating a democratic learning environment, seemed only effective to some extent. Focusing on both teachers’ understanding and practice when developing DT seemed fruitful in explaining differences in practice. Future research could further explore to what extent understanding DT as being a co-learner facilitates professional development. (shrink)

The aim of this study is to investigate students’ conceptions about proof in mathematics and mathematics teaching. A five‐point Likert‐type questionnaire was administered in order to gather data. The sample of the study included 33 first‐year secondary school mathematics students . The data collected were analysed and interpreted using the methods of qualitative and quantitative analysis. The results have revealed that the students think that mathematical proof has an important place in mathematics and (...) class='Hi'>mathematics education. The students’ studying methods for exams based on imitative reasoning which can be described as a type of reasoning built on copying proof, for example, by looking at a textbook or course notes proof or through remembering a proof algorithm. Moreover, they addressed to the differences between mathematics taught in high school and university as the main cause of their difficulties in proof and proving. (shrink)

This article discusses the role of mathematics during physics lessons in upper-secondary school. Mathematics is an inherent part of theoretical models in physics and makes powerful predictions of natural phenomena possible. Ability to use both theoretical models and mathematics is central in physics. This paper takes as a starting point that the relations made during physics lessons between the three entities Reality, Theoretical models and Mathematics are of the outmost importance. A framework has been developed (...) to sustain analyses of the communication during physics lessons. The study described in this article has explored the role of mathematics for physics teaching and learning in upper-secondary school during different kinds of physics lessons. Observations are from three physics classes led by one teacher. The developed analytical framework is described together with results from the analysis of the 7 lessons. The results show that there are some relations made by students and teacher between theoretical models and reality, but the bulk of the discussion in the classroom is concerning the relation between theoretical models and mathematics. The results reported on here indicate that this also holds true for all the investigated organizational forms lectures, problem solving in groups and labwork. (shrink)

The call for evidence-based practice in education emphasizes the need for research to provide evidence for particular fields of educational practice. With this systematic literature review we summarize and analyze aggregated effectiveness information from 41 meta-analyses published between 2004 and 2019 to inform evidence-based practice in a particular field. In line with target specifications in education that are provided for a certain school subject and educational level, we developed and adopted a selection heuristic for filtering aggregated effect sizes specific to (...) both science and mathematics education and the secondary student population. The results include 78 context-specific aggregated effect sizes based on data from over one million students. The findings encompass a multitude of different teaching strategies, most of which offer a measurable advantage to alternatives. Findings demonstrate that context-specific effect size information may often differ from more general effect size information on teaching effectiveness and adherence to quality standards varies in sampled meta-analyses. Thus, although meta-analytic research has strongly developed over the last few years, providing context-specific and high-quality evidence still needs to be a focus in the field of secondary mathematics and science teaching and beyond. (shrink)

The teacher is one of the key factors in the transmission of curricular objectives to their students. Their knowledge and positive attitude toward students' ethnomathematics and its pedagogical approach play an important role to make classroom teaching culture friendly. This paper is intended to explore the in-service teachers’ knowledge of ethnomathematics and perception on the integration of the ethnomathematics approach in their teaching with respect to the demographic factors of in-service teachers such as gender, academic status, teaching (...) experience, teacher training, and nature of institution they associated. This study was conducted among 120 in-service mathematics teachers from 32 secondary schools in Kathmandu District. The explanatory sequential mix method was employed to carry out this study. The perception of in-service teachers on ethnomathematics pedagogy in their classroom teaching was measured with survey instruments in the first stage. For further justification of the results obtained from the quantitative data, six teachers with different demographic factors were interviewed in the second stage of the data collection process. The relationship between in-service teachers’ understanding on ethnomathematics and different demographic factors was investigated using the Pearson Chi-square test. From the statistical data analysis, it was found that there is no relationship between gender, academic qualifications, and teachers' training in their understanding of ethnomathematics. However, a significant relationship was found between the teachers' experience and nature of the institutions they were involved, and their understanding on ethnomathematics. In-service teachers have a similar understanding on ethnomathematics and their perception in the integration of the ethnomathematics approach in the teaching of school mathematics. (shrink)

Microgames are rapidly gaining increased attention and are highly being considered because of the technology-based media that enhances students’ learning interests and educational activities. Therefore, this study aims to develop a new construct through confirmatory factor analysis, to comprehensively understand the factors influencing the use of microgames in mathematics class. Participants of the study were the secondary school teachers in West Java, Indonesia, which had a 1-year training in microgames development. We applied a quantitative approach to collect the (...) data via online questionnaires through google form. Structural Equation Modelling with AMOS software was used to analyze the proposed model. Empirical results confirmed the perceived easy to use and subjective norm influence relationship with teachers’ microgame usage behaviors and intentions. In this condition, SN was found to have the initial significant influence on behavioral intention, as attitude, BI, and facilitating conditions also correlated with the actual use of microgames. Furthermore, the largest influential factor was BI, with the results subsequently showing that TPACK had no significant influence on the actual use of microgames. This report is expected to help bridge the gap across several previous studies, as well as contribute to the explanation and prediction of the factors influencing the teachers’ mathematical utilization of the study’s program. Besides this, it also helps to increase the use of microgames in teaching and learning activities. (shrink)

This book offers an innovative introduction to the psychological basis of mathematics and the nature of mathematical thinking and learning, using an approach that empowers students by fostering their own construction of mathematical structures. Through accessible and engaging writing, award-winning mathematician and educator Anderson Norton reframes mathematics as something that exists first in the minds of students, rather than something that exists first in a textbook. By exploring the psychological basis for mathematics at every level - including (...) geometry, algebra, calculus, complex analysis and more - Norton unlocks students' personal power to construct mathematical objects based on their own mental activity, and illustrates the power of mathematics in organizing the world as we know it. Including reflections and activities designed to inspire awareness of the mental actions and processes coordinated in practicing mathematics, the book is geared towards current and future secondary and elementary mathematics teachers who will empower the next generation of mathematicians and STEM majors. Those interested in the history and philosophy that underpins mathematics will also benefit from this book, as well as those informed and curious minds attentive to the human experience more generally. Anderson Norton is Professor in the Department of Mathematics at Virginia Tech, USA, where he has been teaching mathematics courses for future teachers for 15 years. He is the editor of Constructing Number, and the co-author of Developing Fractions Knowledge and Numeracy for All Learners. (shrink)

A large portion of mathematics education centers heavily around imitative reasoning and rote learning, raising concerns about students’ lack of deeper and conceptual understanding of mathematics. To address these concerns, there has been a growing focus on students learning and teachers teaching methods that aim to enhance conceptual understanding and problem-solving skills. One suggestion is allowing students to construct their own solution methods using creative mathematical reasoning, a method that in previous studies has been contrasted against algorithmic (...) reasoning with positive effects on test tasks. Although previous studies have evaluated the effects of CMR, they have ignored if and to what extent intrinsic cognitive motivation play a role. This study investigated the effects of intrinsic cognitive motivation to engage in cognitive strenuous mathematical tasks, operationalized through Need for Cognition, and working memory capacity. Two independent groups, consisting of upper secondary students, practiced non-routine mathematical problem solving with CMR and AR tasks and were tested 1 week later. An initial t-test confirmed that the CMR group outperformed the AR group. Structural equation modeling revealed that NFC was a significant predictor of math performance for the CMR group but not for the AR group. The results also showed that WMC was a strong predictor of math performance independent of group. These results are discussed in terms of allowing for time and opportunities for struggle with constructing own solution methods using CMR, thereby enhancing students conceptual understanding. (shrink)

The purpose of this paper is the influence of Science- Technology- Engineering- Art- Mathematics education on the Olympic education of primary and middle school students. The research object is the Beijing Olympic model school. The frame structure and educational concept of STEAM education are studied, and a questionnaire survey on the current situation of Olympic education is conducted. Finally, improvement measures based on the survey results are provided combined with STEAM education, and the teaching effect is analyzed. The (...) results show that after the improvement of the educational model, the student’s mastery of the Olympic knowledge has been greatly improved. The proportion of students who can master all knowledge increases from 5.78 to 8.45%, by 2.67%. The proportion of students with knowledge of most Olympic sports increases from 48.6 to 55.67%, by 7.07%. The proportion of students with little or no knowledge drops to 1.54%. Meanwhile, students are increasingly interested in Olympic events, especially after being inspired by the STEAM education model. The proportion of students who are very concerned about Olympic events has increased by 6.45%. The proportion of students who are more concerned about Olympic events has increased by 5.11%. Thus, the Olympic education work has achieved some results. Students gradually become interested in the Olympic events. They begin to actively pay attention to the Olympic events and learn the Olympic spirit. Then, their physical literacy is improved. (shrink)

Autonomous learning capacity is a key competency that supports teachers’ professional development. In this study, a stratified sampling method was used to recruit 396 junior and senior high school mathematics teachers in T city, one of the provincial city in China. A questionnaire with high reliability and validity developed prior to the study by the researchers was employed to measure their autonomous learning capacity and differences across groups. Twelve teachers were then selected for interviews. The results showed satisfactory overall (...) performance. By subdimension, teachers’ performance was the best in the development of study plans, followed by evaluation of learning outcomes, while they needed improvement in learning habit formation and proficiency in using learning methods. Furthermore, the analysis of differences across groups indicated that for autonomous learning capacity, female teachers were significantly better than their male peers; junior high school teachers were better compared to those in senior high school; teachers aged 41–50 underperformed those aged 51 and above; teachers who work in rural areas and townships did not perform worse than urban teachers; and those with doctoral degrees did not demonstrate considerable advantage over others. There were no significant differences in the overall autonomous learning capacity across years of teaching and job title groups. However, in the subdimensions, those with 21–30 years of teaching experience had lower proficiency in using learning methods and evaluation of learning outcomes, and teachers with senior titles did not demonstrate expected advantages in learning habit formation. (shrink)

Studies within mathematics education have shown the existence of a so-called “mathematics anxiety” among secondary and high school students as a global phenomenon and have related it to various social and psychological aspects of the processes of teaching and learning mathematics, which are responsible for the understanding of mathematics. Current research project aims at crystallizing a primary conceptual and theoretical framework necessary for the application of contributions of philosophy of mathematics and of science (...) and epistemology of mathematics in mathematics education from an epistemological perspective. It argues for and discusses applicability of the general principle that teaching about mathematics along with traditionally teaching formal mathematics and ‘doing’ mathematics does not render the process more difficult, but rather, this approach enhances mathematics education, with effects on the mathematics anxiety. -/- . (shrink)

Any study of the 'Scientific Revolution' and particularly Descartes' role in the debates surrounding the conception of nature (atoms and the void v. plenum theory, the role of mathematics and experiment in natural knowledge, the status and derivation of the laws of nature, the eternality and necessity of eternal truths, etc.) should be placed in the philosophical, scientific, theological, and sociological context of its time. Seventeenth-century debates concerning the nature of the eternal truths such as '2 + 2 = (...) 4' or the law of inertia turn on the question of whether these truths were created along with nature, or were uncreated and subsisting in God's mind. One's answer to that question has direct consequences for conceptions of the necessity/contingency of mathematical and natural knowledge, how knowledge of such truths is accomplished by humans, and what grounds these truths. In this paper, I review the positions of four successors to Descartes' philosophy on the question of the eternal truths to illustrate how in specific ways that question with its theological, metaphysical, modal, and epistemological dimensions concerned the objectivity and certainty of the discoveries of the new science. Author Recommends: Clarke, Desmond. Descartes' Philosophy of Science . University Park, Penn State Press, 1982. This work provides an account of Descartes as a practicing scientist whose rationalism is mitigated by reliance on experiment and experience. Author re-examines Descartes' philosophical and scientific works in this new light. Dear, Peter. Revolutionizing the Sciences: European Knowledge and its Ambitions, 1500–1700 . Princeton, Princeton University Press, 2001. This work provides a useful overview of the issues and thinkers of the Scientific Revolution. Of particular relevance is chapter 8 on Cartesian and Newtonian science. Funkenstein, Amos. Theology and the Scientific Imagination from the Middle Ages to the Seventeenth Century . Princeton, Princeton University Press, 1986. This work is an advanced study of the theological and metaphysical foundations of early modern science. Discussions include questions of God's nature, God's knowledge in relation to human knowledge, providence, the laws of nature, and the truths of mathematics. In particular, chapter 3 discusses Descartes' account of the eternal truths and divine omnipotence. Garber, Daniel. Descartes' Metaphysical Physics . Chicago, University of Chicago Press, 1992. This work examines how Descartes' metaphysical doctrines of God, soul, and body set the groundwork for his physics. It includes a study of God and the grounds for the laws of physics (chapter 9). Henry, John. The Scientific Revolution and the Origins of Modern Science . 3rd ed. New York, Palgrave, Macmillan Press, 2008. This work provides a brief, general, and informative overview of the Scientific Revolution, including the themes of method, magic, religion, and culture. Osler, Margaret J. Divine Will and the Mechanical Philosophy: Gassendi and Descartes on Contingency and Necessity in the Created World . Cambridge, Cambridge University Press, 1994. This work is an examination and comparison of the mechanical philosophies of Gassendi and Descartes. It offers in-depth discussion of the issue of voluntarism and intellectualism in the period and how that related to conceptions of laws of nature and the eternal truths. Shapin, Steven. The Scientific Revolution . Chicago, University of Chicago Press, 1996. This work provides a critical synthesis of as well as a guide to recent scholarship in the history of science for a general readership. Online Materials Dr. Robert A. Hatch's Scientific Revolution Website: http://web.clas.ufl.edu/users/rhatch/pages/03-Sci-Rev/SCI-REV-Home/ A compendium of resources for the study of Scientific Revolution. Early English Books Online: http://eebo.chadwyck.com/home Early English Books Online (EEBO) contains digital facsimile page images of virtually every work printed in England, Ireland, Scotland, Wales and British North America and works in English printed elsewhere from 1473 to 1700. Early Modern Resources: http://www.earlymodernweb.org.uk/emr/ Early Modern Resources is a gateway for all those interested in finding electronic resources relating to the early modern period in history. Gallica, the Digital Library of the Bibliothèque Nationale de France: http://gallica.bnf.fr/ An ever-growing digital library which includes numerous primary and secondary texts of relevance to Descartes and his role in Scientific Revolution. Hatfield, Gary, 'René Descartes', The Stanford Encyclopedia of Philosophy. Spring 2009 ed. Ed. Edward N. Zalta; URL: http://plato.stanford.edu/archives/spr2009/entries/descartes/ Slowik, Edward, 'Descartes' Physics', The Stanford Encyclopedia of Philosophy. Winter 2008 ed. Ed. Edward N. Zalta; URL: http://plato.stanford.edu/archives/win2008/entries/descartes-physics/ Syllabus Sample Syllabus: Cartesian Science The following is five weeks covering Cartesian Science in a course on Descartes or the Scientific Revolution, or 17th-century theories of matter, or related themes on early modern truth and method, especially on the continent. This material is best suited to a graduate level audience, but it could be modified to suit an upper-division undergraduate course, as the readings are basically primary texts whose context and background can be explained in lectures. Week 1: Cartesian Revolution in France • Scientific method • Role of mathematics and experiment • Certainty of scientific knowledge Readings: Hatfield, Gary, 'René Descartes', The Stanford Encyclopedia of Philosophy. Spring 2009 ed. Ed. Edward N. Zalta; URL: http://plato.stanford.edu/archives/spr2009/entries/descartes/ Descartes, Discourse on Method , Parts 1–3 Descartes, Meditations on First Philosophy , First Meditation. Week 2: Descartes' Scientific Treatises • Mechanization and mathematization of nature • Primary–secondary quality distinction Readings: Discourse on Method, Parts 4–6 Selections from Descartes' Scientific Essays: The World or Treatise on Light (ATXI 3–48); Treatise on Man (ATXI 119–202); Optics (ATVI 82–147). Slowik, Edward, 'Descartes' Physics', The Stanford Encyclopedia of Philosophy. Winter 2008 ed. Ed. Edward N. Zalta; URL: http://plato.stanford.edu/archives/win2008/entries/descartes-physics/ Henry, John, 'The Mechanical Philosophy,' chapter 5. The Scientific Revolution and the Origins of Modern Science . 3rd ed. Macmillan, 2008. Week 3: Descartes' Theory of Nature • Descartes' derivation of the law of conservation and the three laws of motion • God's role in the metaphysics and physics of nature Readings: Selections from Principles of Philosophy, Preface (all); Letter to Elizabeth; Part I: 1–8; Part II: 1–45, 55, 64; Part III: 1–4, 15–19, 45–47; Part IV: 187–207. John Henry, 'Religion and Science,' chapter 6. The Scientific Revolution and the Origins of Modern Science . 3rd ed. Macmillan, 2008. Week 4: Post-1650 Cartesian Science: Necessity and Contingency in Nature • Debates on God, Creation, and Causes Readings: Easton, Patricia, 'What is at Stake in the Cartesian Debates on the Eternal Truths?' Philosophy Compass 4.2 (2009): 348–62. Malebranche, Nicolas, 'Elucidation 10', from The Search after Truth (1674). Note: All selections available in Nicolas Malebranche (1992). Philosophical Selections , edited by S. Nadler, Hackett. Gottfried Leibniz (1714) Monadology . Week 5: Causes in Nature and Morals • Theodicy as an explanation of defect and evil in a lawful universe: Malebranche v. Leibniz Readings: Nicolas Malebranche, Elucidation XVI (on occasionalism), and Treatise on Nature and Grace, Discourse One, Part 1. Gottfried Leibniz (1706), Theodicy. Focus Questions Weekly questions can be used to focus the readings. This can be done in a web or e-mail discussion thread, as a weekly assignment, or for in class discussion. I require students to post a short paragraph in response to the question or some posting by a classmate on the question. Students are required to post by 10 a.m. the day before we meet for class on a course website. Week 1: According to Descartes, what role does skepticism play in scientific reasoning? Week 2: Comment on the following: 'But I am supposing this machine to be made by the hands of God, and so I think you may reasonably think it capable of a greater variety of movements than I could possibly imagine in it, and of exhibiting more artistry than I could possibly ascribe to it' [ Treatise on Man ; ATXI 120]. Week 3: What is Descartes' conception of the relation between the metaphysics and physics of nature? Week 4: Critically discuss the positions of Descartes, Malebranche, and Leibniz on what provides the foundation for the certitude of natural knowledge? Week 5: Explain why both Malebranche and Leibniz consider moral sin to be analogous to natural defect? Seminar/Project Idea Hold a debate on the question of the status of the eternal truths. The proposition will be Descartes' position: 'Eternal truths must be both created and necessary if certainty in science is to be possible'. Format: 1. At the beginning of the 5-week module, students will be assigned to one of three roles: Team A, Team B, and judge's panel. Students will be given the debate proposition, but will not be told which team will take the affirmative and which team the negative until the time of the debate. 2. Recommend a variation on the Classic Debate Format to encourage the development of argument: sequence begins with affirmative construction (8 minutes), negative construction (8 minutes), second affirmative construction (8 minutes), second negative construction (8 minutes), first negative rebuttal (4 minutes), first affirmative rebuttal (4 minutes), final negative rebuttal (4 minutes) and final affirmative rebuttal (4 minutes). 3. Judges Panel: will consist of 3–4 judges who will assess the performance of Teams A and B. Judgment should be based on the persuasiveness of the team position. 4. Debate will be held at the end of the fifth week, or semester, whichever makes most sense given the course length and structure. Acknowledgements The author gratefully acknowledges the immensely helpful comments and suggestions by the participants in her graduate seminar on the Scientific Revolution: Benjamin Chicka, Sarah Jacques-Ross, Richard Ross, Marcella Stockstill, and Zohra Wolters. (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)

Context: Constructivist teachers who find themselves working within an educational system that adopts a realist epistemology, may find themselves at odds with their own beliefs when they catch themselves paying closer attention to the knowledge authorities intend them to teach rather than the knowledge being constructed by their learners. Method: In the preliminary analysis of the mathematical learning of six low-performing Year 7 boys in a Maltese secondary school, whom one of us taught during the scholastic year 2014-15, we (...) constructed a conceptual framework which would help us analyze the extent to which he managed to be sensitive to constructivism in a typical classroom setting. We describe the development of the framework M-N-L as a viable analytical tool to search for significant moments in the lessons in which the teacher appeared to engage in what we define as “constructivist teaching” during mathematics lessons. The development of M-N-L is part of a research program investigating the way low-performing students make mathematical sense of new notation with the help of the software Grid Algebra. Results: M-N-L was found to be an effective instrument which helped to determine the extent to which the teacher was sensitive to his own constructivist beliefs while trying to negotiate a balance between the mathematical concepts he was expected to teach and the conceptual constructions of his students. Implications: One major implication is that it is indeed possible for mathematics teachers to be sensitive to the individual constructions of their learners without losing sight of the concepts that society, represented by curriculum planners, deems necessary for students to learn. The other is that researchers in the field of education may find M-N-L a helpful tool to analyze CT during typical didactical situations established in classroom settings. (shrink)

In this paper, we discuss a proposal for reform in the teaching of Euclidean geometry that reveals the symbiotic relationship between axiomatics and pedagogy. We examine the role of intuition in this kind of reform, as expressed by Mario Pieri, a prominent member of the Schools of Peano and Segre at the University of Turin. We are well aware of the centuries of attention paid to the notion of intuition by mathematicians, mathematics educators, philosophers, psychologists, historians, and others. (...) To set a context for Pieri’s proposal, we only seek to open a small window on views of the pedagogical role of intuition, from primary education to university study that may have informed early 20th century efforts to improve the teaching of geometry at the secondary school level. Pieri addressed the topic of intuition in many of his axiomatizations, including those in projective geometry which was his main area of concentration in foundations. We focus here primarily on his axiom systems for elementary geometry, which embraced the transformational approach of Felix Klein’s vision for the subject. Our goal is to convey Pieri’s thoughts on how to integrate two types of intuition, denoted as sensible and rational, in his endeavors to improve the teaching of the geometry of Euclid. We show how Pieri’s views on geometric intuition and pedagogical reform were either ignored or misrepresented in several notable publications at the turn of the 20th century. In particular, we give Pieri a voice in response to specific comments made in the early 1900s by Federigo Enriques, Ugo Amaldi, and Florian Cajori in widely circulated publications inspired by Klein. (shrink)

By the end of the 19th century, mathematics teaching underwent a crisis of similar proportion in several European countries. Constrained by the values of classical Antiquity, which dominated in secondary education, mathematics was taught as a formal discipline enhancing elementary geometry and intended as some gymnastics of the mind. Moreover, a specific feature in Germany was that the country was split up into numerous independent states each having a separated educational system of its own. Many of (...) these conferred an only marginal status to mathematics instruction. This complex situation is analyzed for the traditional school system for boys, as well as for the emerging system for girls, and for primary schooling.The study describes the key initiatives by Felix Klein, who forged a dynamic movement of teachers for modernising secondary school teaching. Their culmination in the famous “Meran” reform program is depicted, a program which became the starting point for the first international reform movement in mathematics teaching. (shrink)

The stability of low student math scores has been a concern of education systems around the world for many years. While teaching practices are pointed out as a determining factor in student engagement and the quality of student learning, they are only the tip of the iceberg. Much work has shown that these practices are strongly colored by epistemological beliefs as well as by beliefs related to the teaching and learning of the school discipline under investigation. On this (...) subject, if some research reports a peremptory influence of beliefs on teaching practices, others present more moderate results by underlining, for example, the permeability of beliefs in pre-service training. Based on the results of a quantitative study that identified three belief profiles (pro-traditional, anti-traditionnal, flexible) among 190 future secondary school mathematics teachers in their final year of training, this contribution aims to shed qualitative light on the articulation between beliefs and teaching practices of eleven of them. The analysis of the semi-directive interviews highlights the reconcilable nature of transmissive and constructivist conceptions of teaching-learning. Moreover, the presence of similar pedagogical practices, supported in training, within profiles with contrasting beliefs argues in favor of a belief-practice non-linearity. (shrink)

The purpose of this study was to examine student perceptions of teaching behavior. Additionally the aim was to examine if teacher characteristics (educational level, gender, and teaching experience) could explain differences in student perceptions of their teachers. Teaching behavior was studied from the research on teaching and teacher effectiveness perspective. Secondary students (N = 7,114), taught by 410 teachers in Spain, participated in the study. Survey data were analyzed using non-parametric tests, Kruskal-Wallis, U Mann-Whitney with (...) Bonferroni correction, and the analysis of effect sizes. Furthermore, a regression model was applied. Results showed that teaching behavior was perceived as sufficient to good, depending on the teaching behavior domain. Results indicated interesting differences between lower secondary education, upper secondary education and vocational education and training teachers. The effect size values (rU statistic) ranged between 0.43 and 0.63, highlighting the significant effect of teachers’ teaching experience on the six teaching skills domains: learning climate, efficient classroom management, clarity of instruction, activating teaching, differentiation, and teaching learning strategies. Those teachers with less teaching experience were the ones who showed higher scores. Findings from the regression model showed that educational level had a significant predictive effect on the six teaching skills domains, mainly for male teachers. However, in several domains female teachers were perceived by students to outperform their male counterparts. (shrink)

The incorporation of technology by teachers as a teaching tool is a challenge. Our aim is to analyse whether ICT resources influence the use of pedagogical models and active methodologies by secondary school teachers in Melilla to teach Mathematics. In this cross-sectional quantitative study, a sample n= 61 (83.56% of the population) was evaluated using a validated 20-item questionnaire with three indicators: B.1 Methodologies and Pedagogical Models, B.2 Teaching Practice and C.3 ICT Resources in the (...) class='Hi'>teaching environment. Statistical power allows extrapolation of the findings. (shrink)

This book's activities highlight the important cycle of exploration, conjecture, and justification in all five mathematical strands. Students recognize patterns and make conjectures, learn the value of a counterexample, explore the strengths and weaknesses of visual proofs, discover the power of algebraic representations, and learn that theoretical approaches can substantiate empirical results. The supplemental CD-ROM features interactive electronic activities, master copies of activity pages for students, and additional readings for teachers. --publisher description.

Misconceptions about energy conservation abound due to the gap between physics and secondary school chemistry. This paper surveys this difference and its relevance to the 1690s–2010s Leibnizian argument that mind-body interaction is impossible due to conservation laws. Justifications for energy conservation are partly empirical, such as Joule’s paddle wheel experiment, and partly theoretical, such as Lagrange’s statement in 1811 that energy is conserved if the potential energy does not depend on time. In 1918 Noether generalized results like Lagrange’s and (...) proved a converse: symmetries imply conservation laws and vice versa. Conservation holds if and only if nature is uniform. The rise of field physics during the 1860s–1920s implied that energy is located in particular places and conservation is primordially local: energy cannot disappear in Cambridge and reappear in Lincoln instantaneously or later; neither can it simply disappear in Cambridge or simply appear in Lincoln. A global conservation law can be inferred in some circumstances. Einstein’s General Relativity, which stimulated Noether’s work, is another source of difficulty for conservation laws. As is too rarely realized, the theory admits conserved quantities due to symmetries of the Lagrangian, like other theories. Indeed General Relativity has more symmetries and hence more conserved energies. An argument akin to Leibniz’s finally gets some force. While the mathematics is too advanced for secondary school, the ideas that conservation is tied to uniformities of nature and that energy is in particular places, are accessible. Improved science teaching would serve the truth and enhance the social credibility of science. (shrink)

This study aims to improve the effectiveness of English grammar teaching in high school. Firstly, the Science, Technology, Engineering, Art, and Mathematics (STEAM) educational model is comprehensively discussed. Then, the current situation and difficulties of English grammar teaching in high school are analyzed. Finally, based on the traditional and the STEAM teaching mode, a comprehensive study on the psychological motivation of students is carried out in high school English grammar teaching. The traditional teaching model (...) had little effect on students’ psychological motivation in the process of English grammar teaching. Therefore, students’ satisfaction with English grammar teaching under the traditional teaching model is very low, especially in innovation and practice. The STEAM teaching model has greatly improved the psychological motivation of high school students in the English grammar teaching process. The overall satisfaction rate of the students with the English grammar teaching in the STEAM teaching model is 70%. This teaching model can greatly improve students’ requirements for innovation and practice. This study not only provides a reference for the improvement of English teaching in high schools but also contributes to the reform and development of education. (shrink)

Objective: To assess the impact of Khan Academy on the Mathematics educational process for UNDAC students and determine the relevance and efficacy of its content and strategies within the Mathematics-Physics program. Methodology: This research is framed within the positivist paradigm and takes on a quantitative approach. An experimental pre-experimental design was used with pretest and posttest in a single group. The study is applied in nature and is grounded in the contributions of expert researchers and methodologists on the (...) subject. Analytical tools were developed with a reliability coefficient of 0.90, and the investigated population consisted of students from the Mathematics-Physics program at UNDAC. Results: Students who incorporated Khan Academy exhibited significant improvements in their mathematical performance. Persistence in using the platform led to significant enhancements in academic achievement. 70 % of the students reached high levels of mathematical skill development. The majority expressed high satisfaction with the platform. Conclusions: Khan Academy has a meaningful and positive impact on mathematics learning among university students at UNDAC. The platform not only promotes content mastery but also fosters learning autonomy. These findings advocate for the adoption of innovative educational technologies in teaching, especially within STEM disciplines. (shrink)

Considerable research has demonstrated that teachers’ self-efficacy plays a major role in implementing instructional practices in classroom lessons. Only few studies, however, have examined the interplay between how teachers’ self-efficacy and the challenges that lie outside their influence are related to their implementation of cognitive-activation strategies (CAS), especially in science classrooms. Using the Trends in Mathematics and Science Study (TIMSS) 2015 data from Grades 4, 5, 8, and 9, we explored the extent to which teachers’ self-efficacy in teaching (...) science and the perceived time constraints explained variation in the enactment of general and inquiry-based CAS. Overall, the findings revealed that highly self-efficacious teachers reported more frequent implementation of both general and inquiry-based CAS, whereas those who perceived strong time constraints reported a less frequent use of inquiry-based CAS. We further found that these relations differed significantly between primary- and secondary-school teachers and discuss these findings in light of teachers’ professional competences and the resources for science activities in primary and secondary education. We point to the theoretical implications of this study for enhancing the conceptual understanding of generic and specific aspects of CAS and the practical implications for teacher education, professional development, and educational policy. (shrink)

ArgumentThis paper studies the evolution of mathematics teaching in France and Germany from 1900 to about 1980. These two countries were leading in the processes of international modernization. We investigate the similarities and differences during the various periods, which showed to constitute significant time units and this in a remarkably parallel manner for the two countries. We argue that the processes of reform concerning the teaching of this major school subject are not understandable from within mathematics (...) education or even within the school system. Rather, the evolution of the processes of reform prove to be intimately tied to changing conceptions of modernity according to respective social and cultural values and to changing epistemological conceptions of mathematics. It is particularly novel that we show the key impact of the changing social status of primary schooling for these modernization processes. (shrink)

Long ago, when Alexander the Great asked the mathematician Menaechmus for a crash course in geometry, he got the famous reply ``There is no royal road to mathematics.'' Where there was no shortcut for Alexander, there is no shortcut for us. Still, the fact that we have access to computers and mature programming languages means that there are avenues for us that were denied to the kings and emperors of yore. The purpose of this book is to teach logic (...) and mathematical reasoning in practice, and to connect logical reasoning with computer programming in Haskell. Haskell emerged in the 1990s as a standard for lazy functional programming, a programming style where arguments are evaluated only when the value is actually needed. Haskell is a marvelous demonstration tool for logic and maths because its functional character allows implementations to remain very close to the concepts that get implemented, while the laziness permits smooth handling of infinite data structures. This book does not assume the reader to have previous experience with either programming or construction of formal proofs, but acquaintance with mathematical notation, at the level of secondary school mathematics is presumed. Everything one needs to know about mathematical reasoning or programming is explained as we go along. After proper digestion of the material in this book, the reader will be able to write interesting programs, reason about their correctness, and document them in a clear fashion. The reader will also have learned how to set up mathematical proofs in a structured way, and how to read and digest mathematical proofs written by others. This is the updated, expanded, and corrected second edition of a much-acclaimed textbook. Praise for the first edition: 'Doets and van Eijck's ``The Haskell Road to Logic, Maths and Programming'' is an astonishingly extensive and accessible textbook on logic, maths, and Haskell.' Ralf Laemmel, Professor of Computer Science, University of Koblenz-Landau. (shrink)

Examining the nature of teachers’ emotions and how they are managed and regulated in the act of teaching is crucial to assess the quality of teachers’ instruction. Despite the essential role emotions play in teachers’ lives and instruction, research on teachers’ emotions has not paid much attention on teachers’ emotions in the context of daily teaching. This paper explored elementary teachers’ emotions while preparing for teaching and during teaching mathematics, reasons that underlie these emotions, and (...) the relationship between their emotions and the quality of their mathematics instruction. Participants were seven elementary teachers working in the U.S. who participated in Holistic Individualized Coaching (HIC) professional development that consisted of five cycles of coaching over an year. For each coaching cycle, pre-coaching conversation and post-coaching conversation data were collected regarding emotions teachers felt in anticipation of teaching and during teaching retrospectively. In order to compare teachers’ emotions with instructional quality, coaching sessions were video recorded and analyzed to determine the quality of instruction. Findings of this study showed that teachers reported six categories of emotions (positive, negative, neutral, blended-positive, blended-negative, and mixed), described emotions often in non-typical ways (e.g., “not nervous”, “anxious but in a positive way”), and experienced mixed emotions (co-occcurence of positive and negative emotions) as the most dominant emotion. Teachers also had more positive emotions anticipating teaching than actually teaching the lesson. The reason teachers felt mixed emotions reflected the complex and context-specific nature of teaching, a phenonemenon not currently described in the teacher emotion literature. There were no clear relationships between emotional experiences and instructional quality. This study allowed participants to freely describe their authentic, complex, overlapping, and ambiguous emotions in the context of active teaching, which contributes opening up the possibilities of diversifying teacher emotion research and shows the significance and usefulness of understanding teachers’ emotions related to active instruction. (shrink)

Open peer commentary on the article “Negotiating Between Learner and Mathematics: A Conceptual Framework to Analyze Teacher Sensitivity Toward Constructivism in a Mathematics Classroom” by Philip Borg, Dave Hewitt & Ian Jones. Upshot: My commentary has two general goals. First, I investigate how basic principles of radical constructivism might be used in constructing models of mathematics teaching. Toward that end, I found that I was not in complete intersubjective agreement with Borg et al.’s use of some (...) basic terms. Second, I explore what mathematics teaching might look like in what is construed as constructivist mathematics teaching. Toward that end, I comment on Borg et al.’s use of “negotiation” and explain how a constructivist teacher can establish experiential models of students’ mathematics. (shrink)

The entrepreneurial element is one of the aspects emphasized in the primary school mathematics education curriculum in Malaysia. However, previous studies have found that application of entrepreneurial elements in mathematics teaching is still lacking. This study was therefore conducted to identify the real challenges that mathematics teachers face in applying the entrepreneurial element in mathematics teaching. This study is qualitative case study which involved six primary school mathematics teachers. Semi-structured interviews, observation, document analysis (...) and field notes were used for the data collection process in this study. The data obtained was analyzed using the constant comparative analysis method to determine themes and sub-themes. The findings were that a teacher’s knowledge, teaching style, limited time of teaching, and attitudes, and in-service courses and training are among the identified challenges in applying entrepreneurial elements in mathematics teaching. This study expands knowledge and the literature on the challenges faced by school mathematics teachers in applying entrepreneurial elements in mathematics teaching. Various initiatives need to be taken to ensure that all the challenges can be overcome, and ultimately enable the entrepreneurial elements to be applied to students more widely. (shrink)

In the history of philosophy, Jacques Rohault and Pierre-Sylvain Régis bear a twofold burden. They are professed followers, epigones. Worse yet, the natural philosophy they teach has been consigned to the Tartarus of fable: not a theory that failed, but something that failed even to be a theory. In the years in which they were turning Cartesianism into a system, Newton and Huygens were preparing its demise. Its empirical claims were refuted, its mathematics was rendered obsolete by the calculus, (...) its vortices and channelled magnetic particles met with the same rough justice Descartes meted out to Scholastic forms and qualities. Canonical history has little use for such figures. It prefers originals. Yet if ideas and arguments are not to seem to pass magically from one great mind to the next, we must have some account of the channels through which what was once novel and unique sediments into cliché and common ground. Those channels are not without bias and noise. Inevitably, currents from different streams meet and mix more or less coherently in the works of secondary figures, especially in the competitive intellectual world of the later seventeenth century, with its sometimes ferocious polemics fuelled by religious and political opposition. Cartesianism became a movement and—to use Leibniz’s word—a sect, divided within by disputes over the legacy of its founder, and facing opposition without from steadfast Aristotelians, pious theologians, and the avant garde of the new science. In Régis and Rohault Descartes’ legacy took the outward form of “system”. They present themselves as reworking Cartesian concepts and arguments into something coherent and comprehensive. Rohault, the more modest of the two, aims to reform the teaching of physics, still weighed down by the dead hand of Aristotle. He retains for the old philosophy only what is true and conjoin it with the new physics of Descartes, in whom France is no less fortunate than Greece once was in Aristotle (Rohault 1718, “Præfatio”).. (shrink)

This paper addresses the context of emergence, development, and current status of the use of history and philosophy of science in science education in Brazil. After a short overview of the three areas (history of science, philosophy of science, and science education) in Brazil, the paper focuses on the application of this approach to teaching physics, chemistry, and biology at the secondary school level. The first Brazilian researches along this line appeared more consistently in the decade of 1970. (...) From 1980 onwards, the importance of this approach became widely accepted, and the subject became a common theme of dissertations and theses, appearing in conferences and educational journals. Since 1998, the use of history and philosophy of science was included among the government recommendations for secondary school science teaching in Brazil. Nowadays, this is an important line of research in graduate programs on science and mathematics education. However, the actual use of this approach in secondary education is still a desideratum. (shrink)

Teaching Secondary Science: Theory and Practice provides a dynamic approach to preparing preservice science teachers for practice. Divided into two parts - theory and practice - the text allows students to first become confident in the theory of teaching science before showing how this theory can be applied to practice through ideas for implementation, such as sample lesson plans. These examples span a variety of age levels and subject areas, allowing preservice teachers to adapt each exercise to (...) suit their needs when they enter the classroom.Each chapter is supported by pedagogical features, including learning objectives, reflections, scenarios, key terms, questions, research topics and further readings. Written by leading science education researchers from universities across Australia, Teaching Secondary Science is a practical resource that will continue to inspire preservice teachers as they move from study into the classroom. This book includes a single-use twelve-month subscription to Cambridge Dynamic Science. (shrink)

Research on the nature of science and science education enjoys a longhistory, with its origins in Ernst Mach's work in the late nineteenthcentury and John Dewey's at the beginning of the twentieth century.As early as 1909 the Central Association for Science and MathematicsTeachers published an article – ‘A Consideration of the Principles thatShould Determine the Courses in Biology in Secondary Schools’ – inSchool Science and Mathematics that reflected foundational concernsabout science and how school curricula should be informed by (...) them. Sincethen a large body of literature has developed related to the teaching andlearning about nature of science – see, for example, the Lederman and Meichtry reviews cited below. As well there has been intensephilosophical, historical and philosophical debate about the nature of scienceitself, culminating in the much-publicised ‘Science Wars’ of recent time. Thereferences listed here primarily focus on the empirical research related to thenature of science as an educational goal; along with a few influential philosophicalworks by such authors as Kuhn, Popper, Laudan, Lakatos, and others. Whilenot exhaustive, the list should prove useful to educators, and scholars in otherfields, interested in the nature of science and how its understanding can berealised as a goal of science instruction. The authors welcome correspondenceregarding omissions from the list, and on-going additions that can be made to it. (shrink)

The article is devoted to identifying the value of the phenomenon of aesthetic value and beauty of mathematical knowledge and the beauty of mathematical theory of teaching mathematics. The aesthetic potential of mathematical knowledge allows the use of theater technology in the educational process with the active dialogic interaction between teacher and students. The criteria of beauty in mathematical theories are distinguished: the realization of beauty as the unity of the whole, and in the disclosure of the complex (...) through the elementary; methodological interpretation of the beauty in the community of mathematical structures and optimal information content of the meta-language of mathematics; the practical embodiment of beauty in the formalization of the infinite through the finite. The beauty of mathematics is the force that permeates all the ‘layers of knowledge‘ not along, and across, although the effectiveness of mathematical activity due to aesthetic laws, which do not always lend themselves to unambiguous interpretation. In the article it is stated that, depending on the educational goals of communicative impact on the audience, in fact, ‘mathematical lectures theatricality‘ can have different characteristics, the most important of which are teachers artistry and artistic director’s work of a teacher. This cultural phenomenon that includes the theatrical talent, helps create an atmosphere of cooperation needed in varying degrees of activity for pedagogical interaction. The author believes that such approach, developed on the basis of the Stanislavsky system, allows university professors of mathematics significantly improve mathematical lectures. (shrink)

Feedback encompasses different aspects of learning and enables students tofocus on the goal of the lesson while being conscious of their responsibility fortheir success. It should neither be strictly criticism, nor mere praise; rather, itshould contain a set of information that will guide them in correcting their mistakesand learning. Feedback has a positive impact on student learning (Voerman et al.,2012) and is a key factor in increasing mathematical achievement (Clarke, 2001).The purpose of this action research is to improve elementary (...) class='Hi'>mathematics teachingthrough the provision of feedback. Analysis of the video recordings from the lessonsrevealed the prevalence of feedback aimed at assessing students’ comprehensionand achievement. Additionally, students received feedback on the assignment. Asthe planned activities were implemented, an increase in student motivation, controlover their own learning and student satisfaction was also observed. The findingsdemonstrate that it is possible to improve mathematics education by introducingchanges that align with for the students’ abilities and interests. In doing so, it iscrucial for teachers to be knowledgeable about effective teaching methods thatmust be adapted to the educational context in which they work. (shrink)

Preservice mathematics teachers’ accurate understanding of mathematical problem solving and its teaching is key to the performance of their professional quality. This study aims to investigate preservice mathematics teachers’ understanding of problem solving and its teaching and compares it with the understanding of in-service mathematics teachers. After surveying 326 in-service mathematics teachers, this study constructs a reliable and valid tool for the cognition of mathematical problem solving and its teaching and conducts a questionnaire (...) survey on 26 preservice mathematics teachers. Survey results reveal that preservice mathematics teachers have a good understanding of mathematical problem solving and its teaching and are more confident in the transfer value of problem solving ability. By contrast, in-service teachers are more optimistic that problem solving requires exploration, continuous thinking, and the participation of metacognition. This article concludes that preservice mathematics teachers should focus more on the initiative and creativity of students and put students at the center of education. In addition, teacher educators should provide more teaching practice opportunities for preservice teachers. The findings also show that in-service teachers’ understanding of problem solving and its teaching is inferior to that of preservice teachers on some indicators, implying the importance of post-service training for in-service teachers. (shrink)

This study explored and compared the perspectives of Taiwanese in-service and pre-service high school mathematics teachers regarding ideal teaching behaviours; the perspectives of a nationwide sample of students were taken as the baseline. Fourteen factors contributing to ideal teaching behaviours were identified through exploratory factor analyses. Nine factors, including idea explanation and speedy lecture, were rooted in traditional Chinese culture; five factors, including concrete representation and student activities, were influenced by Western cultures. Three teacher profiles were identified (...) through k-means clustering analysis. The perspectives of in-service teachers were dominated by a painless meaning-emphasised profile; these teachers emphasised meaningful learning for students and avoided the fast pace and demanding requirements that can cause distress in students, whereas pre-service teachers were dominated by an all-round profile, revealing their openness to all factors. Compared with... (shrink)

Over the past few years we have seen an increasing number of legal proceedings related to inappropriately implemented technology. At the same time career paths have diverged from the foundation of statistics out to Data Scientist, Machine Learning and AI. All of these new branches being fundamentally branches of statistics and mathematics. This has meant that formal training has struggled to keep up with what is required in the plethora of new roles. Mathematics as a taught subject is (...) still based in decades old teaching specifications and has not been updated centrally as a curriculum to include new technologies, coding or ethics. This subject area is firmly split between ICT and Mathematics in secondary school, continuing on to be split between Computer Science and Mathematics at University. As we move forwards with technology we see these once seperate fields becoming increasingly intertwined. We propose that existing provision for concepts such as ethics and societal responsibility in analysis currently exist but have not been incorporated into the mainstream curriculum of School or University. This is partially due to the split between fields in an educational setting but also the speed with which education is able to keep up with Industry and its requirements. Principles and frameworks of socially responsible modelling beginning at school level means that ethics and real-life modelling are introduced much earlier than normal. Integrating these concepts with philosophical principles of society and politics ensures a suitable background for future modellers and users of technology to draw on. Modelling is currently undertaken in technical sciences at University but the Subject Benchmark Statements are not current (Subject Benchmark Statements describe the nature of study and the academic standards expected of graduates in specific subject areas. They show what graduates might reasonably be expected to know, do and understand at the end of their studies). Even in 2019 the UK did not yet have Benchmark Statements that discuss the learning to be done in Higher Education around AI and advanced Machine Learning. Where there is provision for AI and Data Science within degree courses ethics is not generally highlighted as a key concept. As such there can be a lack of focus on the teaching of modelling or ethics as specific skills. The skills required to use a basic statistical model, for example would not be sufficient to start from scratch and build an ethical model reflecting real-world scenarios with which to inform policy or organisational decision making. This is a skill in itself and includes such aspects as awareness of data quality, ethics, user implementation problems, context and an understanding of the environment that the model is being created in. As the field of analytics has progressed so quickly in the last decade modules such as those covering AI have simply been bolted on to Maths or Computing degrees rather than being fully detailed as key areas of study or driving new, more integrated courses of study. This paper posits that gaps at primary, secondary and tertiary educational levels need to be addressed. Implementing and integrating key concepts from school level is essential so that areas such as assumptions, caveats, quality assurance and answering the right questions with constructive challenge become a cultural fixture. This not only helps developers of technology but users of rapidly developing technology also. In addition, leadership and soft skills as part of this education will ensure that a cultural shift can take place and promote continuous improvement in analysis within organisations. The addition of key concepts throughout the educational system and the updating of potentially outdated curriculums is key to ensuring a functional society. A society where every citizen is a user of tech and those that develop it can be ethical and socially responsible in its development. (shrink)