Mathematical realism is the doctrine that mathematical objects really exist, that mathematical statements are either determinately true or determinately false, and that the accepted mathematical axioms are predominantly true. A realist understanding of set theory has it that when the sentences of the language of set theory are understood in their standard meaning, each sentence has a determinate truth value, so that there is a fact of the matter whether the cardinality of the continuum is א2 or whether there are (...) measurable cardinals, whether or not those facts are knowable by us. (shrink)

College students struggle with the switch from thinking of mathematics as a calculation based subject to a problem solving based subject. This book describes how the introduction to proofs course can be taught in a way that gently introduces students to this new way of thinking. This introduction utilizes recent research in neuroscience regarding how the brain learns best. Rather than jumping right into proofs, students are first taught how to change their mindset about learning, how to persevere (...) through difficult problems, how to work successfully in a group, and how to reflect on their learning. With these tools in place, students then learn logic and problem solving as a further foundation. Next various proof techniques such as direct proofs, proof by contraposition, proof by contradiction, and mathematical induction are introduced. These proof techniques are introduced using the context of number theory. The last chapter uses Calculus as a way for students to apply the proof techniques they have learned. (shrink)

This study aims to describe the learning motivation of students using virtual media when they are learning mathematics in grade 5. The research design applied in this research is classroom action research. The research is conducted in two phases which involve planning, action and observation and reflection. The results of the study revealed that intrinsic motivation to learn is most prevalent in the form of fun to learn mathematics with virtual media. Other forms of intrinsic motivation (...) include curiosity, need and interests, and satisfaction. Extrinsic motivation in the form of awarded and clarity of learning objectives. The most enjoyable learning experience for students was when they feel like they're learning more than they actually are. This level of enthusiasm increases with the level of learning complexity. The students are more active and motivated to learn. This can be evidenced by the increase in motivation to learn after the first phase. Unfortunately, the lack of bandwidth and time allocated for conducting experiments are some of the obstacles in the way of learning through virtual media. This study revealed that the use of virtual media could improve student's motivation for learning. (shrink)

Differences between the sexes in education is something of particular interest in much research. This study sought to investigate the possible differences between the sexes in math performance, and to deeply examine the causal factors for those differences. Beginning from the administration of the BECOMA-On (Online Evaluation Battery of Mathematics Skills) to 3,795 5th year primary students aged 10-11, in 16 Spanish autonomous communities and the 2 autonomous cities of Ceuta and Melilla. The results for each sex were compared (...) to their perceptions of self-efficacy about completing the test items, and with their interest in and motivation for mathematics. Statistically significant differences were seen in the variables examined. The boys were generally more engaged with science and technical subjects. Generalizing from studies such as this aims to more thoroughly explore, and improve this situation. (shrink)

his study aims to determine whether the application of brain-friendly learning through whole brain teaching gives a positive effect on the creative character of students, to know the response of the students against the application of brain-friendly learning through whole brain teaching, and to find out if the student response against the application of brain-friendly learning through whole brain teaching correlates positively with the creative character of students in learning mathematics. The research method used that (...) is quantitative. The instruments used, namely the now student response related application of brain-friendly learning through whole brain teaching, and sheets of observation about the creativity of students in learning mathematics after implementing this method. The correlation analysis with the results of the study showed that the value of the larger significance of the alpha values (5%) which means accepting and rejecting H0 Ha which means student response against the application of brain-friendly learning through whole brain teaching not correlated positively with the creative character of students in learning mathematics. The average student response score against this method is found on very good i.e. amounted to 85%. As for the observations of the creative character of the students, after this method is applied, the average score of 68% is found quite creative. (shrink)

A beneficial effect of gesture on learning has been demonstrated in multiple domains, including mathematics, science, and foreign language vocabulary. However, because gesture is known to co-vary with other non-verbal behaviors, including eye gaze and prosody along with face, lip, and body movements, it is possible the beneficial effect of gesture is instead attributable to these other behaviors. We used a computer-generated animated pedagogical agent to control both verbal and non-verbal behavior. Children viewed lessons on mathematical equivalence in (...) which an avatar either gestured or did not gesture, while eye gaze, head position, and lip movements remained identical across gesture conditions. Children who observed the gesturing avatar learned more, and they solved problems more quickly. Moreover, those children who learned were more likely to transfer and generalize their knowledge. These findings provide converging evidence that gesture facilitates math learning, and they reveal the potential for using technology to study non-verbal behavior in controlled experiments. (shrink)

I. Prelude -- About this Book -- II. School Mathematics and Its Consequences -- The Foundations of Mathematical Thinking -- Compression, Connection and Blending of Mathematical Ideas -- Set-befores, Met-befores and Long-term Learning -- Mathematics and the Emotions -- The Three Worlds of Mathematics -- Journeys through Embodiment and Symbolism -- Problem-Solving and Proof -- III. Interlude -- The Historical Evolution of Mathematics -- IV. University Mathematics and Beyond -- The Transition to Formal Knowledge (...) -- Blending Knowledge Structures in the Calculus -- Expert Thinking and Structure Theorems -- Contemplating the Infinitely Large and the Infinitely Small -- Expanding the Frontiers through Mathematical Research -- Reflections -- Appendix: Where the Ideas Came From. (shrink)

I develop an account of how mathematized theories in physics represent physical systems, in response to the frequent claim that any such account must presuppose a non-mathematized, and usually linguistic, description of the system represented. The account I develop contains a circularity, in that representation is a mathematical relation between the models of a theory and the system as represented by some other model --- but I argue that this circularity is not vicious, in any case refers in linguistic accounts (...) of meaning and representation, and is simply a consequence of the fact that we have no unmediated, representation-independent access to the world. (shrink)

Mathematical learning difficulties (MLD) refer to a variety of deficits in math skills, typically pertaining to the domains of arithmetic and problem solving. The present study examined the time course of attentional orienting in MLD children with a spatial cueing task, by parametrically manipulating the cue-target onset asynchrony (CTOA). The results of Experiment 1 revealed that, in contrast to typical developing children, the inhibitory aftereffect of attentional orienting—frequently referred to as inhibition of return (IOR)—was not observed in the MLD (...) children, even at the longest CTOA tested (800 ms). However, robust early facilitation effects were observed in the MLD children, suggesting that they have difficulties in attentional disengagement rather than attentional engagement. In a second experiment, a secondary cue was introduced to the cueing task to encourage attentional disengagement and IOR effects were observed in the MLD children. Taken together, the present experiments indicate that MLD children are sluggish in disengaging spatial attention. (shrink)

(I) MATHEMATICAL LECTURES. LECTURE I. Of the Name and general Division of the Mathematical Sciences. BEING about to treat upon the Mathematical Sciences, ...

Hans Hahn's long-neglected philosophy of mathematics is reconstructed here with an eye to his anticipation of the doctrine of logical pluralism. After establishing that Hahn pioneered a post-Tractarian conception of tautologies and attempted to overcome the traditional foundational dispute in mathematics, Hahn's and Carnap's work is briefly compared with Karl Menger's, and several significant agreements or differences between Hahn's and Carnap's work are specified and discussed.

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

Important works of Mostowski and Rasiowa dealing with many-valued logic are analyzed from the point of view of contemporary mathematical fuzzy logic.

Using the work of philosopher Henri Bergson to examine the nature of movement and memory, this article contributes to recent research on the role of the body in learning mathematics. Our aim in this paper is to introduce the ideas of Bergson and to show how these ideas shed light on mathematics classroom activity. Bergson’s monist philosophy provides a framework for understanding the materiality of both bodies and mathematical concepts. We discuss two case studies of classrooms to (...) show how the mathematical concepts of number and function are themselves mobile and full of potentiality, open to deformation and the remapping of the possible. Bergson helps us look differently at mathematical activity in the classroom, not as a closed set of distinct interacting bodies groping after abstract concepts, but as a dynamic relational assemblage. (shrink)

Statistical learning refers to the ability to identify structure in the input based on its statistical properties. For many linguistic structures, the relevant statistical features are distributional: They are related to the frequency and variability of exemplars in the input. These distributional regularities have been suggested to play a role in many different aspects of language learning, including phonetic categories, using phonemic distinctions in word learning, and discovering non-adjacent relations. On the surface, these different aspects share few (...) commonalities. Despite this, we demonstrate that the same computational framework can account for learning in all of these tasks. These results support two conclusions. The first is that much, and perhaps all, of distributional statistical learning can be explained by the same underlying set of processes. The second is that some aspects of language can be learned due to domain-general characteristics of memory. (shrink)

Predictive policing has become a new panacea for crime prevention. However, we still know too little about the performance of computational methods in the context of predictive policing. The paper provides a detailed analysis of existing approaches to algorithmic crime forecasting. First, it is explained how predictive policing makes use of predictive models to generate crime forecasts. Afterwards, three epistemologies of predictive policing are distinguished: mathematical social science, social physics and machine learning. Finally, it is shown that these epistemologies (...) have significant implications for the constitution of predictive knowledge in terms of its genesis, scope, intelligibility and accessibility. It is the different ways future crimes are rendered knowledgeable in order to act upon them that reaffirm or reconfigure the status of criminological knowledge within the criminal justice system, direct the attention of law enforcement agencies to particular types of crimes and criminals and blank out others, satisfy the claim for the meaningfulness of predictions or break with it and allow professionals to understand the algorithmic systems they shall rely on or turn them into a black box. By distinguishing epistemologies and analysing their implications, this analysis provides insight into the techno-scientific foundations of predictive policing and enables us to critically engage with the socio-technical practices of algorithmic crime forecasting. (shrink)

Stereotype threat—a situational context in which individuals are concerned about confirming a negative stereotype—is often shown to impact test performance, with one hypothesized mechanism being that cognitive resources are temporarily co-opted by intrusive thoughts and worries, leading individuals to underperform despite high content knowledge and ability. We test here whether stereotype threat may also impact initial student learning and knowledge formation when experienced prior to instruction. Predominantly African American fifth-grade students provided either their race or the date before a (...) videotaped, conceptually demanding mathematics lesson. Students who gave their race retained less learning over time, enjoyed the lesson less, reported a diminished desire to learn more, and were less likely to choose to engage in an optional math activity. The detrimental impact was greatest among students with high baseline cognitive resources. While stereotype threat has been well documented to harm test performance, the finding that effects extend to initial learning suggests that stereotype threat's contribution to achievement gaps may be greatly underestimated. (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)

Dans son « Découverte d'un nouveau principe de mécanique » (1750) Euler a donné, pour la première fois, une preuve du théorème qu'on appelle aujourd'hui le Théorème d'Euler. Dans cet article je vais me concentrer sur la preuve originale d'Euler, et je vais montrer comment la pratique mathématique d Euler peut éclairer le débat philosophique sur la notion de preuves explicatives en mathématiques. En particulier, je montrerai comment l'un des modèles d'explication mathématique les plus connus, celui proposé par Mark Steiner (...) dans son article « Mathematical explanation » (1978), n'est pas en mesure de rendre compte du caractère explicatif de la preuve donnée par Euler. Cela contredit l'intuition originale du mathématicien Euler, qui attribuait à sa preuve un caractère explicatif spécifique. (shrink)

The purpose of the present study was to investigate the effect and use of distributed practice in the context of self-regulated mathematical learning in high school. With distributed practice, a fixed learning duration is spread over several sessions, whereas with massed practice, the same time is spent learning in one session. Distributed practice has been proven to be an effective tool for improving long-term retention of verbal material and simple procedural knowledge in mathematics, at least when (...) the practice schedule is externally guided. In the present study, distributed practice was investigated in a context that required a higher degree of self-regulation. In total, 158 secondary school students were invited to participate. After motivational and cognitive characteristics of the students were assessed, the students were introduced to basic statistics, a topic of their regular curriculum. At the end of the introduction, the students could sign up for the study to further practice this content. Eighty-seven students did so and were randomly assigned either to the distributed or to the massed practice condition. In the distributed practice condition, they received three practice sets on three different days. In the massed practice condition, they received the same three sets, but all on one day. All exercises were worked in the context of self-regulated learning at home. Performance was tested two weeks after the last practice set. Only 44 students finished the study, which hampered the analysis of the effect of distributed practice. The characteristics of the students who completed the exercises were analyzed exploratory: The proportion of students who finished all exercises was significantly higher in the massed than in the distributed practice condition. Within the distributed practice condition, a significantly larger proportion of female students completed the exercises compared to male students. Additionally, among these female students, a larger proportion showed lower concentration difficulty. No such differential effects were revealed in the massed practice condition. Our results suggest that the use of distributed practice in the context of self-regulated learning might depend on learner characteristics. Accordingly, distributed practice might obtain more reliable effects in more externally guided learning contexts. (shrink)

Although children learn more when teachers gesture, it is not clear how gesture supports learning. Here, we sought to investigate the nature of the memory processes that underlie the observed benefits of gesture on lasting learning. We hypothesized that instruction with gesture might create memory representations that are particularly resistant to interference. We investigated this possibility in a classroom study with 402 second‐ and third‐grade children. Participants received classroom‐level instruction in mathematical equivalence using videos with or without accompanying (...) gesture. After instruction, children solved problems that were either visually similar to the problems that were taught, and consistent with an operational interpretation of the equal sign (interference), or visually distinct from equivalence problems and without an equal sign (control) in order to assess the role of gesture in resisting interference after learning. Gesture facilitated learning, but the effects of gesture and interference varied depending on type of problem being solved and the strategies that children used to solve problems prior to instruction. Some children benefitted from gesture, while others did not. These findings have implications for understanding the mechanisms underlying the beneficial effect of gesture on mathematical learning, revealing that gesture does not work via a general mechanism like enhancing attention or engagement that would apply to children with all forms of prior knowledge. (shrink)

It has been a common belief among scientists, including mathematicians, that young scientists are especially good at bringing about scientific change. A number of studies suggest, however, that older scientists are not more resistant to change than young scientists are. It is nonetheless worth examining why a scientist’s or mathematician’s outsider status – due to age, educational background, or something else – can sometimes be effective in enabling scientific change. This paper focuses on the case of the solving of the (...) Four Color Problem by Wolfgang Haken and Kenneth Appel. Building on Donald MacKenzie’s paper ‘Slaying the kraken: The sociohistory of a mathematical proof’, I argue that Haken’s outsider status is central to understanding his success with the problem. On this basis, I offer an argument against Margaret Gilbert’s account of why a scientist’s outsider status can be effective in enabling scientific change. (shrink)

Using the work of philosopher Henri Bergson to examine the nature of movement and memory, this article contributes to recent research on the role of the body in learning mathematics. Our aim in this paper is to introduce the ideas of Bergson and to show how these ideas shed light on mathematics classroom activity. Bergson’s monist philosophy provides a framework for understanding the materiality of both bodies and mathematical concepts. We discuss two case studies of classrooms to (...) show how the mathematical concepts of number and function are themselves mobile and full of potentiality, open to deformation and the remapping of the possible. Bergson helps us look differently at mathematical activity in the classroom, not as a closed set of distinct interacting bodies groping after abstract concepts, but as a dynamic relational assemblage. (shrink)

Ivor Grattan-Guinness, Routes of Learning: Highways, Pathways and Byways in the History of Mathematics. Baltimore: The John Hopkins University Press, 2009. xii + 372 pp. $75.00. ISBN 10:978-0-8018-...

ObjectivesUnder the COVID-19 prevention and control policy, online learning has been widely used. The current study aimed to identify latent profiles of self-regulated learning in the context of online mathematics learning during the recurrent outbreak of COVID-19, and examine the mechanisms underlying the relationship between self-regulated learning and online mathematics learning engagement among Chinese junior high school students using variable-and person-centered approaches.MethodsA sample of 428 Chinese junior high school students completed questionnaires on self-regulated (...) learning, perceived academic control, and learning engagement. Mplus7.0 was used to analyze the latent classes of self-regulated learning. A mediation model was conducted using the software SPSS PROCESS macro.ResultsThree profiles of self-regulated learning were identified and named as low self-regulated learning, medium self-regulated learning, and high self-regulated learning. In the mediating analysis, results of the variable-centered approach showed that perceived academic control mediated the effects of self-regulated learning on learning engagement. For the person-centered approach, we selected the low self-regulated learning type as the reference profile, and the analysis revealed that compared with the reference profile, perceived academic control partially mediated the link between the medium self-regulated learning profile and learning engagement; perceived academic control partially mediated the relationship between the high self-regulated learning profile and learning engagement.ConclusionThis study showed the heterogeneity in the online mathematics self-regulated learning patterns of Chinese junior high school students during the COVID-19 pandemic, revealing the internal mechanisms of Chinese junior high school students’ online mathematics learning engagement using variable-and person-centered approaches. Furthermore, the findings of the study have important implications for promoting online mathematics learning engagement among junior high students during the pandemic. (shrink)

Despite all methodological efforts made in the last three decades, Western instruction grounds on traditional principles. Most educational programs follow theories that are mentalistic, i.e., they separate the mind from the body. At school, learners sit, watch, listen, and write. The aim of this paper is to present embodied learning as an alternative to mentalistic education. Similarly, this paper wants to describe embodied learning from a neuroscientific perspective. After a brief historical overview, I will review studies highlighting the (...) behavioral effectiveness of embodied instruction in second language learning, mathematics and spatial thinking. On this base, I will discuss some of the brain mechanisms driving embodied learning and describe its advantages, clearly pleading in favor of instructional practice that reunites body and mind. (shrink)

This is a teaching and learning guide to accompany "Explanation in Mathematics: Proofs and Practice".

Mathematics is one of the core STEM (science, technology, engineering, and mathematics) subject disciplines. Engaging students in learning mathematics helps retain students in STEM fields and thus contributes to the sustainable development of society. To increase student engagement, some mathematics instructors have redesigned their courses using the flipped classroom approach. In this review, we examined the results of comparative studies published between 2011 and 2020 to summarize the effects of this instructional approach (vs. traditional lecturing) (...) on students’ behavioral, emotional, and cognitive engagement with mathematics courses. Thirty-three articles in K–12 and higher education contexts were included for analysis. The results suggest that the use of the flipped classroom approach may increase some aspects of behavioral engagement (e.g., interaction and attention/participation), emotional engagement (e.g., course satisfaction), and cognitive engagement (e.g., understanding of mathematics). However, we discovered that several aspects (e.g., students’ attendance, mathematics anxiety, and self-regulation) of student engagement have not been thoroughly explored and are worthy of further study. The results of this review have important implications for future flipped classroom practice (e.g., engaging students in solving real-world problems), and for research on student engagement (e.g., using more objective measures, such as classroom observation) in mathematics education. (shrink)

Technology is an essential aspect of everyday life, thus the growth and integrationof information and communication technology (ICT) in learning and teaching isexpected. Beginning with the origins of technology based on mathematics as a science,we narrow our focus in this scholarly study on the use of ICT in the teaching andlearning of mathematics in elementary education. We focused on studying the impactof ICT on the teaching and learning process, as well as the usage of ICT tools (...) in thedevelopment of knowledge in mathematics. The goal of this systematic analysis is toshow the benefits of integrating ICT into mathematics teaching and learning, as well asthe constraints and challenges that elementary school teachers and students encounter.Our comprehensive analysis of pertinent theoretical and empirical findings ledus to the general conclusion that teacher proficiency with the new technology plays amajor role in the integration of ICT. The positive impact of ICT on mathematics teachingand learning is undeniable, because technology reduces logistics in the planningand implementation of instruction, increases interaction in the classroom, and encouragesmathematics learning through the visualization of abstract concepts and processes,while also allowing for independent study. (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)

Mathematics is an important foundation for the development of science education. In the past, when instructors taught mathematical concepts of geometry shapes, they usually used traditional textbooks and aids to conduct teaching activities, which resulted in students not being able to understand the principles completely. Nowadays, it has become a trend to integrate emerging technologies into mathematics courses and to use digital instructional aids. Emerging technologies can effectively enhance students’ sensory experience while strengthening their impressions and understandings of (...) subject concepts. In this paper, we apply virtual reality immersive technologies to develop a “virtual reality immersive learning mathematics geometry system,” which is used to teach mathematical geometry concepts. Teachers use the system to develop three basic mathematical geometry learning materials: “Triangular pyramid volume = 1/3 prism volume,” “Cone volume calculation,” and “Triangle center of gravity derivation.” In the experimental activity, the teacher uses virtual reality teaching aids to guide students to learn mathematical geometry concepts in a fun way so that they can achieve the effectiveness of immersive learning. This study explores the impact of using the virtual reality immersive learning mathematics geometry system on students’ technology acceptance, learning motivations, and learning performance. The experimental result showed that using the virtual reality immersive learning mathematics geometry system can improve the learning motivation and learning performance of students. The findings indicated that the experimental group had better learning outcomes after completing the learning tasks of three geometric units. The experimental group used the virtual reality immersive learning mathematics geometry system which can lead to better learning outcomes. According to the ARCS questionnaire, students in the experimental group were confident to understand new subjects. At the same time, the mode of completing the game can effectively give students a sense of accomplishment. The use of emerging technologies in the classroom can be an attractive learning mode for students. (shrink)