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 teacher’s instructions. Despite the essential role emotions play in teachers’ lives and instruction, research on teachers’ emotions has not paid much attention on teachers’ state emotions in the context of daily teaching. Significant portion of literature has described teachers’ emotions by foregrounding trait emotions through deductive methodological approaches. This paper explored elementary teachers’ state (...) and trait emotions while preparing for teaching and during teaching, 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 5 cycles of coaching over an year. For each coaching cycle pre-coaching conversation and post-coaching conversation data were collected regarding state emotions teachers felt in anticipation of teaching and during teaching retrospectively. Also, teachers responded adapted Teaching Emotion Scale (TES) that was aimed to measure trait emotions. In order to compare teachers’ emotions with instructional quality, coaching session were video recorded and analyzed to determine the quality of instruction. Findings of this study showed that teachers reported 6 categories of emotions (positive, negative, neutral, mixed-positive, mixed-negative, and mixed-complex), and often in non-typical ways (e.g., “not nervous”, “anxious but in a positive way”), and mixed emotions were the most dominant category. Also, teachers had more positive emotions anticipating teaching, than actually teaching the lesson, and there was no noticeable change of emotions across the 5 coaching cycles. The reason teachers felt mixed emotions reflected complex and context-specific nature of teaching, which was not articulated in the 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’ state emotions. (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.

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)

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)

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)

The theory of objectification offers a perspective to conceptualize learning as a collective cultural-historical process and to transform classrooms into sites of communal life where students make the experience of an ethics of solidarity, plurality, and inclusivity.

This study aims to improve the effectiveness of English grammar teaching in high school. Firstly, the Science, Technology, Engineering, Art, and Mathematics 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)

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)

On the question of whether gambling behavior can be changed as result of teaching gamblers the mathematics of gambling, past studies have yielded contradictory results, and a clear conclusion has not yet been drawn. In this paper, I bring some criticisms to the empirical studies that tended to answer no to this hypothesis, regarding the sampling and laboratory testing, and I argue that an optimal mathematical scholastic intervention with the objective of preventing problem gambling is possible, by providing (...) the principles that would optimize the structure and content of the teaching module. Given the ethical aspects of the exposure of mathematical facts behind games of chance, and starting from the slots case – where the parametric design is missing, we have to draw a line between ethical and optional information with respect to the mathematical content provided by a scholastic intervention. Arguing for the role of mathematics in problem-gambling prevention and treatment, interdisciplinary research directions are drawn toward implementing an optimal mathematical module in cognitive therapies. (shrink)

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

En 1741, la Royal Military Academy de Woolwich est créée par le Board of Ordnance afin d’instruire les futurs artilleurs et ingénieurs militaires. Cette instruction s’appuie dès le départ sur les mathématiques. Dans cet article, nous présentons et étudions les différents programmes sur la longue période. Les évolutions, les changements mais aussi les constances sont évalués et nous donnons les raisons de ceux-ci. L’âge de recrutement, le poids du Board of Ordnance ou encore les diverses guerres ont aussi une influence (...) importante sur la place des mathématiques dans les programmes. Par ailleurs, le peu de turn-over des professeurs entraîne aussi une inertie de l’enseignement des mathématiques. In 1741, the Royal Military Academy at Woolwich was created by the Board of Ordnance in order to instruct artillery officers and military engineers. From the beginning, the teaching was mainly based on mathematics. In this paper, we study different curricula and point out the long-term evolution as well as continuity in mathematical learning and we give reasons for them. The age of recruitment, the authority of the Board of Ordnance and several wars had an influence on the role of mathematics in the syllabus. Furthermore, the low turnover of professors led to inertia in the teaching of mathematics. (shrink)

En 1741, la Royal Military Academy de Woolwich est créée par le Board of Ordnance afin d’instruire les futurs artilleurs et ingénieurs militaires. Cette instruction s’appuie dès le départ sur les mathématiques. Dans cet article, nous présentons et étudions les différents programmes sur la longue période (entre 1741 et les années 1860). Les évolutions, les changements mais aussi les constances sont évalués et nous donnons les raisons de ceux-ci. L’âge de recrutement, le poids du Board of Ordnance ou encore les (...) diverses guerres ont aussi une influence importante sur la place des mathématiques dans les programmes. Par ailleurs, le peu de turn-over des professeurs entraîne aussi une inertie de l’enseignement des mathématiques. In 1741, the Royal Military Academy at Woolwich was created by the Board of Ordnance in order to instruct artillery officers and military engineers. From the beginning, the teaching was mainly based on mathematics. In this paper, we study different curricula and point out the long-term (between 1741 and the 1860s) evolution as well as continuity in mathematical learning and we give reasons for them. The age of recruitment, the authority of the Board of Ordnance and several wars had an influence on the role of mathematics in the syllabus. Furthermore, the low turnover of professors led to inertia in the teaching of mathematics. (shrink)

En 1741, la Royal Military Academy de Woolwich est créée par le Board of Ordnance afin d’instruire les futurs artilleurs et ingénieurs militaires. Cette instruction s’appuie dès le départ sur les mathématiques. Dans cet article, nous présentons et étudions les différents programmes sur la longue période (entre 1741 et les années 1860). Les évolutions, les changements mais aussi les constances sont évalués et nous donnons les raisons de ceux-ci. L’âge de recrutement, le poids du Board of Ordnance ou encore les (...) diverses guerres ont aussi une influence importante sur la place des mathématiques dans les programmes. Par ailleurs, le peu de turn-over des professeurs entraîne aussi une inertie de l’enseignement des mathématiques. In 1741, the Royal Military Academy at Woolwich was created by the Board of Ordnance in order to instruct artillery officers and military engineers. From the beginning, the teaching was mainly based on mathematics. In this paper, we study different curricula and point out the long-term (between 1741 and the 1860s) evolution as well as continuity in mathematical learning and we give reasons for them. The age of recruitment, the authority of the Board of Ordnance and several wars had an influence on the role of mathematics in the syllabus. Furthermore, the low turnover of professors led to inertia in the teaching of mathematics. (shrink)

What is argumentation? -- Building a classroom culture of argumentation -- Structuring classroom discussions to focus on argumentation -- Infusing all instruction with argumentation -- Argumentation for all students -- Argumentation and the mathematical practices -- Technology in teaching and learning argumentation -- Assessing argumentation and proof -- Conclusion.

Glaucon in Plato's _Republic_ fails to grasp intermediates. He confuses pursuing a goal with achieving it, and so he adopts ‘mathematical platonism’. He says mathematical objects are eternal. Socrates urges a seriously debatable, and seriously defensible, alternative centered on the destruction of hypotheses. He offers his version of geometry and astronomy as refuting the charge that he impiously ‘ponders things up in the sky and investigates things under the earth and makes the weaker argument the stronger’. We relate his account (...) briefly to mathematical developments by Plato's associates Theaetetus and Eudoxus, and then to the past 200 years' developments in geometry. (shrink)

Mathematics textbooks teach logical reasoning by example, a practice started by Euclid; while logic textbooks treat logic as a subject in its own right without practical application to mathematics. Stuck in the middle are students seeking mathematical proficiency and educators seeking to provide it. To assist them, the article explains in practical detail how to teach logic-based skills such as: making mathematical reasoning fully explicit; moving from step to step in a mathematical proof in logically correct ways; and (...) checking to make sure inferences are logically correct. The methodology can easily be extended beyond the four examples analyzed. (shrink)

We study the interconnection between Physics and Mathematics in concrete instances, departing from the usual expression for the Coulomb electric field, produced by a point-like charge. It is scrutinized by means of six epistemology-intensive questions and radical answers are proposed, intended to widen one’s understanding of the subject. Our interventions act along two complementary directions. One of them regards ontology, since questions induce one to look closely at the electric charge, from different perspectives, promoting reflections about its nature and (...) reinforcing the corresponding concept. Formal manipulations rely on the identification of concepts with symbols, and the other direction concerns the spatial extension of mathematical structures. Our questions and their somewhat unusual answers help disclosing information which is not present in many textbooks, and show that Mathematics can be used as an efficient epistemological tool in Physics teaching. (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)

One of the criticisms of standard teaching practices is that they support merely ‘ritual’ as opposed to ‘principled’ knowledge, that is, knowledge which is procedural rather than being founded on principled explanation. This paper addresses issues and assumptions in current debate concerning the nature of mathematical knowledge, focusing on the ritual/principle distinction. Taking a discussion of centralism in logic and mathematics as its start-point, it seeks to resolve these issues through an examination of mathematics as a community (...) of practice and the teacher's role as epistemological authority in inducting pupils into such practices. (shrink)

One of the criticisms of standard teaching practices is that they support merely ‘ritual’ as opposed to ‘principled’ knowledge, that is, knowledge which is procedural rather than being founded on principled explanation. This paper addresses issues and assumptions in current debate concerning the nature of mathematical knowledge, focusing on the ritual/principle distinction. Taking a discussion of centralism in logic and mathematics as its start-point, it seeks to resolve these issues through an examination of mathematics as a community (...) of practice and the teacher's role as epistemological authority in inducting pupils into such practices. (shrink)

Much of the neuroimaging research has focused on how mathematical operations are performed. Although this body of research has provided insight for the refinement of pedagogy, there are very few neuroimaging studies on how mathematical operations should be taught. In this article, we describe the teaching of algebra in Singapore schools and the imperatives that led us to develop two neuroimaging studies that examined questions of curricular concerns. One of the challenges was to condense issues from classrooms into tasks (...) suitable for neuroimaging studies. Another challenge, not particular to the neuroimaging method, was to draw suitable inferences from the findings and translate them into pedagogical practices. We describe our efforts and outline some continuing challenges. (shrink)

Abstract:With democracy in mind, promoting students’ cognitive, personal, and social development can inform and shape the mathematics curriculum and classroom practice with the goal of their becoming more capable, self-reflective, and socially aware human beings. Toward that realization, their mathematics experience could include: heuristics, as it provides a natural language for problem solving; habits of mind, so students can think and act with a more developed “reflective intelligence”; and multiple-centers investigations, where collaborations based on shared mathematical interest can (...) be pursued. In this way, their education would be “a fostering, a nurturing, a cultivating process” that supports the development of democratic society. (shrink)

The narrative about the nineteenth century favored by many philosophers of mathematics strongly influenced by either logic or algebra, is that geometric intuition led real and complex analysis astray until Cauchy and Kronecker in one sense and Dedekind in another guided mathematicians out of the labyrinth through the arithmetization of analysis. Yet the use of geometry in most cases in nineteenth century mathematics was not misleading and was often key to important developments. Thus the geometrization of complex numbers (...) was essential to their acceptance and to the development of complex analysis; geometry provided the canonical examples that led to the formulation of group theory; and geometry, transformed by Riemann, lay at the heart of topology, which in turn transformed much of modern mathematics. Using complex numbers as my case study, I argue that the best way to teach students mathematics is through a repertoire of modes of representation, which is also the best way to make mathematical discoveries. (shrink)

The notion of proof is central to mathematics yet it is one of the most difficult aspects of the subject to teach and master. In particular, undergraduate mathematics students often experience difficulties in understanding and constructing proofs. Understanding Mathematical Proof describes the nature of mathematical proof, explores the various techniques that mathematicians adopt to prove their results, and offers advice and strategies for constructing proofs. It will improve students’ ability to understand proofs and construct correct proofs of their (...) own. The first chapter of the text introduces the kind of reasoning that mathematicians use when writing their proofs and gives some example proofs to set the scene. The book then describes basic logic to enable an understanding of the structure of both individual mathematical statements and whole mathematical proofs. It also explains the notions of sets and functions and dissects several proofs with a view to exposing some of the underlying features common to most mathematical proofs. The remainder of the book delves further into different types of proof, including direct proof, proof using contrapositive, proof by contradiction, and mathematical induction. The authors also discuss existence and uniqueness proofs and the role of counter examples. (shrink)

There is more than one way to think. Most people are familiar with the systematic, rule-based thinking that one finds in a mathematical proof or a computer program. But such thinking does not produce breakthroughs in mathematics and science nor is it the kind of thinking that results in significant learning. Deep thinking is a different and more basic way of using the mind. It results in the discontinuous "aha!" experience, which is the essence of creativity. It is at (...) the heart of every paradigm shift or reframing of a problematic situation. The identification of deep thinking as the default state of the mind has the potential to reframe our current approach to technological change, education, and the nature of mathematics and science. For example, there is an unbridgeable gap between deep thinking and computer simulations of thinking. Many people suspect that such a gap exists, but find it difficult to make this intuition precise. This book identifies the way in which the authentic intelligence of deep thinking differs from the artificial intelligence of "big data" and "analytics." Deep thinking is the essential ingredient in every significant learning experience, which leads to a new way to think about education. It is also essential to the construction of conceptual systems that are at the heart of mathematics and science, and of the technologies that shape the modern world. Deep thinking can be found whenever one conceptual system morphs into another. The sources of this study include the cognitive development of numbers in children, neuropsychology, the study of creativity, and the historical development of mathematics and science. The approach is unusual and original. It comes out of the author's lengthy experience as a mathematician, teacher, and writer of books about mathematics and science, such as How Mathematicians Think: Using Ambiguity, Contradiction, and Paradox to Create Mathematics and The Blind Spot: Science and the Crisis of Uncertainty. (shrink)

This paper focuses on the reconceptualization of the mathematics curriculum with Roma students presenting a nomadic exploration of the mathematics teaching. An alternative, methodologically privileged, model is proposed which identifies the opportunity for an alternative curriculum from within rather than from a distance. The main aim of the research is to indicate how researching for a curriculum reconceptualization might be enlightened by thinking rhizomatically where common spaces of in-between are constructed by a confabulative conversation between Roma students (...) and the researcher who takes up a nomadic position. Roma students, educational becoming and subjectivities at the level of individuals and situations involved both interact as elements of an unpredictably evolving rhizome which consists of discontinuity, rupture and constantly emerging multiplicities. (shrink)

It is well known that the largest philosophers differently explain the origin of mathematics. This question was investigated in antiquity, a substantial and decisive role in this respect was played by the Platonic doctrine. Therefore, discussing this issue the problem of interaction of philosophy and mathematics in the teachings of Plato should be taken into consideration. Many mathematicians believe that abstract mathematical objects belong in a certain sense to the world of ideas and that consistency of objects and (...) theories really describes mathematical reality, as Plato quite clearly expressed his views on math, according to which mathematical concepts objectively exist as distinct entities between the world of ideas and the world of material things. In the context of foundations of mathematics, so called ‘Gödel’s Platonism‘ is of particular interest. It is shown in the article how Platonic objectification of mathematical concepts contributes to the development of modern mathematics by revealing philosophical understanding of the nature of abstraction. To substantiate his point of view, the author draws the works of contemporary experts in the field of philosophy of mathematics. (shrink)

The contradiction between the rapid mathematization of health care through the active introduction of modern technologies and methods based on mathematical achievements in the field of medicine and the lack of a system of training medical students corresponding to these scientific successes, which allows them to carry out mathematical modeling of complex physical, chemical and biological processes at the molecular level for the purpose of their analysis and subsequent forecasting,, problems that arise when teaching mathematical analysis in medical Schools, (...) and possible ways to solve them. The solution identified during the research of problematic issues of teaching of mathematical analysis in medical schools is offered by enhancing the quality of learning through the re-allocated for the development of the discipline course hours stimulate the motivation of students, adjusting content and methodical component of teaching and active introduction in educational process of modern information technologies, elimination of excessive mathematical formalism. (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)

Much of the neuroimaging research has focused on how mathematical operations are performed. Although this body of research has provided insight for the refinement of pedagogy, there are very few neuroimaging studies on how mathematical operations should be taught. In this article, we describe the teaching of algebra in Singapore schools and the imperatives that led us to develop two neuroimaging studies that examined questions of curricular concerns. One of the challenges was to condense issues from classrooms into tasks (...) suitable for neuroimaging studies. Another challenge, not particular to the neuroimaging method, was to draw suitable inferences from the findings and translate them into pedagogical practices. We describe our efforts and outline some continuing challenges. (shrink)

This book tells a single story, in many voices, about a serious and sustained set of changes in mathematics teaching practice in a high school and how those efforts influenced and were influenced by a local university. It includes the writings and perspectives of high school students, high school teachers, preservice teacher candidates, doctoral students in mathematics education and other fields, mathematics teacher educators, and other education faculty. As a whole, this case study provides an opportunity (...) to reflect on reform visions of mathematics for all students and the challenges inherent in the implementation of these visions in US schools. It challenges us to rethink boundaries between theory and practice and the relative roles of teachers and university faculty in educational endeavors. (shrink)

Examine how community college faculty in science, technology, engineering, and mathematics partnered with K–12 teachers to mentor community college science and math students and open their minds to pursuing a career in K–12 teaching. This report outlines the lessons learned by the community colleges that participated in AACC’s Teaching Scholar Partnerships, an initiative supported by the National Science Foundation.

In this paper, we discuss the history of the concept of function and emphasize in particular how problems in physics have led to essential changes in its definition and application in mathematical practices. Euler defined a function as an analytic expression, whereas Dirichlet defined it as a variable that depends in an arbitrary manner on another variable. The change was required when mathematicians discovered that analytic expressions were not sufficient to represent physical phenomena such as the vibration of a string (...) and heat conduction. The introduction of generalized functions or distributions is shown to stem partly from the development of new theories of physics such as electrical engineering and quantum mechanics that led to the use of improper functions such as the delta function that demanded a proper foundation. We argue that the development of student understanding of mathematics and its nature is enhanced by embedding mathematical concepts and theories, within an explicit–reflective framework, into a rich historical context emphasizing its interaction with other disciplines such as physics. Students recognize and become engaged with meta-discursive rules governing mathematics. Mathematics teachers can thereby teach inquiry in mathematics as it occurs in the sciences, as mathematical practice aimed at obtaining new mathematical knowledge. We illustrate such a historical teaching and learning of mathematics within an explicit and reflective framework by two examples of student-directed, problem-oriented project work following the Roskilde Model, in which the connection to physics is explicit and provides a learning space where the nature of mathematics and mathematical practices are linked to natural science. (shrink)