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Author: Curtis Franks
Publisher: Cambridge University Press
Published: 2009-10-08
Total Pages: 229
ISBN-13: 0521514371
DOWNLOAD EBOOKThis study reconstructs, analyses and re-evaluates the programme of influential mathematical thinker David Hilbert, presenting it in a different light.
Author: James Robert Brown
Publisher: Routledge
Published: 2013-06-17
Total Pages: 195
ISBN-13: 1136580387
DOWNLOAD EBOOKThis study addresses a central theme in current philosophy: Platonism vs Naturalism and provides accounts of both approaches to mathematics, crucially discussing Quine, Maddy, Kitcher, Lakoff, Colyvan, and many others. Beginning with accounts of both approaches, Brown defends Platonism by arguing that only a Platonistic approach can account for concept acquisition in a number of special cases in the sciences. He also argues for a particular view of applied mathematics, a view that supports Platonism against Naturalist alternatives. Not only does this engaging book present the Platonist-Naturalist debate over mathematics in a comprehensive fashion, but it also sheds considerable light on non-mathematical aspects of a dispute that is central to contemporary philosophy.
Author: Nancy Butler Wolf
Publisher: Heinemann Educational Books
Published: 2015
Total Pages: 0
ISBN-13: 9780325062594
DOWNLOAD EBOOK"Nancy's in-depth look at mathematical modeling offers middle school teachers the kind of practical help they need for incorporating modeling into their classrooms." -Cathy Seeley, Past President of NCTM, author of Faster Isn't Smarter and Smarter Than We Think "This is the book that math teachers and parents have been waiting for. Nancy provides a comprehensive step-by-step guide to modeling in mathematics at the middle school level." -David E. Drew, author of STEM the Tide: Reforming Science, Technology, Engineering, and Math Education in America We all use math to analyze everyday situations we encounter. Whether we realize it or not, we're modeling with mathematics: taking a complex situation and figuring out what we need to make sense of it. In Modeling with Mathematics, Nancy Butler Wolf shows that math is most powerful when it means something to students. She provides clear, friendly guidance for teachers to use authentic modeling projects in their classrooms and help their students develop key problem-solving skills, including: collecting data and formulating a mathematical model interpreting results and comparing them to reality learning to communicate their solutions in meaningful ways. This kind of teaching can be challenging because it is open-ended: it asks students to make decisions about their approach to a scenario, the information they will need, and the tools they will use. But Nancy proves there is ample middle ground between doing all of the work for your students and leaving them to flail in the dark. Through detailed examples and hands-on activities, Nancy shows how to guide your students to become active participants in mathematical explorations who are able to answer the question, "What did I just figure out?" Her approach values all students as important contributors and shows how instruction focused on mathematical modeling engages every learner regardless of their prior history of success or failure in math.
Author: Peter Liljedahl
Publisher: Corwin Press
Published: 2020-09-28
Total Pages: 454
ISBN-13: 1544374844
DOWNLOAD EBOOKA thinking student is an engaged student Teachers often find it difficult to implement lessons that help students go beyond rote memorization and repetitive calculations. In fact, institutional norms and habits that permeate all classrooms can actually be enabling "non-thinking" student behavior. Sparked by observing teachers struggle to implement rich mathematics tasks to engage students in deep thinking, Peter Liljedahl has translated his 15 years of research into this practical guide on how to move toward a thinking classroom. Building Thinking Classrooms in Mathematics, Grades K–12 helps teachers implement 14 optimal practices for thinking that create an ideal setting for deep mathematics learning to occur. This guide Provides the what, why, and how of each practice and answers teachers’ most frequently asked questions Includes firsthand accounts of how these practices foster thinking through teacher and student interviews and student work samples Offers a plethora of macro moves, micro moves, and rich tasks to get started Organizes the 14 practices into four toolkits that can be implemented in order and built on throughout the year When combined, these unique research-based practices create the optimal conditions for learner-centered, student-owned deep mathematical thinking and learning, and have the power to transform mathematics classrooms like never before.
Author: Russell Marcus
Publisher: Lexington Books
Published: 2015-06-11
Total Pages: 259
ISBN-13: 0739173138
DOWNLOAD EBOOKMathematical platonism is the view that mathematical statements are true of real mathematical objects like numbers, shapes, and sets. One central problem with platonism is that numbers, shapes, sets, and the like are not perceivable by our senses. In contemporary philosophy, the most common defense of platonism uses what is known as the indispensability argument. According to the indispensabilist, we can know about mathematics because mathematics is essential to science. Platonism is among the most persistent philosophical views. Our mathematical beliefs are among our most entrenched. They have survived the demise of millennia of failed scientific theories. Once established, mathematical theories are rarely rejected, and never for reasons of their inapplicability to empirical science. Autonomy Platonism and the Indispensability Argument is a defense of an alternative to indispensability platonism. The autonomy platonist believes that mathematics is independent of empirical science: there is purely mathematical evidence for purely mathematical theories which are even more compelling to believe than empirical science. Russell Marcus begins by contrasting autonomy platonism and indispensability platonism. He then argues against a variety of indispensability arguments in the first half of the book. In the latter half, he defends a new approach to a traditional platonistic view, one which includes appeals to a priori but fallible methods of belief acquisition, including mathematical intuition, and a natural adoption of ordinary mathematical methods. In the end, Marcus defends his intuition-based autonomy platonism against charges that the autonomy of mathematics is viciously circular. This book will be useful to researchers, graduate students, and advanced undergraduates with interests in the philosophy of mathematics or in the connection between science and mathematics.
Author: Vladimir G Ivancevic
Publisher: World Scientific
Published: 2017-10-30
Total Pages: 432
ISBN-13: 9813230401
DOWNLOAD EBOOKMathematics of Autonomy provides solid mathematical foundations for building useful Autonomous Systems. It clarifies what makes a system autonomous rather than simply automated, and reveals the inherent limitations of systems currently incorrectly labeled as autonomous in reference to the specific and strong uncertainty that characterizes the environments they operate in. Such complex real-world environments demand truly autonomous solutions to provide the flexibility and robustness needed to operate well within them.This volume embraces hybrid solutions to demonstrate extending the classes of uncertainty autonomous systems can handle. In particular, it combines physical-autonomy (robots), cyber-autonomy (agents) and cognitive-autonomy (cyber and embodied cognition) to produce a rigorous subset of trusted autonomy: Cyber-Physical-Cognitive autonomy (CPC-autonomy).The body of the book alternates between underlying theory and applications of CPC-autonomy including 'Autonomous Supervision of a Swarm of Robots' , 'Using Wind Turbulence against a Swarm of UAVs' and 'Unique Super-Dynamics for All Kinds of Robots (UAVs, UGVs, UUVs and USVs)' to illustrate how to effectively construct Autonomous Systems using this model. It avoids the wishful thinking that characterizes much discussion related to autonomy, discussing the hard limits and challenges of real autonomous systems. In so doing, it clarifies where more work is needed, and also provides a rigorous set of tools to tackle some of the problem space.
Author: Assistant Professor of Philosophy Curtis Franks
Publisher:
Published: 2014-05-14
Total Pages: 229
ISBN-13: 9780511641497
DOWNLOAD EBOOKThis study reconstructs, analyses and re-evaluates the programme of influential mathematical thinker David Hilbert, presenting it in a different light.
Author: Emily Grosholz
Publisher: Springer Science & Business Media
Published: 2013-04-17
Total Pages: 456
ISBN-13: 9401595585
DOWNLOAD EBOOKMathematics has stood as a bridge between the Humanities and the Sciences since the days of classical antiquity. For Plato, mathematics was evidence of Being in the midst of Becoming, garden variety evidence apparent even to small children and the unphilosophical, and therefore of the highest educational significance. In the great central similes of The Republic it is the touchstone ofintelligibility for discourse, and in the Timaeus it provides in an oddly literal sense the framework of nature, insuring the intelligibility ofthe material world. For Descartes, mathematical ideas had a clarity and distinctness akin to the idea of God, as the fifth of the Meditations makes especially clear. Cartesian mathematicals are constructions as well as objects envisioned by the soul; in the Principles, the work ofthe physicist who provides a quantified account ofthe machines of nature hovers between description and constitution. For Kant, mathematics reveals the possibility of universal and necessary knowledge that is neither the logical unpacking ofconcepts nor the record of perceptual experience. In the Critique ofPure Reason, mathematics is one of the transcendental instruments the human mind uses to apprehend nature, and by apprehending to construct it under the universal and necessary lawsofNewtonian mechanics.
Author: Tim Rowland
Publisher: Springer Science & Business Media
Published: 2011-01-06
Total Pages: 300
ISBN-13: 904819766X
DOWNLOAD EBOOKThe quality of primary and secondary school mathematics teaching is generally agreed to depend crucially on the subject-related knowledge of the teacher. However, there is increasing recognition that effective teaching calls for distinctive forms of subject-related knowledge and thinking. Thus, established ways of conceptualizing, developing and assessing mathematical knowledge for teaching may be less than adequate. These are important issues for policy and practice because of longstanding difficulties in recruiting teachers who are confident and conventionally well-qualified in mathematics, and because of rising concern that teaching of the subject has not adapted sufficiently. The issues to be examined in Mathematical Knowledge in Teaching are of considerable significance in addressing global aspirations to raise standards of teaching and learning in mathematics by developing more effective approaches to characterizing, assessing and developing mathematical knowledge for teaching.
Author: Heinz Steinbring
Publisher: Springer Science & Business Media
Published: 2006-03-30
Total Pages: 242
ISBN-13: 0387242538
DOWNLOAD EBOOKMathematics is generally considered as the only science where knowledge is uni form, universal, and free from contradictions. „Mathematics is a social product - a 'net of norms', as Wittgenstein writes. In contrast to other institutions - traffic rules, legal systems or table manners -, which are often internally contradictory and are hardly ever unrestrictedly accepted, mathematics is distinguished by coherence and consensus. Although mathematics is presumably the discipline, which is the most differentiated internally, the corpus of mathematical knowledge constitutes a coher ent whole. The consistency of mathematics cannot be proved, yet, so far, no contra dictions were found that would question the uniformity of mathematics" (Heintz, 2000, p. 11). The coherence of mathematical knowledge is closely related to the kind of pro fessional communication that research mathematicians hold about mathematical knowledge. In an extensive study, Bettina Heintz (Heintz 2000) proposed that the historical development of formal mathematical proof was, in fact, a means of estab lishing a communicable „code of conduct" which helped mathematicians make themselves understood in relation to the truth of mathematical statements in a co ordinated and unequivocal way.
