Visual Thinking In Mathematics

Author: Marcus Giaquinto
Publisher: OUP Oxford
ISBN: 9780199575534
Size: 79.99 MB
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Marcus Giaquinto presents an investigation into the different kinds of visual thinking involved in mathematical thought, drawing on work in cognitive psychology, philosophy, and mathematics. He argues that mental images and physical diagrams are rarely just superfluous aids: they are often a means of discovery, understanding, and even proof.

Proofs Without Words

Author: Malcolm Scott MacKenzie
Publisher: MAA
ISBN: 9780883857007
Size: 75.92 MB
Format: PDF, Docs
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Proofs without words are generally pictures or diagrams that help the reader see why a particular mathematical statement may be true, and how one could begin to go about proving it. While in some proofs without words an equation or two may appear to help guide that process, the emphasis is clearly on providing visual clues to stimulate mathematical thought. The proofs in this collection are arranged by topic into five chapters: Geometry and algebra; Trigonometry, calculus and analytic geometry; Inequalities; Integer sums; and Sequences and series. Teachers will find that many of the proofs in this collection are well suited for classroom discussion and for helping students to think visually in mathematics.

Visible Thinking In The K 8 Mathematics Classroom

Author: Ted H. Hull
Publisher: Corwin Press
ISBN: 1452269408
Size: 48.80 MB
Format: PDF, Mobi
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Seeing is believing with this interactive approach to math instruction Do you ever wish your students could read each other’s thoughts? Now they can—and so can you! This newest book by veteran mathematics educators provides instructional strategies for maximizing students’ mathematics comprehension by integrating visual thinking into the classroom. Included are numerous grade-specific sample problems for teaching essential concepts such as number sense, fractions, and estimation. Among the many benefits of visible thinking are: Interactive student-to-student learning Increased class participation Development of metacognitive thinking and problem-solving skills

Math Made Visual

Author: Claudi Alsina
Publisher: MAA
ISBN: 9780883857465
Size: 61.56 MB
Format: PDF
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A book describing how visualization techniques can be used in the teaching of mathematics.

Mathematical Mindsets

Author: Jo Boaler
Publisher: John Wiley & Sons
ISBN: 1118418271
Size: 53.35 MB
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Banish math anxiety and give students of all ages a clear roadmap to success Mathematical Mindsets provides practical strategies and activities to help teachers and parents show all children, even those who are convinced that they are bad at math, that they can enjoy and succeed in math. Jo Boaler—Stanford researcher, professor of math education, and expert on math learning—has studied why students don't like math and often fail in math classes. She's followed thousands of students through middle and high schools to study how they learn and to find the most effective ways to unleash the math potential in all students. There is a clear gap between what research has shown to work in teaching math and what happens in schools and at home. This book bridges that gap by turning research findings into practical activities and advice. Boaler translates Carol Dweck's concept of 'mindset' into math teaching and parenting strategies, showing how students can go from self-doubt to strong self-confidence, which is so important to math learning. Boaler reveals the steps that must be taken by schools and parents to improve math education for all. Mathematical Mindsets: Explains how the brain processes mathematics learning Reveals how to turn mistakes and struggles into valuable learning experiences Provides examples of rich mathematical activities to replace rote learning Explains ways to give students a positive math mindset Gives examples of how assessment and grading policies need to change to support real understanding Scores of students hate and fear math, so they end up leaving school without an understanding of basic mathematical concepts. Their evasion and departure hinders math-related pathways and STEM career opportunities. Research has shown very clear methods to change this phenomena, but the information has been confined to research journals—until now. Mathematical Mindsets provides a proven, practical roadmap to mathematics success for any student at any age.

Toward A Visually Oriented School Mathematics Curriculum

Author: Ferdinand Rivera
Publisher: Springer Science & Business Media
ISBN: 9789400700147
Size: 62.29 MB
Format: PDF, ePub
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What does it mean to have a visual representation of a mathematical object, concept, or process? What visualization strategies support growth in mathematical thinking, reasoning, generalization, and knowledge? Is mathematical seeing culture-free? How can information drawn from studies in blind subjects help us understand the significance of a multimodal approach to learning mathematics? Toward a Visually-Oriented School Mathematics Curriculum explores a unified theory of visualization in school mathematical learning via the notion of progressive modeling. Based on the author’s longitudinal research investigations in elementary and middle school classrooms, the book provides a compelling empirical account of ways in which instruction can effectively orchestrate the transition from personally-constructed visuals, both externally-drawn and internally-derived, into more structured visual representations within the context of a socioculturally grounded mathematical activity. Both for teachers and researchers, a discussion of this topic is relevant in the history of the present. The ubiquity of technological tools and virtual spaces for learning and doing mathematics has aroused interest among concerned stakeholders about the role of mathematics in these contexts. The book begins with a prolegomenon on the author’s reflections on past and present visual studies in mathematics education. In the remaining seven chapters, visualization is pursued in terms of its role in bringing about progressions in mathematical symbolization, abduction, pattern generalization, and diagrammatization. Toward a Visually-Oriented School Mathematics Curriculum views issues surrounding visualization through the eyes of a classroom teacher-researcher; it draws on findings within and outside of mathematics education that help practitioners and scholars gain a better understanding of what it means to pleasurably experience the symmetric visual/symbolic reversal phenomenon – that is, seeing the visual in the symbolic and the symbolic in the visual."

Making Thinking Visible

Author: Ron Ritchhart
Publisher: John Wiley & Sons
ISBN: 9781118015032
Size: 45.92 MB
Format: PDF, Docs
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A proven program for enhancing students' thinking and comprehension abilities Visible Thinking is a research-based approach to teaching thinking, begun at Harvard's Project Zero, that develops students' thinking dispositions, while at the same time deepening their understanding of the topics they study. Rather than a set of fixed lessons, Visible Thinking is a varied collection of practices, including thinking routines?small sets of questions or a short sequence of steps?as well as the documentation of student thinking. Using this process thinking becomes visible as the students' different viewpoints are expressed, documented, discussed and reflected upon. Helps direct student thinking and structure classroom discussion Can be applied with students at all grade levels and in all content areas Includes easy-to-implement classroom strategies The book also comes with a DVD of video clips featuring Visible Thinking in practice in different classrooms.

Proofs Without Words Iii

Author: Roger B. Nelsen
Publisher: The Mathematical Association of America
ISBN: 0883857901
Size: 24.84 MB
Format: PDF, ePub
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Proofs without words (PWWs) are figures or diagrams that help the reader see why a particular mathematical statement is true, and how one might begin to formally prove it true. PWWs are not new, many date back to classical Greece, ancient China, and medieval Europe and the Middle East. PWWs have been regular features of the MAA journals Mathematics Magazine and The College Mathematics Journal for many years, and the MAA published the collections of PWWs Proofs Without Words: Exercises in Visual Thinking in 1993 and Proofs Without Words II: More Exercises in Visual Thinking in 2000. This book is the third such collection of PWWs. The proofs in the book are divided by topic into five chapters: Geometry & Algebra; Trigonometry, Calculus & Analytic Geometry; Inequalities; Integers & Integer Sums; and Infinite Series & Other Topics. The proofs in the book are intended primarily for the enjoyment of the reader, however, teachers will want to use them with students at many levels: high school courses from algebra through precalculus and calculus; college level courses in number theory, combinatorics, and discrete mathematics; and pre-service and in-service courses for teachers.

Visible Learning For Mathematics Grades K 12

Author: John Hattie
Publisher: Corwin Press
ISBN: 1506362974
Size: 58.54 MB
Format: PDF, ePub
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Rich tasks, collaborative work, number talks, problem-based learning, direct instruction…with so many possible approaches, how do we know which ones work the best? In Visible Learning for Mathematics, six acclaimed educators assert it’s not about which one—it’s about when—and show you how to design high-impact instruction so all students demonstrate more than a year’s worth of mathematics learning for a year spent in school. That’s a high bar, but with the amazing K-12 framework here, you choose the right approach at the right time, depending upon where learners are within three phases of learning: surface, deep, and transfer. This results in “visible” learning because the effect is tangible. The framework is forged out of current research in mathematics combined with John Hattie’s synthesis of more than 15 years of education research involving 300 million students. Chapter by chapter, and equipped with video clips, planning tools, rubrics, and templates, you get the inside track on which instructional strategies to use at each phase of the learning cycle: Surface learning phase: When—through carefully constructed experiences—students explore new concepts and make connections to procedural skills and vocabulary that give shape to developing conceptual understandings. Deep learning phase: When—through the solving of rich high-cognitive tasks and rigorous discussion—students make connections among conceptual ideas, form mathematical generalizations, and apply and practice procedural skills with fluency. Transfer phase: When students can independently think through more complex mathematics, and can plan, investigate, and elaborate as they apply what they know to new mathematical situations. To equip students for higher-level mathematics learning, we have to be clear about where students are, where they need to go, and what it looks like when they get there. Visible Learning for Math brings about powerful, precision teaching for K-12 through intentionally designed guided, collaborative, and independent learning.

Teaching Maths To Pupils With Different Learning Styles

Author: Tandi Clausen-May
Publisher: SAGE
ISBN: 9781412903592
Size: 58.42 MB
Format: PDF, Kindle
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'Tackles an area of the curriculum many teachers lack confidence in.' 'Lots of good starting points'. 'Spans a lot of material and is strong on diverse learning styles.' 'Clear explanation and good visual layour, very innovative in approach.' - Judging Panel for NASEN/TES Book Award 'The book is rich in lively teaching suggestions and in insights into the impact of different forms of explanation' - Debate '[C]arries us away from narrow views of ability and special needs and into the consideration of difference. The author takes us through lively discussions of many aspects of mathematics learning. Each section offers learning and teaching ideas involving visual and kinaesthetic approaches. The book is a compendium of sound ideas rather than a collection of startlingly new approaches. But throughout it has the great strength of being exceptionally clear in its arguments, descriptions and drawings. The design is generally helpful with plenty of illustrations, as befits the book's message. There are handy pages of photocopiable resources. This is a lively and often passionate account of ways of ensuring that multi-sensory approaches infect mathematics learning. As the author says, "pictures in the mind can help all pupils". We might add, "They help all teachers too"' - TES Extra for Special Needs 'If you have found pupils struggling to understand some aspects of mathematics at any age then this book is for you. It is a very readable book that would interest all those who work in classrooms, whether as a teacher or support worker with all ages and abilities, for those who work with older pupils as it gives possible approaches to use with those for whom basic skills are weak or have difficulty in understanding some of the concepts required of GCSE examinations' - Alison Parish, Second in Mathematics Department, Stowmarket High School, Suffolk Read the full review as posted on the Association of Teachers of Mathematics website! 'It is a highly practical book. One strength is the way that it develops a topic from the very basics through to the harder concepts. There are a large number of activities that are 'ready to run' but these really are just a starting point for teachers to begin thinking about teaching topics in a different way, and from these teachers will be able to develop their own approach. Although this book is focusing on pupils who are visual and kinaesthetic learners, the great majority of learners adopt a mixture of learning styles, so this approach will benefit the entire class. Worth a read!' - Maths Coordinator's File 'This excellent and very informative teaching resource is about teaching mathematics to pupils who have learning differences. [It] is very practical and easy to read. A really nice feature is the inclusion of photocopiable resource sheets allowing readers to try out easily the ideas suggested in the book. This resource is highly recommended and will be very suitable for maths teachers in primary and secondary schools, SENCOs and teaching assistants' - British Journal of Special Education 'This book is about making mathematics visible and tangible -- not something that just lies flat on the page. Dipping into it will provide instantly usable suggestions across a variety of topics at different levels: from early number concepts through to fractions and ratios, algebra, aspects of geometry (including angles and circles), and data handling. When you get a chance to read it more thoroughly you will find arguments for using these approaches, consideration of some of the pitfalls to avoid, and inspiration to develop different ways of helping students to achieve deep and connected understandings. For any teacher who wants to provide students with opportunities for visual and kinaesthetic learning in mathematics' - The Australian Association of Mathematics Teachers Inc. 'A very good book, offering teachers, SENCOs and teaching assistants guidelines, strategies and practical activities to access the thought processes of pupils with different learning styles. It has an easy-to-read format giving suggestions, rather than dictat, on the use of "models to think" and is a unique document for those who have input into the furthering of the teaching and learning of mathematics' - Mathematics in School How can you make maths exciting and meaningful for all your pupils? Some pupils find even basic concepts in mathematics difficult to grasp and it can be a challenge to make lessons accessible to all. This book offers practising teachers a range of approaches to making maths clear for struggling students. It looks at the different ways in which maths can be taught so that pupils with different learning styles can be stimulated. Maths is visible and tangible - not something that just lies flat on the page. Included are: - ideas to be used in lessons - suggestions for exciting, visual ways to teach basic concepts - lots of practical advice and guidance. The book shows teachers how to unlock mathematics for all their learners, and it encourages the use of a variety of methods to teach the subject. It provides a valuable resource for maths teachers in both primary and secondary schools, for SENCOs and teaching assistants, and for those delivering initial teacher training or inservice courses. Tandi Clausen-May is an educational researcher responsible for the development of a range of mathematics curriculum and assessment materials. She delivers popular workshops on teaching mathematics around the United Kingdom. She also writes regular articles on mathematics teaching for educational journals and newspapers.