New Arrivals · Mathematics & Physics

November 4, 2017
These titles were recently added to the collection of the Public Library of Cincinnati and Hamilton County.

The quantum labyrinth : how Richard Feynman and John Wheeler revolutionized time and reality

November 3, 2017
Halpern, Paul, 1961- author.
©2017
ix, 311 pages : illustrations ; 25 cm
Wheeler's watch -- The only particle in the universe -- All the roads not to paradise -- The hidden paths of ghosts -- The island and the mountains: mapping the particle landscape -- Life as an amoeba in the foamy sea of possibilities -- Time's arrow and the mysterious Mr. X -- Minds, machines, and the cosmos -- Conclusion: The way of the labyrinth.
"In Fall 1939, Richard Feynman, a brash and brilliant recent graduate of MIT, arrived in John Wheeler's Princeton office to report for duty as his teaching assistant. The prim and proper Wheeler timed their interaction with a watch placed on the table. Feynman caught on, and for the next meeting brought his own cheap watch, set it on the table next to Wheeler's, and also began timing the chat. The two had a hearty laugh and a lifelong friendship was born. At first glance, they would seem an unlikely pair. Feynman was rough on the exterior, spoke in a working class Queens accent, and loved playing bongo drums, picking up hitchhikers, and exploring out-of-the way places. Wheeler was a family man, spoke softly and politely, dressed in suits, and had the manners of a minister. Yet intellectually, their roles were reversed. Wheeler was a raging nonconformist, full of wild ideas about space, time, and the universe. Feynman was very cautious in his research, wanting to prove and confirm everything himself. Yet when Feynman saw merit in one of Wheeler's crazy ideas and found that it matched experimental data, their joint efforts paid off phenomenally"--Provided by publisher.

Chancing it : the laws of chance and how they can work for you

November 1, 2017
Matthews, Robert, 1959-
New York, NY : Skyhorse Publishing, 2017.
xii, 290 pages : illustrations ; 24 cm
First published in Great Britain in 2016 by Profile Books Ltd.
Foreword / by Larry Gonick -- The coin-tossing prisoner of the Nazis -- What the Law of Averages really means -- The dark secret of the Golden Theorem -- What are the chances of that? -- Thinking independently is no yolk -- Random lessons from the lottery -- Warning : there's a lot of X about -- Why the amazing so often turns ho-hum -- If you don't know, go random -- Doing the right thing isn't always ethical -- How a lot of bull sparked a revolution -- How to beat casinos at their own game -- Where wise-guys go wrong -- The Golden Rule of Gambling -- Insure it-- or chance it? -- Making better bets in the Casino of Life -- Tell me straight doc-- what are my chances? -- This is not a drill! : Repeat: this is not a drill! -- The miraculous formula of Reverend Bayes -- When Dr. Turing met Reverend Bayes -- Using Bayes to be a better judge -- A scandal of significance -- Dodging the Amazing Baloney Machine -- Making use of what you already know -- I'm sorry professor, I just don't buy it -- The Amazing Curve for Everything -- The dangers of thinking everything's Normal -- Ugly sisters and evil twins -- Going to extremes -- See a Nicolas Cage movie and die -- We've got to draw the line somewhere -- Playing the markets isn't rocket science -- Beware geeks bearing models.
"Over the years, some very smart people have thought they understood the rules of chance--only to fail dismally. Whether you call it probability, risk, or uncertainty, the workings of chance often defy common sense. Fortunately, advances in math and science have revealed the laws of chance, and understanding those laws can help in your everyday life. In Chancing It, award-winning scientist and writer Robert Matthews shows how to understand the laws of probability and use them to your advantage. He gives you access to some of the most potent intellectual tools ever developed and explains how to use them to guide your judgments and decisions. By the end of the book, you will know: how to understand and even predict coincidences; when an insurance policy is worth having; why "expert" predictions are often misleading; how to tell when a scientific claim is a breakthrough or baloney; when it makes sense to place a bet on anything from sports to stock markets. A groundbreaking introduction to the power of probability, Chancing It will sharpen your decision-making and maximize your luck."--Jacket.

Arithmetic

November 1, 2017
Lockhart, Paul, author.
Cambridge, Massachusetts : The Belknap Press of Harvard University Press, 2017.
223 pages : illustrations ; 22 cm
Includes index.
Things -- Language -- Repetition -- Three tribes -- Egypt -- Rome -- China and Japan -- India -- Europe -- Multiplication -- Division -- Machines -- Fractions -- Negative numbers -- The art of counting.
Educator Paul Lockhart's goal is to demystify arithmetic: to bring the subject to life in a fun and accessible way, and to reveal its profound and simple beauty, as seen through the eyes of a modern research mathematician. The craft of arithmetic arises from our natural desire to count, arrange, and compare quantities. Over the centuries, humans have devised a wide variety of strategies for representing and manipulating numerical information: tally marks, rocks and beads, marked-value and place-value systems, as well as mechanical and electronic calculators. Arithmetic traces the history and development of these various number languages and calculating devices and examines their comparative advantages and disadvantages, providing readers with an opportunity to develop not only their computational skills but also their own personal tastes and preferences. The book is neither a training manual nor an authoritative history, but rather an entertaining survey of ideas and methods for the reader to enjoy and appreciate. Written in a lively conversational style, Arithmetic is a fun and engaging introduction to both practical techniques as well as the more abstract mathematical aspects of the subject.-- Provided by publisher.

The formative years of relativity : the history and meaning of Einstein's Princeton lectures

September 28, 2017
Gutfreund, Hanoch, author.
Princeton : Princeton University Press, [2017]
xiv, 415 pages : illustrations ; 27 cm
"Featuring Einstein's Classic Text, The Meaning of Relativity, in its Historical Context" -- title page.
Einstein's first trip to America -- Structure and contents of the meaning of relativity -- Physics and geometry -- The principles of general relativity -- The first solutions and the challenge of their interpretation -- Einstein and astronomers -- The genesis of relativistic cosmology -- The controversy over gravitational waves -- Philosophical debates on general relativity -- The quest for a unified field theory -- Early monographs on relativity -- Beyond the formative years -- Pre-relativity physics -- The theory of special relativity -- The general theory of relativity -- The general theory of relativity (continued).

Ripples in spacetime : Einstein, gravitational waves, and the future of astronomy

September 20, 2017
Schilling, Govert, author.
©2017
xi, 339 pages : illustrations ; 22 cm
A spacetime appetizer -- Relatively speaking -- Einstein on trial -- Wave talk and bar fights -- The lives of stars -- Clockwork precision -- Laser quest -- The path to perfection -- Creation stories -- Cold case -- Gotcha -- Black magic -- Nanoscience -- Follow-up questions -- Space invaders -- Surf's up for Einstein wave astronomy.
It has already been called the scientific breakthrough of the century: the detection of gravitational waves. Einstein predicted these tiny ripples in the fabric of spacetime nearly a hundred years ago, but they were never perceived directly until now. Decades in the making, this momentous discovery has given scientists a new understanding of the cataclysmic events that shape the universe and a new confirmation of Einstein's theory of general relativity. Ripples in Spacetime is an engaging account of the international effort to complete Einstein's project, capture his elusive ripples, and launch an era of gravitational-wave astronomy that promises to explain, more vividly than ever before, our universe's structure and origin. The quest for gravitational waves involved years of risky research and many personal and professional struggles that threatened to derail one of the world's largest scientific endeavors. Govert Schilling takes readers to sites where these stories unfolded--including Japan's KAGRA detector, Chile's Atacama Cosmology Telescope, the South Pole's BICEP detectors, and the United States' LIGO labs. He explains the seeming impossibility of developing technologies sensitive enough to detect waves from two colliding black holes in the very distant universe, and describes the astounding precision of the LIGO detectors. Along the way Schilling clarifies concepts such as general relativity, neutron stars, and the big bang using language that readers with little scientific background can grasp.-- Provided by publisher

Number talks. Fractions, decimals, and percentages : a multimedia professional learning resource

September 12, 2017
Parrish, Sherry.
©2016
xliv, 404 pages : illustrations ; 24 cm
This resource was created in response to the requests of teachers--those who want to implement number talks but are unsure of how to begin, and those with experience who want more guidance in crafting purposeful problems.

Fields of color : the theory that escaped Einstein

September 8, 2017
Brooks, Rodney Allen, author publisher.
Silver Spring, Maryland : Universal Printing, LLC.
v, 149 pages : illustrations (some color) ; 23 cm

Mass : the quest to understand matter from Greek atoms to quantum fields

August 30, 2017
Baggott, J. E., author.
©2017
xvi, 346 pages : illustrations ; 24 cm
Part I. Atom and void -- The quiet citadel -- Things-in-themselves -- An impression of force -- The sceptical chymists -- Part II. Mass and energy -- A very interesting conclusion -- Incommensurable -- The fabric -- In the heart of darkness -- Part III. Wave and particle -- An act of desperation -- The wave equation -- The only mystery -- Mass bare and dressed -- Part IV. Field and force -- The symmetries of nature -- The Goddamn particle -- The standard model -- Mass without mass.
Everything around us is made of 'stuff', from planets, to books, to our own bodies. Whatever it is, we call it matter or material substance. It is solid; it has mass. But what is matter, exactly? We are taught in school that matter is not continuous, but discrete. As a few of the philosophers of ancient Greece once speculated, nearly two and a half thousand years ago, matter comes in 'lumps', and science has relentlessly peeled away successive layers of matter to reveal its ultimate constituents.

Zapped : from infrared to x-rays, the curious history of invisible light

August 1, 2017
Berman, Bob, author.
New York : Little, Brown and Company, 2017.
vi, 261 pages : illustrations ; 22 cm
Light fantastic -- Now you see it, now you don't -- The green planet and the red heat -- Hot rays -- Ultraviolet brings the blues -- Danger beyond the violet -- Energy rhythms -- The exploding sun -- No soap -- Turning on and tuning in -- The speed that destroyed space and time -- Microwaves everywhere -- The man with the x-ray vision -- Röntgen rays for everyone -- What's in your basement? -- The atomic quartet -- Gamma rays: the impossible light -- Cell-phone radiation -- Cosmic rays -- Beams from the universe's birth -- Energy from our minds -- Ray guns -- The next frontier: zero-point and dark energies -- Total solar eclipse: when the rays stop -- ET's may be broadcasting, but what's their number? -- Does light have a bright future?
"Zapped tells the story of all the light we cannot see, tracing microwaves, X-rays, gamma rays, radio waves, infrared, ultraviolet, and other forms of radiation from their historic, world-altering discoveries in the nineteenth century to their central role in modern life"--Provided by publisher.

Cracking the AP physics C exam.

July 31, 2017
New York, NY : Random House
At head of title: The Princeton Review.

Cracking the AP calculus AB exam.

July 31, 2017
New York, NY : Penguin Random House, Inc.
volumes : illustrations ; 28 cm
At head of title: The Princeton Review.

Cracking the AP physics 2 exam.

July 31, 2017
New York, NY : Penguin Random House, c2015-
At head of title: The Princeton Review.

Cracking the AP physics 1 exam.

July 31, 2017
New York, NY : Penguin Random House, c2014-
At head of title: The Princeton Review.

Cracking the AP statistics exam.

July 28, 2017
New York, NY : Random House,
v. : ill. ; 28 cm.
At head of title: The Princeton Review.

Cracking the AP calculus BC exam.

July 28, 2017
New York, NY : Penguin Random House, Inc.
volumes : illustrations ; 28 cm
At head of title: The Princeton Review.

Errors, blunders, and lies : how to tell the difference

July 18, 2017
Salsburg, David, 1931- author.
Boca Raton, FL : CRC Press, Taylor & Francis Group, [2017]
xiii, 154 pages : illustrations ; 22 cm.
The transit of Venus -- Probability versus likelihood -- The central limit conjecture -- Measuring disease -- Other uses of multilinear models -- When multilinear regression is not adequate -- Correlation versus causation -- Regression and big data -- Contaminated distributions -- The Princeton robustness study -- When the blunder is what you want -- Parsing "blunders" -- The reigns of kings -- Searching for the "real" Davy Crockett -- Detecting falsified counts -- Uncovering secrets -- Errors, blunders, or curbstoning?
We live in a world that is not quite 'right'. The central tenet of statistical inquiry is that Observation = Truth + Error because even the most careful of scientific investigations have always been bedevilled by uncertainty. Our attempts to measure things are plagued with small errors. Our attempts to understand our world are blocked by blunders. And, unfortunately, in some cases, people have been known to lie. In this follow-up to his acclaimed best-seller, The Lady Tasting Tea, David Salsburg opens a door to the widespread use of statistical methods by looking at historical examples of errors, blunders and lies from areas as diverse as archaeology, law, economics, medicine, psychology, sociology, Biblical studies, history, and war-time espionage. In doing so, he shows how, upon closer statistical investigation, errors and blunders often lead to useful information. And how statistical methods have been used to uncover falsified data. Beginning with Edmund Halley's examination of the Transit of Venus and ending with discussions of how many tanks Rommel had during the Second World War and whether modern African censuses contain falsified data, the author invites the reader to come along on this easily-accessible and fascinating journey of how to identify the nature of errors, minimize the effects of blunders, and figure out who the liars are.

Bad choices : how algorithms can help you think smarter and live happier

June 22, 2017
Almossawi, Ali, author.
New York, New York : Viking, [2017]
xi, 145 pages ; 24 cm
Includes index.
Algorithmic thinking in everyday life -- Match those socks -- Find your size -- Pop to the shops -- Bring him home -- Sort that post -- Be more hip -- Update that status -- Get the jobs done -- Fix that necklace -- Locate that box -- Fill those shelves -- Navigate those aisles -- Final thoughts.
An introduction to algorithms and their problem-solving potential in the everyday world outlines alternative methods for managing twelve different scenarios using the same systems that underline a word processor, a Google search engine, or a Facebook ad.

Learning to love math : teaching strategies that change student attitudes and get results

June 16, 2017
Willis, Judy.
Alexandria, Va. : ASCD, ©2010.
x, 195 pages : illustrations ; 23 cm
Reversing math negativity with an attitude makeover -- Understanding and planning achievable challenge -- Examples of differentiated planning for achievable challenge -- Reducing mistake anxiety -- Change your intelligence? Yes, you can! -- Motivating all of your students -- Bringing the real world to the math classroom -- Creating student goals for motivation.
Dr. Judy Willis presents this informative guide to getting better results in math class. Willis presents a practical approach for how we can improve academic results by demonstrating certain behaviors and teaching students in a way that minimizes negativity.

The mathematics lover's companion : masterpieces for everyone

June 12, 2017
Scheinerman, Edward R., author.
©2017
xx, 274 pages : illustrations ; 24 cm
Part I. Number : 1. Prime numbers ; 2. Binary ; 3. 0.999999999999... ; 4. [Square root of]2 ; 5. i ; 6. [Pi] ; 7. e ; 8. [Infinity] ; 9. Fibonacci numbers ; 10. Factorial! ; 11. Benford's law ; 12. Algorithm ; -- Part II. Shape: 13. Triangles ; 14. Pythagoras and Fermat ; 15. Circles ; 16. The platonic solids ; 17. Fractals ; 18. Hyperbolic geometry -- Part III. Uncertainty: 19. Nontransitive dice ; 20. Medical probability ; 21. Chaos ; 22. Social choice and Arrow's theorem ; 23. Newcomb's paradox.
In bite-sized chapters that require only high school algebra, [Edward Scheinerman] invites recreational mathematicians and neophytes alike to try their hands at solving mathematical puzzles and provides an engaging and friendly tour of numbers, shapes, and uncertainty. The result is an unforgettable introduction to the fundamentals and pleasures of thinking mathematically. -- From book jacket.

True genius : the life and work of Richard Garwin, the most influential scientist you've never heard of

May 26, 2017
Shurkin, Joel N., 1938- author.
Amherst, New York : Prometheus Books, 2017.
308 pages, 16 unnumbered pages of plates : illustrations ; 24 cm
The tinkerers -- Fermi -- The super -- Garwin's design -- Garwin, Lederman, and the Marx Brothers -- IBM & LampLight -- Advising presidents-or not -- JASONS -- Vietnam and McNamara's Wall -- Super sonic transport -- Offense -- The great gap -- Treaty -- Star wars -- Gravity -- Health, pandemics -- Far out -- Rumpled -- Decline of influence.
"The first biography of Richard Garwin, a physicist whose work has had wide-ranging impacts on modern life from well-known technical innovations to progress in nuclear disarmament"-- Provided by publisher.

Finding Fibonacci : the quest to rediscover the forgotten mathematical genius who changed the world

May 26, 2017
Devlin, Keith J., author.
©2017
vi, 241 pages : illustrations, facsimiles, portraits ; 23 cm
Sputnik and calculus -- The flood plain -- The manuscript -- First steps -- The statue -- A walk along the Pisan riverbank -- A very boring book? -- Franci -- Publishing Fibonacci: from the cloister to Amazon.com -- Translation -- Reading Fibonacci -- Manuscript hunting, part I (failures) -- Manuscript hunting, part II (success at last) -- The missing link -- This will change the world -- Leonardo and the birth of modern finance -- Reflections in a medieval mirror -- Guide to the chapters of Liber abbaci.
In 2000, Keith Devlin set out to research the life and legacy of the medieval mathematician Leonardo of Pisa, popularly known as Fibonacci, whose book Liber abbaci has quite literally affected the lives of everyone alive today. Although he is most famous for the Fibonacci numbers--which, it so happens, he didn't invent--Fibonacci's greatest contribution was as an expositor of mathematical ideas at a level ordinary people could understand. In 1202, Liber abbaci--the "Book of Calculation"--introduced modern arithmetic to the Western world. Yet Fibonacci was long forgotten after his death, and it was not until the 1960s that his true achievements were finally recognized. Finding Fibonacci is Devlin's compelling firsthand account of his ten-year quest to tell Fibonacci's story. Devlin, a math expositor himself, kept a diary of the undertaking, which he draws on here to describe the project's highs and lows, its false starts and disappointments, the tragedies and unexpected turns, some hilarious episodes, and the occasional lucky breaks. You will also meet the unique individuals Devlin encountered along the way, people who, each for their own reasons, became fascinated by Fibonacci, from the Yale professor who traced modern finance back to Fibonacci to the Italian historian who made the crucial archival discovery that brought together all the threads of Fibonacci's astonishing story. Fibonacci helped to revive the West as the cradle of science, technology, and commerce, yet he vanished from the pages of history. This is Devlin's search to find him. -- Back cover.

Numbers and the making of us : counting and the course of human cultures

May 26, 2017
Everett, Caleb, author.
Cambridge, Massachusetts : Harvard University Press, 2017.
viii, 297 pages : illustrations ; 22 cm
Prologue: On the success of our species -- Part One. Numbers pervade the human experience: Numbers woven into our present -- Numbers carved into our past -- A numerical journey around the world today -- Beyond number words: other kinds of numeric language -- Part Two. Worlds without numbers: Anumeric people today -- Quantities in the minds of young children -- Quantities in the minds of animals -- Part Three. Numbers and the shaping of our lives: Inventing numbers and arithmetic -- Numbers and culture: subsistence and symbolism -- Transformative tools.
Numbers and the Making of Us examines the origins and effects of numbers--words and other symbols for quantities. It focuses on the influence that numbers have had on human thought. As a result of this influence, the book claims, numbers transformed the human narrative. This transformation is supported by data from many disciplines: archaeology, linguistics, psychology, and primatology. The book surveys the types of number systems that have been innovated independently in languages around the world, most of which (like our own decimal system) owe themselves in one way or another to the shape of our hands. Furthermore, the book examines evidence from anumeric humans, such as those the author has conducted research with in Amazonia, as it advances the following claim: Numbers served as a pivotal cognitive invention, an underappreciated tool whose usage ultimately resulted in the societies most of us now live in. In short, the book suggests that verbal and written numbers served as a cognitive foundation of sorts, helping to establish the ground floor of all sorts of distinctly human behaviors. These include elaborate agriculture, writing, the telling of time, and many other aspects of the human experience that are all ultimately dependent on the simple invention of numbers.-- Provided by publisher

The physics of everyday things : the extraordinary science behind an ordinary day

May 22, 2017
Kakalios, James, 1958- author.
New York : Crown, 2017.
245 pages : illustrations ; 22 cm
You begin your day -- You drive into the city -- You go to the doctor -- You go to the airport -- You take a flight -- You give a business presentation -- You go to a hotel.
"Easy-to-follow and imaginative explanations about the extraordinary physics that invisibly guides our daily lives"--Provided by publisher.

The greatest story ever told--so far : why are we here?

March 29, 2017
Krauss, Lawrence M., 1954- author.
New York : Atria Books, 2017.
xii, 322 pages : illustrations ; 24 cm
Includes index.
Part 1: Genesis. From the armoire to the cave ; Seeing in the dark ; Through a glass, lightly ; There, and back again ; A stitch in time ; The shadows of reality ; A universe stranger than fiction ; A wrinkle in time ; Decay and rubble ; From here to eternity : shedding light on the sun -- Part 2: Exodus. Desperate times and desperate measure ; March of the titans ; Endless forms most beautiful : symmetry strikes back ; Cold, stark reality : breaking bad or beautiful? ; Living inside a superconductor ; The bearable heaviness of being : symmetry broken, physics fixed -- Part 3: Revelation. The wrong place at the right time ; The fog lifts ; Free at last ; Spanking the vacuum ; Gothic cathedrals of the twenty-first century ; More questions than answers ; From a beer party to the end of time ; Cosmic humility.
An award-winning theoretical physicist traces the dramatic discovery of the counterintuitive world of reality, explaining how readers can shift their perspectives to gain greater understandings of our individual roles in the universe. --Publisher.

Visible learning for mathematics, grades K-12 : what works best to optimize student learning

March 20, 2017
Hattie, John, author.
Thousand Oaks, California : Corwin Mathematics, [2017]
xxvii, 269 pages : illustrations ; 24 cm
Make learning visible in mathematics. Forgetting the past -- What makes for good instruction? -- The evidence base -- Noticing what does and does not work -- Direct and dialogic approaches to teaching and learning -- The balance of surface, deep, and transfer learning -- Surface, deep, and transfer learning working in concert -- Making learning visible starts with teacher clarity. Learning intentions for mathematics -- Success criteria for mathematics -- Preassessments -- Mathematical tasks and talk that guide learning. Making learning visible through appropriate mathematical tasks -- Making learning visible through mathematical talk -- Surface mathematics learning made visible. The nature of surface learning -- Selecting mathematical tasks that promote surface learning -- Mathematical talk that guides surface learning -- Mathematical talk and metacognition -- Strategic use of vocabulary instruction -- Strategic use of manipulatives for surface learning -- Strategic use of spaced practice with feedback -- Strategic use of mnemonics -- Deep mathematics learning made visible. The nature of deep learning -- Selecting mathematical tasks that promote deep learning -- Mathematical talk that guides deep learning -- Mathematical thinking in whole class and small group discourse -- Small group collaboration and discussion strategies -- Whole class collaboration and discourse strategies -- Using multiple representations to promote deep learning -- Strategic use of manipulatives for deep learning -- Making mathematics learning visible through transfer learning. The nature of transfer learning -- The paths for transfer: low-road hugging and high-road bridging -- Selecting mathematical tasks that promote transfer learning -- Conditions necessary for transfer learning -- Metacognition promotes transfer learning -- Mathematical talk that promotes transfer learning -- Helping students connect mathematical understandings -- Helping students transform mathematical understandings -- Assessment, feedback, and meeting the needs of all learners. Assessing learning and providing feedback -- Meeting individual needs through differentiation -- Learning from what doesn't work -- Visible mathematics teaching and visible mathematics learning -- Appendices. A. Effect sizes -- B. Standards for mathematical practice -- C. A selection of international mathematical practice or process standards -- D. Eight effective mathematics teaching practices -- E. Websites to help make mathematics learning visible.
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. -- Provided by publisher.

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