My math classes focus on teaching math for understanding, not rote memorization of processes. For sure knowing basic facts and how to compute is important, but understanding strategies and why procedures work is just as valuable.
This quote highlights one problem teachers using this approach run into.
"Since I’ve begun to focus on the development of conceptual understanding, I’ve run into some challenges. Students and parents alike want me to teach short cuts and algorithms. Yet, when I hear students talking and thinking mathematically, I’m certain that this struggle will prepare them to be risk-takers, not only in my classroom but in their daily lives, too."
It's a post worth reading. You can also see her in action teaching students about inequalities, with a focus on concepts FIRST and notation last.
Click on the video title to be taken to Teaching Channel for a transcript of the lesson, copy of the lesson plan, and more.
Based on fossil evidence, how can scientists know what color dinosaurs were? In this TEDEd lesson, Len Bloch shows how making sense of the evidence requires careful examination of the fossil and a good understanding of the physics of light and color.
In this terrific video you'll learn a bit about the microraptor, a four-winged carnivorous dinosaur with iridescent black feathers, as well as the work that scientists do in determining how dinosaurs actually looked.
September 25th is Math Storytelling Day. Today (every day!) is the perfect opportunity to encourage and nurture the love of mathematics through reading about math. There are many terrific books that include mathematical content or challenging puzzles to solve. Here are some titles that will encourage children to stretch their mathematical muscles in a different way.
The Phantom Tollbooth by Norton Juster - Take a journey with Milo, a young boy who drives through a magic tollbooth into the Lands Beyond and embarks on a quest to rescue the maidens Rhyme and Reason from exile and reconcile the estranged kingdoms of Dictionopolis and Digitopolis. This is a great book for kids enamored of words and/or numbers.
Grandfather Tang's Story: A Tale Told With Tangrams by Ann Tompert and The Warlord's Puzzle by Virginia Pilegard are both stories that revolve around an ancient Chinese puzzle made from a large square cut into seven pieces. The seven shapes include a small square, two small triangles, a medium-sized triangle, two large triangles and a parallelogram. Kids can read the stories and follow along with their own set of tangrams!
The Number Devil: A Mathematical Adventure by Hans Magnus Enzensberger - With full color illustrations, this book tells the story of a twelve year old boy and math hater named Robert, who meets the Number Devil in his dreams. Over the course of twelve nights, the Number Devil illustrates different mathematical ideas using things like coconuts and furry calculators. Along the way he also takes Robert to Number Paradise where he meets different mathematicians.
Chasing Vermeer by Blue Balliett - Petra and Calder are preoccupied with Vermeer. When a Vermeer painting is stolen in transit from the National Gallery in Washington D.C. to the Chicago Institute of Art, they become intent on finding the painting and solving the mystery. Clues and mysteries abound.
Calder carries a set of pentominoes in his pocket at all times, so be sure to print your own set to use while reading this one!
Learn more about the book, the author, and the other books in the series at the Scholastic site.
Brown Paper School Math Books by Marilyn Burns - Don't let the publication dates fool you into thinking these are out of date (one was first published in 1975!). These are great books for helping kids see that math is fun and for everyone.
Books by Greg Tang - Greg Tang has written a series of books that encourage children to look for patterns in math and find more "economical" ways of solving problems.
The Man Who Counted: A Collection of Mathematical Adventures by Malba Tahan - Orginally published in 1949 as O Homem que Calculava, this book of mathematical puzzles was written by Júlio César de Mello e Souza and published under the pen name Malba Tahan. The book is an enjoyable series of "Arabian nights"-style tales, with each story built around a classic mathematical puzzle. In each tale, Beremiz Samir uses his mathematical powers to "settle disputes, give wise advice, overcome dangerous enemies, and win for himself fame and fortune."
The Puzzling World of Winston Breen by Eric Berlin - Winston sees puzzles everywhere. Imagine his dismay when he gives his sister a box for her birthday, only to learn that it has a secret compartment containing four wood sticks with puzzle clues. Readers will solve puzzles right along with Winston and his sister Katie as they try to solve the mystery. The sequel to this book, The Potato Chip Puzzles, is also highly entertaining.
Books by Theoni Pappas - Written in the same vein as the Brown Paper School Books, Pappas has written many books about math, my favorites of which are those where a cat explores the math in and around his house.
The Origami Master by Nathaniel Lachenmeyer, Lissy's Friends by Grace Lin (picture books), and Fold Me a Poem by Kristine O'Connell George (poetry) are all books about origami. Paper folding is a great visual and spatial puzzler for kids and adults. It's also fun!
If you have trouble reading origami directions in print, try following along with this video on Origami for Children.
Do the Math: Secrets, Lies, and Algebra by Wendy Lichtman - Tess is an eighth grade girl experiencing typical middle school problems--friends breaking promises, peers cheating on tests, the boy that may-or-may not be interested--as well negotiating some drama at home. Tess examines everything logically and views her world through the lens of mathematics.
"The way Sammy spoke about her mother made me think of what Venn diagrams look like when the two sets have nothing in common--like, for example, the set of odd numbers and the set of even numbers. Their intersection is called an empty set, because there's nothing in it. There's not one number that can be both odd and even. I didn't like thinking of Sammy and her mother like that--like an empty set." (p.49)
While the book isn't necessarily about math, Tess has many interesting mathematical insights and how they relate to the world we live in.
That's it for now. Do you have a favorite book that offers something mathematical to puzzle over? If so, please share. I would love to add your ideas to this list.
Last week on Science Friday there was a segment entitled The Science of Story Time. While it begins with a discussion of a study that shows reading with kids has positive effects ranging from increased vocabulary to greater success reading independently, it ends with experts and callers sharing favorite books for science-curious kids.
This is an old (2010) video, but I keep returning to it because I love the notion of a math salon.
A while back I wrote a post describing the components of elementary school homework I believed were important. Here's a description of one of those components.
Puzzle - When was the last time you sat down to solve a puzzle and did it for fun? I do this all the time. Sudoku, crossword puzzles, jigsaw puzzles, logic problems, tangrams ... I could go on. Puzzles are good for the brain. They develop critical thinking and problem solving skills. They teach kids to persevere, guess and check, collaborate with others, and try a whole host of new strategies. Can you think of a better training ground for mathematical thinking than puzzling? Now imagine if your teacher encouraged you to do this for homework.
This is exactly the kind of think happening in this math salon--kids exploring ideas in meaningful ways that just happen to touch on aspects of mathematical thinking. Just imagine what you could do with this idea in a classroom.
Last week I was interested in a story I heard on NPR. It began with the question "How many trees are there on the planet?" I started to think about how one would make such an estimation. My guess was 100 billion. Was I close? No. The actual answer is closer to 3 trillion. That's TRILLION, or 3 x 1,000,000,000,000.
And while this sounds like a huge amount (no, we don't have enough), the researchers found that the Earth has lost nearly half its trees since the start of human civilization. We also know that we are losing 10 Billion trees every year. All of this is pretty disturbing.
These numbers teach us a lot about habitat loss, how much carbon dioxide is being absorbed from the atmosphere, how water is recycled in an ecosystem, and how we can preserve and replenish our forests. Take a minute to learn more in this video.
In a letter to Robert Hooke in 1676, Isaac Newton wrote "If I have seen further it is by standing on ye sholders of Giants." Newton, just like the scientists of today, relied on the work of scientists and mathematicians who came before him.
Below you will find a list of books on scientists before and including Newton. I've also thrown in a couple of important mathematicians. Titles are roughly arranged in chronological order.
The Life and Times of Aristotle (2006), written by Jim Whiting - This biography from the Biography from Ancient Civilizations series provides a compelling look at Aristotle and his influence across history in a wide range of subjects. Though Aristotle was a philosopher, he was for many centuries considered the world's greatest scientist. Whiting explores Aristotle's contributions to science, as well as history and politics. Back matter includes a chronology, selected works, timeline in history, chapter notes, glossary, and further reading ideas.
The Librarian Who Measured the Earth (1994), written by Kathryn Lansky and illustrated by Kevin Hawkes - This biography of the Greek philosopher and scientist Eratosthenes, who compiled the first geography book and accurately measured the globe's circumference, tells the story of his life from his birth over two thousand years ago in northern Africa (modern Libya) to his work as the chief librarian at the great library of Alexandria in ancient Egypt.
Of Numbers and Stars: The Story of Hypatia (2006), written by D. Anne Love and illustrated by Pamela Paparone - The daughter of Theon, a mathematician, philosopher, and the last director of the Library at Alexandria, Hypatia was educated in the ways of many young men of her time and was one of the first women to study math, science, and philosophy. This book provides a nice overview of the time and place in which Hypatia lived. The artwork evokes both Egyptian and Greek styles and nicely incorporates images that reflect the subjects Hypatia studied. This is a carefully crafted picture book biography on a woman that little is known of. Despite this, her story is one that will inspire. Included are an author's note and bibliography, as well as some additional notes about mathematics.
Blockhead: The Life of Fibonacci (2009), written by Joseph D'Agnese and illustrated by John O'Brien - Medieval Italian mathematician Leonardo Fibonacci is introduced in this first person biography. In traveling with this father, Fibonacci learned geometry in Greece, fractions from the Egyptians, and Hindu-Arabic numerals in India. Largely responsible for converting Europe from Roman numerals to Hindu-Arabic numerals, he also realized that many things in nature followed a certain pattern, today known as the Fibonacci sequence.
Leonardo: Beautiful Dreamer (2012), written and illustrated by Robert Byrd - In this gorgeously illustrated picture book biography, Byrd provides a wealth of information about da Vinci's life and work. In addition to the traditional narrative, da Vinci's own words, anecdotes, and journal excerpts are found in sidebars and small panel illustrations. Byrd clearly and concisely explains da Vinci's theories in a way all readers can understand.
Leonardo da Vinci: Giants of Science (2008), written by Kathleen Krull and illustrated by Boris Kulikov - An extensive biography for older students (middle grades and up), this engaging work in the Giants of Science series focuses on the life of da Vinci while exploring his study the natural world, including aerodynamics, anatomy, astronomy, botany, geology, paleontology, and zoology. Special attention is given to da Vinci's notebooks and their meaning.
Leonardo da Vinci for Kids: His Life and Ideas: 21 Activities (1998), written by Janis Herbert - This biography of da Vinci is interspersed with activities readers can try on their own, including observing nature, painting birds, growing an herb garden, making minestrone soup, building a kite, and more. Includes extensive reproductions of da Vinci's sketches and paintings. Includes a list of related Web sites.
Neo Leo: The Ageless Ideas of Leonardo da Vinci (2009), written and illustrated by Gene Barretta - This biography for younger students focuses on the ideas and inventions found in the more than 20,000 pages of da Vinci's notes. Readers learn how many inventions that came centuries after da Vinci's time were actually imagined and described in his notes.
Galileo For Kids: His Life and Ideas: 21 Activities (2005), written by Richard Panchyk - This biography of Galileo is interspersed with activities readers can try on their own, including letter writing, observing the moon, playing with gravity and motion, making a pendulum, painting with light and shadow, and more. Back matter includes glossaries of key terms, people, and places in Italy, helpful web sites, and a list of planetariums and space museums.
Galileo's Telescope (2009), written by Gerry Bailey and Karen Foster and illustrated by Leighton Noyes - Every Saturday morning, Digby Platt and his sister Hannah visit Knicknack Market to check out the interesting and unique “antiques” for sale. In finding a telescope, the children learn about the life of mathematician, physicist and astronomer Galileo Galilei. Back matter includes a glossary.
I, Galileo (2012), written and illustrated by Bonnie Christensen - This first person biography opens with Galileo imprisoned and remembering his life from childhood onward, highlighting his education and scientific discoveries. In the Afterword, Christensen explains that it took nearly 400 years for the Catholic Church to admit they were wrong to condemn Galileo. Back matter includes a glossary, chronology, and descriptions of his experiments, inventions, improvements, and astronomic discoveries.
Starry Messenger: Galileo Galilei (2000), written and illustrated by Peter Sis - In this Caldecott honor book, gorgeous illustrations take center stage in telling the story of Galileo. Sis creates for readers images of the things Galileo saw in his observations of space, including sunspots, planets revolving around Jupiter, valleys and chasms on the moon, and more. Though not a detailed treatment of his life, the text is enhanced by notes and quotes from Galileo's own writings, scrawled throughout the pages.
Summer Birds: The Butterflies of Maria Merian (2010), written by Margarita Engle and illustrated by Julie Paschkis. Maria Merian was an artist and scientist who studied plants and animals in their natural habitat and then captured them in her art. This book is based on the true story of how Merian secretly observed the life cycle of summer birds (a medieval name for butterflies) and documented it in her paintings. Focusing on her young life, this book shows readers how curiosity at a young age can lead to a lifelong pursuit.
Isaac Newton: Giants of Science (2008), written by Kathleen Krull and illustrated by Boris Kulikov - An extensive biography for older students (middle grades and up), this engaging work in the Giants of Science series focuses on the life of Newton, a boy who was incredibly curious. Though he lived a solitary life, he attended Cambridge, worked for an apothecary, served in Parliament, and so much more. Despite his successes in the fields of math and science, Newton was also "secretive, vindictive, withdrawn, obsessive, and, oh, yes, brilliant."
Isaac Newton and Physics for Kids: His Life and Ideas with 21 Activities (2009), written by Kerrie Logan Hollihan - This biography of Newton is interspersed with activities readers can try on their own, including making a waste book, building a water wheel, making ink, creating a 17th century plague mask, tracking the phases of the moon, testing Newton’s Three Laws of Motion, and more. Back matter includes a list of useful books and web sites.
World History Biographies: Isaac Newton: The Scientist Who Changed Everything (2013), written by Philip Steele - This book in the National Geographic World History Biographies series profiles Newton as more than just a physicist, but also as an acclaimed mathematician, astronomer, alchemist, philosopher, and inventor as well.
Online Resources
The Timeline Index of Scientists provides a quick chronology and overview of important figures in the history of science. (Note that mathematicians are integrated into this list.)
The video Carl Sagan on Eratosthenes discusses the contributions to science by Eratosthenes. This clip comes from Carl Sagan's Cosmos episode 1, The Shores of the Cosmic Ocean.
Learn more about Eratosthenes' contributions to mathematics by learning about the Sieve of Eratosthenes.
Universal Leonardo is a program aimed at deepening our understanding of Leonardo da Vinci through a series of European exhibitions (2006), scientific research and web-based resources.
The NOVA production Newton's Dark Side explores the strange and complex mind of Isaac Newton and explores how he was searching out hidden meanings in the Bible and pursuing the covert art of alchemy.
That's it for this list. Coming up next is a list of biographies for scientists from the 18th and 19th centuries.
