I discovered this gem in my blog reading today. One of the secrets of great teaching:
Audrey seemed, for once, at a loss for words. She was thinking about the question.
I try to stay focused on being silent after I ask young children questions, even semi-serious accidental ones. Unlike most adults, they actually take time to think about their answers and that often means waiting for a response, at least if you want an honest answer.
If you’re only looking for the “right” answer, it’s fairly easy to gently badger a child into it, but I’m not interested in doing that.
This was a fun activity from Moebius Noodles for our PK-1st grade Homeschool Math in the Park group. The children take turns making a maze and setting a dinosaur inside. Then the other dinosaurs (parents or siblings) try to guess whether their friend is on the land or in the water.
(1) First, draw a big circle on the white board. This is your lake.
(2) With a finger or a bit of cloth, erase a small section of the circle to create the opening for your maze.
(3) Starting at one edge of the opening, draw a random squiggle inside the circle. Make your squiggle end at the other edge of the opening.
(4) Set your dinosaur anywhere inside the maze.
(1) Now it’s your turn to guess. Is the dinosaur standing on the land? Is it swimming in the water?
(2) How will you figure out if you guessed right?
(3) Check by jumping across the lines of the maze. Each jump takes you across a boundary: Splash! (Into the water.) Thump! (Back on the land.) Splash! Thump! … Until you reach the dinosaur inside.
(4) Or go to the maze entrance and walk your dinosaur along the path. Can you find your way?
It’s a short book with plenty of great stories, advice, and conversation-starters. While Danielson writes directly to parents, the book will also interest grandparents, aunts & uncles, teachers, and anyone else who wants to help children notice and think about math in daily life.
You don’t need special skills to do this. If you can read with your kids, then you can talk math with them. You can support and encourage their developing mathematical minds.
You don’t need to love math. You don’t need to have been particularly successful in school mathematics. You just need to notice when your children are being curious about math, and you need some ideas for turning that curiosity into a conversation.
In nearly all circumstances, our conversations grow organically out of our everyday activity. We have not scheduled “talking math time” in our household. Instead, we talk about these things when it seems natural to do so, when the things we are doing (reading books, making lunch, riding in the car, etc) bump up against important mathematical ideas.
The dialogues in this book are intended to open your eyes to these opportunities in your own family’s life.
During off-times, at a long stoplight or in grocery store line, when the kids are restless and ready to argue for the sake of argument, I invite them to play the numbers game.
“Can you tell me how to get to twelve?”
My five year old begins, “You could take two fives and add a two.”
“Take sixty and divide it into five parts,” my nearly-seven year old says.
“You could do two tens and then take away a five and a three,” my younger son adds.
Eventually we run out of options and they begin naming numbers. It’s a simple game that builds up computational fluency, flexible thinking and number sense. I never say, “Can you tell me the transitive properties of numbers?” However, they are understanding that they can play with numbers.
photo by Mike Baird via flickr
I didn’t learn the rules of baseball by filling out a packet on baseball facts. Nobody held out a flash card where, in isolation, I recited someone else’s definition of the Infield Fly Rule. I didn’t memorize the rules of balls, strikes, and how to get someone out through a catechism of recitation.
As for mathematics itself, it’s one of the most adventurous endeavors a young child can experience. Mathematics is exotic, even bizarre. It is surprising and unpredictable. And it can be more exciting, scary and dangerous than sailing the high seas!
But most parents and educators don’t present math this way. They just want the children to develop their mathematical skills rather than going for something more nebulous, like the mathematical state of mind.
Children marvel as snowflakes magically become fractals, inviting explorations of infinity, symmetry and recursion. Cookies offer gameplay in combinatorics and calculus. Paint chips come in beautiful gradients, and floor tiles form tessellations. Bedtime routines turn into children’s first algorithms. Cooking, then mashing potatoes (and not the other way around!) humorously introduces commutative property. Noticing and exploring math becomes a lot more interesting, even addictive.
Unlike simplistic math that quickly becomes boring, these deep experiences remain fresh, because they grow together with children’s and parents’ understanding of mathematics.
Check out my newest home decor item, a hundred chart. The amount of work I put into it, I consider getting it framed to be proudly displayed in the living room. The thing is monumental in several ways:
1. It is monumentally different from my usual approach to choosing math aids. My rule is if it takes me more than 5 minutes to prepare a math manipulative, I skip it and find another way.
2. It is monumentally time-consuming to create from scratch all by yourself.
It began with a humble list of seven things in the first (now out of print) edition of my book about teaching home school math. Over the years I added new ideas, and online friends contributed, too, so the list grew to become one of the most popular posts on my blog:
Most young children are not developmentally ready to master abstract, pencil-and-paper rules for manipulating numbers. But they are eager to learn about and explore the world of ideas. Numbers, patterns, and shapes are part of life all around us. As parent-teachers, we have many ways to feed our children’s voracious mental appetites without resorting to workbooks.
To delay formal arithmetic does not mean that we avoid mathematical topics — only that we delay math fact drill and the memorization of procedures. Notice the wide variety of mathematics Benezet’s children explored through books and through their own life experiences:
There is no formal instruction in arithmetic. In connection with the use of readers, and as the need for it arises, the children are taught to recognize and read numbers up to 100. This instruction is not concentrated into any particular period or time but comes in incidentally in connection with assignments of the reading lesson or with reference to certain pages of the text.
Meanwhile, the children are given a basic idea of comparison and estimate thru [sic] the understanding of such contrasting words as: more, less; many. few; higher, lower; taller, shorter; earlier, later; narrower, wider; smaller, larger; etc.
As soon as it is practicable the children are taught to keep count of the date upon the calendar. Holidays and birthdays, both of members of the class and their friends and relatives, are noted.
It’s counter-intuitive, but true: Our children will do better in math if we delay teaching them formal arithmetic skills. In the early years, we need to focus on conversation and reasoning — talking to them about numbers, bugs, patterns, cooking, shapes, dinosaurs, logic, science, gardening, knights, princesses, and whatever else they are interested in.
In the fall of 1929 I made up my mind to try the experiment of abandoning all formal instruction in arithmetic below the seventh grade and concentrating on teaching the children to read, to reason, and to recite – my new Three R’s. And by reciting I did not mean giving back, verbatim, the words of the teacher or of the textbook. I meant speaking the English language.
Notice that subtraction is not defined independently of addition. It must be taught along with addition, as an inverse (or mirror-image) operation. The basic question of subtraction is, “What would I have to add to this number, to get that number?”
Inverse operations are a very fundamental idea in mathematics. The inverse of any math operation is whatever will get you back to where you started. In order to fully understand a math operation, you must understand its inverse.
The basic idea of addition is that we are combining similar things. Once again, we meet the counting models from lesson 1.1: sets, measurement, and the numberline. As homeschooling parents, we need to keep our eyes open for a chance to use all of these models — to point them out in the “real world” or to weave them into oral story problems — so our children gain a well-rounded understanding of math.
Addition arises in the set model when we combine two sets, and in the measurement model when we combine objects and measure their total length, weight, etc.
One can also model addition as “steps on the number line”. In this number line model the two summands play different roles: the first specifies our starting point and the second specifies how many steps to take.
My favorite playful math lessons rely on adult/child conversation — a proven method for increasing a child’s reasoning skills. What better way could there be to do math than snuggled up on a couch with your little one, or side by side at the sink while your middle-school student helps you wash the dishes, or passing the time on a car ride into town?
As soon as your little ones can count past five, start giving them simple, oral story problems to solve: “If you have a cookie and I give you two more cookies, how many cookies will you have then?”
The fastest way to a child’s mind is through the taste buds. Children can easily visualize their favorite foods, so we use mainly edible stories at first. Then we expand our range, adding stories about other familiar things: toys, pets, trains.
As anyone who has taught or raised young children knows, mathematical education for little kids is a real mystery. What are they capable of? What should they learn ﬁrst? How hard should they work? Should they even “work” at all? Should we push them, or just let them be?
There are no correct answers to these questions, and Zvonkin deals with them in classic math-circle style: He doesn’t ask and then answer a question, but shows us a problem — be it mathematical or pedagogical — and describes to us what happened. His book is a narrative about what he did, what he tried, what worked, what failed, but most important, what the kids experienced.
This book is not a guidebook. It does not purport to show you how to create precocious high achievers. It is just one person’s story about things he tried with a half-dozen young children. On the other hand, if you are interested in running a math circle, or homeschooling children, you will ﬁnd this book to be an invaluable, inspiring resource. It’s not a “how to” manual as much as a “this happened” journal. … Just about every page contains a really clever teaching idea, a cool math problem, and an inspiring and funny story.
Krista at the LivingMathForum wrote, “We’ve used these for several years. My son even made a bunch of them a few years ago and sold them at a homeschool resource fair. We always have one in most of our board games to help younger children add up their die rolls. I find them relaxing for some reason, just moving the beads along the cord, and my son will sometimes sit and listen to me reading, etc., and just manipulate the beads.”
Would you love to introduce your preschoolers to the mind-stretching wonder of math? Check out Moebius Noodles, an open community of mathematical resources for parents and educators of young children.
Moebius Noodles activities and games use stuff you already have around the house to explore symmetry, fractals, functions, transformations, topology, and more, in a way that is accessible to babies and toddlers (and their parents!) and easily adapted to include older siblings.
Even a small donation will help this amazing project get off the ground — and Maria is offering a variety of “secret rewards” (like free books, or a secret-message game designed especially for your child) to those who chip in now.
Would you like to do math with your babies and toddlers — real math, not just counting and simple shapes — but don’t know how? For the last few years, Maria Droujkova has been teaching parents to do advanced, fun math with young children. I had the chance to lurk on a discussion group this summer and thoroughly enjoyed it. Now she is taking the next step — and you are invited to join in the fun!
Cool math concepts.
Symmetry, fractals, coordinate planes, functions, transformations, topology, and more.
Math that grows with your child.
Games and activities are adaptable to a wide range of ages, from babies to teens.
Math your child owns.
Plenty of variations and suggestions to help kids bring math into their own worlds.
Active and open support group.
A community of parents who share stories and photos, discuss examples, modify games, and ask and answer questions.
Math therapy for parents.
For those who feared and disliked math in school, this project offers a second chance as you experience the math-rich world around you through playing with your child.
Maria is offering a variety of “Secret Rewards” (including free copies of the Moebius Noodles book) to those who chip in toward the project. Check out her Moebius Noodles blog post for more information.
Homeschool Freebie of the Day is offering a pdf math guidebook for grades 1-2. This is not a workbook, but offers suggestions for how to weave oral math discussions with young children into your daily life.
When you think of math do you think of a light-hearted fairy tale?
No? Then come and meet some of the delightful characters who live in Arithmetic Village.
Polly Plus collects jewels slowly and methodically, Linus Minus is carefree and loses his. Tina Times and King David Divide… well you’ll see.
The first book offers the overview of the math concepts. These are then demonstrated through the lives of each character. The books are designed to be supported by a manipulative kit [homemade: see video below] with 100 jewels, 10 golden bags, and a treasure chest…
Update: World Maths Day for 2011 is over. Congratulations to the 5.3 million registered students from 218 countries, who answered a grand total of 428,598,214 problems correctly. Wow!
It’s time to register for World Maths Day, which will take place on March 1, 2011. Last year, more than two million students from 235 countries combined to correctly answer 479,732,631 World Maths Day questions.
Would you like to help break the record this year? Don’t delay:
Registrations close February 28!
About World Maths Day
Play with students from schools all around the world. Individuals and homeschoolers are welcome, too.
The competition is designed for ages 4-18 and all ability levels. Teachers, parents and media can also register and play.
It’s simple to register and participate. Start practicing as soon as you register.
Well, that was longer ago than I care to admit. But of course, it takes quite a bit of daily use before one can be absolutely sure of one’s opinion about a homeschool program — or at least, it does for me. Too many times a homeschool resource will look great in the catalog, and we’ll start it with high hopes only to bog down in the day-to-day grind and abandon it after a few weeks or months. So I wanted to give Math Mammoth a thorough workout before I wrote this review.
My aim is to help parents and teachers teach math so our children and students can really understand what is going on. I’ve strived to explain the concepts so that both the teacher and the student can “get it” by reading the explanations in the books.
The book covers the basic arithmetic from addition and subtraction to simple multiplication and division, with quite a bit of measuring and fraction work, too. The concepts are taught through stories, and then the teacher is to give the student plenty of hands-on experience with measurements. Each lesson concludes with a page of practice problems.
If you miss the daily freebie, you may still be able to get the book through Google (but I’m not sure this works if you are outside the United States):
The States by the Numbers series features 4th-6th grade “math adventures” about your favorite states (the freebie is Wisconsin). The workbook covers place values, rounding, estimation, fractions and percentages using data from the Census Bureau’s 2008 Statistical Abstract of the United States. The workbook includes basic instruction and 80 practice problems, plus several “What’s the big idea?” journal pages which ask learners to reflect on the things they’ve learned. You may want to let your students use a calculator, since real data often means large, unwieldy numbers.
The question came from a homeschool forum, though I’ve reworded it to avoid plagiarism:
My student is just starting first grade, but I’ve been looking ahead and wondering: How will we do big addition problems without using pencil and paper? I think it must have something to do with number bonds. For instance, how would you solve a problem like 27 + 35 mentally?
The purpose of number bonds is that students will be comfortable taking numbers apart and putting them back together in their heads. As they learn to work with numbers this way, students grow in understanding — some call it “number sense” — and develop a confidence about math that I often find lacking in children who simply follow the steps of an algorithm.
["Algorithm" means a set of instructions for doing something, like a recipe. In this case, it means the standard, pencil and paper method for adding numbers: Write one number above the other, then start by adding the ones column and work towards the higher place values, carrying or "renaming" as needed.]
For the calculation you mention, I can think of three ways to take the numbers apart and put them back together. You can choose whichever method you like, or perhaps you might come up with another one yourself…
In addition to all the funny Google searches, I get plenty of normal inquiries about math topics. People come here looking for help with fractions, wordproblems, and mathclubactivities — no surprise, those — but I would never have predicted the popularity of the search topic “writing in math class.”
Last year, I compiled a variety of math journal resources, but I’ve found many more since then, especially for older (high school and college) students. So if you’re looking for new ways to get your math students writing…
Are you looking for creative ways to help your children study math? Even without a workbook or teacher’s manual, your kids can learn a lot about numbers. Just spend an afternoon playing around with a hundred chart (also called a hundred board or hundred grid).
[This download is available for one day only. If you missed it, see the end of this post for other ways to get the book.]
From the Preface
“These are the stories that were really told in the crisp autumn evenings, the Story Teller sitting by the fire that burned in the great fireplace in the cottage by the sea. These are the stories as he told them to the Tease and the rest of the circle of friends known as the Crowd. Sitting by the fire and listening to the stories, in the lights and shadows of the dancing flames they could see the forms of Ching and Lugal and all the rest with their curious dress of long ago…”