HandsOn Math for Young Children: Exploring Numbers and Operations With Whole Numbers by Stephanie Marshall Ward
This article explores teaching whole number concepts to young kids (preschool  about grade 3). Working with children in the middle grades (about grades 46) will be discussed in a separate article.
This is the first in a series of articles on helping children understand math. Other topics will include Fractions and Decimals, Laying the Groundwork for Algebraic Thinking (patterns, equivalence, variables, and graphing), Measurements, Geometry, Logic, and Probability and Statistics.
Young children tend to learn best through "living books," handson activities, and "real life learning." Though most of us agree with this statement, few of us were actually taught mathematics this way. This is an ongoing learning process for many of us. I hope the simple ideas offered in this article will help you have more fun on this journey, and generate new ideas of your own.
Learning About Numbers
Learning about numbers and operations usually begins with counting. At some point, young children make the crucial developmental leap from rote counting (reciting 1,2,3...) to actually counting objects. Most children enjoy counting toys, manipulatives, objects from the natural world, such as plants, stones, or pine cones, and other things. There are also many wonderful counting picture books. While these books are not a substitute for counting concrete objects, they are great experiences. Eric Carle’s The Very Hungry Caterpillar and Rooster’s off to See the World are reliable favorites. Counting games are also fun for young children. These can be simple board games (such a Candyland), playground games, or other activities. Some parents like to make a simple counting activity with a used egg carton. Write the numerals 112, or another representation of these numbers, on the twelve compartments of an egg carton. Then let the child place the specified number of objects (such as beans, buttons, or pennies) in each compartment. It might be a good challenge for an older sibling to prepare this activity, deciding how many buttons (or whatever object you choose) the child will need (1+2+3+4+5+6+7+8+9+10+11+12).
What Time is it Mr. Fox?
A game to practice counting:
The parent starts the game by having the kids line up side by side on one end of the play area. The parent is "Mr. Fox" and stands about half way from the children.
The students ask: "What time is it, Mr. Fox?"
Mr. Fox calls out a time (ex. 10:00 and the kids take 10 steps forward).
This continues until Mr. Fox says it is "Supper Time!" At that point the children try to run to the other end of the gym without Mr. Fox "tagging" them. When Mr. Fox tags a player, he becomes the next Mr. Fox.

A concept related to counting, which begins in toddlerhood, is the understanding of quantities: "more," "less," and "the same amount." Offer your child many opportunities to estimate whether there are more toys in one box than another or which bucket has more sand. When your child begins using a number line, show him how to use it to determine whether a number is more or less than another. This will help him learn to "visualize" these concepts, and gradually bridge the gap between a concrete and an abstract understanding of quantities.
Around age 5 or so, kids begin to "see" a group of objects as a whole. If you show him five blocks and ask him "How Many?" he can immediately identify it as a group of five, without counting them one by one. Math teachers sometimes all this "conserving for five." This is a developmental milestone which cannot be forced. However, playing with dominoes (the traditional ones with sets of dots) is a wonderful way to encourage this skill.
Children's songs and finger plays are a important way to introduce math concepts, including counting, to young children. Even babies and toddlers benefit from these experiences. Rhymes like "Ten Little Monkeys (jumping on the bed)" introduce the idea of counting backwards. Rooster’s off to See the World also offers an experience with both forward and backward counting. For more practice in backward counting: you might enjoy Ten Sly Piranhas: A Counting Story in Reverse: A Tale of Wickedness and Worse by William Wise, in which ten fish are devoured one by one. (My son enjoyed this book though my daughter, at the same age, would have found it disturbing).
Ten little monkeys
jumping on the bed
One fell off
And bumped his head
Mama called the Doctor
And the Doctor Said
No more monkeys
Jumping on the bed!
You and your child might enjoy 5 Little Monkeys Jumping on the Bed by Eileene Christlow.

Don't forget to help your child learn skip counting, by 5s, 10, and 2s. Of course, it is helpful to practice this with concrete objects. Skip counting lays the groundwork for multiplication and opens the door to exploring mathematical patterns.
Exploring Patterns
Another critical component of early math learning is developing an understanding of patterns. The concept of patterns is the common thread interweaving all branches of mathematics, and is a wonderful tool for helping a child develop critical thinking skills.
"You see a pattern when something happens again," I explained, simply, to my young son. Many traditional children's stories, songs, and fingerplays intuitively teach this concept. What do you hear again and again in these stories: "The Three Little Pigs," "The Three Bears," "The Gingerbread Man," and "The Little Red Hen?" Hearing and reading these tales is a wonderful introduction to patterns for young learners. You can expand this activity in many ways. I sometimes sketch or photocopy important parts of the story and ask my son to put them in order, encouraging him to retell the story's plot as he does. Many caregivers use flannel boards with characters and objects from the story. The child is encouraged to retell the story using the flannelboard pictures.
Patterns can be experienced in many other ways, with sounds, ideas, and concrete objects. What happens when you play simple musical sequences together? Try showing your child a simple musical pattern like this: three quick shakes of the morocca or bells, followed by one long, slow shake, three quick shakes, one long slow shake. Then let him develop a rhythm for you to follow. Try teaching each other clapping patterns, perhaps through a game of "Follow the Leader." (e.g. clap  pause  clap, clap, clap; clap  pause  clap, clap, clap) How many patterns can you make with toys or manipulatives? examples  red bear, blue bear/red bear, blue bear
 big car, little car, little car/big car, little car, little car
 big green dinosaur, little green dinosaur,big green dinosaur, little blue dinosaur/big green dinosaur, little green dinosaur, big green dinosaur, little blue dinosaur
Ask your child to tell you what comes next, then let him challenge you with a sequence to finish. Explore natural rhythms with your child, also. What is the normal pattern of your daily routine? Does it repeat each day? What about the seasons of the year or an animal's life cycle? These explorations can enrich your child's learning and development in many ways, and deepen his appreciation of nature.
A related skill is the ability to group things by certain attributes. For instance, some animals have four legs, and some have two. Some of these buttons are round and some are star shaped. Help your child sort groups of objects, challenging him to find out how many ways (by how many attributes) they can be sorted. These blocks can be sorted by color: some are blue and some are red. Or these blocks can be sorted by shape: some are square and some are rectangular. Or they can be sorted by size: some are large and some are small. You can create many fun sorting activities. For instance, you may want to create a "sorting box" with a used egg carton. I have seen pricier versions of this in catalogs. Very cool, but  as you may have guessed  I am a big fan of "free" things such as recycled egg cartons. Here is an example. Offer your child a set of buttons of different colors, sizes, and with different numbers of holes in each one. Let him sort the collection by color (a set of red, a set of brown, and a set of blue). Then let him sort each set by size (large and small). He now has six separate sets of buttons. Now let him sort each set by number of holes (two holes, four holes). He now has twelve sets, each in a separate compartment of the egg carton.
Practice identifying and discriminating things by attributes often. The popular game "I Spy" is a fun way to do this. The leader says "I spy something blue." Then he offers another hint "I spy something blue and square shaped." He keeps providing clues, one at a time, until a player guesses the object he has in mind.
Learning How Numbers and Operations Work
As a child gets older, and is becoming proficient at counting and understanding patterns, you have a wonderful opportunity to help him develop his “number sense,” a feeling for numbers and how they work. It is also a chance to begin developing the ability to “see” how math works, in his mind. These will be valuable assets later, when he delves into more complicated and abstract concepts. Here are several commonly used tools:
 Number Line:
This helps the child visualize numbers and counting, and easily make the transition from counting to addition, subtraction, multiplication, and division. I have seen the idea of a “living number line” offered in several places. Using sidewalk chalk, draw a number line on the sidewalk or driveway. Make it large enough that the child can comfortably step from one to two, and so on.
 Encourage him to count concrete objects using the number line. For example, he can step from one number to the next as he counts the cars parked on your street.
 Try a variation of the game “Mother May I.” Have each child stand at the zero point of your number line, or let each child draw his own number line and stand at zero. The leader says “take three steps forward” or “take five steps forward by twos.” The child asks “Mother May I?” The leader says yes, and each player may perform the action. If any player forgets to ask “Mother May I?” he forfeits his turn, or has to return to start (depending on your rules). When a player completes a turn, have him call out what number he has reached. You might also want to have him verbalize the operation he has performed: “zero plus three is three” or “five twos is ten.” The first player to reach the “finish line” (the end of the number line) wins. Can your children tackle a number line going up to 100?
Recipe for Homemade Sidewalk Chalk
Materials:
 1 cup plaster of Paris
 1 cup water
 Powdered tempera paint
 Mold for chalk (small paper cups, ice cube trays, tissue rolls, etc.)
 Mixing bowl
Directions:
 In a large bowl, mix the water and plaster of Paris together.
 Add the powdered tempera paint to the mixture.
 Once the paint has been mixed in well, set it aside for a few minutes.
 Pour the mixture into the mold and let it dry. This can take anywhere from several hours to a day (or maybe longer), depending on the size of the mold. Remember: The bigger the mold, the longer it will take to dry.
 Once the mold is dry, remove the chalk. If the chalk is still moist, let it air dry for another 24 hours.

 100 Number Chart: This is a chart divided equally into 100 squares, numbered either 099 or 1100. You can easily make or download a 100 number chart (For downloadable charts, see donnayoung.org. She has a terrific assortment of math tools and other forms!). My Kindergartner made his own 100 number chart by filling in the numbers 099 on a blank grid, and decorating it with racing car pictures and stickers. (you can provide cut out numbers to spare him the difficulty of writing all the numerals).
 Encourage him to count concrete objects using the chart.
 Ask him whether he notices certain patterns on the chart. (Examples: all the numbers in the tens column end in zero (10, 20, 30 …), all the numbers in the fives column end in 5, the first digit increases by one as you go vertically down a column).
 Have him color all the numbers ending in five, ending in seven, beginning in two, etc. Let him notice the horizontal and vertical patterns this makes.
Writing and Recognizing Numerals
When you feel your child is ready to begin writing and recognizing numerals, help him practice. Expose him to numerals through books, and through sometimes writing the numerals that you talk about during your math play.
Do not be concerned if your kindergartner or first grader still reverses numerals: writing them backwards or writing 32 for twenty three. In order to help him practice doing it correctly, you might want to write the numerals for him, and let him copy them. Reversals are a common developmental issue at this age. Physical activities which build bilateral coordination (coordinating both sides of the body) can help.
If you want your child to practice recognizing numerals or the written words which represent numbers, try making it a game. You might try a variation of the popular playground game "What Time is it Mister Fox" (above), in which "Mr. Fox" holds up a piece of paper with a numeral or number word printed on it (such as "10" or "ten"), instead of calling out the number. The children read the number on the paper, and call it out: "ten o'clock!"
Practice With Operations
When a child is becoming comfortable with adding numbers together, you might want to introduce him to all four operations: addition, subtraction, multiplication, and division. While this is not the traditional way mathematics is taught, it allows the young learner to see how all the operations relate to one another. It helps to use simple language to your child, such as  ”You have some, then add some more.”
 ”Subtraction is taking away."
 "When you multiply, it’s a speedy way of adding." (3 + 3 + 3 is three three times: 3 x 3) (Amanda Bean’s Amazing Dream by by Cindy Neuschwander is a good way to introduce this concept to young learners.
 When you divide something, you divide it between you: sharing. Try The Doorbell Rang by Pat Hutchins to illustrate this concept. Then help your kids bake a dozen cookies, and see how many ways you can divide them equally (2 people sharing them: 6 each, 3 people sharing them, 4 each, etc.).
Recipe for Chewy Peanut Butter Cookies
Ingredients:
 1/2 cup shortening
 1/2 cup softened margarine
 1 1/2 cups peanut butter
 1 cup sugar
 2/3 cup brown sugar  packed
 2 eggs  beaten
 2 1/2 cups flour
 1 teaspoon baking powder
 1 1/2 teaspoons baking soda
 1 teaspoon salt
Directions:
 In large bowl, cream shortening, margarine and peanut butter, add eggs.
 In another bowl mix dry ingredients. Add this gradually to moist mixture and mix well.
 Pinch and form golf ball size balls rolled in hands.
 Press floured fork crosswise on top to flatten.
 Bake at 375 degrees for 10 minutes just until golden, longer for crispier cookies. Makes 3 1/2 dozen.
Cooking is a mathematical gold mine! Show your children how you measure different capacities: 1 cup, 1/2 cip, etc. Let them help you count out 5 1/2 cups of flour, etc.
Challenge for Older Kids: How many is 3 1/2 dozen? Do we really need 42 cookies (Well  why not? Yum!) How could we change the recipe to make 21 cookies instead? What if we wanted 84 cookies? Then how would we change the recipe?

More Practice With Operations
Let the child experiment with many ways of adding, subtracting, multiplying, and dividing numbers, including manipulatives (cubes, rods, small animals, beans, pennies, or whatever else you have on hand), the number line, and the hundred number chart.
There are many games on the market which reinforce these skills in a dynamic and fun way. One can also devise games from things already around the house. Here are a few basic examples: We play a simple game to help my son become comfortable with the operations and prepare him for the concept of place value. It is sort of a variation of Marilyn Burns’ game “Race to 100.” We use a pair of dice, a third die with an operation symbol (such as a multiplication or division sign) on each side , a jar of pennies, and a jar of dimes. Each player rolls the three dice, and puts them together to form a math problem. If it is a division problem, we always divide the smaller number into the larger number. We use pennies to concretely solve the problem.  “Your problem is 4 divided by 3. We divide 4 pennies among three people … one for you, one for you, and one for you. There’s one left over. We have to throw out the remainder. So how many will you get? One.”
 “Your problem is 4 times 3. Four three times. Each of these stacks of pennies has four pennies. There are three stacks. How many do you get? 12.”
After solving the problem, each player keeps his pennies (the number of pennies that comprise the answer to his problem).
When he accumulates ten pennies, he may trade them for a dime. This does not change the amount of his winnings, but enables him to “travel light.” This introduces the concept of place value. It is not forced. He must come to an understanding of this concept on his own. It helps if you model it for him. The game ends when a player reaches a dollar; he wins. As an extra incentive, my kids get to keep their winnings. They like this very much, as they do not get an allowance.
Playing cards are a wonderful math tool. Try playing the traditional game of War. Then add a variation. Remove the face cards from the deck, then play. Each player lays down two cards at each turn instead of one. Then add the values (aces are 1) to see whose sum is the greatest. For an added challenge, if you feel your children are ready for a gentle introduction to the concept of negative numbers, let the red cards be negative (subtract this number) and the black cards be positive (add this number). If you have a red 9 and a black 10, for example, 109=1.
Description of the Card Game "War"
War for two players
In the basic game there are two players and you use a standard 52 card pack. Cards rank as usual from high to low: A K Q J T 9 8 7 6 5 4 3 2. Suits are ignored in this game.
Deal out all the cards, so that each player has 26. Players do not look at their cards, but keep them in a packet face down. The object of the game is to win all the cards.
Both players now turn their top card face up and put them on the table. Whoever turned the higher card takes both cards and adds them (face down) to the bottom of their packet. Then both players turn up their next card and so on.
If the turned up cards are equal there is a war. The tied cards stay on the table and both players play the next card of their pile face down and then another card faceup. Whoever has the higher of the new faceup cards wins the war and adds all six cards facedown to the bottom of their packet. If the new faceup cards are equal as well, the war continues: each player puts another card facedown and one faceup. The war goes on like this as long as the faceup cards continue to be equal. As soon as they are different the player of the higher card wins all the cards in the war.
The game continues until one player has all the cards and wins. This can take a long time.
Most descriptions of War are not clear about what happens if a player runs out of cards during a war. There are at least two possibilities:
 If you don't have enough cards to complete the war, you lose. If neither player has enough cards, the one who runs out first loses. If both run out simultaneously, it's a draw. Example: Players A and B both play sevens, so there is a war. Each player plays a card face down, but this is player B's last card. Player A wins, since player B does not have enough cards to fight the war.
 If you run out of cards during a war, your last card is turned face up and is used for all battles in that war. If this happens to both players in a war and their last cards are equal, the game is a draw. Example: Players A and B both play sevens, so there is a war. Player A plays a card face down, but player B has only one card, so it must be played face up. It is a queen. Player A plays a card face up and it is also a queen, so the war must continue. Player B's queen stays (B's last card) while player A plays a card face dowmn and one face up, which is a nine. Player B wins the war and takes all these seven cards (the five cards that A played and the two cards that B played) and the game continues normally.
War for three or four players
War can also be played by three or more players in much the same way. Deal out as many as possible of the cards so that everyone has an equal number (17 for 3 players, 13 for 4).
All players simultaneously turn over a card and the highest wins all the cards tuned up. If two or more players tie for highest there is a war  everyone plays their next card facedown and then turns up a third card. This continues until one of the faceup cards is higher than all the others, and then that player wins all the cards in a war.
Note that all players take part in a war, not only the ones who had the highest cards.
A player who runs out of cards drops out. The game goes on until only one player has cards, and that player wins.
"Borrowed" from http://www.pagat.com

Story Problems
Encourage your child to invent story problems, using toys and manipulatives. For example, I’ve used toy dinosaurs with my son.  “Two dinosaurs are at the water hole. Then two more come. One hears a noise, so he gets scared and runs away. How many are left? 2+21=3.”
 Young children sometimes have difficulty with the concept of zero. You might try drawing a circle to represent an empty set. “Five dinosaurs are at the watering hole, and you have nobody over here – in this circle. How many in all? 5+0=5.
Making the transition from sums and differences to completing equations like this 2 + ___ = 5 can be confusing for kids. Many young children will answer “7.” (This will be discussed further in the section in patterns/functions/algebra). Try some concrete story problems using this concept. “There are five dinosaurs in all, but all but two are hiding (actually hide the other 3 dinosaurs). How many are hiding?”
We sometimes enjoy drawing/writing math stories. My son and I take turns drawing the stories, and I write in the words. "There were two spiders, then three more came to visit. How many altogether? With five spiders, what is the total number of legs? Let's see." Instead of drawing, a child could use stickers or stampers with paint or stamp pads to represent the objects for his story. For additional fun, try making a book of your child's math stories.
Making Books with Children
Materials
 Copier paper or typing paper (I prefer copier paper because it is thicker)
 Ruler
 Pen
 sharp craft knife
 Adhesive shelf paper
 Cardboard
 Packaging tape
 Button thread or other thick thread
 Large needle
 Rubber Cement or glue
Directions:

For something different, invent story problems while making simple things from clay. “Here are three snakes (snakes and worms are fun and easy to make from clay). Then two more come. How many snakes are here now? 3+2=5.” “This kind of worm can actually be split in half, and each half will become a new worm. Here are two worms. They split in half – see? Watch! How many worms are here now? 2x2=4.”
Recipe for Baking Soda Clay
Materials:
 1 cup cornstarch
 2 cups baking soda
 1 1/4 cups water
 Food coloring if desired
Directions:
 Combine ingredients in a pan.
 Cook over medium heat, stirring constantly until a clay is formed (7  10 min.).
 Spoon clay onto a cookie sheet & knead.
 Cover with a damp towel until mixture is cool.
 Store unused clay in an airtight container.
 Finished clay products can be airdried completely or fired.

Help your child explore different ways of getting the same sum. For example, using 12 Hotwheels cars, see how many ways you can add them together to get 12 (12+0, 1+11, 2+10, 3+9, 4+8, 5+7, 6+6). I did this activity by making two parking lots for my son's Hotwheels using two pieces of black construction paper. We treated the two parking lots as two sets to add together.
Explore fact families. Expand on the last activity by showing your child how subtraction relates to addition. "There are 12 cars, but 5 drove off and went into the other parking lot. Now, 7 are left in this lot. But when you add together the cars in both parking lots, you still have 12."
Relate multiplication to division. When your child enjoys experimenting with multiplication, and understands that it is a process of repeated addition, show him how multiplication and division are opposite sides of the same coin. "We are going to divide the 12 hotwheels among 3 parking lots. Now, 4 are in this lot. But look at all 3 parking lots: when you put the 3 fours together, you still have 12."
Once your child has a strong understanding of these basics, he will have more confidence in his math abilities, and will see the fun and logic in mathematics. This will lay a firm foundation on which he will build his future math education.
Recommended Resources:
Books on Teaching Math:
 Burns, Marilyn (1992) About Teaching Mathematics : A K8 Resource Sausalito, CA : Marilyn Burns Education Associates ; White Plains, NY : Distributed by Cuisenaire.
This book is an excellent guide to teaching math for comprehension, not by rote methods. It offers handson techniques for assessing a child's level of skill and many fun activities. (Note to local subscribers: this book is available in the ParentTeacher section of the Staunton Public Library 372.4 B)
 Stenmark, Jean Kerr, Thompson, Virginia, and Cossey, Ruth (1986) Family Math Berkeley, CA : Regents, Univ. of CA.
This popular book offers many handson math activities that the family can do together, incorporating learners of different ages.
(Note to local subscribers: this book is available in the ParentTeacher section of the Staunton Public Library 372.7 S and in the Waynesboro Library Adult NonFiction 510 Ste)
 Kohl, MaryAnn F. (1996) Matharts: Exploring Math Through Art for 3 to 6 Year Olds. Beltsville, MD : Gryphon House, 1996. .(Note to local subscribers: this book is available in the ParentTeacher section of the Staunton Public Library 372.7 K)
 Kaye, Peggy (1987) Games for Math: Playful Ways to Help Your Child Learn Math from Kindergarten to Third Grade. New York : Pantheon Books.
A wonderful collection of handson math activities, grouped by concept.
Math Books for Kids
 Books by Greg Tang: discussed in a separate article.
 Books by Mitsumasa Anno: wonderful picture books full of math; Anno's Counting Book, Anno's Counting House, Anno's Math Games, and others.
 Books by Demi: more wonderful mathematical picture books; Demi's Count the Animals 123, One Grain of Rice: A Mathematical Folktale.
 The Hello Math Readers, distributed by Scholastic Books.
Copyright 2004 Stephanie Marshall Ward
About the Author:
Stephanie Marshall Ward has been a journalist, a professional counselor working with children in agencies and public schools, a "reader advisor" for library patrons, and a daycare provider. She is a home educating mother of three children: Sarah Stephanie, James Ernest, and Patricia Elizabeth and editor of Growing Together Family Learning Newsletter.
Growing Together Family Learning Newsletter, Vol. 1, No. 1, page 13
Download this article in PDF format (seventeen pages)
