When you are walking, count your steps out loud with your child. Count stairs. Count the things you put in your grocery basket. Count the plates at the dinner table. Count the candies in your hand, and count again after you eat each one.
Get 10 heavy magnets and a clean cookie sheet. Hold the sheet so your child can't see one side, and slam a few magnets onto it, one at a time. Have him guess how many are on the sheet and then count to make sure. Repeat.
Count as if you were the original Count Dracula. Count forward, backward, fast, slow, loud and quiet. It will never get old for your 3-year-old.
Cut the numbers zero to 10 out of sandpaper and glue each number down on its own large index card. Small children will enjoy feeling the sandpaper numbers, and it can help them remember the shape of each number more easily.
Gather 55 small objects (like pennies) and a long strip of paper with the numbers zero to 10 written out in order from left to right. Make the paper long enough to line up the number cards under each corresponding number on the strip of paper.
Lay the strip of paper in front of the child. Then, scatter the numbers above it out of order. Point to the first number on the strip, and tell the child it is zero. Ask them to find the "0" card and put it under the "0" on the strip. Tell your child zero means there is nothing; it is a big hole full of nothing, like an empty mouth (make your mouth look like a zero).
Point to the "1" and have your child find the "1" card and place it correctly. Then, count out one penny into your child's hand, and have her put it under the "1." Continue by counting two pennies out and lining them up vertically under the "2." Then, count three pennies and put them under the "3," etc.
When all the pennies are lined up under the numbers, ask your child to find different numbers and check them by counting.
Kids love doing this activity, which is adapted from Maria Montessori's "Numerals and Counters" math presentation. They'll do it over and over with different sizes and types of numbers (plastic magnet numbers, numbers cut out from construction paper, numbers written with pretty colors... You don't always have to use the sandpaper numbers) and different types of objects. For more information, or to get the original presentation, read "The Montessorri Method" by Maria Montessori. You can also check out "Teaching Montessori in the Home: The Pre-School Years" by Elizabeth G. Hainstock and Lee Havis for similar ideas.
Roll dice and count the dots. Use big dice, little dice, colored dice---each change makes a new game. After you count the dots, count out the same number of objects to extend the activity.
To get really crazy, get two dice, one for you and one for your child. Roll the dice. The one with more dots wins!
The same games can be played with dominoes, drawing them out of a box or bag, or turning them over rather than rolling them.
Seeing patterns is important to mathematical understanding and should not be dismissed as less important than counting and number recognition.
Create patterns by clapping or stomping simple rhythms and having your child repeat them, Simon Says style.
Create patterns with everyday objects like food or toys. Dinner is captivating when you line up your veggies---pea, carrot, pea, carrot---and then eat them down the row. Snacks are educational arranged as raisin, peanut, M&M, raisin, peanut, M&M.
You can also create a pattern with the objects and have your child continue the pattern where you left off. You can use more types of objects and more complicated patterns as your child gets older.
Play with money. Dump out a piggy bank and let your child replace the coins, telling her what each one is as she goes. Repeat. Hunt for the pennies. Sort the coins into piles.
Put a penny, a nickel, a dime and a quarter in a row in front of your child, and then cover one up. Ask him to name the one he can't see.
Drop coins into a jar one at a time and count the clinks. See if your child can count them without looking.
For more ideas for awesome math games you can play with your young child, look to the following sources:
"Montessori Play and Learn: A Parent's Guide to Purposeful Play from Two to Six," by Lesley Britton
"Math Activities & Games for Early Learners," by LaRose
"Marshmallow Math; Early Math for Toddlers, Preschoolers, and Primary School Children," by Trevor Schindeler
"Early Childhood Number Games: Teachers Reinvent Math Instruction, Pre-K through 3rd Grade," by Alice Wakefield
Early childhood games, Everyday Mathematics, media.emgames.com/emgames/demosite/demolevel1.html
Activities for Your Day, U.S. Department of Education, www.ed.gov/pubs/EarlyMath/activities1.html
Math activities, members.tripod.com/patricia_f/mathscience.html