A four year old may easily compare sets of objects to know which is more, but have difficulty figuring out how many he actually has. He may recognize that how many? means to count, but may struggle to do this. There are hurdles in successful counting: (1) each object is counted only once; (2) the name of a number corresponds to each one counted; and (3) there's a logic to the sequence: base ten has a predictable pattern. Four year olds are also fascinated with collecting and sorting objects.

Observe and listen while your child is playing to understand his mathematical knowledge. How does she count? Is it a sing-song or meaningful? Does she touch each object once? Is her voice in sync with her tag? Does she keep track of what's been counted? Ask questions to help her develop counting strategies. Try to resist showing your child what to do so you don't rob her of figuring it out on her own! When walking, collect objects she likes. At home find different ways to sort this collection.

A five year old is less depen­dent on matching strategies to determine one-to-one correspondence, and knows that for 5 kids she'll need 5 pencils. When she counts she knows how many? but may not know that the last number counted means the total quantity. Once they can count on, five year olds may know which set is more and may sequence quantities from smallest to largest. But the question how much more? can be difficult. They may struggle with how much larger one quantity is than another.

If your child counts accurately you can help him think about the permanence of a set of objects. Put six pennies in a row, then change the arrangement. Will he think the quantity changed? Conservation of number is a big idea needed for addition and subtraction. Five year olds love repetitive patterns, which help develop mathematical thinking. Clapping patterns can help him discern sequences and predict what comes next. Recognizing the unit in a pattern is an important tool in his mathematical toolbox.

The six year old is devel­oping a more complex understanding of number. He knows that 6 can be 5 and 1, 3 and 3, etc. He knows that all sets of 6, no matter what objects, are equivalent. And the last number counted is the number of the set. These big ideas underpin more effi­cient counting strategies such as counting on from the larger number. He's also developing the idea that "nothing" is represented by 0, and that any number in the system can be written with the digits 0-9.

Dice, cards, and board games are fun and can help a six year old gain fluency with addition combinations. Engage her thinking by playing "Hidden Counters" in which part of a set is hidden. Count out eight pennies (making sure she knows there are eight). Hide four under a cup, leaving the rest visible. Ask, "How many are hidden?" Notice her strate­gies for figuring this out. Does she know automatically that four are under the cup because she knows 4 + 4 are 8? Does she use her fingers to figure it out?

Because seven year olds can better understand space and quantity, a broader range of mathematical ideas become more acces­sible. They now have a repertoire of basic addition and subtraction combinations that they can use as tools in computing. For example, to solve 19 + 21, a child might think of a related combination, 20 + 20. Some may grasp reversibility, a big developmental shift needed to under­stand how subtraction is the inverse of addition (50 – 25 can be thought of as 25 + ? = 50).

Find ways to help your seven year old build confidence in her mathemati­cal reasoning ability. Asking, how much money is six quarters? might help her realize that if she knows that four quarters is one dollar, she can figure out what six quarters is. To help prepare for multiplication, you can pose questions that help her think in groups. A question like how many fingers do five people have? may be hard or easy for her to solve. Your challenge will be to find ques­tions that support her reasoning.

At eight, children are using numbers and quantitative methods in advanced ways, such as reversibility — they understand subtraction as the inverse of addition. A major developmental shift occurs when beginning multiplication. In addition, 130 is two or more addends that make a whole, but in multiplication 130 is related to a unit that can shift. If the unit is ten, 130 means 13 tens But if the unit is 100, 130 means 1.3 hundreds. This is unitizing, the basis for future work with fractions, decimals, and percents.

Eight year olds may take on a lot, then feel frustrated when accomplishments don't come easily. Help her break a task down: (1) what do you know? (2) identify the problem to solve; (3) identify one way to solve it. Help her develop confidence by not correcting a wrong answer, but try to follow her reasoning. Focus on the process, not the answer, to help her take risks. Eight year olds are developing more complex ways of reasoning — they like strategic thinking games like checkers, chess, Monopoly, and Clue.