i. Count out pennies to buy items that together cost less than 30 cents.

j. Make fair trades using combinations involving pennies and nickels and pennies and dimes.

Quite frankly, I really don't understand why money and clock reading are in the mathematics curriculum. Those ideas should be learned in everyday contexts in which they mean something. The names and the values of each coin aren't mathematical concepts. However, whether or not I like these topics in the mathematics curriculum really matters much as teachers are expected to teach them. I heard many teachers say that money is a difficult idea for children. There may be a number of reasons for children to have difficulty with money. For one thing, being able to exchange a merchandise with a piece of metal or paper cannot be really "natural" to children. Another major source of difficulty for young children is the notion of exchanging coins. It's not quite logical why 5 shiny pennies can be exchanged with one, slightly larger coin, for example. Neither of these is mathematical ideas, and I'm not really sure how you can help young children with these ideas.

However, I do want to say something about exchanging coins. A part of the difficulty for young children is, I believe, because they have yet to develop a very sophisticated understanding of numbers. Kindergarten teachers are familiar with an exchange like the following:

Child: (counting to herself) One, two, three, four. Four !

Teacher: How many red counters do you see?

Child: (counting to herself) One, two, three. Three!

Teacher: So, how many counters are there altogether?

Child: One, two, three,...

Teacher: (interrupting). Wait a second. How many blue counters?

Child: One, two, three, four. Four.

Teacher: How many red counters?

Child: One, two, three. Three.

Teacher: So, how many counters altogether?

Child: One, two, three, four, five, six, seven. Seven!

Many adults are simply puzzled why a child will have to count all the counters when they know that there are four blue counters and three more red ones. However, for many young children, numbers exist only after they count - "four" cannot exist by itself. Furthermore, for many of them, four simply means four ones. Children must develop an understanding that four can be considered as an entity, or a unit, in itself before they can count on. In the previous post, I discussed the idea of five and ten as a benchmark. Children cannot think of five as a benchmark unless they can think of five as five ones and, simultaneously, one five. For adults, this is so obvious, and it is difficult to even fathom anyone (including children) not understanding it. However, research clearly shows that children don't automatically understand this idea.

So, if a child is still "counting all" stage, it is probably not reasonable to expect him/her to be able to make an exchange of five pennies with one nickel with understanding. One way to help children overcome difficulty with money is to help them develop good number sense - the ability to see numbers flexibly. Without number sense, money is much more difficulty to make sense.

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