## Saturday, April 28, 2007

### MKN1a, c, d & e - Numbers

MKN1. Students will connect numerals to the quantities they represent.
a. Count a number of objects up to 30.
c. Write numerals through 20 to label sets.
d. Sequence and identify using ordinal numbers (1st-10th).
e. Compare two or more sets of objects (1-10) and identify which set is equal to, more than, or less than the other.
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Helping children understand numbers is one of the major focus of primary grades mathematics instruction. However, what does it mean to understand numbers? Careful reading of this GPS standard suggest different aspects of coming to “understand numbers.”

1. Numbers are used to represent quantities.
Children encounter “numbers” everyday in many different contexts. Here are some examples: I weigh 53 pounds; and I have 24 baseball cards. In each of these examples, “numbers” indicate “how many” or “how much” of something. Because some quantities are more than, less than, or equal to other quantities, numbers may be more than, less than, or equal to others (indicator e).

2. Understanding numbers is more than counting, but counting is important.
The previous discussion suggests that understanding numbers is much more than counting. However, counting plays an important role in understanding numbers as well (indicator a). After all, unless you can count, how would you know that you have “24” baseball cards? So, what are important ideas related to learning to count? First, children need to be able to recite the number words sequence in the correct order, i.e., “one, two, three, four, …”.

Second, children must make one-to-one correspondence between the number words and the objects they are counting. Young children often point/touch objects as they count, but some children point/touch in one rhythm while saying number words in a completely different rhythm. So, sometimes they point/touch more than one object while saying only one number word.

Third, children must understand that the result of counting indicates how many objects there are. To do so, each object must be counted once and exactly once. Therefore, this understanding helps children realize the necessity of making one-to-one correspondence. This understanding also creates the need to keep track of objects that are already counted so that no object is skipped or counted more than once.

Moreover, children must understand that the result of counting, i.e., the last number word produced, indicates how many objects in the whole set. Children sometimes think that a particular number word is a “name” of a particular object. So, after they counted five objects, they will point to the last object when asked to show “five.” They must understand “five” is the amount of objects in the whole group, not the name for a particular objects. Without this understanding, children will not be able to truly understand, for example, 7 is 5 and 2 more, or 9 is 1 less than 10.

3. Numbers are expressed using numerals.
We saw that numbers represent quantities, and counting plays an important role in determining quantities as well as connecting quantities with numbers, or more accurately, number words. In addition to number words, we can also represent numbers using numerals (indicator c). Learning to write numerals is like learning to write letters of alphabets. Moreover, learning to write 2-digit numbers at this level is like learning to spell simple words. Only when children understand our number system, they can make sense of the logic of writing “thirteen” as “13,” but that goes beyond the Kindergarten GPS.

4. Because numbers may be more than, less than, or equal to others, we can use numbers to represent an order (indicator d).
Another way children might see numerals used in their everyday life includes examples like these: My classroom is room 14; My parents watch news on channel 11, and My soccer jersey number is 7. In each of these cases, the numerals do not indicate any particular quantities. Rather, because numbers can be sequenced (1, 2, 3, …) based on the size relationship of the quantities they represent, we can use then to represent order and to label objects uniquely. Unlike a number, which indicates how many objects are in a whole group, an ordinal number is a label for a particular object.

In closing...
I hope this short discussion gave you a better sense of what this GPS is trying to address. Kindergarteners need a lot of experiences in counting objects, making groups, splitting groups, ordering groups, etc. These are activities they encounter throughout a day, not just during the math lesson. Because much, if not all, of elementary mathematics derives out of everyday phenomena, we should try to take advantage of those everyday situations where numbers play important roles. We cannot teach children mathematics by simply giving them worksheets after worksheets.

## Monday, April 23, 2007

### As I begin...

The new Georgia Performance Standards for mathematics is modeled after (based on?) the 1989 Japanese Course of Study. One of my research interests as a mathematics educator is mathematics education in Japan, where I am originally from.

From time to time, I plan to post my own interpretation of the GPS based on my study of the Japanese curriculum (and my interactions with Japanese mathematics educators at all levels).

I was involved in the translation of a Japanese elementary mathematics textbook series. My colleagues and I have also translated an elaboration document of the Japanese standards that the Ministry of Education published. Both of those materials may be obtained from Global Education Resources (http://www.globaledresources.com/).