## Monday, September 5, 2011

### Number and Operations in Base Ten (K.NBT.1 & 1.NBT.2)

Number and Operations in Base Ten (K.NBT.1 & 1.NBT.2)
At the end of the last post, I briefly touched upon the idea of composing and decomposing numbers 11 through 19. This idea is discussed in both Kindergarten and Grade 1.

K.NBT.1 Compose and decompose numbers from 11 to 19 into ten ones and some further ones, e.g., by using objects or drawings, and record each composition or decomposition by a drawing or equation (e.g., 18 = 10 + 8); understand that these numbers are composed of ten ones and one, two, three, four, five, six, seven, eight, or nine ones.

1.NBT.2 Understand that the two digits of a two-digit number represent amounts of tens and ones. Understand the following as special cases:
a. 10 can be thought of as a bundle of ten ones—called a “ten.”
b. The numbers from 11 to 19 are composed of a ten and one, two, three, four, five, six, seven, eight, or nine ones.
c. The numbers 10, 20, 30, 40, 50, 60, 70, 80, 90 refer to one, two, three, four, five, six, seven, eight, or nine tens (and 0 ones).

What is important to note here is the slight difference in these two standards. In Kindergarten, students are thinking of numbers 11 through 19 as "ten ones and some further ones" while in Grade 1, students need to develop an understanding of 10 as "a 'ten.'" In other words, in Grade 1, students need to develop 10 as a unit - at the same time it is a collection of ten ones. Research has shown that this understanding is a major shift, and some might argue that this expectation is not developmentally appropriate for most first graders. Children can easily learn to recite the number word sequence, "ten, twenty, thirty, ... ninety," but just as simply reciting "one, two, three, four, ..." does not necessarily indicate an understanding of numbers, the ability to recite the decade number words in order does not indicate the understanding of ten as a unit (1.NBT.2.c).

In historical numeration systems, the idea of grouping by 10's, 100's, etc. appears fairly early. In those systems, 20, 30, 40, ... were recorded with multiple symbols of 10's instead of saying how many 10's. Even in the systems that utilized place values like the Babylonian System, 20, 30 , 40 were recorded with multiple symbols of 10's just as simpler additive systems did. Thus, even in those systems, 30, for example, meant 10+10+10, not three 10's (or 3x10). This shift, although it might look rather simple for those of us who already understand the base-10 numeration system, is not that obvious for children. For them, "10" doesn't naturally mean 1 tens and 0 ones. Rather it is just like a word "cat" spelled with multiple letters. "10" is just "ten" spelled with 2 numerals "1" and "0." Thus, it is not logical that twenty should be spelled as "20" - even if they understand twenty is made up of 2 tens. After all, there is no logical connection (in how they are written) going from 1 to 2 ones. Although this standard puts this understanding of ten as a unit in focus, we should keep in mind that students will not develop this understanding in one single lesson. In fact, this understanding will probably take months to develop - perhaps stretching into Grades 2 and 3. We should keep this in mind as we look at other NBT standards in Grade 1 - they are, in part, serving to achieve this standard even though their focus may be elsewhere.