## Wednesday, February 23, 2011

Kindergarten: Operations and Algebraic Thinking (1)

In the domain of Operations and Algebraic Thinking in Kindergarten, there is only one cluster - "Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from." This cluster statement makes it quite clear what meaning Kindergarteners are to give to the arithmetic operations of addition and subtraction. The current GPS (MKN2a) states, "Use counting strategies to find out how many items are in two sets when they are combined, separated, or compared." Table 1 of the CCSS explain what is meant by "putting together," "adding to," "taking apart," and "taking from." In my previous post on MKN2a (link), I discussed how the GPS's classification was based on the framework developed by the Cognitively Guided Instruction (CGI). The categories used by the CCSS are comparable to the CGI categories as well, but labeled differently. Thus, "adding to" is equivalent to "combine," "taking from" is equivalent to "separate." "Putting together" and "taken apart" are the "part-part-whole" category of the CGI - with the "putting together," the whole is unknown while in "taken apart," a part is unknown.

At this point, one major difference between the CCSS and the current GPS should be obvious. In the CCSS there is no comparison meaning of subtraction is addressed in Kindergarten. Instead, the CCSS includes the part-unknown case of the part-part-whole structure for subtraction. How significant is this difference? This might turn out to be a pretty significant difference. One of the findings from the CGI research is that children approach these word problems using different strategies - usually counting and/or direct modeling of the problem situations as the first step. Gradually, children will move toward the strategies that involve more advanced counting or the use of previously learned facts.

An example of "taken apart" problem included in the Appendix is this:
Five apples are on the table. Three are red and the rest are green. How many apples are green?From an adult's perspective, we might think it is simple to use counting or direct modeling for this problem. You just need to count on from 3 till you reach 5, or start counting from 5 down to 3. However, I discussed in the previous post that counting-on requires a major cognitive development. Moreover, the CGI research seems to show that "counting down to" is a more advanced counting strategy (than simply counting back 3 times). In order to model the situation, children must be able to anticipate the result - you can't start with 5 because it is made up of the known quantity and the unknown quantity. Thus, that is a more advanced thinking as well.

Comparison problems, on the other hand, are easier to model. Children can model both quantities, and they can make one-to-one correspondence between the two groups. The ones without matches are the difference. So, from a developmental perspective, comparison situations seem to be more "primitive" type. When mathematics educators discuss subtraction, we often talk about three different ways we can think of subtraction: subtraction as a take away, subtraction as comparison, and subtraction as missing addend. The "taken apart" (or part-part-whole with part unknown) seems to relate more to the last type, and we can see, from other standards, the CCSS emphasizes that way of thinking subtraction. Perhaps a careful and thoughtful teaching with that focus might help students make the necessary cognitive advances. But, I think it is critical teachers are aware of the non-trivial challenges students are expected to overcome.

## Friday, February 4, 2011

### Kindergarten: Counting and Cardinality (K.CC)

Kindergarten: Counting and Cardinality (K.CC)

In the Counting and Cardinality domain in Kindergarten, there are 7 standards in 3 clusters (Know number names and count sequence; Count to tell the number of objects; and Compare numbers). Those standards are as follows:1. Count to 100 by ones and by tens.
2. Count forward beginning from a given number within the known sequence (instead of having to begin at 1).
3. Write numbers from 0 to 20. Represent a number of objects with a written numeral 0-20 (with 0 representing a count of no objects).
4. Understand the relationship between numbers and quantities; connect counting to cardinality.a. When counting objects, say the number names in the standard order, pairing each object with one and only one number name and each number name with one and only one object.
b. Understand that the last number name said tells the number of objects counted. The number of objects is the same regardless of their arrangement or the order in which they were counted.
c. Understand that each successive number name refers to a quantity that is one larger.
5. Count to answer “how many?” questions about as many as 20 things arranged in a line, a rectangular array, or a circle, or as many as 10 things in a scattered configuration; given a number from 1–20, count out that many objects.
6. Identify whether the number of objects in one group is greater than, less than, or equal to the number of objects in another group, e.g., by using matching and counting strategies.
7. Compare two numbers between 1 and 10 presented as written numerals.

Compared to the current GPS, there are some similarities, but there are some differences, too. Like the current GPS, the CCSS expects students to write numbers up to 20 and be able to compare two (or more) sets - the CCSS has an additional expectation that students be able to compare two written numbers (between 1 and 10) without actual objects. One difference that might stand out is that the CCSS expects students to be able to count up to 100 by ones and tens while the current GPS expects students to be able to count up to 30 objects in Kindergarten. In the current GPS, the range of numbers is expanded to 100 in Grade 1, as well as counting by ones and tens. In contrast, in the CCSS the range of numbers are expanded to 120 in Grade 1. On the surface, this difference (up to 30 or up to 100) appears rather significant. On the other hand, there is an obvious number word patterns in counting from 20 through 99. So, from a language perspective, this difference might not be too significant - other than learning additional number words for 40 through 90 and 100.

Perhaps a bigger question is what is meant by the phrase, "by ones and tens." The CCSS does not provide any elaboration, but if this is limited to simply knowing the decade number words (ten, twenty, thirty, ... ninety) in sequence, it is probably not a major concern. However, the CCSS expects students to be able to count beginning with numbers other than 1. If this expectation also applies to counting "by tens," then that may not be developmentally appropriate. This idea (start counting from number other than one, or counting on) involves a major cognitive development. For many young children, numbers exist only as a result of counting. Thus, numbers do not exist without counting from 1. In order to start counting from numbers other than 1 meaningfully, or to count on from a given number, require a different way of understanding of numbers. Moreover, research seems to be clear that understanding of ten as an iterable unit is a major step that even some 2nd graders are not ready to make. I hope that there will be further elaboration and articulation of what these standards are expecting in terms of children's understanding of ten.

The CCSS seems to articulate various aspects of counting much more explicitly and in details (Standard 3). These ideas are implicit in the GPS as I discussed this matter previously (here). However, the CCSS does not appear to place much emphasis on counting (other than expanding the range of numbers to 120) in Grade 1. However, I believe counting is not something children just "master" in one grade level. Rather, it should be an important activity in primary grades for children to build number concepts. Although we do not want children to become dependent on counting to complete simple arithmetic, counting is nevertheless an important foundational activity for children to construct their number concepts. So, I hope primary grade teachers will continue to engage their students in appropriate counting activities.