Monday, July 25, 2011

Kindergarten: Operations and Algebraic Thinking (2)

Kindergarten: Operations and Algebraic Thinking (2)
In the previous post, I discussed the difference in the meaning of subtraction between the CCSS and the current GPS and its potential implications. In this post, I would like to begin the discussion of the five specific standards in the cluster. Those standards are as follows:

1. Represent addition and subtraction with objects, fingers, mental images, drawings, sounds (e.g., claps), acting out situations, verbal explanations, expressions, or equations.
2. Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem.
3. Decompose numbers less than or equal to 10 into pairs in more than one way, e.g., by using objects or drawings, and record each decomposition by a drawing or equation (e.g., 5 = 2 + 3 and 5 = 4 + 1).
4. For any number from 1 to 9, find the number that makes 10 when added to the given number, e.g., by using objects or drawings, and record the answer with a drawing or equation.
5. Fluently add and subtract within 5.

There is a footnote for the term "drawings" in Standards 1. The footnote states, "Drawings need not show details, but should show the mathematics in the problem." It is easy to read this statement and think that we are making things simpler for children since we are asking them to do less (not showing details). However, for those of us who work with primary students know that this is not quite as simple as it may sound. In fact, many of us have experiences of watching children draw very detailed drawings when they are asked to "draw pictures" to help them solve word problems. In order for children to draw pictures that "show the mathematics in the problem," children must understand first what features of problems are and are not relevant to the mathematics in the problem. If children are drawing pictures to help them solve word problems, they may not understand what the mathematics in the problem is. If so, how can they know what features are or are not relevant to the mathematics? Thus, helping children become able to draw pictures that "show the mathematics in the problem" is itself a major teaching goal in Kindergarten. At the same time, we also want to help students develop an understanding/disposition that drawings are useful thinking tools. So, how might we achieve this goal? One potentially useful strategy used in many Japanese elementary school mathematics textbooks is to use problem contexts in which objects in the problems are fairly simple objects. Thus, when children draw their pictures, drawings will not be overly complicated. Moreover, it will be useful for children to share their drawings. By examining and reflecting on different drawings their friends made, children can begin to think what features of their drawings are essential for doing mathematics.

Another major change from the GPS to the CCSS is the idea of representing with equations. The GPS does not emphasize the formal representations with numerals and mathematical symbols in Kindergarten. However, the CCSS begins the use of the formal/symbolic representations in Kindergarten. Some people may disagree that such an expectation is developmentally appropriate. However, the expectation is there, and we must teach Kindergarteners about the formal representations. As we do, I hope we will emphasize both representing and interpreting. Thus, not simply asking children to represent addition or subtraction situations using equations, we should ask them to come up with different situations for a given equation. Moreover, even from the beginning, we should remember that the "=" sign indicate that the two quantities on both sides are equal, not "calculate." Thus, from time to time, we should write "5 = 3 + 2," not just "3 + 2 = 5."