Estimation

Rationale

The study of place value and order of magnitude is essential to constructing a solid understanding of our decimal notation system. A substantial grasp of decimal notation is the foundation for rounding, comparing numbers and computing. In Virginia, the Standards of Learning and benchmark tests put a strong emphasis on students being able to successfully round a value, compare two values, and perform the 4 basic operations with multiple numbers. I feel that a key factor for helping our students achieve success in these areas is to approach building a more thorough sense of number in early childhood education. We must help young children start to comprehend the complexity of the way we communicate mathematically before we can expect them to manipulate and compute within that system. A key factor in their success in simple computations is an appreciation of exactly what each digit in a number stands for and that where you put a digit can make a huge impact on the value of the number.

In second grade, students start to explore the way we order digits to represent different values. However, it is not often presented to them from this perspective. Instead, seemingly out of the blue, we ask them to answer questions like "Which digit is in the tens place?" or "What is the value of 4 in 402?" As adults, we see these concepts as transparent. However, from the viewpoint of a child who has been counting groups of objects from 1-10 since he or she could talk, but has not gone much beyond ten, the fact that our numbers are communicated in specific combinations using decimal notation is NOT readily apparent. Most of the students I have worked with do not understand the concept of "digit" when we begin mathematics in September. They were not introduced to "base-10" as a concept before they got to my class. When they think of 10, they see it as a complete picture representing ten objects. Most often, they do not see 10 as representing 1 ten and 0 ones. I think we are doing a disservice to our students when we don't emphasize that 10 is a two-digit number well before 2 ^nd grade. Our students learn that there are 26 letters by learning a nifty song. Then they learn that when you combine those letters they begin to represent specific sounds that hold meaning (words). Why do we not do the same thing in early mathematics teaching? Why is there not a digit song that all children are encouraged to learn at a very young age? (See Appendix D.)

Once a student understands the concept of "digit," he or she can begin to look at the pieces of a multi-digit number. However, they first need to be able to explain the relationships between single-digit numbers. Most of my students do not enter second grade with this ability. They cannot readily explain the relationship between a set of numbers. If I asked them, "How are 7, 3 and 4 related? Can you show me with a picture?" they wouldn't be able to successfully answer. This is an indication that they don't have sufficient experience composing and decomposing numbers. If they can't pull apart and recompose a number, then they can't begin to manipulate its pieces. Before I can ask them to manipulate a two or three-digit number, they need to be able to compose and decompose smaller numbers with accuracy and fluency. (Activities for building these skills are included in Appendix B.)

The way in which we see base-10 numbers determines our success in doing computations with them. "The fact that base ten numbers are sums has a pervasive influence on the methods for computing with them…the procedures for carrying out the four arithmetic operations with base ten numbers are largely determined by the fact that base ten numbers are sums of their place value components." ² We must begin to see base 10 numbers as sums of their components. When we begin to do this, we can manipulate them with greater efficiency and accuracy.

After a second grader is able to see the relationships between pieces of numbers, the activities integrating order of magnitude can be implemented. As we start to talk about multi-digit numbers in terms of their place value components, the activities in this unit become relevant. Subsequently, we can begin to use "very round numbers" to estimate quantities and lengths.

If we don't give students the opportunity to find the proof for the rules of math on their own, then how can we expect them to truly understand mathematical processes at the elementary level? Sure, we could tell them the age-old adage "Because I said so," but we would be leaving them with a very superficial concept of number and our decimal notation system. Children need to repeatedly see the difference in magnitude for each place value component so that they have an internal understanding of the significance of position of the decimal notation system. It is only when children build a solid concept of digits, place value and order of magnitude that they will truly be successful in computation, estimation and the further manipulation of numbers.