Estimation

Rationale

"According to the Environmental Protection Agency, the average American produces about 4.4 pounds (2 kg) of garbage a day, or a total of 29 pounds (13 kg) per week and 1,600 pounds (726 kg) a year. This only takes into consideration the average household member and does not count industrial waste or commercial trash…with the garbage produced in America alone, you could form a line of filled-up garbage trucks and reach the moon. Or cover the state of Texas two and a half times. Or bury more than 990,000 football fields under six-foot high (1.8 meter high) piles of waste." ^a1a

Whereas in literature we often enjoy or contemplate the experience of a subject with which we have no relationship, when discussing mathematical terms or quantities we have to have some background knowledge in order to make logical sense of the message. These days a great deal of media reporting focuses on "green" issues, i.e., environmental health, global warming, the food chain, and pollution. Such global issues naturally bring with them global numbers, which may be easy to understand, but may not be so easy to comprehend. Compounding this problem is the nature of mass media which strives to make the greatest impact regardless of the legitimacy of the statement. Take the above paragraph as an example. First of all, 29 x 52 = 1508, which would round to 1500 pounds per year instead of the more impressive 1600. Likewise, 13 x 52 = 676 kg per year, not 726. And about that layer of trash burying Texas…how deep would it be? Two and a half millimeters or two and a half feet? There's quite a difference in those two quantities, yet we are given no information with which we can make a reasonable assumption. Ironically, in the one instance where a recognizable number (one million) could have been appropriately used, the author chose to use the very ambiguous 990,000.

Children in particular may not have the life experiences that enable them to conceive of these great quantities. In addition, the life experiences they have had may lead them to misconceptions based on their concrete interpretation of observations; i.e., the sun and the moon appear to be approximately the same size. How do children make sense of very great numbers? How can we help them evaluate their ideas and correct misconceptions? Students need a firm understanding of the magnitude of numbers, place value and estimation in order to make sense of these kinds of statements and to be able to critically evaluate numeric statements to see if they are reasonable. The literacy strategies so heavily emphasized in primary grade pedagogy can also be used as instructional tools to help them develop their capabilities in reading and understanding mathematical statements. Another strategy I frequently use is to establish a line of questioning which will guide students toward a conclusion that I have already determined to be the main idea of the lesson. The conversations included here are merely examples of how I would structure that questioning.

One event that is consistent in the daily lives of students that can provide a context for our studies is lunch. The cafeteria provides multiple "units" that are consistently uniform and therefore easy to evaluate; For the purpose of this unit, we will focus on polystyrene trays. Students will use estimation to arrive at a variety of quantities which they could then extrapolate through different units of measurement. For example, students could examine the number of trays discarded by each person; classroom; grade; the entire school, or even all of the schools in the district. Then they could analyze how those quantities increase over time; days, weeks, months, and the school year. Students would also predict, analyze and estimate a variety of comparative measurements: How far would that stretch? What area would it cover? How large a container would it fill? How tall would it be? How much would it weigh? Lesson 4 is an example of what this instructional sequence could look like in the classroom.

When all of the data has been collected, computed, and checked for reasonableness, students will evaluate their data and create two reports; one non-text, such as a chart or graph, and one text statement. To reconnect to the starting point of the unit, students will create a logical comparison in order to develop an appropriate voice for the text statement. Depending upon the intended use of the data and the impact the student authors would like to make, the voice could be one of relief, shock, pride, etc. In order to establish a context and reference for this voice, we will be working on a parallel unit which will teach students about the processes involved in waste management, the impact on the environment, and responsible practices.

I have found the 4MAT System, developed by Bernice McCarthy at About Learning, to be an appropriate framework to use in developing units in which all components are related to a central idea. In simple terms, the 4MAT concept states that "…teachers as well as students need to understand the reasons for doing what they do." ^a2a I am also intrigued by Liping Ma's "four properties of understanding - basic ideas, connectedness, multiple representations, and longitudinal coherence - [as] a powerful framework for grasping the mathematical content necessary to understand and instruct the thinking of schoolchildren." ^a3a Therefore, I have attempted to structure this unit in a way that emphasizes the connections between the various topics.

Ideally this unit would be started early in the fourth grade year to review and reinforce place value concepts taught in third grade and establish strong understandings of whole number place value in order to better grasp the decimal and fraction concepts which will be introduced later in fourth grade. When I taught fifth and sixth grade, I was surprised at the difficulty many students had with computation, particularly long division and any operations with fractions and decimals. I hope that a "side effect" of this unit will give students the foundation they need in manipulating and estimating numbers so that later curriculum will be supported.