Part I (A): Content Goals Paper The two main goals that the National Council of Teachers of Mathematics (NCTM) have created for measurement standards for prekindergarten through grade twelve are that all students should understand measurable characteristics of objects and the units, systems and processes of measurement. The other goal for measurement is that all students will be able to apply appropriate techniques, tools, and formulas to determine measurements. There are expectations for different grade levels that correlate with these two goals. In this paper we will focus on the measurement standards and expectations that the NCTM has outlined for grades third through fifth. First, it is important to acknowledge why measurement is important for students to learn. “Measurement is a process that students in grades 3-5 use every day as they explore questions related to their school or home environment.” (NCTM Chp. 5). Learning about measurement also helps students in other areas of math like describing data, fractions, and geometric shapes. The main focus of the measurement standards for grades 3-5 is to deepen their understanding and use of measurement, such as area, angles, accuracy, formulas, and measurement tools (NCTM Chp. 5). Students can gain an understanding for measurement through exploring every day objects and spaces that they are familiar with. “Length, capacity, mass, area, volume, time, temperature, and angle are expressed ideas that we, as humans, have developed to make sense of our environment and to communicate meaningfully to one another about those experiences.” (Reynolds, 1997, p. 166). One expectation that the NCTM has for students in grades 3-5 is that they gain an understanding of the characteristics of length, area, weight, volume, and size of angles. Also with this understanding of the characteristics they need to determine what would be the best unit for measuring each characteristics. Students will begin to use benchmarks to estimate the size objects (NCTM Chp. 5). An example of using benchmarks can been seen through third grader, Jim’s idea. “Joe stood against the tree trunk while Jim positioned himself about five meters away. Jim then used his hand as sighting instrument and found that at that distance, it took two of his hands to mark off the height of Joe standing against the tree. He then continued using his hands to measure to the top of the tree. Each time he counted off two hands, he converted those units to the unit of Joe’s body length; he reported that the height of the tree was five times Joe’s height.” (Reynolds, 1997, p. 169). Through exploring this concept of units they discover how standard units are so valuable to measurement. During an experiment project, the teacher asked Who can make it through the tires in the shortest amount of time? (Reynolds, 1997, p.168). The students were not allowed to use stopwatches or clocks. After some groups tried the experiment out, the students posed the question “How can we tell that we are both counting at the same rate.” (Reynolds, 1997, p. 168.). “Several pairs of students were observed rehearsing their counting rhythms so they could guarantee fairness. These students were thus considering the issue of a “standard” unit as they invented ways of dealing with the task.” (Reynolds, 1997, p. 168.) . The NTCM states that children should become familiar with customary and metric system and be able to carry out simple unit conversions within a system of measurement (NCTM Chp. 5). An example of a simple unit conversion would to change 250 centimeters to 2.5 meters. They should also be able to compare metric and customary systems, since the United States uses both systems for measurement. An example of comparing the two systems would be to know that a kilogram is about two pounds (NCTM Chp. 5). It is also important for students to recognize the difference between approximations and precise measurements. Through exploring this concept students will realize that all measurements are approximations. During an activity were students put thermometers in refrigerators and graphed the different temperatures, the teacher asked the students “Why do some graphs start and stop at different numbers than others.” (Moore, 1999, p. 541). Through activities like this one, students can discover what factors lead to measurements being estimations. Some examples would be human error reading the scale or the errors in the instruments being used (NCTM Chp. 5). Another expectation that the NCTM has for understanding measurable characteristics is what happens to a two-dimensional shape’s perimeter and area when the shape is changed in some way (NCTM Chp. 5). Students need to develop an understanding of how area and perimeter are related. An activity for students to explore this relationship is giving them twenty straws to build as many different rectangles as they can. The students then can explore how rectangles with the same area have different perimeters. (NCTM Chp. 5). The next category of measurement standards that the NCTM has develop deal with applying appropriate techniques, tools, and formulas to determine measurements (NCTM Chp. 5). Students in grades 3-5 should learn how to use measurement tools correctly. The need to know where the beginning point is located and understand the markings on a tool. They also need to when it would be appropriate to use a certain measuring tool. “Using string or some other flexible object to outline the clock face and then measuring the string” would be a good strategy for measuring something that wasn’t a straightedge (NCTM Chp. 5.). “Students should develop strategies for estimating the perimeters, areas, and volumes of irregular shapes.”(NCTM Chp. 5.) In a playground measurement activity, the teacher posed the question Find some way to measure the length of the playground. How long is it? (Reynolds, 1997, p. 169.) The playground was not perfectly rectangular. “To complete this task using a conventionally accepted notion of “length,” students had to mentally reconstruct this shape so that it approximated a rectangle.” (Reynolds, 1997, p. 169). So a strategy that students used in this case was estimating the length of the playground’s irregular shape as if it were a regular rectangle, which is the shape that it most resembled. Benchmarks can also be used as an estimation strategy. A good example of this was explained earlier when Jim compared his friend’s height to find the tree’s approximate height by using his hands. The next expectation is for students to “select and apply appropriate standard units and tools to measure length, area, volume, weight, time, temperature, and the size of angles.” (NCTM Chp. 5). An example of this is seen through an activity dealing with angles called “Experimenting with Crazy Cars.” “Students must understand that if an object is turned so that it faces the opposite direction, it has rotated 180 degrees.” (Whitman, 2000, p. 70.) The teacher builds upon the students understanding of 180 degrees, by estimating the size each angle inside a triangle, and that all three interior angles add up to 180. After the students had estimated how much their car rotated at each angle, they checked their angles by using a protractor. (Whitman, 2000, p. 70). Using the idea of crazy cars, and rotating their cars at each angle help them understand what exactly are they measuring when the measure an angle, which is how much turn or rotation is it. Students also need to develop and understand formulas to find the area of rectangles (NCTM Chp. 5). They develop these formulas through looking at patterns. They must first figure out the area for a set of rectangles, which would be on a grid. After they have figured out the area of the first few rectangles, “they realize that counting all the squares is not necessary once the length and width of the rectangles are determined with the grid.” (NCTM Chp. 5). They then can test their new formula, Area=Length x Width, on a different rectangles. The students must explore why this formula works for all rectangles. The final expectation is for all students to develop strategies to determine the surface areas and volumes of rectangular solids (NCTM Chp. 5). Students develop strategies by using real objects, such as cubes and tiles. One example of this would be filling a hollow cube full of smaller cubes and then counting how many smaller cubes fit into the large cube, to find the volume of the large cube. Once they do this, then they might move on to just counting how many cubes across is the length, height, and width of the cube, and then multiplying those values together to get the volume of the large cube. This strategy would be building on their knowledge of the formula for area, and that volume also involves height. Students need to have experiences with discovering how and what volume and surface area are before they find the formulas for these measurements. “With concrete experiences, students can easily gain this understanding and then use measurement to represent the phenomena of everyday experiences.” (Moore, 1999, p. 538).