Measurement - Mathematic Reform

Most students enter grade 3 with enthusiasm for, and interest in, learning mathematics. In fact, nearly three-quarters of U. S. fourth graders report liking mathematics (NCTM, 143). This can be a very critical time in keeping children interested in what they are learning. If the work turns too monotonous and uninteresting it can have a negative effect on their perceptions of the subject later in life. If students in grades three through five are given mathematic material that is interesting it can help keep their enthusiasm toward the subject. One of the major content areas that is covered at this time is measurement.

Measurement is one of the ways that teachers can introduce students to the usefulness and practicality of mathematics. Measurement requires the comparison of an attribute (distance, surface, capacity, mass, time, temperature) between two objects or to a known standard. Measurement also introduces students to the important concepts of precision, approximation, tolerance, error and dimension. Instructional programs from prekindergarten through grade twelve should enable students to understand measurable attributes of objects and the units, systems, and processes of measurement.

Also, apply the appropriate techniques, tools, and formulas to determine measurements (NCTM, 171). This paper will describe how those ideas are developed in grades three through five. The first and most basic standard for measurement at this level is being able to understand measurement attributes that we use on a daily basis. Some of these attributes include length, area, weight, volume, and size of an angle. Knowledge of these variables is very important because they are ideas that will be used regularly throughout their lives.

When students attain a better understanding of these measurement variables the next objective is to have them decipher the correct way to measure them. Choosing the appropriate unit to measure variables such as length, area, and weight can be just as important as knowing their meaning. For example, knowing that length is the distance between two points is irrelevant if a student tries to measure it with an angle or area. Knowing the proper way to measure a variable is very important.

This idea also brings into perspective the standard of measurement that deals with understanding the need for standard units, or a basic way to describe an attribute. This requires students to become familiar with standard units in the customary and metric systems. At this time students move from using objects to measure length (this desk is ten toy cars long), to using inches, feet, centimeters, and meters. This type of knowledge will allow the students to describe measured variables in understandable terms. The best way to expand children’s knowledge of measurement systems is to have them work first-hand with them.

A good example of how to do this is found in the December 2004/ January 2005, Teaching Children Mathematics Journal. The activity is called “The Hanging Plate Problem”. It involves a person wanting to hang six plates, each eight inches in diameter, on a 104-inch wall. The exercise can be done on paper or actually performed on a classroom wall. It is a great real-life measurement question because it allows the students to work with new terms such as length and diameter, while also having them explore one of the units of measurement (inches).

Simple exercises similar to this are an easy way to show students the many uses for measurement in their everyday lives. Once a student starts to get a grasp on the different measurement systems and the units within them, the next step is to learn unit conversion within a system. This is where the use of the customary and metric system can become difficult. The students must realize that only units of measurement within each system can be converted easily to one another. For example, inches to feet for the customary system, and centimeters to meters in the metric system.

This understanding of the smaller and larger units within a system is a very important standard. It involves being able to relate the number of units to the size of the units used to measure an object. For example, comparing the number of cups to fill a pitcher in relation to the number of quarts to fill the same pitcher. It will take more cups than quarts to fill the pitcher, making the cups a more precise form of measurement. Measurements are approximations, and students will learn that the smaller the unit used for measurement the closer the approximation.

This will help the student’s to realize what type of unit gives a better representation of an object. An example of this is how a student measuring their desk using inches will get a closer approximation to the actual length of the desk than a student measuring it with feet. Making sure that the student’s understand the correct tools to perform measurement is a very important process. They must be able to understand basic tools such as rulers, tape measures, and clocks. Students must also be able to develop methods of measuring objects that cannot be measured by basic tools or when these tools are not available.

One problem they will face is measuring round objects. To perform this measurement students will have to use a piece of string to wrap around the object and then measure the length of the string with a ruler. Problems like this are great because they let the student’s interact with the question and help develop problem-solving skills. But, what will children do when no measurement tools are available? One way to do this is to use benchmarks as measurement estimates. An example of a benchmark is saying that the width of a pinky is close to one centimeter.

These ideas can be easy for children to remember and lets them relate a measurement to something familiar (their pinky). After students start to get an understanding of characteristics that the units of measurement have and how to identify them, they can start to explore how these characteristics can be changed. Students can explore what happens to measurements of a two-dimensional shape, such as its perimeter and area, when the shape is changed in some way (NCTM, 171). An excellent activity that demonstrates this standard can be found in the February 2005 Teaching Children Mathematics Journal.

It gives the students a square on a piece of graph paper that measures six centimeters by six centimeters with an area of 36 square centimeters. It then asks them how many shapes they can find that have the same area (triangles, rectangles, trapezoids, and parallelograms). Along with finding the area of these shapes the students are asked to look at their differences in perimeter. They will find that with some shapes whose area will remain at 36 square centimeters and the perimeter has changed.

While exploring this activity students are also encouraged to make assumptions about formulas for areas of different shapes. Students might begin with a rectangle whose area is 36 square centimeters and then cut it in half to make a triangle whose area is 18 square centimeters. Without even knowing it this student is making a huge discovery. That the area of triangle is half of the area of the rectangle that it was made from. With more exploration students can begin to build a solid foundation on which to derive the formulas for area (Make It 36, 330).

The last standard that student’s will need to learn is how to develop strategies to determine the surface areas and volumes of rectangular solids. This can be a difficult topic to understand because it moves a student from thinking two-dimensionally to three-dimensionally. Lessons that teach this strategy must be very interactive to fully develop the topic. The November 2003 Teaching Children Mathematics Journal provides a ten day lesson that analyzes and compares attributes of two and three-dimensional shapes. The lesson has the student’s designing and creating their own scaled-down version of a “Dream Clubhouse”.

First, the students create their own representations of the clubhouses by designing a scale drawing on grid paper (1inch = 1 square foot). Then, the teacher provided cardboard to begin construction of three-dimensional scale models, with the children having to then transfer the image from the graph paper to an actual model. Unlike textbook problems, this problem-based project allowed for many explorations with student-generated problems. That is, the problems that students encountered were not prefabricated but genuinely created by students based on the task of designing the clubhouse (Junior Architects, 175).


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