However, in the most recent international assessment of mathematics achievement, 13-year-olds in the United States performed very poorly in comparison to other nations. The results of this study indicated that, while students are given instruction on measurement, they do not learn the concepts well. For example, some students have difficulty recognizing two fundamental ideas of measurement: unit and the iteration of units. A common error is counting number marks on a ruler rather than counting the intervals between the marks. Another difficult concept is that the size of the unit and the number of units needed to measure an object are inversely related; as one increases the other decreases. In the fifth and sixth grades, as students encounter both very small and very large measurement units (such as milligrams or tons), these ideas become increasingly critical to understanding measurement.

Students must be involved in the act of measurement; they must have opportunities to use measurement skills to solve real problems if they are to develop understanding. Textbooks by themselves can only provide symbolic activities. Teachers must take responsibility for furnishing hands-on opportunities that reinforce measurement concepts with all common measures.

The development of measurement formulas is an important part of middle grade mathematics, but being exposed to formulas in the abstract arena of a textbook does not promote understanding. Formulas should be the result of exploration and discovery; they should be developed as an appropriate means of counting iterated units. Less than half of the seventh grade students tested could figure out the area of a rectangle drawn on a sheet of graph paper, and only a little more than half could compute the area given the dimensions of length and height. Often length and width are taught as separate entities and the overlapping of the two to form the square units of area is not emphasized in instruction. In addition, area and perimeter are often confused by middle grade students. Limiting experience to the printed page restricts flexibility so that understanding cannot be generalized.

In order to further strengthen students' understanding of measurement concepts, it is important to connect measurement to other ideas in mathematics. Students should manipulate objects (e.g., measure), represent the information gathered visually (e.g., graph), model the situation with symbols (e.g., formulas), and apply what they have learned to real-world events. For example, they might collect information about waste in the school lunchroom and present their results to the principal, with suggestions for reducing waste. Integrating across mathematical topics helps to organize instruction and generate useful ideas for teaching the important content of measurement.

In summary, measurement activities should require a dynamic interaction between students and their environment, as students encounter measurement outside of school as well as inside school. Students must use measuring instruments until that use becomes second nature. The curriculum should focus on the development of understanding of measurement rather than on the rote memorization of formulas. This can be reinforced by teaching students to estimate and to be aware of context in their estimates, such as estimating too high when buying carpeting. Students must be given the opportunity to extend their learning to new situations and new applications.

Bright, G. W. & Hoeffner, K. "Measurement, probability, statistics, and graphing," in D. T. Owens (Ed.), Research Ideas for the Classroom: Middle Grades Mathematics. New York: Macmillan Publishing Company, 1993.

LaPoint, A. E., Mead, N. A., & Phillips, G. W. A World of Differences: An International Assessment of Mathematics and Science. Princeton, N. J. : Educational Testing Service, 1989.

Building upon K-4 expectations, experiences in grades 5-6 will be such that all students:

H. estimate, make, and use measurements to describe and compare phenomena.

• Students estimate the number of square centimeters in a triangle. Then they enclose the triangle in a rectangle and use a transparent square centimeter grid to find the area of the rectangle. They also count squares to find the area of the triangle and that of any other triangles formed by the rectangle. They look for a pattern in their results and compare their results to their estimates.
• Students explain why the following is or is not reasonable: An average person can run one kilometer in one minute.
• Students measure how long it takes to go 10 meters, first using "baby steps," then using normal steps, and finally using "giant steps." They then compare their rates.
• Students measure the area of their foot by tracing around it on centimeter graph paper, measuring its length and width, and counting how many squares are covered by the foot.

• Students use a given scale to compute the actual length of a variety of illustrated dinosaurs.
• Students make a scale drawing of their classroom.
• Students use a map to find the distance between two cities.

• Students plan a vegetable garden, determining the unit of measure appropriate for the garden, estimating its size, and then computing the perimeter (for fencing) and area (for fertilizer).
• Groups of students use a scale drawing of an apartment (1 cm = 1 foot) to find out how many square yards of carpeting are needed for the rectangular (9' x 12') living room.

• Students measure a given line and compare their results, focussing on the idea that any measurement is approximate.
• Before a cotton ball toss competition, students discuss what units should be used to measure the tosses. They decide that measuring to the nearest centimeter should be close enough, even though the actual tosses could be slightly more or less.

• Students complete a worksheet showing several rectangles on grids that are partially obscured by inkblots. In order to find the area of each rectangle, they must use a systematic procedure involving multiplying the length of the rectangle by its width.
• Students develop the formula for finding the volume of a rectangular prism by constructing and filling boxes of various sizes with centimeter cubes and looking for patterns in their results. They describe the "shortcut" way to find the volume in their journals.

• Students work in groups to estimate the number of bricks needed to build the school building. They explain their results in a class presentation, describing the strategies they used.
• Students are asked to estimate how many heads tall they are. Then they work in groups to develop a procedure for finding out how many heads tall each student is.
• Students construct a measuring tool that they can use to find the height of trees, flagpoles, and buildings when they are standing a fixed distance from the object to be measured.

• Students are given a ring and asked to find the height of the person who lost the ring. They measure their own fingers and their heights, plotting the data on a coordinate graph. They use a piece of spaghetti to fit a straight line to the plotted points and make a prediction about the height of the person who owns the ring, based on the data they have collected.
• Students are asked to find how many pumpkin seeds there are in a kilogram. They decide to measure how much 50 seeds weigh and use this result to help them find the answer.
• Students use approximate "rules of thumb" to help them convert units. For example:
  
  1000 ml weighs 1 kg
  1 km is about 6/10 of a mile
  1 liter is a little bigger than a quart
  1 meter is a little bigger than a yard
  1 kg is about 2 pounds
  20 degrees C is about 70 degrees F (room temperature)

• Students estimate and then develop a plan to find out how many pieces of popped popcorn will fit in their locker.
• Students work in pairs to design a birdhouse that can be made from a single sheet of wood (posterboard) that is 22" x 28". The students use butcher paper to lay out their plans so that the birdhouse is as large as possible. Each pair of students must show how the pieces can be laid out on the wood (posterboard) before cutting.
• Students compare the measurements of an object to that of its shadow on a wall as the distance between the object and the wall increases.

• Students use pattern blocks to see how the area of a square changes when the length is doubled. They then repeat the experiment using equilateral triangles.
• Students use cubes to explore how the volume of a cube changes when the length of the side is doubled.
• Students use graph paper to draw as many rectangles as they can that have a perimeter of 16 units. They then find the area of each rectangle, look for patterns, and summarize their results.