units of measure: results and implications from national assessment

Units of Measure: Results and Implications from National AssessmentAuthor(s): James HiebertSource: The Arithmetic Teacher, Vol. 28, No. 6 (February 1981), pp. 38-43Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41191805 .

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Units of Measure:

Results and Implications from

National Assessment By James Hiebert, University of Kentucky, Lexington, Kentucky

Measurement skills provide children with a powerful link between the ab- stract world of numbers and the con- crete world of physical objects. Before young children learn to measure, they can only describe objects or quantities with relatively vague, uninformative terms such as "big" or "many." As they learn to measure, children acquire the skills to describe quantities in more precise and more useful terms. They can now talk about the size of quantities as the number of units measured.

The unit of measure serves as the bridge between the object and the number used to describe its size. The concept of a unit is a central, unifying idea underlying all measurement. In fact, children's ability to measure depends to a large extent upon their understanding of the unit in measurement situations. The mathematics assessment of the National Assessment of Educational Progress (NAEP) in- cluded several exercises which provide soirie insight into elementary school students9 understanding of several basic properties of units. The degree to which children understand these properties provides an indication of how well they understand some of the underlying concepts of measurement.

What Is a Unit? An important property of a unit of measure is that it may be composed of more, or less, than a single object or entity. Although an individual object, such as a yardstick or a cup, is often used as the unit of measure, children need to understand that a unit does not have to correspond with a single object. In certain cases one-half of a yardstick, or three cupfuls, might serve as the unit of measure. The results of several NAEP exercises highlight the difficulty students have with this concept. One graphing item symbolized a popu- lation of students by using a stick figure to represent 10 students. In com- pleting the task, 60 percent of the 9- year-olds and 29 percent of the 13- year-olds ignored this description and simply counted each figure as one student. This suggests that many elementary school pupils do not fully understand that a unit of measure may represent more than a single entity.

The difficulty that this may cause in common measurement situations is il- lustrated by the performance on the exercise shown in figure 1. About 80 percent of the 9-year-olds and 90 percent of the 13-year-olds correctly

38 Arithmetic Teacher



Fig. 1. Reading a thermometer

solved a related exercise for which the temperature corresponded to a label on the scale like 20°. But in the exercise shown in figure 1, students nee4ed to recognize that the unit of measure rep- resented by each mark was 2°. Few 9- year-olds responded correctly, and even by 13 years of age most students still did not attend to the fact that the unit of measure was something other than a single entity. Almost all of the errors in this exercise can be accounted for by students assuming that the unit of measure indicated by the markings was Г rather than 2°.

A related concept is that a unit can be broken into parts to facilitate the measurement of a quantity. If students believe that units must be single, dis- crete entities, they might either ignore the fractions of units or count them as whole units. One exercise assessed students' understanding of this concept by requiring them to work with fractional parts of units. As shown in figure 2, students needed to recombine the units

that had been partitioned to estimate the area of the shaded region. About two-thirds of the 13-year-olds selected the correct response (this item was not given to 9-year-olds). Almost one-third of the students either ignored the unit pieces or counted them as whole units. This represents a substantial number of 13-year-olds who apparently believe that a unit of measure must be a single entity. The fact that this belief was evi- denced in a variety of exercises points to a rather pervasive misunderstanding of a basic measurement concept.

Relationship between Size of Unit and Number of Units

Another important property of the unit concept is the relationship between the size of the unit and the number of units measured. The larger the size of the unit, the fewer the number of units re-

quired to measure a particular quantity. Since young children tend to focus on the number of units measured, they must learn that the size of the unit is equally important in describing the measure of a quantity. Results on an exercise like the one shown in figure 3 indicate that most 9- and 13-year-old students understand this concept. It is important to note that students were asked to predict whether more or less units would be needed to measure the quantity with a smaller unit. Since it is easier for students to predict the relationship between unit number and unit size than to apply this knowledge in a measurement situation (Carpenter and Lewis 1976), the level of performance may have been somewhat lower on an application item. Nevertheless, the performance on this exercise is encourag- ing in light of the fact that the inverse relationship between unit size and unit number is one of the most difficult measurement concepts for younger children to grasp (Carpenter 1976).

February 1981 39

Fig. 2. Working with fractional parts of units.

z - i square cm What temperature is shown -= r- 1 1 0 i-±-i 1 I I I I i | i i on this thermometer?* 1 00 4 r-

4 |- 90 80 4 h-

2o|f30 ZZI^^ë-ZZZ

1 Estimate the area of the Shaded region above.

Response Percent responding Response Present responding

Age 9 Age 13 Age 13

О 19° 5 1 О 10 square cm. 19 О 24° 27 13 • 15 square cm. 63 • 28° 12 46 О 20 square cm. 10 О 29° 53 40 О 23 square cm. 3 О I don't know 4 1 О I don't know 3

*This exercise is similar to an unreleased exercise.



Measuring Is Covering with Units A third important property of a unit deals with the process of applying the unit to measure a quantity. The units must "cover" the quantity exactly - there can be no overlap between units and no part can be left uncovered. When measuring volume or capacity, this means that the units must fill exactly the volume to be measured. A se- ries of exercises dealing with length, area, and volume focused on the no- tion of unit-covering/ In one exercise students were given a paper clip and were asked to find the number of paper clips it took to measure the length of a line segment that was drawn on the page. Only one-half of the 9-year-olds and 62 percent of the 13-year-olds achieved an accurate solution, with an additional 30 percent in each age group providing responses that were within one unit of the correct answer. Although some of these errors are

probably explained by a lack of eye- hand coordination required to move the paper clip accurately, the rate of failure suggests that some children do not recognize the care with which a unit must be moved to achieve such a covering.

A multiple-choice item assessed children's ability to interpret unit covering in an area context. Students were asked to find the area of a rectangle like the one shown in figure 4. Even though students only needed to count the unit squares covering the rectangle to find its area, more than two-thirds of the 9- year-olds were unsuccessful. Almost one-half of the students simply gave one of the dimensions as the answer or added the dimensions together. This suggests that most students of this age do not yet understand that the measure of a quantity, within an area context, is given by the number of units that exactly cover the region.

Three open-ended exercises assessed students' understanding of the unit-

covering idea in a volume context. As shown in figure 5, the exercises asked students to determine the volume of rectangular solids which had been partitioned into unit cubes. Apparently, by 9 years of age, most children have some concept of volume and can determine the volume of a solid if all the units are visible. But their knowledge is quite superficial. If all the units are not visible, many students resort to alter- native solutions that do not involve unit cubes.

It seems that when students do not understand the concept involved in a problem, or are unsure of how to pro- ceed, they often select the most imme- diate or most salient response. In the previous area exercise, many students simply manipulated the numbers in the drawing to generate a response. The volume exercises did not include the dimensions of the solids so children who did not understand the problem needed to look elsewhere for alterna- tive solutions. A tempting, but incor-

Fig. 3. Fig. 4. Measuring a quantity with different size units. Finding the measure of an area that has been covered by units.

3 cm

С Q S1 5cm==z Box В block

Using the A blocks, it took 8 blocks to fill the box. What is the area of this rectangle?* If the В blocks were used, how many would it take?*

Response Percent responding

Response Percent responding" - ! ^ ^1*_ Ã « Ã 7^ ^ 3 square cm. 8 1

*&* Ã « Age Ã 13 7^

o 3

5sJuarecm> square cm.

15 8

3 1

О Less than 8 15 8 О 8 square cm. 27 12 О 8 14 4 О 16 square cm. 6 10 • More than 8 67 87 • 15 square cm. 28 71 Q I don't know 4 1 Q I don't know 1j> 3

*This exercise is similar to an unreleased exercise. *This exercise is similar to an unreleased exercise.




Fig. 5. Finding the volume of rectangular solids. f

s' 1. This is one unit:

What is the volume of each rectangular solid below?


Age 9 Age 13

4* 76 78

8 3 1

f

f f f Hf7! 9 5 4 18 1 2

I I don't know 8 5 1 l ' ' ^ Other 7 10


^x ^ >» » Age 9 Age 1 3 ^ ^s ^ ^^^ ~~~ Г

Г Г

^ Г 12* 35 58 6 26 11

У 16 10 6 32 1 2

U I don't know 8 6 I 1 ! ^ Other 20 17

2. A rectangular solid is cut into cubes as shown. How many cubes are there?

^^^^^^^^l Response Percent responding

^^^¿^^^^^^^^^^y^ A9e 9 A9e 1 3

^^^s^^-^s' y^ 36* 7 24

Чччч Г4^^/! ^S 33 45 26 N.

^ ^р ^^ ^,66 1 10 . ^

s4^ . ̂ <г | don't know • 3 1 ^ч

s4^ ^^у^ у^ Other 44 39

* Correct response

February 1981 41



rect, strategy is just to count the surface squares shown in the drawing. This strategy increased in frequency for 9-year-olds from 5 percent in the first volume exercise to 10 percent in the second exercise to 45 percent in the third exercise. A similar pattern is evi- dent for the 13-year-olds. The increase in this type of error corresponds with an increase in the difficulty of the task and an increase in the degree to which the surface squares become the most dominant visual characteristic in the drawing.

The results of these exercises suggest that it is not appropriate to think of children as completely possessing or lacking a measurement concept, but rather as being able to apply the understanding they have in particular situations. Some children demonstrate knowledge of a concept in simple tasks but appear to abandon this knowledge in more complex settings. The degree to which they do so likely depends upon the salience or appeal of alterna- tive solutions. As children acquire more mature concepts, they are able to demonstrate knowledge of them in in- creasingly difficult contexts, even when faced with attractive competing (and erroneous) solutions.

Recommendations for Instruction An important conclusion from the first mathematics assessment was that 9- and 13 -year-old students are quite pro- ficient at simple measuring skills but do not yet understand certain underlying concepts of measurement (Car- penter, Coburn, Reys, and Wilson 1975a; 1975b). The exercises presented here, from the second assessment, focused on several of these underlying concepts dealing with units of measure. Performance on these items confirms a similar lack of understanding by students of today and isolates two unit concepts that are in particular need of attention.

A substantial number of children of each age experienced difficulty with problems in which the unit of measure did not correspond to a single entity, and with problems which required them to recognize that the measure of a quantity can be found by counting the

units which exactly cover (or fill) the quantity. A priority of instruction in measurement should be to provide experiences which help to deepen children's understanding of these basic unit concepts.

It is instructive for children to en- counter problem situations in which they must select a unit of measure and partition the quantity or object to be measured into congruent units. This kind of activity requires them to focus on the unit of measure - to consider its size and to recognize whether it is more, less, or the same as an individual entity. The activity also forces children to think about the idea of a unit covering. They must recognize that the measure of a quantity is found by pre- cisely covering or filling the quantity with equal-size units. Many textbooks present measurement problems with the quantity already partitioned into units. According to the NAEP results, it is exactly these kinds of problems that students seem to solve with a superficial understanding. Consequently the textbook should be supplemented with problems in which the quantities are not yet partitioned into units. Stu- dents themselves can select the unit of measure, construct the unit covering, and carry out the measurement.

The selection of a unit is an espe- cially important part of this process. Children should have substantial experience working with various non- standard units of measure. Sometimes the unit should be more than a single entity and sometimes it should be less than a single entity. For example, when determining the area of a large region traced on graph paper, students might use four adjacent squares form- ing a larger square as the unit of measure. To find the width of their desks, students might use one-half of a straw as the unit of measure. The teacher will need to suggest these as acceptable units of measure.

Students also need experience with partitioning quantities into congruent units and using these unit coverings to determine the measures of the quantities. These experiences should pre- cede the introduction of formulas for calculating these measures. Perform- ance on the area exercises indicates that many students do not understand

that the measure of an area means the number of units which exactly cover it. As Hirstein, Lamb, and Osborne (1978) point out, it is difficult to see how children can attach meaning to a formula of length times width if they do not yet understand that the area of a region can be found by counting the units which partition or cover it.

To develop this kind of understanding, students need experience with constructing unit coverings in a variety of situations. Partitioning regions into unit squares, filling solids with unit cubes, and scaling thermometers with unit markings are all useful activities. Children can also benefit from working with nonsquare units to cover a region. Any triangle or rectangle can be used as the unit of measure since either of these fit together to completely cover a region without overlapping one another. After regions and solids are partitioned into congruent units they might then be rearranged to form different figures with equivalent area or volume. For example, students could cut out unit squares or rectangles and make differently shaped regions which have the same area, recording on graph paper the regions which are con- structed. This type of activity focuses attention on the fact that the number and size of the units, not the shape of the region, determine the area. Addi- tional activities which involve unit coverings are presented in a recent issue of the Arithmetic Teacher (Leutzinger and Nelson 1980).

Initial experience with constructing unit coverings should include work with physical materials. The results of the NAEP exercises indicate that elementary school students have an espe- cially difficult time interpreting two-di- mensional drawings of solid figures. Meaning for unit-of-volume concepts can be developed by providing substantial hands-on experience in fiUing solids with unit cubes, and rearranging the units to create other solids with the same volume.

Another type of activity which may help students focus on the ideas of unit size and unit covering is estimation. Students can be asked to estimate the number of units of particular size needed to cover or equal a given length, area, volume, or weight before




actually measuring the quantity. The process of estimating encourages students to think about the unit of measure to be used and the process of constructing a unit cover before proceeding with a measurement which could otherwise be performed in a rote, mechanical way. Estimating also helps students avoid unreasonable answers, which often result from mechanically applying a measuring procedure without considering whether it is an appropriate procedure to solve the problem. Some instructive estimating activities are described by Bright (1976) in an NCTM Yearbook on measurement.

The kinds of activities identified in the preceding paragraphs do not ex- haust all of the measuring experiences children should have; they merely il- lustrate the types of experiences children need with units of measure. The NAEP results indicate that many 9-

year-olds and some 13-year-olds still have a primitive understanding of unit concepts, an understanding which is abandoned in all but the simplest measuring contexts. Consequently it is important for the teacher to supplement the textbook program with measuring activities, like those described here, which are designed to facilitate children's understanding of the basic unit measurement concepts.

References

Bright, George W. "Estimation as Part of Learn- ing to Measure." In Measurement in School Mathematics. The 1976 Yearbook of the Na- tional Council of Teachers of Mathematics. . Reston: The Council, 1976.

Carpenter, Thomas P. "Analysis and Synthesis of Existing Research on Measurement." In Number and Measurement, edited by Richard A. Lesh. Columbus, Ohio: ERIC, 1976.

Carpenter, Thomas P., Terrence G. Coburn, Robert E. Reys, and James W. Wilson, (a) "Notes from National Assessment: Basic Con-

cepts of Area and Volume.*' Arithmetic Teacher 22 (October 1975):501-7.

(b) "Notes from National Assessment: Perimeter and Area." Arithmetic Teacher 22 (November 1975):586-90.

Carpenter, Thomas P., and Ruth Lewis. "The Development of the Concept of a Standard Unit of Measure in Young Children." Journal for Research in Mathematics Education 7 (Jan- uary 1976):53-58.

Hirstein, James J., Charles E. Lamb, and Alan Osborne. "Student Misconceptions about Area Measure." Arithmetic Teacher 25 (March 1978): 10-18.

Leutzinger, Larry P., and Glenn Nelson. "Mean- ingful Measurements." Arithmetic Teacher 27 (March 1980):6-ll.

This article is part of the results of a grant (SED 7920086) from the National Science Foun- dation to the National Council of Teachers of Mathematics. Any opinions, findings, and con- clusions or recommendations expressed in this publication are those of the authors and do not necessarily reflect the views of the National Sci- ence Foundation. W

February 1981 43

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