length, perimeter, area, and volume

LENGTH, PERIMETER, AREA, AND VOLUME

Leibniz was in many respects a mathematical amateur. He studied philosophy in college, and like many seventeenth-century mathematicians, was a lawyer. He worked his entire adult life as a courtier for various German principalities. Although his diplomatic travels brought him into contact with mathematicians in Paris, London, and the Netherlands, he learned mathematics virtually on his own. See also Calculus, Overview; Logic; Newton

SELECTED REFERENCES Aiton, Eric. LeibnizJ A Biography. Bristol, England: Hilgar, 1985. Boyer, Carl B. A History of Mathematics. 2nd ed., rev. Revised by Uta C. Merzbach. New York: Wiley, 1991, pp. 391-414. Hall, Alfred Rupert. Philosophers at war: The Quarrel Between Newton and Leibniz. Cambridge, England: Cambridge University Press, 1980. Hofmann, Joseph. Leibniz in ParisJ 1672-1676. Cambridge, England: Cambridge University Press, 1974. Hofmann, Joseph, et al. "Gottfried Wilhelm Leibniz." In Dictionary of Scientific Biography (pp. 149-168). Vol 8. Charles C. Gillispie (ed.). New York: Scribner's, 1973. Katz, Victor. A History ofMathematics:An Introduction. New York: HarperCollins, 1993, pp. 428-493. Kline, Morris. Mathematical Thought from Ancient to Modern Times. New York: Oxford University Press, 1972, pp.342-383. DANIEL S. ALEXANDER

Only about 10% of seventh graders and 52% of en tering secondary students can find the area of square given the length of one of its sides. Miscon ceptions are common. Many students believe tha the angle sum of a quadrilateral is the same as it area and that the area of a quadrilateral can be ob tained by transforming it into a rectangle with th same perimeter. Research has not just identifiel problems. Studies also have contributed to our un derstanding of the critical mathematical aspects ( measurement. More important, they have helped u understand how students think about measuremer concepts and thus can guide educational efforts t ameliorate weaknesses in students' performanct These are discussed in the following sections. Th final section discusses implications for teaching.

BACKGROUND: MATHEMATICAL CONCEPTS OF MEASUREMENT

There is a fundamental difference between sc entific and mathematical measurement. Scientifi measurement is observational and always includt some degree of error. Mathematical measurement based on certain foundational concepts. We will prt sent these concepts briefly in the context of lengt measurement (for a discussion, see Wilson and 0: borne 1988). 1.

2.

LENGTH, PERIMETER, AREA, AND VOLUME Measurement is one of the principal real-world applications of mathematics. It bridges two critical realms of mathematics: geometry or spatial relations and real numbers. Done well, education in measurement can connect these two realms, each providing conceptual support to the other. Indications are, however, that this potential is usually not realized.

STUDENT PERFORMANCE ON MEASUREMENT TASKS Many students use measurement instruments or count units in a rote fashion and apply formulas to attain answers without meaning (Clements and Battista 1992). For example, less than 50% of seventh graders can determine the length of a line segment when the beginning of the ruler used to measure it is not aligned at the beginning of the line segment.

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Number assignment: Given a pair of points. and B J there is exactly one nonnegative numbe d (A,B) > 0, that is the length of segment AB. Comparison: If segment AB is contained in se~ ment AC then d(A,B) < d(A,C). Congruence: Segment AB congruent to segmeI CD means that d(A,B) = d(A,C). Unit: There is a line segment that can be a: signed the length 1. Additivity: A line segment made by joining tv. distinct line segments has a length equal to tt sum of the lengths of the joined segments. This important, as now we know that the structure l addition can guide children's thinking abOl measurement, and vice versa. Archimedean iteration: If a point B is betwee points A and C on a line, then a counting nun ber n can be found so that n X d (A,B) . d(A,C). Thus, some whole number of copies ( iterations of AB laid end-to-end is all that needed to get beyond point C. This notion is tt basis for developing number lines and ruler uSt J

3. 4. 5.

6.

These ideas should be encountered by children in a informal, significant setting, not introduced formall


The history of measurement is the story of a continuing effort to achieve standardization of measures and measurement processes; such understandings must be constructed by children over time.

LENGTH An early emphasis in research on measurement focused on conservation-the idea that a physical quantity does not change during certain transformations. Jean Piaget and his collaborators (Piaget and Inhelder 1967; Piaget et al. 1960) found that children younger than five years would judge length in terms of endpoints only; therefore, a line segment and a bent path with the same endpoints would be judged to have the same length. These researchers claimed that children achieve an understanding of linear and area measurement only at about age nine. Such understandings depend on mental operations such as subdivision and ordering. If two equal-length strips are cut as shown in Figure 1, younger children will view one segment as longer. They may judge the left strip to be longer, because it has longer pieces, or they may judge the right strip to be longer, because it has more pieces (Carpenter and Lewis 1976). When they can coordinate both the subdivision and reconnection of the parts and the ordering of the positions of the parts, they can understand linear measurement, according to the Piagetian position. When first to third graders were given number line estimation problems to be solved with specially designed rulers, they progressively altered their estimation strategies from strictly sequential ones (that neglect the matter of distance between points on the number line) to ones that incorporate elements of proportional reasoning (Petitto 1990). Their acquisition of a concept of equal

DD D DD DD Figure 1

intervals required learning both the convention of equal intervals and conservation of length. Children develop, for example, three types of strategies for solving different length problems: (1) some students, rather than segmenting lengths and connecting the number for the measure with the length of the line segment, applied general strategies such as visual guessing of measures; (2) most students drew hash marks, dots, or line segments to partition lengths; they needed to have perceptible units such as these to quantify the length; and (3) a few other students did not use physical partitioning; however, they did use quantitative concepts in discussing the problems, drew proportional figures, integrating the number for the measures with the lengths of the segments, and sighted along line segments to assign them a length measure. These students have created an abstract unit of length, a "conceptual ruler" that they can project onto unsegmented objects. Thus, children must create an abstract unit oflength (Clements et al. 1997; Steffe 1991). This is not a static image, but rather an interiorization of the process of moving (visually or physically) along an object, segmenting it, and counting the segments. This study also has implications for teaching. Working with length activities using the computer program Logo helped these students develop more sophisticated strategies (Clements et al. 1997; Clements and Meredith 1994). Other research indicates that Logo can help young children learn about measurement, because Logo's turtle graphics provides an arena in which young children may use units of varying size, define and create their own units, maintain or predict unit size, and create length rather than endpoint representations through either iterative or numeric distance commands (Clements and Battista 1992). On and off computer, teachers cannot assume that children understand measure fully, even if they can complete textbook exercises involving reading a pictured ruler aligned with a pictured object. More worthwhile experiences include measuring with different size units and different materials, from paper clips to inch cubes to rulers. Students should also compare the measures of different lines, measure the same length with different units, and use computer programs that present measurement tasks with varying unit sizes (Clements and Meredith 1994).

AREA, PERIMETER, AND VOLUME Understanding of area measure, according to the Piagetian position, involves coordinating many ideas.

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Children must understand that subtracting equal parts from equal wholes results in equal remainders and that decomposing and rearranging shapes does not affect their area. Important is the ability to coordinate two linear dimensions to build the idea of a two-dimensional space. This explains why understanding of area is often not fully developed until twelve to thirteen years of age. Recent research has revealed that just structuring a two-dimensional region is not a simple task. For example, while most elementary students have no difficulty covering a region with tiles and finding area by counting individual tiles, many cannot represent the results of such actions in a drawing. These children do not interpret arrays in terms of rows and columns, which obstructs their learning about area measurement and probably makes formula use next to meaningless. Some elementary students can make an array drawing; however, many of these students still cannot apply their multiplication and linear measurement skills to determine the area of an array (Outhred and Mitchelmore 1992). Research has identified "rules" that children use to make area judgments. For example, four- and fiveyear-olds match one side of figures to match their areas. They also use height and width rules to make area judgments. Children from six to eight years use a linear extent rule, such as the diagonal of a rectangle. Only after this age do most children move to multiplicative rules (Forman 1993). Elementary school children often confuse perimeter and area. For example, they believe that counting the units around a figure gives its area. Many adolescents can conserve area, but believe that the perimeter of the figure is also conserved. Differentiating and coordinating these two measures is a difficult task. Teachers help when they offer many experiences comparing areas, encouraging children to use their own strategies (even one-by-one counting) rather than teaching rote rules. Children should also build different shapes (e.g., rectangles) with the same area, checking the perimeter for each. They should guess and check how many unit squares fit into various rectangular frames. When three dimensions are involved, it is no surprise that people have similar and sometimes greater difficulties with volume concepts. Again, informal educational opportunities, starting with counting, are important.

PEDAGOGICAL IMPLICATIONS While researchers and educators have taken a variety of positions, most agree that understanding

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measurement is an important conceptual area, as well as a skill. Teaching of measurement should build on children's intuitive spatial understanding and help children establish connections between this understanding and number. Counting preassigned units or multiplying dimensions in formulas alone will not result in quality learning. As we have seen, many students apply formulas without understanding how they work or what they mean. Such rote practice is especially harmful to students' development of understanding of the nature of units of measurement in one, two, and three dimensions. Experiences with qualitative comparison are indicated. Young children should have a variety of experiences comparing the size of objects in various dimensions; for example, finding all the objects in the room that are as long as their forearm. Connections should be made to number through counting, and then to arithmetic, including simple proportional relations such as doubling or halving, using real-world materials and problems. Close observation of children's strategies for solving measurement problems is pedagogically useful. For example, teachers presenting children with length tasks such as sketching a rectangle with particular dimensions should observe if students are partitioning the lengths (Clements et al. 1997). Many students draw hash marks, dots, or line segments to partition or segment lengths but do not maintain equal-length parts. They need to have perceptible units to quantify the length. Continued presentation of such tasks, such as drawing a 10 X 5 cm rectangle, with an emphasis on equal internal partitioning and the creation of different units of length, will help these students. Eventually, students should use a variety of nonstandard and standard units to discover that the results of counting to yield a measure depend on the unit. (Note that some research indicates that standard instruments such as rulers may support reasoning more than nonstandard units such as paper clips. The traditional "nonstandard then standard" sequence may be overly simplistic.) Students who do not partition lengths may be using one of two quite distinct strategies. Some can mentally partition the lengths using a "conceptual ruler" (i.e., they are using the third type of strategy identified previously). These students should be challenged with more difficult tasks (see Figure 2). If they do not show signs of this ability, they may be using the first kind of strategy; such students need to engage in partitioning and iterating lengths, continually tying the results of that activity to their counting.


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Figure 2 Students fill in all the missing measures (Clements et al. 1995).

For example, they might draw a favorite toy, measuring the toy and drawing it using the same (and, later, a smaller) measure. The research on area and volume suggest that even young children can base their judgments of quantity on several dimensions, but in many contexts, such as judging the amount of water in a glass, they use only one. When they do attend to multiple dimensions, they often use an additive rather than a multiplicative rule. In middle school, students start to use multiplicative reasoning, though even older students are not consistent in this regard; many still calculate the perimeter when they need to find the area. Instruction on formulas for area and perimeter may actually hinder the development of both concepts and skills in measurement. Unfortunately, such bare-bones instruction on formulas is all most mathematics textbooks include (Fuys et a1. 1988). Thus, for example, simple counting of units to find area (achievable by preschoolers) leading directly to teaching formulas is a recipe for instructional disaster. Instead, educators should build upon young children's initial spatial intuitions and appreciate the need for students to: (a) construct the idea of measurement units (including development of a measurement sense for standard units; e.g., finding common objects in the environment that have a unit measure); (b) have many experiences covering quantities with appropriate measurement units and counting those units; (c) spatially structure the object they are to measure (e.g., linking counting by groups to the structure of rectangular arrays; building two- and three-dimensional multiplicative concepts); (d) construct the inverse relationship between the size of a unit and the number of units used in a particular measurement; and (e) construct two- (and later three-) dimensional space and corresponding multiplicative relations. For example, ask students to

make boxes out of nets (2D patterns that fold into 3D shapes) and determine how many cubes fit in the boxes. Students also might determine how many tiles cover a given rectangle (sometimes the tiles could be square, other times they could be nonsquare rectangles). Students could make buildings (not prisms, but columns with different heights) and be asked to draw the buildings from different perspectives. For middle school students, these basic ideas can be expanded to include explicit generalizations of measurement concepts and processes across different types of quantities (e.g., the foundational concepts of measurement); differences between scientific and mathematical measurement (scientific measurement is based on observation and always contains error); and the relationships between length, perimeter, area, and volume (e.g., what happens to area when the length of a side is doubled). For most students, connections between geometric forms and numerical ideas are tenuous at best, even in situations designed to emphasize and develop these connections. Such lack of linkages would appear to limit the growth of number sense, geometric knowledge, and problem-solving ability. Studying more measurement and geometry may ameliorate this situation. Using situations that help students forge such links is also indicated. Students' integration of number and geometry was especially potent and synergistic in Logo environments (especially mathematically oriented versions of Logo, e.g., Clements and Meredith 1994). In most tasks, students should use measurement as a means for achieving a goal, not only as an end in itself. See also Geometry, Instruction; Logo; Measurement; Systems of Measurement

SELECTED REFERENCES 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(1976):53-58. Clements, Douglas H., and Michael T. Battista. "Geometry and Spatial Reasoning." In Handbook of Research on Mathematics Teaching and Learning (pp. 420-464). Douglas A. Grouws (ed.). New York: Macmillan, 1992. - - - , Joan Akers, Virginia Woolley, Julie Sarama Meredith, and Sue McMillen. Turtle Paths. Palo Alto, CA: Seymour, 1995. Clements, Douglas H., Michael T. Battista, Julie Sarama, Sudha Swaminathan, and Sue McMillen. "Students' Development of Length Measurement Concepts in a Logo-based Unit on Geometric Paths." Journal for Research in Mathematics Education 28(1997):70-95.

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LESSON PLANS

Clements, Douglas H., and Julie Sarama Meredith. Turtle Math. Montreal, Quebec, Canada: Logo Computer Systems (LCSI), 1994.

Forman, Ellice. "Middle School Students' Understanding of Area, Perimeter, Surface Area, and Volume." Unpublished manuscript, University of Pittsburgh, 1993. Fuys, David, Dorothy Geddes, and Rosamond Tischler. The van Hiele Model of Thinking in Geometry among Adolescents. Reston, VA: National Council of Teachers of Mathematics, 1988. Outhred, Lynne, and Michael Mitchelmore. "Repre-

sentation ofArea: A Pictorial Perspective." In Proceedings of the Sixteenth PME Conference (pp. 194-201). Vol. II. William Geeslin and Karen Graham (eds.). Durham, NH: Program Committee of the Sixteenth PME Conference, 1992.

Petitto, Andrea L. "Development of Numberline and Measurement Concepts." Cognition and Instruction 7(1990):55-78.

Piaget, Jean, and Barbellnhelder. The Child's Conception of Space. NewYork: Norton, 1967. - - - , and Alina Szeminska. The Child's Conception of Geometry. London, England: Routledge and Kegan Paul, 1960. Steffe, Leslie P. "Operations that Generate Quantity." Learning and Individual Differences 3(1991):61-82.

Wilson, Patricia S., and Alan Osborne. "Foundational Ideas in Teaching about Measure." In Teaching Mathematics in Grades K-8: Research Based Methods (pp. 78-110). Thomas R. Post (ed.). Boston, MA: Allyn and Bacon, 1988.

DOUGLAS H. CLEMENTS MICHAEL T. BATTISTA

LESSON PLANS A form of instructional planning, which is a primary responsibility of every classroom teacher. Often, the success or failure of the instructional experience for both teacher and student is directly related to the thoroughness of such planning from broad-based goals to seemingly trivial, procedural issues. A lesson plan is written to organize the flow of information among students and teachers. It is not designed to disseminate information exclusively from the teacher to the students; rather, a lesson plan can, and should, be designed with many potential forms of information flow in mind. The development of goals is an important first step in the lessonplanning process. When writing lesson plans with a mathematics focus, teachers may find it helpful to consider the five goals of the National Council of Teachers of Mathematics (NCTM 1989). These goals provide a framework for the development of

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long-term instructional planning as well as individual lesson plans.

CURRICULUM MAPPING Curriculum mapping includes both horizontal planning, within the grade level, among the disciplines; and vertical planning, the sequence of curriculum from grade level to grade level. As a part of long-term curriculum mapping, the need exists to develop individual lesson plans. Individual lesson plans provide a vehicle for the teacher to plan daily, short-term lessons either exclusive to a particular discipline or across two or more disciplines.

A LESSON PLAN FORMAT Critical Elements of the Daily Lesson Plan With the daily lesson plan in mind, the critical elements to include are: (a) objectives/outcomes for the lesson; (b) key concepts, skills, and/or strategies to be taught; (c) essential questions to be asked; (d) methodology to be used; (e) resources needed; (f) procedure for implementing instruction; (g) closure; (h) assessment; and (i) lesson adaptations. Furthermore, it is important that teachers have a clear rationale for choosing all the elements of the lesson plan and relating the learning to real-life needs and experiences. The following sections include a more in-depth explanation of the critical elements of the daily, lesson-plan format and an example that can guide teachers in creating teacher-directed plans, studentcentered plans, multiple grouping scenarios, minilessons, and so forth. It also can be modified for discovery and experiential learning. A continuous cycle of the critical lesson elements can be instantaneous or can occur over long periods of time. "For all of us, the ultimate goal is more coherent, organic, and integrated schooling for American young people" (Zemelman, Daniels, and Hyde 1993, 17).

o bjectiveslOutcomes Many state boards of education and school districts have mandated adherence to the achievement of outcomes rather than merely "covering the curriculum." In this entry, we use both the terms objective and outcome cautiously, and refer to objectives with intended learning outcomes (Gronlund 1995). We define "outcomes-based education" (OBE) as "focusing curriculum, instruction, and assessments on student success in achieving the knowledge skills and attitudes through standards-based outcomes"