Teaching Area, Perimeter, and Volume to 5th Graders

Introduction

In an era of increased demands for teacher and student accountability, identifying better ways of delivering educational services represents a timely and worthwhile endeavor. There are some significant constraints involved in teaching young learners about mathematics concepts that must be taken into account when devising such approaches. Nevertheless, the mandate is clear: state-level high-stakes testing regimens across the country currently require that all students achieve minimum performance standards on reading and mathematics tests in order to be promoted to the next grade. In this environment, identifying what works best — and what does not — is an important first step, and these issues are discussed further below.

Today, primary school teachers are faced with a three-fold challenge when it comes to providing their pupils with quality educational services. The first challenge is the increasingly multicultural and diverse nature of the student body itself. This challenge is compounded by a second: the mandates of various high-stakes testing regimens, inclusion requirements, and accountability standards established by the No Child Left Behind Act (NCLB) of 2001 and the Individuals with Disabilities Education Act (IDEA), which make success in mathematics and reading for all students absolutely essential. For instance, in their book American Standards: Quality Education in a Complex World, the Texas Case, Horn and Kincheloe (2001) report that in Texas, "District/school approval is being linked to student performance rather than compliance to regulations; accountability is focused more on schools as the unit of improvement; continuous improvement strategies involving school-level decisions around specific performance targets are being adopted; new approaches to classroom inspection are being developed; more categories or levels are being developed; school-level test scores are being publicly reported; and more consequences are being attached to performance levels" (p. 124).

The third challenge is the existence of significant gender differences that must be taken into account among young learners. For example, "By the second grade, students already have identified mathematics and science as 'male' domains. By third grade, females rate their own competence in mathematics lower than that of their male classmates, even when they received the same or better grades" (Plymate & Ashley, 2003, p. 162). Because resources are by definition scarce, it is important for educators and administrators alike to identify what works best. Because schools are not factories and young learners are not products, there is absolutely no room for false starts and very little room for experimentation.

Problem Statement and Needs Analysis

In their study "The Effects of Task Demands and Additive Interspersal Ratios on Fifth-Grade Students' Mathematics Accuracy," Hawkins, Skinner, and Oliver (2005) report that a number of careers require a keen grasp of mathematics, including engineering, architecture, and financial planning. Mathematics and geometry involve several important measurement concepts such as length and width, perimeter, area, volume, and angles (Baroody & Coslick, 1998). These authors note that, "In the traditional skills approach, instruction focuses on memorizing measurement procedures and formulas. Unfortunately, premature use of instruments or formulas leaves children without the understanding necessary for solving measurement problems. Moreover, a focus on memorizing facts (e.g., metric equivalents such as 1 meter = 39.37 inches), procedures (e.g., metric conversion methods), and formulas (e.g., area formulas) turns off many students" (p. 15-4). Because so much rides on successful academic outcomes for mathematics instruction in Texas public schools today, providing young learners with the level of instruction they need to become more proficient in measurement concepts represents a timely and worthwhile initiative.

The Texas Education Agency reported that Texas students achieved the highest passing rate on the state's 5th-grade mathematics test to date, with 85% of students mastering the English-language version of the exam and 39% achieving the prestigious "Commended Performance" level (Passing rates on 5th grade math, 2007). To receive the "Commended Performance" rating on the 5th-grade Texas Assessment of Knowledge and Skills (TAKS) math test, Texas public school students must answer at least 40 of the 44 questions correctly (Passing rates on 5th grade math, 2007).

There have been indications that English-speaking Texas students are becoming more proficient at passing these testing regimens; however, whether this reflects teachers "teaching to the test" or actual gains in knowledge and mathematics expertise remains unclear. The Texas Education Agency (2007) reports that, "The passing rate has risen from 79% in 2005, when the passing standards were first fully phased in, to 81% in 2006 to 85% in 2007. This year, 299,337 students took the English-language math test, and 254,233 passed it. An additional 5,834 students took the math test in Spanish, and 2,895, or 50%, passed it. This was also the highest passing rate ever achieved on the Spanish-language math test" (Passing rates on 5th grade math, 2007, p. 2).

By sharp contrast, while the passing rate for Spanish-speaking students increased from 44% in 2005 to 47% in 2006, the percentage of Spanish-speaking students who earned a "Commended Performance" designation was 11% in Spring 2007, representing a decrease from 12% in 2006 (Passing rates on 5th grade math, 2007). Students were also required to answer one fewer question correctly (39 or more questions on the Spanish-version test) to earn the commended label (Passing rates on 5th grade math, 2007).

According to the Texas Education Agency Accountability Manual (2007), "For grades 3 and 5 reading and grade 5 mathematics performance, a cumulative percent passing is calculated by combining the first and second administrations of the TAKS. The results include performance on the Spanish versions of these tests" (p. 3). All fifth-grade students in Texas public schools must successfully pass the TAKS reading and math tests in order to be promoted to sixth grade under Texas law, although they are provided with additional opportunities to take the tests each year (Passing rates on 5th grade math, 2007). Texas state law also requires that these students receive intensive instruction immediately to help improve their skills, and many have already received extra instruction such as small-group tutoring (Passing rates on 5th grade math, 2007).

In cases where students fail to pass one of the high-stakes tests after three attempts, the student is retained in fifth grade; however, a student's parents have the authority to appeal the retention, and a committee composed of the child's parents, teachers, and principal must be convened in response (Passing rates on 5th grade math, 2007). These committees, termed "Grade Placement Committees," are responsible for reviewing students' class work and any other relevant information. According to the Texas Education Agency, "If all committee members agree that the student can be successful if promoted and given extra instructional assistance, then the child may be promoted to the next grade. In 2006, 93% of the 5th-grade students tested in English ultimately passed the math test after three attempts, as did 74% of those tested in Spanish" (Passing rates on 5th grade math, 2007, p. 3).

Many careers in which fifth-grade students may be interested require a good grasp of mathematical fundamentals. Moreover, the requirements of high-stakes testing regimens have made mathematics a high priority for educators. According to Chanter and Welsh (2000), "The goals set forth in the National Council of Teachers of Mathematics' Principles and Standards for School Mathematics (2000) recommend that students learn to value mathematics, become confident in their abilities, become mathematical problem solvers, and learn to reason and communicate mathematically" (p. 236). Furthermore, the existing NCTM standards recommend that fifth graders become acquainted with a number of geometry concepts at this point in their academic careers.

NCTM Standard 13 for grades 5–8 recommends that "the mathematics curriculum should include extensive concrete experiences using measurement so that students can:

The rationale for this exercise relates to the need to provide young learners with hands-on opportunities to learn about measurement concepts that go beyond what is offered in public school textbooks. Baroody and Coslick emphasize that, "Instruction that does not actively involve children in measuring and reflection can impede conceptual development. Unfortunately, the traditional textbook-based skills approach focuses on memorizing by rote measurement facts (e.g., equivalent measures such as 12 inches = 1 foot) and measurement procedures (e.g., how to use a ruler)" (1998, p. 15-9).

Absent hands-on exercises, many young learners will not have an opportunity to construct an understanding of the process of measurement or a concept of measurement unit, which can frequently result in mechanical and inappropriate applications of measurement knowledge and tools. Baroody and Coslick point out that many elementary-level children tend to confuse area with perimeter and vice versa. Some common types of errors made by young learners when using a ruler include the following:

These authors also emphasize that, "Children should be encouraged to look for patterns and to use what they know to reinvent area and perimeter formulas. Deriving these formulas themselves can promote mathematical power in three ways: increase their confidence that they can make sense of mathematics, engage them in genuine mathematical thinking, and foster understanding" (Baroody & Coslick, 1998, p. 15-18). The rationale for using these exercises also relates to improving long-term retention and comprehension of important measurement concepts: "Promoting adaptive expertise makes it less likely children will forget the formulas, more likely they can reconstruct them if they do, and far more likely they will be able to devise new formulas on their own" (Baroody & Coslick, 1998, p. 15-18).

The resources available for this exercise include typical fifth-grade classroom materials available in Texas public schools, the TAKS Study Guides provided by the Texas Education Agency (designed to help students strengthen the skills taught in class and tested on TAKS), and some inexpensive hands-on materials such as pie pans, cardboard boxes, paper towel cardboard tubes, Frisbees, string, plastic tumblers, and Styrofoam cups. These study guides are designed for students to use on their own or for students and families to work through together; concepts are presented in a variety of ways that will help students review the information and skills they need to be successful on the TAKS.

The goal of this initiative speaks directly to the role of schools in providing young learners with the knowledge and skills they will need to succeed in school and in their professional careers. Based on the mandates established in Chapter 111, Texas Essential Knowledge and Skills for Mathematics, Subchapter A, Elementary (5.10): Measurement, the goal of this exercise is to provide fifth-grade public school students in the Texas primary school in question with a superior approach to learning the concepts of area, perimeter, and volume, and to improve their performance on the state-mandated high-stakes testing regimens currently in place.

The Texas Education Agency (2007) provides the following student demographic categories used in Texas public schools.

Table 1. Texas public school student demographics.

Economic Status: A student may be identified as economically disadvantaged if he or she meets eligibility requirements for the federal free or reduced price lunch programs; Title II of the Job Training Partnership Act (JTPA); Food Stamp benefits; Temporary Assistance to Needy Families (TANF) or other public assistance; received a Pell grant or funds from other comparable state programs of needs-based financial assistance; or is from a family with an annual income at or below the official federal poverty line.

Ethnicity: Districts assign student ethnicity from one of the following categories: American Indian or Alaskan Native; Asian or Pacific Islander; Black, not of Hispanic origin; Hispanic; and White, not of Hispanic origin.

At Risk: A student is identified as at risk of dropping out of school based on state-defined criteria (TEC 29.081(d)). Statutory criteria for at-risk status include students who were not advanced from one grade level to the next for one or more school years; students in grades 7–12 who did not maintain an average equivalent to 70 on a scale of 100 in two or more foundation curriculum subjects; students who did not perform satisfactorily on a required assessment instrument; students in prekindergarten through grade 3 who did not perform satisfactorily on a readiness test; students who are pregnant or are parents; students placed in an alternative education program; students who have been expelled; students currently on parole, probation, or other conditional release; students previously reported as having dropped out; students of limited English proficiency; students in the custody of the Department of Protective and Regulatory Services; homeless students; and students residing in residential placement facilities including detention facilities, substance abuse treatment facilities, emergency shelters, psychiatric hospitals, halfway houses, or foster group homes.

Special Education Status: Special education status indicates the student is participating in a special education instructional and related services program or a general education program using special education support services, supplementary aids, or other special arrangements.

Source: Appendix D — Data Sources, Texas Education Agency Accountability Manual, 2007.

There are currently 29 pupils in the 5th-grade class in question; of these, 15 are male (51.72%) and 14 are female (48.27%). Approximately three-quarters of the class (21 students, or 72.41%) are English-speaking pupils, with the remaining 8 students (27.58%) being Spanish-speaking pupils.

Learner Analysis

Many fifth-grade pupils bring a good grasp of measurement concepts with them to school, but in some cases these concepts have been based on erroneous processes or are otherwise flawed in their rationale.

For the purposes of this exercise, all fifth-grade pupils will be assumed to possess the requisite knowledge and skills required to achieve promotion from the fourth grade.

Attitudinal problems with learning mathematics can be confirmed by virtually any primary classroom teacher (Kenschaft, 1997). There are also some problems associated with the way math tests are designed. According to Hawkins and his colleagues (2005), many American fifth-grade pupils fail to achieve satisfactory results on mathematics tests because of the manner in which the tests are designed. These researchers report that many pupils react negatively to lengthy problems, a reaction compounded by the paucity of opportunities for success. To overcome these constraints, Hawkins and colleagues recommend including shorter math problems interspersed with more difficult ones to maintain interest and commitment in young learners. They suggest that a ratio of one-to-one is preferable: "The current results demonstrated that interspersing briefer, easier problems following each target problem (1:1 ratio) did increase target problem accuracy on written assignments" (p. 543).

According to Kenschaft (1997), "Area, like length, is another form of measurement. Like length, area plays a more vital role in advanced mathematics than, say, temperature and weight. Area measures two-dimensional things; we might say these things have both length and width. Or we might say they have length in two dimensions, one of which we choose to call 'width.' For beginners, it sounds confusing. For the rest of us, it's just the words that are confusing" (p. 117). A useful and cost-effective method of teaching fifth-grade students about the concept of area — one that can be used after pupils are introduced to measuring the area of rectangles and squares — is described below.

Distribute pie pans, Frisbees, or other circular objects to students and ask them to estimate the area of each object. If pupils have already learned the formula for the area of a circle, instruct them to devise their own non-algorithmic method. Then consider how pupils can apply their knowledge about finding the area of rectangles and squares to estimate the area of the pie pan (Baroody & Coslick, 1998, p. 15-9).

Area: Questions for Reflection

How might the strategy pupils used be refined to give a more accurate estimate? Discuss the relative accuracy of the strategies used by different people in the group or class. How might beans be used to estimate the area of a circle? How might weight be used?

When presented with the problem above, one pupil proposed: "Wrap a string around the pie tin and then fashion the string into a square. Compute the area of the square, and this will tell you the area of the pie tin." (a) Is this pupil's conjecture correct or not? Do polygons with the same perimeter have the same area? (b) How could this conjecture be tested?