Absent hands-on exercises, though, 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. For instance, Baroody and Coslick point out that many elementary-level children tend to confuse area with perimeter and vice versa; some common types of errors that are made by these young learners when using a ruler, for example, include the following:
Counting the number of has marks starting with 0 rather than the units between the marks;
Treating the 1 hashmark as the starting point instead of the 0 hashmark and counting the 2 hashmark as one; and,
Placing the edge of an object at the 1-inch hashmark rather than the 0 hashmark (Baroody & Coslick, 1998, p. 15-9).
Finally, these authors 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 these 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 the typical fifth-grade classroom materials available in Texas public schools, the TAKS Study Guides provide by the Texas Education Agency designed to help students strengthen the skills that are taught in class and tested on TAKS (the study guides are designed for students to use on their own or for students and families to work through together and 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), as well as some inexpensive materials (i.e., pie pans, cardboard boxes, paper towel cardboard tubes, Frisbees, string, plastic tumblers, Styrofoam cups and so forth which were donated by the author).
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. Therefore, 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 improve their performance on the state-mandated high-stakes testing regimens that are currently in place.
The Texas Education Agency (2007) provides the following student demographic categories used in Texas public schools.
Texas public school student demographics.
Economic Status student may be identified as economically disadvantaged by the district 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 program of needs-based financial assistance; or, is from a family with an annual income at or below the official federal poverty line.
Districts assign student ethnicity from one of the following categories: American Indian or Alaskan Native (not evaluated separately for accountability); Asian or Pacific Islander (not evaluated separately for accountability); Black, not of Hispanic origin; Hispanic; and White, not of Hispanic origin.
At Risk student is identified as at risk of dropping out of school based on state-defined criteria (TEC 29.081(d).) the statutory criteria for at risk status include each student who is under 21 years of age and who: was not advanced from one grade level to the next for one or more school years; is in grades 7, 8, 9, 10, 11, or 12 and did not maintain an average equivalent to 70 on a scale of 100 in two or more subjects in the foundation curriculum during a semester in the preceding or current school year or is not maintaining such an average in two or more subjects in the foundation curriculum in the current semester; did not perform satisfactorily on an assessment instrument administered to the student under TEC Subchapter B, Chapter 39, and who has not in the previous or current school year subsequently performed on that instrument or another appropriate instrument at a level equal to at least 110% of the level of satisfactory performance on that instrument; is in prekindergarten, kindergarten or grades 1, 2, or 3 and did not perform satisfactorily on a readiness test or assessment instrument administered during the current school year; is pregnant or is a parent; has been placed in an alternative education program in accordance with TEC 37.006 during the preceding or current school year; has been expelled in accordance with TEC 37.007 during the preceding or current school year; is currently on parole, probation, deferred prosecution, or other conditional release; was previously reported through the PEIMS to have dropped out of school; is a student of limited English proficiency, as defined by TEC 29.052; is in the custody or care of the Department of Protective and Regulatory Services or has, during the current school year, been referred to the department by a school official, officer of the juvenile court, or law enforcement official; is homeless, as defined by 42 U.S.C. Section 11302 and its subsequent amendments; or resided in the preceding school year or resides in the current school year in a residential placement facility in the district, including a detention facility, substance abuse treatment facility, emergency shelter, psychiatric hospital, halfway house, or foster group home.
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, http://www.tea.state.tx.us/perfreport/account/2007/manual/app_d.html.
Relevant group characteristics
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%); about three-quarters of the class (21 or 72.41%) are English-speaking pupils, with the remaining eight students (or 27.58%) being Spanish-speaking pupils.
Prior knowledge of topic
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.
Entry level knowledge and skills
For the purposes of this exercise, all fifth-grade pupils will be assumed to possess the requisite knowledge and skills that are required to achieve promotion from the fourth grade.
Attitudes and/or motivation toward the subject
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 of these pupils react negatively to lengthy problems, a reaction that is compounded by the paucity of opportunities for success. To overcome these constraints, Hawkins and his colleagues recommend including some shorter math problems interspersed with the more difficult ones to maintain interest and commitment to succeed in these young learners. In this regard, Hawkins et al. (2005) suggests 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 that can be used after pupils are introduced to measuring the area of rectangles and squares is described further below.
Distribute pie pans, Frisbees or other circular objects to students and ask them to estimate their area; if pupils have already learned the formula for the area of a circle, instruct them to devise their own (nonalgorithmic) method.…