Promoting percent as a proportion in eighth-grade mathematics

School Science and Mathematics, Nov 2000 by Dole, Shelley

The literature provides many and varied suggestions for promoting conceptual understanding of percent and performing percent calculations. The diversity of ideas provides a wide selection but offers little clarity on the true nature of percent. From the premise that percent is fundamentally a proportion, this study incorporated a proportional approach for percent problem solving within an instructional program on percent. Classroom research with eighth-grade students indicated that the method was readily adopted by students and helped them experience success in percent problem solving, with percent problem solving proficiency maintained over a delayed period It is hypothesized that the method has the potential to promote students' conceptual knowledge of percent as a proportion and the multiplicative structure of percent, as well as to build proportional knowledge.

Background

The use ofthe word percent and the percent symbol are common in Western society. Being culturally embedded, percent is a valuable topic within the school mathematics curriculum that translates directly to the real world. Despite its pervasiveness, however, percent is often misused or misunderstood when applied to the real world (Watson, 1994), and within the mathematics curriculum, it is both difficult to teach and to learn (Cole & Weissenfluh, 1974; Parker & Leinhardt, 1995; Smart, 1980).

Student Performance

Research has indicated that the concept of percent as abase of 100 is generally well understood by students in the middle school years, and they can competently apply common percent benchmarks (such as 50% is 1/2, 25% is 1/4 10% is 1/10, and so on) (Dole, 1999; Dole, Cooper, Baturo, & Conoplia, 1997; Lembke & Reys, 1994). However, research has also indicated that students generally perform poorly on mathematical tasks and problems involving percents. For example, the fourth National Assessment of Educational Performance (NAEP) of mathematics (Kouba et al., 1988) provided evidence that students at the seventh-grade level had difficulty with percent calculations and appeared to lack understanding of the concepts of percent underlying calculations - a trend continuing through to students in the 1I th grade. In this study, only 32% of 7th-grade students and 62% of 1I th-grade students could calculate 4% of 75. Only 9% of 7th-grade students and 37% of 11th-grade students could solve a two-step word problem involving simple interest calculations. In a more recent study (Dole, 1999), similar results were found. Of 117 eighth-grade students presented with similar percent problems, 57% could calculate 4% of 75, but only 5% could solve a two-step word problem.

Dole et al., (1997) explored the cognitions of 8th, 9th, and 10th-grade students who performed well on percent tasks and reported that such students appeared to utilize a well-developed "sense of percent" (or percent-sense) to interpret and operate in percent situations. These students were seen to draw upon their percent-sense, frequently referring to percent benchmarks to estimate and check reasonableness of calculations. However, the researchers also reported that percent-sense was not evident in the majority of the 90 students in this study.

Approaches for Teaching Percent

A review of the literature reveals a range of ideas, strategies, and methods for both developing percent conceptual knowledge and for assisting percent problem solving and calculations. The picture that emerges, however, is unclear. For developing percent conceptual knowledge, the various approaches appear to suggest the building of percent knowledge from students' knowledge of other topics, particularly common and decimal fractions and, to a lesser extent, ratio and proportion. For percent computation, a variety of arithmetic procedures were suggested, including whole and decimal-fraction multiplication and division and conversions of percent, common, and decimal fractions. The various models and strategies appear to derive from a particular perspective of percent as a decimal fraction, a ratio, or a proportion.

Implications for Instruction

The diversity of suggestions for teaching percent reveal the multidirectional way in which percent instruction can occur, but also serve to highlight the lack of consensus in teaching approaches, From a comprehensive review of percent literature, Parker and Leinhardt (1995) stated that, to date, there is no single best method for teaching percent. They also stated that, although percent meanings are diverse and multiple, the essence of percent is proportionality; it is "an alternative language used to describe proportional relationships" (p. 444).

If it is accepted that percent is a proportion, then instruction in percent must focus on developing students' understanding of percent as a proportion. Building upon students' prior proportion knowledge to build and link to percent knowledge, however, is not an easy task. Students typically do not have a well-developed concept of proportion at the time they meet instruction in percent (Lo & Watanabe, 1997). The development of the proportion concept and proportional reasoning skills takes a long time and is dependent upon the consolidation ofmany other prior mathematics topics (Post, Behr, & Lesh, 1988). The challenge for educators is to find the means to take a holistic approach to percent so that students internalize percent's proportional nature.