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The mathematics assessment results of the National Assessment of Educational Progress (NAEP) as reported by the National Council of Teachers of Mathematics (NCTM, 2004) indicated that the performance of fourth-, eighth-, and 12th-grade students in the U. S. was below proficiency level during the past two decades. The international assessments such as the Program for International Student Assessment (PISA) also showed U.S. students performed lower than students from most other countries (Lemke et al., 2004). Those assessments use mathematical word problems for measuring student achievement. This study was motivated by the low performance of U.S. students on the 2007 NAEP reported by the National Center for Education Statistics (NCES, 2007), international studies such as the 2007 PISA, the 1965 First International Mathematics Study (FIMS), the 1982 Second International Mathematics Study (SIMS), the 2003 Trends in Mathematics and Science Study (TIMSS), and the prevailing low mathematics performance of high school students in one urban school district in Northern New Jersey. Both national and international assessments continue to document U.S. students' low achievement in mathematics.
The 2007 NAEP reported improved test results in mathematics for fourth-grade and eighth-grade students. Those improvements over the span of 17 years (1990 to 2007), as well as current scores, remained low on a scale of 0 to 500 as shown in Figure 1.1. Fourth-grade students improved from 213/500 to 240/500 and eighth-grade students improved from 263/500 to 283/500 during this period. However, the average score of 12th-graders remained stagnant for a decade (1990-2000) at around 300/500.
Figure 1.1. National Assessment of Educational Progress in Math
(1990-2007). Scores are on a scale of 0 to 500.
The NCTM expressed its serious concern about the discouraging scenario of mathematics education in the U.S. According to its Principles and Standards for School Mathematics (NCTM, 2000), "many students are not learning the mathematics they need or are expected to learn" due to the lack of engaging curriculum and learning opportunity (p. 5). Many students continue to be taught mathematics by teachers with limited preparation in that content and pedagogical area. As a result, as noted by the NCTM, instruction in mathematics differs widely.
Mathematical problem solving is the central issue that many educators and researchers have been addressing for quite some time. Empirical investigations support the finding that students' varying mathematics achievement in different countries may be a function of the amount and kind of instruction they receive in mathematics (Stevenson, 1992; Stevenson, Chuansheng & Lee, 1993; Stevenson, Lee, Chen, Stigler, Hsu, & Kitamura, 1990; Stevenson & Stigler, 1992; Stigler, Lee, & Stevenson, 1990). These reports, as well as the assessment results of the PISA, FIMS, SIMS, and TIMSS and the concern raised by the NCTM, strongly suggest that students' mathematics achievement depends on the quality of instruction.
For most public high school students in New Jersey, passing the state high-stakes test--the High School Proficiency Assessment (HSPA)--in either the 11th or 12th grade is a requirement for graduation. In addition, college-bound students in New Jersey and other states consider taking the Scholastic Aptitude Test (SAT) or the American College Test (ACT). At the research site, an inner city high school in Northern New Jersey, the results of the HSPA and SAT are much lower than the state average, according to the district's 2006 annual report. During his teaching for the last six years at this school, the researcher has observed that the mathematics ability of most students is generally low. According to the 2006 district report, the percent of students who passed the HSPA during the previous two years is only 20.7% as compared to the state passing average of 85.8%. Similarly, the average SAT scores in mathematics for students at the study site has been hovering around 370/800 during the previous three years. This is an alarmingly low performance compared to the state average of 518/800 for the same time period.
The mathematics sections of both the HSPA (as evident from the released 2001 test by the New Jersey Board of Education) and the SAT (as evident from The College Board SAT, 2005, 2006) are primarily comprised of word problems. For example, 67% (24 out of 36 questions) of the problems on the 2001 HSPA are word problems. Similarly, 59 % (32 out of 54 questions) of the problems on the seven SAT practice tests published by the College Board are word problems. Therefore, the skill for solving word problems may be the most important factor that affects success on the mathematics portions of those tests. These problems require students to read with complete understanding, to create a representation when a diagram or any other form of representation is not given, and to present a solution.
This study focused on a diagramming method of solving word problems similar to those found on the HSPA, SAT, and ACT. The method stressed the importance of vocabulary and diagramming as two necessary skills for successful problem solving. It involved training teachers, 6-week instructional unit, and pre- and post- written problem solving tests of student participants. Student achievement on word problems was measured as the change from their pretest and posttest scores.
This study was guided by the following research question: What is the effect of a 6-week unit that focuses on diagramming as a representational technique on high school students' achievement in solving math word problems?
Justification of the Research
Many research studies in mathematics education use problem solving and word problems interchangeably (Branca, 1987; Brown, Cronin, & McEntire, 1994; Kilpatrick, 1985). According to the National Council of Supervisors of Mathematics, "learning to solve problems is the principal reason of studying mathematics" (1977, p. 2). The NCTM (1980) also suggested that problem solving be considered as the major goal of learning mathematics in school from 1980 to 1989. Student achievement in mathematics was also identified as a major concern in the U. S. with the publication of A Nation at Risk (U.S. Department of Education, 1983). The commission that authored the report recommended that the teaching of mathematics should focus on geometry, algebra and real-life application of mathematics. The latest mathematics advisory committee expressed similar concern about student achievement in mathematics (The National Mathematics Advisory Panel, 2008). Thus, for the last three decades, students' problem solving ability has been the main topic of discussion in mathematics education.
According to Suydam (1980), most of the research studies on problem solving during the last century used word problems that are essentially repetitive exercises-"especially at the elementary school level" (p. 35). However, the extent to which research recommendations are implemented in classrooms is a matter of debate, as student achievement in problem solving has been a national concern for at least the last three decades (Lemke et al., 2004).
As noted in the introduction of this chapter, international reports, such as PISA and TIMSS highlighted U.S. students' poor mathematical achievement. According to the NCTM (2004), the PISA measures students' numerical and problem solving abilities whereas the TIMSS measures students' performance on concepts. The NCTM also reported that low-stakes tests such as NAEP, TIMSS, and PISA generate performance results for group of students, not for individual students.
Based on the results of PISA (2000, 2003), Lemke et al. (2004) reported that U.S. students performed below the average of those from most of the 41 participating Organization for Economic Cooperation and Development (OECD) countries in mathematical ability and problem solving. Lemke et al. found that U.S. participants demonstrated poor performance in mathematical concepts such as volume, geometric figures, functions, problem solving, in 2000 and 2003 assessments. Following that report, high school reform was considered a national importance (U.S. Department of Education, 2005). The latest report of PISA (2007) also indicated that U.S. students' mathematical achievement remains below average at the international level.
As noted previously, U.S. students' performance in problem solving has been a major concern at the national and international level. Problems in those assessments are mostly traditional, textbook type word problems and it is generally accepted that students have difficulties solving them. Whether or not students have to choose the correct answers from a list or to explain their answers, a response to a word problem demands comprehension, representation, and solution.
A review of the scholarly literature showed that educators and researchers have examined the effectiveness of various teaching methods and representational techniques for solving math word problems (Baker, 1992; Bovenmyer, 1989; Delinda, 2002; Grossen & Carnine, 1990; Hutchinson, 1989; Jitendra & Hoff, 1996; Simon, 1986; Toppel, 1997; Walker & Poteet, 1989; Waters, 2004; Zawaiza & Gerber, 1993). However, these studies primarily focused on students with learning challenges.
The purpose of this study was to investigate the effect of diagramming as a representational technique for solving word problems on the achievement of regular, nonclassified, high school students. The review of the literature indicated that further research is needed to understand the reasons for the low level of U.S. student achievement and to develop teaching methods that can improve students' skills and achievement in solving math word problems. This research was an effort to address that research need.
The researcher has been teaching mathematics at the secondary level for the last six years. The researcher views mathematics as a natural, essential, and invisible structure that coexists with and improves human life when applied effectively. In addition, he strives to reach many students with this idea, so they can experience the existence of mathematics and use mathematics as a tool to enhance their lives.
The researcher embraced mathematics as an interesting subject while studying Mechanical Engineering and had a lot of fun using mathematics. Throughout his career the researcher applied mathematical concepts to everyday tasks for improved efficiency at work and in every other field of his life. Even now he gets excited when his students understand and connect mathematical concepts while solving problems and discovering higher level concepts.
Word problems in mathematics have always been interesting to the researcher who used to solve word problems in elementary grades using pictures and diagrams. This imagery remained dormant in his mind, unexplored for quite some time, until he revisited and taught mathematics, particularly word problems. The researcher now knows why he liked mathematics and why it has become both his vocation and avocation. According to the researcher, diagramming is an effective way to solve many mathematical word problems and real life problems.
Based on his study of the scholarly literature in the field of problem solving, the researcher contends that diagrams are the rudiments of problem solving. Furthermore, in his opinion, diagrams are useful aids for connecting mathematical concepts and thus may improve student achievement in word problem solving. Students who can read and understand the words used to describe a situation, can use images and diagrams to visualize, analyze, and solve related problems in mathematics, as well as in areas such as physics, chemistry, mechanics, engineering, and medicine.