The importance of geometry

This chapter includes the importance of geometry and the importance of learning how to solve traditional word problems by students in school mathematics. The concerns of mathematics education stakeholders about word problem solving based on national and international assessments and the suggestions provided by researchers and educators to improve students’ performance when solving word problems are also reviewed. The theories and empirical studies that focus on comprehension, representation, and solution of word problems are summarized.

Although using mathematics, and in particular geometry, to model situations from work places has been part of education for centuries, the review of the literature starts with the beginning of the late nineteenth century, with the exception of René Descartes’ (1596-1650) doctrine of problem solving (Encyclopedia Britannica, 1983). The review includes recommendations from important publications that inform mathematics education. Research-based theoretical and conceptual frameworks that support the solution process of mathematics word problems are used to develop a research hypothesis for examination in this study. Problem Solving and Solving Word Problems Some mathematics educators and researchers believe that a problem lies as an obstruction between two ends, the problem and the solution, without any clearly defined ways to traverse (Brownell, 1942; Mayer, 1985; Polya, 1980). This definition may also be applied to word problems because many researchers include math word problems in problem solving research (Kilpatrick, 1985). The logic behind this definition can be traced back to René Descartes’ (1596-1650) philosophy which suggests that method is necessary to uncover the truth of nature. The following excerpt from Encyclopedia Britannica (1983) on Descartes’ Discourse on Method is worth mentioning as part of his doctrine of problem solving [1] [The Discourse] is a philosophical classic. [It] hides the fundamental assertion that the human mind is basically sound and the only means of attaining truth … “never to accept anything as true which I [you] did not clearly and distinctly see to be so.” Descartes thus implies the rejection of all accepted ideas and opinions, the determination to doubt until convinced of the contrary by self-evident facts. The second rule is an instruction to analyze the problem to be solved. Once cleared of its prejudices, the mind, using the example set by mathematicians, “must divide each of the difficulties under examination into as many parts as possible”; that is, discover what is relevant to the problem and reduce it as far as possible to its simplest data. The third rule is “to conduct my thoughts in order, beginning with objects that are the simplest and easiest to know and so proceed, gradually, to knowledge of the more complex.” The fourth rule is a warning to recapitulate the “chains of reasoning” to be certain that there are no omissions. These simple rules are not to be considered a mere automatic formula; they are to be regarded as a mental discipline, based on the example of mathematical practice. (p. 600) Schoenfeld (1987) summarized the four phases of Descartes’ problem solving plan. The idea in phase I is to reduce an algebra problem to a single variable equation for solving. Phase II suggests reducing a mathematics problem to an algebra problem and solving it according to phase I. In phase III, any problem situation is converted to a mathematics problem by “mathematizing”. In phase IV, the problem is then solved using the ideas in phase I and II. In two of his many rules (rules XIV and XV), Descartes suggested the drawing of diagrams as an aid to solving problems (pp. 29-36). It is noted from the above excerpt of Descartes’ problem solving process that a problem should be broken down to its parts before attempting to solve it. Each part should also be understood separately. For example, a word problem can usually be solved if one can understand the words (vocabulary), their meaning, their interconnection, the objects they represent, and the relevance of those objects in the problem. Solving a word problem is also sometimes referred to as problem solving. According to Branca (1987), “problem solving” is an alternative meaning of applying mathematics to different circumstances (p. 72). That means if a situation is explained in words, or in a word problem, then applying mathematics as a tool to solve that problem situation may be treated as problem solving. Also, Brown, Cronin, and McEntire (1994) stated that assessment on word problems has different names, including “math reasoning”, “problem solving”, “word problems”, as well as “story problems” (p. 32). Although word problems have been extensively used in problem solving research, the similarity and differences between word problems and problem solving should be clarified. A word problem is also a problem to solve, according to the definitions previously mentioned. Many educators think solving word problems require the problem solving skills. For this dissertation, word problems will refer to problems of the type that appear in standardized assessments and tests such as the NAEP, the New Jersey HSPA, the SAT, and the ACT. They are not problems related to everyday human life without unstated facts where students have to wander, collect facts for “mathematizing” the situation before solving them. The problems in this study can be attempted using general heuristics (Polya, 1945; Schoenfeld, 1985), as well as through the application of Descartes’ problem solving principle and other methods based on Descartes’ philosophy. According to Kilpatrick (1987), in recent years, some researchers in mathematics education have used problems with increasing level of difficulty and learning opportunity that require the novel combination of rules and reasoning. A few similar problems were used in this research. (See Appendix K for sample problems) However, these problems are infrequently found outside of tests or class assignments. Solving Word Problems: A Goal of Mathematics Education “Learning to solve problems is the principal reason of studying mathematics” (National Council of Supervisors of Mathematics, 1977, p. 2). The NCTM (Krulik & Reys, 1980) also suggested that problem solving be regarded as the major goal of learning school mathematics from 1980 to 1989 and repeated that recommendation more recently (NCTM, 2000). Mathematics accomplishment of students, which includes problem solving, became a major concern in the U. S. with the release of A Nation at Risk (U. S. Department of Education, 1983). This publication recommended focusing on the teaching of geometric and algebraic concepts and real-life importance of mathematics in solving problems. The low word problem solving ability of U.S. students of 9, 13, and 17 years of age was verified by the first data from the NAEP conducted in 1973. While analyzing the results of that assessment, Carpenter, Coburn, Reys, and Wilson (1976) concluded: It is most disturbing to ascertain the suggestion that many students receive very little opportunity to learn to solve world problems. The assessment results are so poor, however, that we wonder whether this is not the case. A commitment to working and thinking about word problems is needed for teachers and their students. (p. 392) Table 2.1 shows the scale scores of NAEP on mathematics obtained by U.S. students in grades 4, 8, and 12, on a 0 to 500 scale, from 1990 to 2007. Table 2.2 shows the percent of different types of word problems correctly answered by the students in grades 8 and 12. According to Braswell et al. (2001), the achievement levels of 249, 299, and 336 are considered proficient levels for fourth-, eighth-, and 12th-grade students, respectively. Table 2.1 indicates very small improvements in the NAEP test scores for fourth-grade and eighth-grade students over the span of 17 years (1990 to 2007). However, these scores are below the suggested proficiency levels. It may be noted from Tables 2.1 and 2.2 that improvement, either in overall performance or in word problem solving skills for all participating U.S. students, is trivial. Also the scores that hover around 230 for grade 4, 275 for grade 8, and 300 for grade 12 on a 0 to 500 scale are too low. Of particular concern is an average of only 4% correctly answered questions for the years 1990 to 2000 (Table 2.2) by U.S. grade 12 students on volume and surface area related problems. International assessments such as the FIMS in 1965, the SIMS in 1982, the PISA in 2003 and 2007, and the TIMSS in 1995 and 2003 further attested U.S. students’ poor problem solving skills and highlighted their low mathematical achievement in comparison to students from other participating countries. The FIMS and SIMS conducted mathematics assessment of 13year-old students and high school seniors (National Council of Educational Statistics, 1992). According to the NCTM (2004), the PISA measures the numerical skills and problem solving aptitude of 15-year-old students on a scale of 0 to 500whereas the TIMSS measures fourth and eighth grade students’ ability on concepts on a scale of 0 to 1000. The NCTM also reported that the NAEP, TIMSS, and PISA, which are low-stakes tests, generate group performance results of students. High-stakes tests, like New Jersey’s HSPA or other state mandated tests, as well as the SAT and ACT, focus on the performance of individual students. Of the three assessments, NAEP, TIMSS, and PISA, TIMSS and NAEP have the most in common in terms of mathematical concepts and cognitive necessity (NCTM). The findings from the mathematics results of the PISA of 2000 and 2003 reported by Lemke et al. (2004) indicated that U. S. performance in algebra and geometry was lower than two-third of the participating OECD countries. Even the top 10% of the participants in the U.S. were outperformed by more than half of their OECD counterparts in solving problems. The then U.S. Education Secretary emphasized the need to reform high schools on top priority basis (U.S. Department of Education, 2005). The latest PISA (2007) results indicated that the mathematical accomplishment of U.S students is lower than the international average. According to TIMSS (2003), U.S. students of fourth and eighth grades scored on average 518 and 504, respectively in mathematics. These scores were higher than the average score of 495 of the fourth-grade students in the 25 participating countries and the average score of 466 of the eighth-grade students in the 45 participating countries. However, these scores were lower than the 4 Asian countries and 7 European countries for fourth grade and lower than the 5 Asian countries and 4 European countries for eighth grade. Although the average score of U.S. eighth-grade students improved by only 12 points from 492 in 1995 to 504 in 2003, there was no change reported by TIMSS in their score from 1999 to 2003. Overall, these scores on a scale from 0 to 1000 indicate that students in grades four and eight in the U.S. only achieved about 50% mastery of the concepts tested. National (NAEP, 2007) and international (FIMS, 1965; SIMS, 1982; TIMSS, 1995, 1999, 2003) assessments indicate that student achievement in mathematics remains a major educational concern. Those assessments use multiple choice, short-response, and open-ended word problems which are similar to those on the New Jersey HSPA, SAT, and ACT. Since students’ mathematical skills are measured using one or more of the above assessments, learning to solve word problems must be considered a major goal of mathematics education and a major component of assessing student achievement in mathematics. Further, learning to solve word problems related to real-life situations using mathematical concepts also helps students to be successful at work and in their lives. Geometry as a Cornerstone of Mathematics-History of Problem Solving and Geometry “In ancient India, the rudiments of Geometry, called Rekha-Ganita, were formulated and applied” to solve architectural problems for building temple motifs (Srivathsa, Narasimhan, & Saṃsat 2003, p. 218). The 4000 years old mathematics that emerged in India during The Indus Civilization (2500 BC-1700 BC) proposed for the first time, the ideas of zero, algebra, and finding square and cube roots in Indian Vedic literature (Birodhkar, 1997; O’Connor & Robertson, 2000; Singh, 2004). The significance of studying geometry is evident from the past mathematical records. The book, A History of Mathematics (Suzuki, 2002) provides the mathematical innovations made by the most brilliant mathematicians from ancient times until the 20th century. Some of the mathematical developments presented in this book that are related to problem solving and geometry are discussed next. According to Suzuki (2002), the ancient Egyptians (3000 B.C.) demonstrated their skills in solving word problems by an Egyptian scribe on the mathematical papyri using the concepts of ‘linear and nonlinear equations” without any mathematical notations. That is, “every problem solved by an Egyptian scribe was a word problem” (p. 13). In order to redraw property lines after the yearly flooding of the Nile, the Egyptians developed realistic geometry related geometric figures, “but not their abstract properties”. Also, their geometry is “filled with problems relating to pyramids” (p. 16). The Babylonians (1700 B.C.) also “routinely solved more complicated and complex problems … entirely verbally” (Suzuki, 2002, p. 28) without any system of mathematical notations. Their ways of solving interest relate problems show their advanced mathematical skills. According to Suzuki, the Babylonians also developed methods for calculating the area of triangles, trapezoids and other polygons. Before Pythagoras (580-500 B.C.), the Pythagorean Theorem was “well known to the Babylonians” (p. 31). The development of pre-Euclidean geometry goes back to the age of Plato (427-347 B.C.). It is said that the entrance plaque to Plato’s school in Athens read, “Let No One Unversed In Geometry Come Under My Roof” (Suzuki, 2002, p. 74). According to Suzuki, Plato had probably discovered the word mathematics from “the mathema”, meaning the three liberal arts, arithmetic, geometry, and astronomy (p. 74). Later, Euclid (300 B.C.), who lived in Alexandria, Egypt, wrote the Elements, a conglomeration of 300 years of Greek geometrical development. The Elements was so important for “the next two thousand years of mathematics” that Euclidean geometry became an essential part of learning mathematics until it faced “the first serious mathematical challenges” (p. 86) in the 19th century. The significance of understanding geometry for high school students has been a part of recommendations of the committees on mathematics education in the U.S. since 1894 (Commission on Mathematics, 1959; National Education Association, 1894, National Committee on Mathematical Requirements, 1923; Progressive Education Association (PEA) Committee and the Joint Commission, 1940; The National Committee of Fifteen, 1912). An account of these committees’ reports may be found in the 1970 yearbook of the NCTM, A History of Mathematics Education in the United States and Canada. A brief of the recommendations of these committees are presented below. The first national group of experts that addressed mathematics education was the subcommittee on mathematics of the Committee of Ten (National Education Association, 1894). They considered the goals and curriculum for mathematics education and recommended preparatory work on algebra and geometry in the upper elementary school curriculum. On “demonstrative geometry”, the committee stressed on “the importance of elegance and finish in geometrical demonstration” (p. 25). About “demonstrative geometry”, the committee further stated, “there is no student whom it will not brighten and strengthen intellectually as few other exercises can” (p. 116). This suggests all mathematics teachers engage their students in using the geometric concepts to visualize their surroundings and to geometrically demonstrate what they visualize. The final report of The National Committee of Fifteen on the Geometry Syllabus (National Education Association, 1912) recommended using realistic approaches to exercises in mathematics instruction. Eleven years later, its final report, The Reorganization of Mathematics in Secondary Education (The National Committee on Mathematical Requirements, 1923) also stressed the importance of the studying geometry. The commission advocated that “the course of study in mathematics during the seventh, eighth, and ninth years contain the fundamental notions of arithmetic, of algebra, of intuitive geometry, of numerical trigonometry, and at least an introduction to demonstrative geometry” (p. 1). One of the practical aims of this ecommendation was to encourage familiarity with geometric forms common in nature and life, as well as the elementary properties and relations of these forms, including their measurement, the development of space-perception, and the exercise of spatial imagination.